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SIXTH EDITION
Introduction to Heat Transfer THEODORE L. BERGMAN Department of Mec...

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SIXTH EDITION

Introduction to Heat Transfer THEODORE L. BERGMAN Department of Mechanical Engineering University of Connecticut

ADRIENNE S. LAVINE Mechanical and Aerospace Engineering Department University of California, Los Angeles

FRANK P. INCROPERA College of Engineering University of Notre Dame

DAVID P. DEWITT School of Mechanical Engineering Purdue University

JOHN WILEY & SONS, INC.

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This book was typeset in 10.5/12 Times Roman by MPS Limited, a Macmillan Company and printed and bound by R. R. Donnelley (Jefferson City). The cover was printed by R. R. Donnelley (Jefferson City). Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations. Our company is built on a foundation of principles that include responsibility to the communities we serve and where we live and work. In 2008, we launched a Corporate Citizenship Initiative, a global effort to address the environmental, social, economic, and ethical challenges we face in our business. Among the issues we are addressing are carbon impact, paper specifications and procurement, ethical conduct within our business and among our vendors, and community and charitable support. For more information, please visit our website: www.wiley.com/go/citizenship. The paper in this book was manufactured by a mill whose forest management programs include sustained yield harvesting of its timberlands. Sustained yield-harvesting principles ensure that the number of trees cut each year does not exceed the amount of new growth. This book is printed on acid-free paper. Copyright © 2011, 2007, 2002 by John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008. Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. If you have chosen to adopt this textbook for use in your course, please accept this book as your complimentary desk copy. Outside of the United States, please contact your local representative.

ISBN 13 978-0470-50196-2 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Preface

In the Preface to the previous edition, we posed questions regarding trends in engineering education and practice, and whether the discipline of heat transfer would remain relevant. After weighing various arguments, we concluded that the future of engineering was bright and that heat transfer would remain a vital and enabling discipline across a range of emerging technologies including but not limited to information technology, biotechnology, pharmacology, and alternative energy generation. Since we drew these conclusions, many changes have occurred in both engineering education and engineering practice. Driving factors have been a contracting global economy, coupled with technological and environmental challenges associated with energy production and energy conversion. The impact of a weak global economy on higher education has been sobering. Colleges and universities around the world are being forced to set priorities and answer tough questions as to which educational programs are crucial, and which are not. Was our previous assessment of the future of engineering, including the relevance of heat transfer, too optimistic? Faced with economic realities, many colleges and universities have set clear priorities. In recognition of its value and relevance to society, investment in engineering education has, in many cases, increased. Pedagogically, there is renewed emphasis on the fundamental principles that are the foundation for lifelong learning. The important and sometimes dominant role of heat transfer in many applications, particularly in conventional as well as in alternative energy generation and concomitant environmental effects, has reaffirmed its relevance. We believe our previous conclusions were correct: The future of engineering is bright, and heat transfer is a topic that is crucial to address a broad array of technological and environmental challenges. In preparing this edition, we have sought to incorporate recent heat transfer research at a level that is appropriate for an undergraduate student. We have strived to include new examples and problems that motivate students with interesting applications, but whose solutions are based firmly on fundamental principles. We have remained true to the pedagogical approach of previous editions by retaining a rigorous and systematic methodology for problem solving. We have attempted to continue the tradition of providing a text that will serve as a valuable, everyday resource for students and practicing engineers throughout their careers.

iv

Preface

Approach and Organization Previous editions of the text have adhered to four learning objectives: 1. The student should internalize the meaning of the terminology and physical principles associated with heat transfer. 2. The student should be able to delineate pertinent transport phenomena for any process or system involving heat transfer. 3. The student should be able to use requisite inputs for computing heat transfer rates and/or material temperatures. 4. The student should be able to develop representative models of real processes and systems and draw conclusions concerning process/system design or performance from the attendant analysis. Moreover, as in previous editions, specific learning objectives for each chapter are clarified, as are means by which achievement of the objectives may be assessed. The summary of each chapter highlights key terminology and concepts developed in the chapter and poses questions designed to test and enhance student comprehension. It is recommended that problems involving complex models and/or exploratory, whatif, and parameter sensitivity considerations be addressed using a computational equationsolving package. To this end, the Interactive Heat Transfer (IHT) package available in previous editions has been updated. Specifically, a simplified user interface now delineates between the basic and advanced features of the software. It has been our experience that most students and instructors will use primarily the basic features of IHT. By clearly identifying which features are advanced, we believe students will be motivated to use IHT on a daily basis. A second software package, Finite Element Heat Transfer (FEHT), developed by F-Chart Software (Madison, Wisconsin), provides enhanced capabilities for solving two-dimensional conduction heat transfer problems. To encourage use of IHT, a Quickstart User’s Guide has been installed in the software. Students and instructors can become familiar with the basic features of IHT in approximately one hour. It has been our experience that once students have read the Quickstart guide, they will use IHT heavily, even in courses other than heat transfer. Students report that IHT significantly reduces the time spent on the mechanics of lengthy problem solutions, reduces errors, and allows more attention to be paid to substantive aspects of the solution. Graphical output can be generated for homework solutions, reports, and papers. As in previous editions, some homework problems require a computer-based solution. Other problems include both a hand calculation and an extension that is computer based. The latter approach is time-tested and promotes the habit of checking a computer-generated solution with a hand calculation. Once validated in this manner, the computer solution can be utilized to conduct parametric calculations. Problems involving both hand- and computer-generated solutions are identified by enclosing the exploratory part in a red rectangle, as, for example, (b) , (c) , or (d) . This feature also allows instructors who wish to limit their assignments of computer-based problems to benefit from the richness of these problems without assigning their computer-based parts. Solutions to problems for which the number is highlighted (for example, 1.19 ) are entirely computer based.

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v

What’s New in the Sixth Edition In the previous edition, Chapter 1 Introduction was modified to emphasize the relevance of heat transfer in various contemporary applications. Responding to today’s challenges involving energy production and its environmental impact, an expanded discussion of the efficiency of energy conversion and the production of greenhouse gases has been added. Chapter 1 has also been modified to embellish the complementary nature of heat transfer and thermodynamics. The existing treatment of the first law of thermodynamics is augmented with a new section on the relationship between heat transfer and the second law of thermodynamics as well as the efficiency of heat engines. Indeed, the influence of heat transfer on the efficiency of energy conversion is a recurring theme throughout this edition. The coverage of micro- and nanoscale effects in Chapter 2 Introduction to Conduction has been updated, reflecting recent advances. For example, the description of the thermophysical properties of composite materials is enhanced, with a new discussion of nanofluids. Chapter 3 One-Dimensional, Steady-State Conduction has undergone extensive revision and includes new material on conduction in porous media, thermoelectric power generation, and micro- as well as nanoscale systems. Inclusion of these new topics follows recent fundamental discoveries and is presented through the use of the thermal resistance network concept. Hence the power and utility of the resistance network approach is further emphasized in this edition. Chapter 4 Two-Dimensional, Steady-State Conduction has been reduced in length. Today, systems of linear, algebraic equations are readily solved using standard computer software or even handheld calculators. Hence the focus of the shortened chapter is on the application of heat transfer principles to derive the systems of algebraic equations to be solved and on the discussion and interpretation of results. The discussion of Gauss–Seidel iteration has been moved to an appendix for instructors wishing to cover that material. Chapter 5 Transient Conduction was substantially modified in the previous edition and has been augmented in this edition with a streamlined presentation of the lumpedcapacitance method. Chapter 6 Introduction to Convection includes clarification of how temperature-dependent properties should be evaluated when calculating the convection heat transfer coefficient. The fundamental aspects of compressible flow are introduced to provide the reader with guidelines regarding the limits of applicability of the treatment of convection in the text. Chapter 7 External Flow has been updated and reduced in length. Specifically, presentation of the similarity solution for flow over a flat plate has been simplified. New results for flow over noncircular cylinders have been added, replacing the correlations of previous editions. The discussion of flow across banks of tubes has been shortened, eliminating redundancy without sacrificing content. Chapter 8 Internal Flow entry length correlations have been updated, and the discussion of micro- and nanoscale convection has been modified and linked to the content of Chapter 3. Changes to Chapter 9 Free Convection include a new correlation for free convection from flat plates, replacing a correlation from previous editions. The discussion of boundary layer effects has been modified. Aspects of condensation included in Chapter 10 Boiling and Condensation have been updated to incorporate recent advances in, for example, external condensation on finned tubes. The effects of surface tension and the presence of noncondensable gases in modifying

Chapter-by-Chapter Content Changes

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Preface

condensation phenomena and heat transfer rates are elucidated. The coverage of forced convection condensation and related enhancement techniques has been expanded, again reflecting advances reported in the recent literature. The content of Chapter 11 Heat Exchangers is experiencing a resurgence in interest due to the critical role such devices play in conventional and alternative energy generation technologies. A new section illustrates the applicability of heat exchanger analysis to heat sink design and materials processing. Much of the coverage of compact heat exchangers included in the previous edition was limited to a specific heat exchanger. Although general coverage of compact heat exchangers has been retained, the discussion that is limited to the specific heat exchanger has been relegated to supplemental material, where it is available to instructors who wish to cover this topic in greater depth. The concepts of emissive power, irradiation, radiosity, and net radiative flux are now introduced early in Chapter 12 Radiation: Processes and Properties, allowing early assignment of end-of-chapter problems dealing with surface energy balances and properties, as well as radiation detection. The coverage of environmental radiation has undergone substantial revision, with the inclusion of separate discussions of solar radiation, the atmospheric radiation balance, and terrestrial solar irradiation. Concern for the potential impact of anthropogenic activity on the temperature of the earth is addressed and related to the concepts of the chapter. Much of the modification to Chapter 13 Radiation Exchange Between Surfaces emphasizes the difference between geometrical surfaces and radiative surfaces, a key concept that is often difficult for students to appreciate. Increased coverage of radiation exchange between multiple blackbody surfaces, included in older editions of the text, has been returned to Chapter 13. In doing so, radiation exchange between differentially small surfaces is briefly introduced and used to illustrate the limitations of the analysis techniques included in Chapter 13. Problem Sets Approximately 225 new end-of-chapter problems have been developed for this edition. An effort has been made to include new problems that (a) are amenable to short solutions or (b) involve finite-difference solutions. A significant number of solutions to existing end-of-chapter problems have been modified due to the inclusion of the new convection correlations in this edition.

Classroom Coverage The content of the text has evolved over many years in response to a variety of factors. Some factors are obvious, such as the development of powerful, yet inexpensive calculators and software. There is also the need to be sensitive to the diversity of users of the text, both in terms of (a) the broad background and research interests of instructors and (b) the wide range of missions associated with the departments and institutions at which the text is used. Regardless of these and other factors, it is important that the four previously identified learning objectives be achieved. Mindful of the broad diversity of users, the authors’ intent is not to assemble a text whose content is to be covered, in entirety, during a single semester- or quarter-long course. Rather, the text includes both (a) fundamental material that we believe must be covered and (b) optional material that instructors can use to address specific interests or that can be

Preface

vii

covered in a second, intermediate heat transfer course. To assist instructors in preparing a syllabus for a first course in heat transfer, we have several recommendations. Chapter 1 Introduction sets the stage for any course in heat transfer. It explains the linkage between heat transfer and thermodynamics, and it reveals the relevance and richness of the subject. It should be covered in its entirety. Much of the content of Chapter 2 Introduction to Conduction is critical in a first course, especially Section 2.1 The Conduction Rate Equation, Section 2.3 The Heat Diffusion Equation, and Section 2.4 Boundary and Initial Conditions. It is recommended that Chapter 2 be covered in its entirety. Chapter 3 One-Dimensional, Steady-State Conduction includes a substantial amount of optional material from which instructors can pick-and-choose or defer to a subsequent, intermediate heat transfer course. The optional material includes Section 3.1.5 Porous Media, Section 3.7 The Bioheat Equation, Section 3.8 Thermoelectric Power Generation, and Section 3.9 Micro- and Nanoscale Conduction. Because the content of these sections is not interlinked, instructors may elect to cover any or all of the optional material. The content of Chapter 4 Two-Dimensional, Steady-State Conduction is important because both (a) fundamental concepts and (b) powerful and practical solution techniques are presented. We recommend that all of Chapter 4 be covered in any introductory heat transfer course. The optional material in Chapter 5 Transient Conduction is Section 5.9 Periodic Heating. Also, some instructors do not feel compelled to cover Section 5.10 Finite-Difference Methods in an introductory course, especially if time is short. The content of Chapter 6 Introduction to Convection is often difficult for students to absorb. However, Chapter 6 introduces fundamental concepts and lays the foundation for the subsequent convection chapters. It is recommended that all of Chapter 6 be covered in an introductory course. Chapter 7 External Flow introduces several important concepts and presents convection correlations that students will utilize throughout the remainder of the text and in subsequent professional practice. Sections 7.1 through 7.5 should be included in any first course in heat transfer. However, the content of Section 7.6 Flow Across Banks of Tubes, Section 7.7 Impinging Jets, and Section 7.8 Packed Beds is optional. Since the content of these sections is not interlinked, instructors may select from any of the optional topics. Likewise, Chapter 8 Internal Flow includes matter that is used throughout the remainder of the text and by practicing engineers. However, Section 8.7 Heat Transfer Enhancement, and Section 8.8 Flow in Small Channels may be viewed as optional. Buoyancy-induced flow and heat transfer is covered in Chapter 9 Free Convection. Because free convection thermal resistances are typically large, they are often the dominant resistance in many thermal systems and govern overall heat transfer rates. Therefore, most of Chapter 9 should be covered in a first course in heat transfer. Optional material includes Section 9.7 Free Convection Within Parallel Plate Channels and Section 9.9 Combined Free and Forced Convection. In contrast to resistances associated with free convection, thermal resistances corresponding to liquid-vapor phase change are typically small, and they can sometimes be neglected. Nonetheless, the content of Chapter 10 Boiling and Condensation that should be covered in a first heat transfer course includes Sections 10.1 through 10.4, Sections 10.6 through 10.8, and Section 10.11. Section 10.5 Forced Convection Boiling may be material appropriate for an intermediate heat transfer course. Similarly, Section 10.9 Film Condensation on Radial Systems and Section 10.10 Condensation in Horizontal Tubes may be either covered as time permits or included in a subsequent heat transfer course.

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Preface

We recommend that all of Chapter 11 Heat Exchangers be covered in a first heat transfer course. A distinguishing feature of the text, from its inception, is the in-depth coverage of radiation heat transfer in Chapter 12 Radiation: Processes and Properties. The content of the chapter is perhaps more relevant today than ever, with applications ranging from advanced manufacturing, to radiation detection and monitoring, to environmental issues related to global climate change. Although Chapter 12 has been reorganized to accommodate instructors who may wish to skip ahead to Chapter 13 after Section 12.4, we encourage instructors to cover Chapter 12 in its entirety. Chapter 13 Radiation Exchange Between Surfaces may be covered as time permits or in an intermediate heat transfer course.

Acknowledgments We wish to acknowledge and thank many of our colleagues in the heat transfer community. In particular, we would like to express our appreciation to Diana Borca-Tasciuc of the Rensselaer Polytechnic Institute and David Cahill of the University of Illinois UrbanaChampaign for their assistance in developing the periodic heating material of Chapter 5. We thank John Abraham of the University of St. Thomas for recommendations that have led to an improved treatment of flow over noncircular tubes in Chapter 7. We are very grateful to Ken Smith, Clark Colton, and William Dalzell of the Massachusetts Institute of Technology for the stimulating and detailed discussion of thermal entry effects in Chapter 8. We acknowledge Amir Faghri of the University of Connecticut for his advice regarding the treatment of condensation in Chapter 10. We extend our gratitude to Ralph Grief of the University of California, Berkeley for his many constructive suggestions pertaining to material throughout the text. Finally, we wish to thank the many students, instructors, and practicing engineers from around the globe who have offered countless interesting, valuable, and stimulating suggestions. In closing, we are deeply grateful to our spouses and children, Tricia, Nate, Tico, Greg, Elias, Jacob, Andrea, Terri, Donna, and Shaunna for their endless love and patience. We extend appreciation to Tricia Bergman who expertly processed solutions for the end-ofchapter problems. Theodore L. Bergman ([email protected]) Storrs, Connecticut Adrienne S. Lavine ([email protected]) Los Angeles, California Frank P. Incropera ([email protected]) Notre Dame, Indiana

Preface

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Supplemental and Web Site Material The companion web site for the texts is www.wiley.com/college/bergman. By selecting one of the two texts and clicking on the “student companion site” link, students may access the Answers to Selected Exercises and the Supplemental Sections of the text. Supplemental Sections are identified throughout the text with the icon shown in the margin to the left. Material available for instructors only may also be found by selecting one of the two texts at www.wiley.com/college/bergman and clicking on the “instructor companion site” link. The available content includes the Solutions Manual, PowerPoint Slides that can be used by instructors for lectures, and Electronic Versions of figures from the text for those wishing to prepare their own materials for electronic classroom presentation. The Instructor Solutions Manual is copyrighted material for use only by instructors who are requiring the text for their course.1 Interactive Heat Transfer 4.0/FEHT is available either with the text or as a separate purchase. As described by the authors in the Approach and Organization, this simple-to-use software tool provides modeling and computational features useful in solving many problems in the text, and it enables rapid what-if and exploratory analysis of many types of problems. Instructors interested in using this tool in their course can download the software from the book’s web site at www.wiley.com/college/bergman. Students can download the software by registering on the student companion site; for details, see the registration card provided in this book. The software is also available as a stand-alone purchase at the web site. Any questions can be directed to your local Wiley representative.

This mouse icon identifies Supplemental Sections and is used throughout the text. Excerpts from the Solutions Manual may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of the contents of the Solutions Manual beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. 1

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Contents

CHAPTER

Symbols

xxi

1 Introduction

1

1.1 1.2

1.3

1.4 1.5

What and How? Physical Origins and Rate Equations 1.2.1 Conduction 3 1.2.2 Convection 6 1.2.3 Radiation 8 1.2.4 The Thermal Resistance Concept 12 Relationship to Thermodynamics 1.3.1 Relationship to the First Law of Thermodynamics (Conservation of Energy) 13 1.3.2 Relationship to the Second Law of Thermodynamics and the Efficiency of Heat Engines 31 Units and Dimensions Analysis of Heat Transfer Problems: Methodology

2 3

12

36 38

xii

Contents

1.6 1.7

CHAPTER

2 Introduction to Conduction 2.1 2.2

2.3 2.4 2.5

CHAPTER

Relevance of Heat Transfer Summary References Problems

The Conduction Rate Equation The Thermal Properties of Matter 2.2.1 Thermal Conductivity 70 2.2.2 Other Relevant Properties 78 The Heat Diffusion Equation Boundary and Initial Conditions Summary References Problems

3 One-Dimensional, Steady-State Conduction 3.1

The Plane Wall 3.1.1 Temperature Distribution 112 3.1.2 Thermal Resistance 114 3.1.3 The Composite Wall 115 3.1.4 Contact Resistance 117 3.1.5 Porous Media 119 3.2 An Alternative Conduction Analysis 3.3 Radial Systems 3.3.1 The Cylinder 136 3.3.2 The Sphere 141 3.4 Summary of One-Dimensional Conduction Results 3.5 Conduction with Thermal Energy Generation 3.5.1 The Plane Wall 143 3.5.2 Radial Systems 149 3.5.3 Tabulated Solutions 150 3.5.4 Application of Resistance Concepts 150 3.6 Heat Transfer from Extended Surfaces 3.6.1 A General Conduction Analysis 156 3.6.2 Fins of Uniform Cross-Sectional Area 158 3.6.3 Fin Performance 164 3.6.4 Fins of Nonuniform Cross-Sectional Area 167 3.6.5 Overall Surface Efficiency 170 3.7 The Bioheat Equation 3.8 Thermoelectric Power Generation 3.9 Micro- and Nanoscale Conduction 3.9.1 Conduction Through Thin Gas Layers 189 3.9.2 Conduction Through Thin Solid Films 190 3.10 Summary References Problems

41 45 48 49

67 68 70

82 90 94 95 95

111 112

132 136

142 142

154

178 182 189

190 193 193

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Contents

CHAPTER

4 Two-Dimensional, Steady-State Conduction 4.1 4.2 4.3 4.4

4.5

4.6

229

Alternative Approaches The Method of Separation of Variables The Conduction Shape Factor and the Dimensionless Conduction Heat Rate Finite-Difference Equations 4.4.1 The Nodal Network 241 4.4.2 Finite-Difference Form of the Heat Equation 242 4.4.3 The Energy Balance Method 243 Solving the Finite-Difference Equations 4.5.1 Formulation as a Matrix Equation 250 4.5.2 Verifying the Accuracy of the Solution 251 Summary References Problems

4S.1 The Graphical Method 4S.1.1 Methodology of Constructing a Flux Plot W-1 4S.1.2 Determination of the Heat Transfer Rate W-2 4S.1.3 The Conduction Shape Factor W-3 4S.2 The Gauss–Seidel Method: Example of Usage References Problems CHAPTER

5.4 5.5

5.6

5.7 5.8

The Lumped Capacitance Method Validity of the Lumped Capacitance Method General Lumped Capacitance Analysis 5.3.1 Radiation Only 288 5.3.2 Negligible Radiation 288 5.3.3 Convection Only with Variable Convection Coefficient 5.3.4 Additional Considerations 289 Spatial Effects The Plane Wall with Convection 5.5.1 Exact Solution 300 5.5.2 Approximate Solution 300 5.5.3 Total Energy Transfer 302 5.5.4 Additional Considerations 302 Radial Systems with Convection 5.6.1 Exact Solutions 303 5.6.2 Approximate Solutions 304 5.6.3 Total Energy Transfer 304 5.6.4 Additional Considerations 305 The Semi-Infinite Solid Objects with Constant Surface Temperatures or Surface Heat Fluxes 5.8.1 Constant Temperature Boundary Conditions 317 5.8.2 Constant Heat Flux Boundary Conditions 319 5.8.3 Approximate Solutions 320

250

256 257 257 W-1

W-5 W-9 W-10

5 Transient Conduction 5.1 5.2 5.3

230 231 235 241

279 280 283 287

289 298 299

303

310 317

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Contents

5.9 Periodic Heating 5.10 Finite-Difference Methods 5.10.1 Discretization of the Heat Equation: The Explicit Method 5.10.2 Discretization of the Heat Equation: The Implicit Method 5.11 Summary References Problems

327 330 330 337 345 346 346

5S.1 Graphical Representation of One-Dimensional, Transient Conduction in the Plane Wall, Long Cylinder, and Sphere 5S.2 Analytical Solution of Multidimensional Effects References Problems CHAPTER

6 Introduction to Convection 6.1

6.2

6.3

6.4

6.5

6.6 6.7 6.8

377

The Convection Boundary Layers 6.1.1 The Velocity Boundary Layer 378 6.1.2 The Thermal Boundary Layer 379 6.1.3 Significance of the Boundary Layers 380 Local and Average Convection Coefficients 6.2.1 Heat Transfer 381 6.2.2 The Problem of Convection 382 Laminar and Turbulent Flow 6.3.1 Laminar and Turbulent Velocity Boundary Layers 383 6.3.2 Laminar and Turbulent Thermal Boundary Layers 385 The Boundary Layer Equations 6.4.1 Boundary Layer Equations for Laminar Flow 389 6.4.2 Compressible Flow 391 Boundary Layer Similarity: The Normalized Boundary Layer Equations 6.5.1 Boundary Layer Similarity Parameters 392 6.5.2 Functional Form of the Solutions 393 Physical Interpretation of the Dimensionless Parameters Momentum and Heat Transfer (Reynolds) Analogy Summary References Problems

6S.1 Derivation of the Convection Transfer Equations 6S.1.1 Conservation of Mass W-25 6S.1.2 Newton’s Second Law of Motion W-26 6S.1.3 Conservation of Energy W-29 References Problems CHAPTER

The Empirical Method The Flat Plate in Parallel Flow 7.2.1 Laminar Flow over an Isothermal Plate: A Similarity Solution 7.2.2 Turbulent Flow over an Isothermal Plate 424

378

381

383

388

392

400 402 404 405 405 W-25

W-35 W-35

7 External Flow 7.1 7.2

W-12 W-16 W-22 W-22

415 416 418 418

Contents

7.3 7.4

7.5 7.6 7.7

7.8 7.9

CHAPTER

7.2.3 Mixed Boundary Layer Conditions 425 7.2.4 Unheated Starting Length 426 7.2.5 Flat Plates with Constant Heat Flux Conditions 427 7.2.6 Limitations on Use of Convection Coefficients 427 Methodology for a Convection Calculation The Cylinder in Cross Flow 7.4.1 Flow Considerations 433 7.4.2 Convection Heat Transfer 436 The Sphere Flow Across Banks of Tubes Impinging Jets 7.7.1 Hydrodynamic and Geometric Considerations 456 7.7.2 Convection Heat Transfer 458 Packed Beds Summary References Problems

8 Internal Flow 8.1

8.2

8.3

8.4

8.5 8.6 8.7 8.8

8.9

Hydrodynamic Considerations 8.1.1 Flow Conditions 490 8.1.2 The Mean Velocity 491 8.1.3 Velocity Profile in the Fully Developed Region 492 8.1.4 Pressure Gradient and Friction Factor in Fully Developed Flow 494 Thermal Considerations 8.2.1 The Mean Temperature 496 8.2.2 Newton’s Law of Cooling 497 8.2.3 Fully Developed Conditions 497 The Energy Balance 8.3.1 General Considerations 501 8.3.2 Constant Surface Heat Flux 502 8.3.3 Constant Surface Temperature 505 Laminar Flow in Circular Tubes: Thermal Analysis and Convection Correlations 8.4.1 The Fully Developed Region 509 8.4.2 The Entry Region 514 8.4.3 Temperature-Dependent Properties 516 Convection Correlations: Turbulent Flow in Circular Tubes Convection Correlations: Noncircular Tubes and the Concentric Tube Annulus Heat Transfer Enhancement Flow in Small Channels 8.8.1 Microscale Convection in Gases (0.1 m ⱗ Dh ⱗ 100 m) 530 8.8.2 Microscale Convection in Liquids 531 8.8.3 Nanoscale Convection (Dh ⱗ 100 nm) 532 Summary References Problems

xv

428 433

443 447 455

461 462 464 465

489 490

495

501

509

516 524 527 530

535 537 538

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CHAPTER

Contents

9 Free Convection 9.1 9.2 9.3 9.4 9.5 9.6

Physical Considerations The Governing Equations for Laminar Boundary Layers Similarity Considerations Laminar Free Convection on a Vertical Surface The Effects of Turbulence Empirical Correlations: External Free Convection Flows 9.6.1 The Vertical Plate 573 9.6.2 Inclined and Horizontal Plates 576 9.6.3 The Long Horizontal Cylinder 581 9.6.4 Spheres 585 9.7 Free Convection Within Parallel Plate Channels 9.7.1 Vertical Channels 587 9.7.2 Inclined Channels 589 9.8 Empirical Correlations: Enclosures 9.8.1 Rectangular Cavities 589 9.8.2 Concentric Cylinders 592 9.8.3 Concentric Spheres 593 9.9 Combined Free and Forced Convection 9.10 Summary References Problems CHAPTER

10 Boiling and Condensation 10.1 Dimensionless Parameters in Boiling and Condensation 10.2 Boiling Modes 10.3 Pool Boiling 10.3.1 The Boiling Curve 622 10.3.2 Modes of Pool Boiling 623 10.4 Pool Boiling Correlations 10.4.1 Nucleate Pool Boiling 626 10.4.2 Critical Heat Flux for Nucleate Pool Boiling 628 10.4.3 Minimum Heat Flux 629 10.4.4 Film Pool Boiling 629 10.4.5 Parametric Effects on Pool Boiling 630 10.5 Forced Convection Boiling 10.5.1 External Forced Convection Boiling 636 10.5.2 Two-Phase Flow 636 10.5.3 Two-Phase Flow in Microchannels 639 10.6 Condensation: Physical Mechanisms 10.7 Laminar Film Condensation on a Vertical Plate 10.8 Turbulent Film Condensation 10.9 Film Condensation on Radial Systems 10.10 Condensation in Horizontal Tubes 10.11 Dropwise Condensation

561 562 565 566 567 570 572

586

589

595 596 597 598

619 620 621 622

626

635

639 641 645 650 655 656

Contents

CHAPTER

10.12 Summary References Problems

657 657 659

11 Heat Exchangers

671

11.1 11.2 11.3

11.4

11.5 11.6 11.7

Heat Exchanger Types The Overall Heat Transfer Coefficient Heat Exchanger Analysis: Use of the Log Mean Temperature Difference 11.3.1 The Parallel-Flow Heat Exchanger 678 11.3.2 The Counterflow Heat Exchanger 680 11.3.3 Special Operating Conditions 681 Heat Exchanger Analysis: The Effectiveness–NTU Method 11.4.1 Definitions 688 11.4.2 Effectiveness–NTU Relations 689 Heat Exchanger Design and Performance Calculations Additional Considerations Summary References Problems

11S.1 Log Mean Temperature Difference Method for Multipass and Cross-Flow Heat Exchangers 11S.2 Compact Heat Exchangers References Problems CHAPTER

xvii

12 Radiation: Processes and Properties 12.1 12.2 12.3

12.4

12.5 12.6

Fundamental Concepts Radiation Heat Fluxes Radiation Intensity 12.3.1 Mathematical Definitions 739 12.3.2 Radiation Intensity and Its Relation to Emission 740 12.3.3 Relation to Irradiation 745 12.3.4 Relation to Radiosity for an Opaque Surface 747 12.3.5 Relation to the Net Radiative Flux for an Opaque Surface 748 Blackbody Radiation 12.4.1 The Planck Distribution 749 12.4.2 Wien’s Displacement Law 750 12.4.3 The Stefan–Boltzmann Law 750 12.4.4 Band Emission 751 Emission from Real Surfaces Absorption, Reflection, and Transmission by Real Surfaces 12.6.1 Absorptivity 768 12.6.2 Reflectivity 769

672 674 677

688

696 705 713 714 714 W-38 W-42 W-47 W-48

733 734 737 739

748

758 767

xviii

Contents

12.6.3 Transmissivity 771 12.6.4 Special Considerations 771 12.7 Kirchhoff’s Law 12.8 The Gray Surface 12.9 Environmental Radiation 12.9.1 Solar Radiation 785 12.9.2 The Atmospheric Radiation Balance 12.9.3 Terrestrial Solar Irradiation 789 12.10 Summary References Problems CHAPTER

776 778 784 787

13 Radiation Exchange Between Surfaces 13.1

13.2 13.3

13.4 13.5 13.6

13.7

The View Factor 13.1.1 The View Factor Integral 828 13.1.2 View Factor Relations 829 Blackbody Radiation Exchange Radiation Exchange Between Opaque, Diffuse, Gray Surfaces in an Enclosure 13.3.1 Net Radiation Exchange at a Surface 843 13.3.2 Radiation Exchange Between Surfaces 844 13.3.3 The Two-Surface Enclosure 850 13.3.4 Radiation Shields 852 13.3.5 The Reradiating Surface 854 Multimode Heat Transfer Implications of the Simplifying Assumptions Radiation Exchange with Participating Media 13.6.1 Volumetric Absorption 862 13.6.2 Gaseous Emission and Absorption 863 Summary References Problems

792 796 796

827 828

838 842

859 862 862

867 868 869

APPENDIX

A Thermophysical Properties of Matter

897

APPENDIX

B Mathematical Relations and Functions

927

APPENDIX

C Thermal Conditions Associated with Uniform Energy

Generation in One-Dimensional, Steady-State Systems APPENDIX

D The Gauss–Seidel Method

933

939

Contents

APPENDIX

E The Convection Transfer Equations E.1 Conservation of Mass E.2 Newton’s Second Law of Motion E.3 Conservation of Energy

APPENDIX

F Boundary Layer Equations for Turbulent Flow

APPENDIX

G An Integral Laminar Boundary Layer Solution for

xix 941 942 942 943

945

Parallel Flow over a Flat Plate

949

Index

953

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Symbols

A Ab Ac Ap Ar a Bi Bo CD Cf Ct Co c cp cv D Db Dh d E E tot Ec E˙ g E˙ in E˙ out E˙ st e F Fo Fr

area, m2 area of prime (unfinned) surface, m2 cross-sectional area, m2 fin profile area, m2 nozzle area ratio acceleration, m/s2; speed of sound, m/s Biot number Bond number drag coefficient friction coefficient thermal capacitance, J/K Confinement number specific heat, J/kg 䡠 K; speed of light, m/s specific heat at constant pressure, J/kg 䡠 K specific heat at constant volume, J/kg 䡠 K diameter, m bubble diameter, m hydraulic diameter, m diameter of gas molecule, nm thermal plus mechanical energy, J; electric potential, V; emissive power, W/m2 total energy, J Eckert number rate of energy generation, W rate of energy transfer into a control volume, W rate of energy transfer out of control volume, W rate of increase of energy stored within a control volume, W thermal internal energy per unit mass, J/kg; surface roughness, m force, N; fraction of blackbody radiation in a wavelength band; view factor Fourier number Froude number

f G Gr Gz g H h hfg h⬘fg hsf hrad I i J Ja jH k kB L M ˙ in M ˙ out M ˙ st M ᏹi Ma m m˙ N NL, NT Nu

friction factor; similarity variable irradiation, W/m2; mass velocity, kg/s 䡠 m2 Grashof number Graetz number gravitational acceleration, m/s2 nozzle height, m; Henry’s constant, bars convection heat transfer coefficient, W/m2 䡠 K; Planck’s constant, J 䡠 s latent heat of vaporization, J/kg modified heat of vaporization, J/kg latent heat of fusion, J/kg radiation heat transfer coefficient, W/m2 䡠 K electric current, A; radiation intensity, W/m2 䡠 sr electric current density, A/m2; enthalpy per unit mass, J/kg radiosity, W/m2 Jakob number Colburn j factor for heat transfer thermal conductivity, W/m 䡠 K Boltzmann’s constant, J/K length, m mass, kg rate at which mass enters a control volume, kg/s rate at which mass leaves a control volume, kg/s rate of increase of mass stored within a control volume, kg/s molecular weight of species i, kg/kmol Mach number mass, kg mass flow rate, kg/s integer number number of tubes in longitudinal and transverse directions Nusselt number

xxii

Symbols

NTU ᏺ P PL , PT Pe Pr p Q q q˙ q⬘ q⬙ q* R Ra Re Re Rf Rm,n Rt Rt,c Rt,f Rt,o ro r, , z r, , S

Sc SD, SL, ST St T t U u, v, w V v W ˙ W We X Xtt X, Y, Z x, y, z xc xfd,h xfd,t Z

number of transfer units Avogadro’s number power, W; perimeter, m dimensionless longitudinal and transverse pitch of a tube bank Peclet number Prandtl number pressure, N/m2 energy transfer, J heat transfer rate, W rate of energy generation per unit volume, W/m3 heat transfer rate per unit length, W/m heat flux, W/m2 dimensionless conduction heat rate cylinder radius, m; gas constant, J/kg 䡠 K universal gas constant, J/kmol 䡠 K Rayleigh number Reynolds number electric resistance, ⍀ fouling factor, m2 䡠 K/W residual for the m, n nodal point thermal resistance, K/W thermal contact resistance, K/W fin thermal resistance, K/W thermal resistance of fin array, K/W cylinder or sphere radius, m cylindrical coordinates spherical coordinates shape factor for two-dimensional conduction, m; nozzle pitch, m; plate spacing, m; Seebeck coefficient, V/K solar constant, W/m2 diagonal, longitudinal, and transverse pitch of a tube bank, m Stanton number temperature, K time, s overall heat transfer coefficient, W/m2 䡠 K; internal energy, J mass average fluid velocity components, m/s volume, m3; fluid velocity, m/s specific volume, m3/kg width of a slot nozzle, m rate at which work is performed, W Weber number vapor quality Martinelli parameter components of the body force per unit volume, N/m3 rectangular coordinates, m critical location for transition to turbulence, m hydrodynamic entry length, m thermal entry length, m thermoelectric material property, K⫺1

Greek Letters ␣ thermal diffusivity, m2/s; accommodation coefficient; absorptivity

 ⌫ ␥ ␦ ␦p ␦t f f o mfp e ⌽ Subscripts abs am atm b C c cr cond conv CF D e f fc fd g H h i L l lat lm

volumetric thermal expansion coefficient, K⫺1 mass flow rate per unit width in film condensation, kg/s 䡠 m ratio of specific heats hydrodynamic boundary layer thickness, m thermal penetration depth, m thermal boundary layer thickness, m emissivity; porosity; heat exchanger effectiveness fin effectiveness thermodynamic efficiency; similarity variable fin efficiency overall efficiency of fin array zenith angle, rad; temperature difference, K absorption coefficient, m⫺1 wavelength, m mean free path length, nm viscosity, kg/s 䡠 m kinematic viscosity, m2/s; frequency of radiation, s⫺1 mass density, kg/m3; reflectivity electric resistivity, ⍀/m Stefan–Boltzmann constant, W/m2 䡠 K4; electrical conductivity, 1/⍀ 䡠 m; normal viscous stress, N/m2; surface tension, N/m viscous dissipation function, s⫺2 volume fraction azimuthal angle, rad stream function, m2/s shear stress, N/m2; transmissivity solid angle, sr; perfusion rate, s⫺1

absorbed arithmetic mean atmospheric base of an extended surface; blackbody Carnot cross-sectional; cold fluid; critical critical insulation thickness conduction convection counterflow diameter; drag excess; emission; electron fluid properties; fin conditions; saturated liquid conditions forced convection fully developed conditions saturated vapor conditions heat transfer conditions hydrodynamic; hot fluid; helical inner surface of an annulus; initial condition; tube inlet condition; incident radiation based on characteristic length saturated liquid conditions latent energy log mean condition

xxiii

Symbols

m max o p ph R r, ref rad S s sat sens sky

mean value over a tube cross section maximum center or midplane condition; tube outlet condition; outer momentum phonon reradiating surface reflected radiation radiation solar conditions surface conditions; solid properties; saturated solid conditions saturated conditions sensible energy sky conditions

ss sur t tr v x 앝

steady state surroundings thermal transmitted saturated vapor conditions local conditions on a surface spectral free stream conditions

Superscripts * dimensionless quantity Overbar surface average conditions; time mean

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C H A P T E R

Introduction

1

2

Chapter 1

䊏

Introduction

F

rom the study of thermodynamics, you have learned that energy can be transferred by interactions of a system with its surroundings. These interactions are called work and heat. However, thermodynamics deals with the end states of the process during which an interaction occurs and provides no information concerning the nature of the interaction or the time rate at which it occurs. The objective of this text is to extend thermodynamic analysis through the study of the modes of heat transfer and through the development of relations to calculate heat transfer rates. In this chapter we lay the foundation for much of the material treated in the text. We do so by raising several questions: What is heat transfer? How is heat transferred? Why is it important? One objective is to develop an appreciation for the fundamental concepts and principles that underlie heat transfer processes. A second objective is to illustrate the manner in which a knowledge of heat transfer may be used with the first law of thermodynamics (conservation of energy) to solve problems relevant to technology and society.

1.1 What and How? A simple, yet general, definition provides sufficient response to the question: What is heat transfer? Heat transfer (or heat) is thermal energy in transit due to a spatial temperature difference.

Whenever a temperature difference exists in a medium or between media, heat transfer must occur. As shown in Figure 1.1, we refer to different types of heat transfer processes as modes. When a temperature gradient exists in a stationary medium, which may be a solid or a fluid, we use the term conduction to refer to the heat transfer that will occur across the medium. In contrast, the term convection refers to heat transfer that will occur between a surface and a moving fluid when they are at different temperatures. The third mode of heat transfer is termed thermal radiation. All surfaces of finite temperature emit energy in the form of electromagnetic waves. Hence, in the absence of an intervening medium, there is net heat transfer by radiation between two surfaces at different temperatures.

Conduction through a solid or a stationary fluid

T1

T1 > T2

T2

Convection from a surface to a moving fluid

Net radiation heat exchange between two surfaces

Ts > T∞

Surface, T1

Moving fluid, T∞

q"

Surface, T2

q"

q"1 Ts

FIGURE 1.1

q"2

Conduction, convection, and radiation heat transfer modes.

1.2

3

Physical Origins and Rate Equations

䊏

1.2 Physical Origins and Rate Equations As engineers, it is important that we understand the physical mechanisms which underlie the heat transfer modes and that we be able to use the rate equations that quantify the amount of energy being transferred per unit time.

1.2.1

Conduction

At mention of the word conduction, we should immediately conjure up concepts of atomic and molecular activity because processes at these levels sustain this mode of heat transfer. Conduction may be viewed as the transfer of energy from the more energetic to the less energetic particles of a substance due to interactions between the particles. The physical mechanism of conduction is most easily explained by considering a gas and using ideas familiar from your thermodynamics background. Consider a gas in which a temperature gradient exists, and assume that there is no bulk, or macroscopic, motion. The gas may occupy the space between two surfaces that are maintained at different temperatures, as shown in Figure 1.2. We associate the temperature at any point with the energy of gas molecules in proximity to the point. This energy is related to the random translational motion, as well as to the internal rotational and vibrational motions, of the molecules. Higher temperatures are associated with higher molecular energies. When neighboring molecules collide, as they are constantly doing, a transfer of energy from the more energetic to the less energetic molecules must occur. In the presence of a temperature gradient, energy transfer by conduction must then occur in the direction of decreasing temperature. This would be true even in the absence of collisions, as is evident from Figure 1.2. The hypothetical plane at xo is constantly being crossed by molecules from above and below due to their random motion. However, molecules from above are associated with a higher temperature than those from below, in which case there must be a net transfer of energy in the positive x-direction. Collisions between molecules enhance this energy transfer. We may speak of the net transfer of energy by random molecular motion as a diffusion of energy. The situation is much the same in liquids, although the molecules are more closely spaced and the molecular interactions are stronger and more frequent. Similarly, in a solid, conduction may be attributed to atomic activity in the form of lattice vibrations. The modern T

xo

x

q"x

T1 > T2

q"x

T2

FIGURE 1.2 Association of conduction heat transfer with diffusion of energy due to molecular activity.

4

Chapter 1

䊏

Introduction

T

T1 q"x

T(x) T2 L

x

FIGURE 1.3 One-dimensional heat transfer by conduction (diffusion of energy).

view is to ascribe the energy transfer to lattice waves induced by atomic motion. In an electrical nonconductor, the energy transfer is exclusively via these lattice waves; in a conductor, it is also due to the translational motion of the free electrons. We treat the important properties associated with conduction phenomena in Chapter 2 and in Appendix A. Examples of conduction heat transfer are legion. The exposed end of a metal spoon suddenly immersed in a cup of hot coffee is eventually warmed due to the conduction of energy through the spoon. On a winter day, there is significant energy loss from a heated room to the outside air. This loss is principally due to conduction heat transfer through the wall that separates the room air from the outside air. Heat transfer processes can be quantified in terms of appropriate rate equations. These equations may be used to compute the amount of energy being transferred per unit time. For heat conduction, the rate equation is known as Fourier’s law. For the one-dimensional plane wall shown in Figure 1.3, having a temperature distribution T(x), the rate equation is expressed as q⬙x ⫽ ⫺ k dT dx

(1.1)

The heat flux q⬙x (W/m2) is the heat transfer rate in the x-direction per unit area perpendicular to the direction of transfer, and it is proportional to the temperature gradient, dT/dx, in this direction. The parameter k is a transport property known as the thermal conductivity (W/m 䡠 K) and is a characteristic of the wall material. The minus sign is a consequence of the fact that heat is transferred in the direction of decreasing temperature. Under the steady-state conditions shown in Figure 1.3, where the temperature distribution is linear, the temperature gradient may be expressed as dT ⫽ T2 ⫺ T1 dx L and the heat flux is then q⬙x ⫽ ⫺k

T2 ⫺ T1 L

or q⬙x ⫽ k

T1 ⫺ T2 ⫽ k ⌬T L L

(1.2)

Note that this equation provides a heat flux, that is, the rate of heat transfer per unit area. The heat rate by conduction, qx (W), through a plane wall of area A is then the product of the flux and the area, qx ⫽ q⬙x 䡠 A.

1.2

䊏

5

Physical Origins and Rate Equations

* EXAMPLE 1.1 The wall of an industrial furnace is constructed from 0.15-m-thick fireclay brick having a thermal conductivity of 1.7 W/m 䡠 K. Measurements made during steady-state operation reveal temperatures of 1400 and 1150 K at the inner and outer surfaces, respectively. What is the rate of heat loss through a wall that is 0.5 m ⫻ 1.2 m on a side?

SOLUTION Known: Steady-state conditions with prescribed wall thickness, area, thermal conductivity, and surface temperatures. Find: Wall heat loss. Schematic: W = 1.2 m H = 0.5 m k = 1.7 W/m•K T2 = 1150 K

T1 = 1400 K

qx qx''

Wall area, A

x

L = 0.15 m

x

L

Assumptions: 1. Steady-state conditions. 2. One-dimensional conduction through the wall. 3. Constant thermal conductivity. Analysis: Since heat transfer through the wall is by conduction, the heat flux may be determined from Fourier’s law. Using Equation 1.2, we have q⬙x ⫽ k ⌬T ⫽ 1.7 W/m 䡠 K ⫻ 250 K ⫽ 2833 W/m2 L 0.15 m The heat flux represents the rate of heat transfer through a section of unit area, and it is uniform (invariant) across the surface of the wall. The heat loss through the wall of area A ⫽ H ⫻ W is then qx ⫽ (HW ) q⬙x ⫽ (0.5 m ⫻ 1.2 m) 2833 W/m2 ⫽1700 W

䉰

Comments: Note the direction of heat flow and the distinction between heat flux and heat rate. *This icon identifies examples that are available in tutorial form in the Interactive Heat Transfer (IHT) software that accompanies the text. Each tutorial is brief and illustrates a basic function of the software. IHT can be used to solve simultaneous equations, perform parameter sensitivity studies, and graph the results. Use of IHT will reduce the time spent solving more complex end-of-chapter problems.

6

Chapter 1

1.2.2

䊏

Introduction

Convection

The convection heat transfer mode is comprised of two mechanisms. In addition to energy transfer due to random molecular motion (diffusion), energy is also transferred by the bulk, or macroscopic, motion of the fluid. This fluid motion is associated with the fact that, at any instant, large numbers of molecules are moving collectively or as aggregates. Such motion, in the presence of a temperature gradient, contributes to heat transfer. Because the molecules in the aggregate retain their random motion, the total heat transfer is then due to a superposition of energy transport by the random motion of the molecules and by the bulk motion of the fluid. The term convection is customarily used when referring to this cumulative transport, and the term advection refers to transport due to bulk fluid motion. We are especially interested in convection heat transfer, which occurs between a fluid in motion and a bounding surface when the two are at different temperatures. Consider fluid flow over the heated surface of Figure 1.4. A consequence of the fluid–surface interaction is the development of a region in the fluid through which the velocity varies from zero at the surface to a finite value u앝 associated with the flow. This region of the fluid is known as the hydrodynamic, or velocity, boundary layer. Moreover, if the surface and flow temperatures differ, there will be a region of the fluid through which the temperature varies from Ts at y ⫽ 0 to T앝 in the outer flow. This region, called the thermal boundary layer, may be smaller, larger, or the same size as that through which the velocity varies. In any case, if Ts ⬎ T앝, convection heat transfer will occur from the surface to the outer flow. The convection heat transfer mode is sustained both by random molecular motion and by the bulk motion of the fluid within the boundary layer. The contribution due to random molecular motion (diffusion) dominates near the surface where the fluid velocity is low. In fact, at the interface between the surface and the fluid (y ⫽ 0), the fluid velocity is zero, and heat is transferred by this mechanism only. The contribution due to bulk fluid motion originates from the fact that the boundary layer grows as the flow progresses in the x-direction. In effect, the heat that is conducted into this layer is swept downstream and is eventually transferred to the fluid outside the boundary layer. Appreciation of boundary layer phenomena is essential to understanding convection heat transfer. For this reason, the discipline of fluid mechanics will play a vital role in our later analysis of convection. Convection heat transfer may be classified according to the nature of the flow. We speak of forced convection when the flow is caused by external means, such as by a fan, a pump, or atmospheric winds. As an example, consider the use of a fan to provide forced convection air cooling of hot electrical components on a stack of printed circuit boards (Figure 1.5a). In contrast, for free (or natural) convection, the flow is induced by buoyancy forces, which are due to density differences caused by temperature variations in the fluid. An example is the free convection heat transfer that occurs from hot components on a vertical array of circuit

y

y

Fluid

u∞

Velocity distribution u(y)

q"

T∞

Temperature distribution T(y) Ts x

u(y)

Heated surface

T(y)

FIGURE 1.4 Boundary layer development in convection heat transfer.

1.2

䊏

7

Physical Origins and Rate Equations

boards in air (Figure 1.5b). Air that makes contact with the components experiences an increase in temperature and hence a reduction in density. Since it is now lighter than the surrounding air, buoyancy forces induce a vertical motion for which warm air ascending from the boards is replaced by an inflow of cooler ambient air. While we have presumed pure forced convection in Figure 1.5a and pure natural convection in Figure 1.5b, conditions corresponding to mixed (combined) forced and natural convection may exist. For example, if velocities associated with the flow of Figure 1.5a are small and/or buoyancy forces are large, a secondary flow that is comparable to the imposed forced flow could be induced. In this case, the buoyancy-induced flow would be normal to the forced flow and could have a significant effect on convection heat transfer from the components. In Figure 1.5b, mixed convection would result if a fan were used to force air upward between the circuit boards, thereby assisting the buoyancy flow, or downward, thereby opposing the buoyancy flow. We have described the convection heat transfer mode as energy transfer occurring within a fluid due to the combined effects of conduction and bulk fluid motion. Typically, the energy that is being transferred is the sensible, or internal thermal, energy of the fluid. However, for some convection processes, there is, in addition, latent heat exchange. This latent heat exchange is generally associated with a phase change between the liquid and vapor states of the fluid. Two special cases of interest in this text are boiling and condensation. For example, convection heat transfer results from fluid motion induced by vapor bubbles generated at the bottom of a pan of boiling water (Figure 1.5c) or by the condensation of water vapor on the outer surface of a cold water pipe (Figure 1.5d).

Buoyancy-driven flow Forced flow

q'' Hot components on printed circuit boards

Air

q''

Air (a)

(b)

Moist air

q'' Cold water Vapor bubbles

q" Water

Hot plate (c)

(d)

FIGURE 1.5 Convection heat transfer processes. (a) Forced convection. (b) Natural convection. (c) Boiling. (d) Condensation.

Water droplets

8

Chapter 1

䊏

Introduction

TABLE 1.1 Typical values of the convection heat transfer coefficient h (W/m2 䡠 K)

Process Free convection Gases Liquids Forced convection Gases Liquids Convection with phase change Boiling or condensation

2–25 50–1000 25–250 100–20,000 2500–100,000

Regardless of the nature of the convection heat transfer process, the appropriate rate equation is of the form q⬙ ⫽ h(Ts ⫺ T앝)

(1.3a)

where q⬙, the convective heat flux (W/m2), is proportional to the difference between the surface and fluid temperatures, Ts and T앝, respectively. This expression is known as Newton’s law of cooling, and the parameter h (W/m2 䡠 K) is termed the convection heat transfer coefficient. This coefficient depends on conditions in the boundary layer, which are influenced by surface geometry, the nature of the fluid motion, and an assortment of fluid thermodynamic and transport properties. Any study of convection ultimately reduces to a study of the means by which h may be determined. Although consideration of these means is deferred to Chapter 6, convection heat transfer will frequently appear as a boundary condition in the solution of conduction problems (Chapters 2 through 5). In the solution of such problems we presume h to be known, using typical values given in Table 1.1. When Equation 1.3a is used, the convection heat flux is presumed to be positive if heat is transferred from the surface (Ts ⬎ T앝) and negative if heat is transferred to the surface (T앝 ⬎ Ts). However, nothing precludes us from expressing Newton’s law of cooling as q⬙ ⫽ h(T앝 ⫺ Ts)

(1.3b)

in which case heat transfer is positive if it is to the surface.

1.2.3

Radiation

Thermal radiation is energy emitted by matter that is at a nonzero temperature. Although we will focus on radiation from solid surfaces, emission may also occur from liquids and gases. Regardless of the form of matter, the emission may be attributed to changes in the electron configurations of the constituent atoms or molecules. The energy of the radiation field is transported by electromagnetic waves (or alternatively, photons). While the transfer of energy by conduction or convection requires the presence of a material medium, radiation does not. In fact, radiation transfer occurs most efficiently in a vacuum. Consider radiation transfer processes for the surface of Figure 1.6a. Radiation that is emitted by the surface originates from the thermal energy of matter bounded by the surface,

1.2

䊏

9

Physical Origins and Rate Equations

and the rate at which energy is released per unit area (W/m2) is termed the surface emissive power, E. There is an upper limit to the emissive power, which is prescribed by the Stefan–Boltzmann law Eb ⫽ T s4

(1.4)

where Ts is the absolute temperature (K) of the surface and is the Stefan– Boltzmann constant ( ⫽ 5.67 ⫻ 10⫺8 W/m2 䡠 K4). Such a surface is called an ideal radiator or blackbody. The heat flux emitted by a real surface is less than that of a blackbody at the same temperature and is given by E ⫽ T 4s

(1.5)

where is a radiative property of the surface termed the emissivity. With values in the range 0 ⱕ ⱕ 1, this property provides a measure of how efficiently a surface emits energy relative to a blackbody. It depends strongly on the surface material and finish, and representative values are provided in Appendix A. Radiation may also be incident on a surface from its surroundings. The radiation may originate from a special source, such as the sun, or from other surfaces to which the surface of interest is exposed. Irrespective of the source(s), we designate the rate at which all such radiation is incident on a unit area of the surface as the irradiation G (Figure 1.6a). A portion, or all, of the irradiation may be absorbed by the surface, thereby increasing the thermal energy of the material. The rate at which radiant energy is absorbed per unit surface area may be evaluated from knowledge of a surface radiative property termed the absorptivity ␣. That is, Gabs ⫽ ␣G (1.6) where 0 ⱕ ␣ ⱕ 1. If ␣ ⬍ 1 and the surface is opaque, portions of the irradiation are reflected. If the surface is semitransparent, portions of the irradiation may also be transmitted. However, whereas absorbed and emitted radiation increase and reduce, respectively, the thermal energy of matter, reflected and transmitted radiation have no effect on this energy. Note that the value of ␣ depends on the nature of the irradiation, as well as on the surface itself. For example, the absorptivity of a surface to solar radiation may differ from its absorptivity to radiation emitted by the walls of a furnace.

G

Gas

Gas

T, h

T, h

E q"conv

Surface of emissivity , absorptivity ␣, and temperature Ts (a)

Surroundings at Tsur

q"rad

Surface of emissivity = ␣ , area A, and temperature Ts

q"conv

Ts > Tsur, Ts > T

(b)

FIGURE 1.6 Radiation exchange: (a) at a surface and (b) between a surface and large surroundings.

10

Chapter 1

䊏

Introduction

In many engineering problems (a notable exception being problems involving solar radiation or radiation from other very high temperature sources), liquids can be considered opaque to radiation heat transfer, and gases can be considered transparent to it. Solids can be opaque (as is the case for metals) or semitransparent (as is the case for thin sheets of some polymers and some semiconducting materials). A special case that occurs frequently involves radiation exchange between a small surface at Ts and a much larger, isothermal surface that completely surrounds the smaller one (Figure 1.6b). The surroundings could, for example, be the walls of a room or a furnace whose temperature Tsur differs from that of an enclosed surface (Tsur ⫽ Ts). We will show in Chapter 12 that, for such a condition, the irradiation may be approximated by emission from 4 . If the surface is assumed to be one for which a blackbody at Tsur, in which case G ⫽ T sur ␣ ⫽ (a gray surface), the net rate of radiation heat transfer from the surface, expressed per unit area of the surface, is q⬙rad ⫽

q 4 ) ⫽ Eb(Ts ) ⫺ ␣G ⫽ (T 4s ⫺ Tsur A

(1.7)

This expression provides the difference between thermal energy that is released due to radiation emission and that gained due to radiation absorption. For many applications, it is convenient to express the net radiation heat exchange in the form qrad ⫽ hr A(Ts ⫺ Tsur) (1.8) where, from Equation 1.7, the radiation heat transfer coefficient hr is 2 hr (Ts ⫹ Tsur)(Ts2 ⫹ Tsur )

(1.9)

Here we have modeled the radiation mode in a manner similar to convection. In this sense we have linearized the radiation rate equation, making the heat rate proportional to a temperature difference rather than to the difference between two temperatures to the fourth power. Note, however, that hr depends strongly on temperature, whereas the temperature dependence of the convection heat transfer coefficient h is generally weak. The surfaces of Figure 1.6 may also simultaneously transfer heat by convection to an adjoining gas. For the conditions of Figure 1.6b, the total rate of heat transfer from the surface is then 4 ) q ⫽ qconv ⫹ qrad ⫽ hA(Ts ⫺ T앝) ⫹ A(Ts4 ⫺ Tsur (1.10)

EXAMPLE 1.2 An uninsulated steam pipe passes through a room in which the air and walls are at 25⬚C. The outside diameter of the pipe is 70 mm, and its surface temperature and emissivity are 200⬚C and 0.8, respectively. What are the surface emissive power and irradiation? If the coefficient associated with free convection heat transfer from the surface to the air is 15 W/m2 䡠 K, what is the rate of heat loss from the surface per unit length of pipe?

SOLUTION Known: Uninsulated pipe of prescribed diameter, emissivity, and surface temperature in a room with fixed wall and air temperatures.

1.2

䊏

11

Physical Origins and Rate Equations

Find: 1. Surface emissive power and irradiation. 2. Pipe heat loss per unit length, q⬘. Schematic:

Air

q'

T∞ = 25°C h = 15 W/m2•K

E L Ts = 200°C ε = 0.8 G Tsur = 25°C

D = 70 mm

Assumptions: 1. Steady-state conditions. 2. Radiation exchange between the pipe and the room is between a small surface and a much larger enclosure. 3. The surface emissivity and absorptivity are equal. Analysis: 1. The surface emissive power may be evaluated from Equation 1.5, while the irradiation 4 corresponds to G ⫽ Tsur . Hence E ⫽ Ts4 ⫽ 0.8(5.67 ⫻ 10⫺8 W/m2 䡠 K4)(473 K)4 ⫽ 2270 W/m2 G⫽

4 T sur

⫺8

⫽ 5.67 ⫻ 10

W/m 䡠 K (298 K) ⫽ 447 W/m 2

4

4

2

䉰 䉰

2. Heat loss from the pipe is by convection to the room air and by radiation exchange with the walls. Hence, q ⫽ qconv ⫹ qrad and from Equation 1.10, with A ⫽ DL, 4 ) q ⫽ h(DL)(Ts ⫺ T앝) ⫹ (DL)(T 4s ⫺ Tsur

The heat loss per unit length of pipe is then q⬘ ⫽

q ⫽ 15 W/m2 䡠 K( ⫻ 0.07 m)(200 ⫺ 25)⬚C L ⫹ 0.8( ⫻ 0.07 m) 5.67 ⫻ 10⫺8 W/m2 䡠 K4 (4734 ⫺ 2984) K4 q⬘ ⫽ 577 W/m ⫹ 421 W/m ⫽ 998 W/m

䉰

Comments: 1. Note that temperature may be expressed in units of ⬚C or K when evaluating the temperature difference for a convection (or conduction) heat transfer rate. However, temperature must be expressed in kelvins (K) when evaluating a radiation transfer rate.

12

Chapter 1

䊏

Introduction

2. The net rate of radiation heat transfer from the pipe may be expressed as q⬘rad ⫽ D (E ⫺ ␣G) q⬘rad ⫽ ⫻ 0.07 m (2270 ⫺ 0.8 ⫻ 447) W/m2 ⫽ 421 W/m 3. In this situation, the radiation and convection heat transfer rates are comparable because Ts is large compared to Tsur and the coefficient associated with free convection is small. For more moderate values of Ts and the larger values of h associated with forced convection, the effect of radiation may often be neglected. The radiation heat transfer coefficient may be computed from Equation 1.9. For the conditions of this problem, its value is hr ⫽ 11 W/m2 䡠 K.

1.2.4 The Thermal Resistance Concept The three modes of heat transfer were introduced in the preceding sections. As is evident from Equations 1.2, 1.3, and 1.8, the heat transfer rate can be expressed in the form q ⫽ q⬙A ⫽ ⌬T Rt

(1.11)

where ⌬T is a relevant temperature difference and A is the area normal to the direction of heat transfer. The quantity Rt is called a thermal resistance and takes different forms for the three different modes of heat transfer. For example, Equation 1.2 may be multiplied by the area A and rewritten as qx ⫽ ⌬T/Rt,c , where Rt,c ⫽ L /kA is a thermal resistance associated with conduction, having the units K/W. The thermal resistance concept will be considered in detail in Chapter 3 and will be seen to have great utility in solving complex heat transfer problems.

1.3 Relationship to Thermodynamics The subjects of heat transfer and thermodynamics are highly complementary and interrelated, but they also have fundamental differences. If you have taken a thermodynamics course, you are aware that heat exchange plays a vital role in the first and second laws of thermodynamics because it is one of the primary mechanisms for energy transfer between a system and its surroundings. While thermodynamics may be used to determine the amount of energy required in the form of heat for a system to pass from one state to another, it considers neither the mechanisms that provide for heat exchange nor the methods that exist for computing the rate of heat exchange. The discipline of heat transfer specifically seeks to quantify the rate at which heat is exchanged through the rate equations expressed, for example, by Equations 1.2, 1.3, and 1.7. Indeed, heat transfer principles often enable the engineer to implement the concepts of thermodynamics. For example, the actual size of a power plant to be constructed cannot be determined from thermodynamics alone; the principles of heat transfer must also be invoked at the design stage. The remainder of this section considers the relationship of heat transfer to thermodynamics. Since the first law of thermodynamics (the law of conservation of energy) provides a useful, often essential, starting point for the solution of heat transfer problems, Section 1.3.1 will provide a development of the general formulations of the first law. The ideal

1.3

䊏

13

Relationship to Thermodynamics

(Carnot) efficiency of a heat engine, as determined by the second law of thermodynamics will be reviewed in Section 1.3.2. It will be shown that a realistic description of the heat transfer between a heat engine and its surroundings further limits the actual efficiency of a heat engine.

1.3.1 Relationship to the First Law of Thermodynamics (Conservation of Energy) At its heart, the first law of thermodynamics is simply a statement that the total energy of a system is conserved, and therefore the only way that the amount of energy in a system can change is if energy crosses its boundaries. The first law also addresses the ways in which energy can cross the boundaries of a system. For a closed system (a region of fixed mass), there are only two ways: heat transfer through the boundaries and work done on or by the system. This leads to the following statement of the first law for a closed system, which is familiar if you have taken a course in thermodynamics: ⌬Esttot ⫽ Q ⫺ W

(1.12a)

where ⌬Esttot is the change in the total energy stored in the system, Q is the net heat transferred to the system, and W is the net work done by the system. This is schematically illustrated in Figure 1.7a. The first law can also be applied to a control volume (or open system), a region of space bounded by a control surface through which mass may pass. Mass entering and leaving the control volume carries energy with it; this process, termed energy advection, adds a third way in which energy can cross the boundaries of a control volume. To summarize, the first law of thermodynamics can be very simply stated as follows for both a control volume and a closed system. First Law of Thermodynamics over a Time Interval (⌬t) The increase in the amount of energy stored in a control volume must equal the amount of energy that enters the control volume, minus the amount of energy that leaves the control volume.

In applying this principle, it is recognized that energy can enter and leave the control volume due to heat transfer through the boundaries, work done on or by the control volume, and energy advection. The first law of thermodynamics addresses total energy, which consists of kinetic and potential energies (together known as mechanical energy) and internal energy. Internal energy can be further subdivided into thermal energy (which will be defined more carefully later) W Q •

tot ∆ Est

E in

•

•

E g, E st •

E out

(a)

(b)

FIGURE 1.7 Conservation of energy: (a) for a closed system over a time interval and (b) for a control volume at an instant.

14

Chapter 1

䊏

Introduction

and other forms of internal energy, such as chemical and nuclear energy. For the study of heat transfer, we wish to focus attention on the thermal and mechanical forms of energy. We must recognize that the sum of thermal and mechanical energy is not conserved, because conversion can occur between other forms of energy and thermal or mechanical energy. For example, if a chemical reaction occurs that decreases the amount of chemical energy in the system, it will result in an increase in the thermal energy of the system. If an electric motor operates within the system, it will cause conversion from electrical to mechanical energy. We can think of such energy conversions as resulting in thermal or mechanical energy generation (which can be either positive or negative). So a statement of the first law that is well suited for heat transfer analysis is: Thermal and Mechanical Energy Equation over a Time Interval (⌬t) The increase in the amount of thermal and mechanical energy stored in the control volume must equal the amount of thermal and mechanical energy that enters the control volume, minus the amount of thermal and mechanical energy that leaves the control volume, plus the amount of thermal and mechanical energy that is generated within the control volume.

This expression applies over a time interval ⌬t, and all the energy terms are measured in joules. Since the first law must be satisfied at each and every instant of time t, we can also formulate the law on a rate basis. That is, at any instant, there must be a balance between all energy rates, as measured in joules per second (W). In words, this is expressed as follows: Thermal and Mechanical Energy Equation at an Instant (t) The rate of increase of thermal and mechanical energy stored in the control volume must equal the rate at which thermal and mechanical energy enters the control volume, minus the rate at which thermal and mechanical energy leaves the control volume, plus the rate at which thermal and mechanical energy is generated within the control volume.

If the inflow and generation of thermal and mechanical energy exceed the outflow, the amount of thermal and mechanical energy stored (accumulated) in the control volume must increase. If the converse is true, thermal and mechanical energy storage must decrease. If the inflow and generation equal the outflow, a steady-state condition must prevail such that there will be no change in the amount of thermal and mechanical energy stored in the control volume. We will now define symbols for each of the energy terms so that the boxed statements can be rewritten as equations. We let E stand for the sum of thermal and mechanical energy (in contrast to the symbol Etot for total energy). Using the subscript st to denote energy stored in the control volume, the change in thermal and mechanical energy stored over the time interval ⌬t is then ⌬Est. The subscripts in and out refer to energy entering and leaving the control volume. Finally, thermal and mechanical energy generation is given the symbol Eg. Thus, the first boxed statement can be written as: ⌬Est ⫽ Ein ⫺ Eout ⫹ Eg

(1.12b)

Next, using a dot over a term to indicate a rate, the second boxed statement becomes: dE E˙ st st ⫽ E˙ in ⫺ E˙ out ⫹ E˙ g dt

(1.12c)

1.3

䊏

Relationship to Thermodynamics

15

This expression is illustrated schematically in Figure 1.7b. Equations 1.12b,c provide important and, in some cases, essential tools for solving heat transfer problems. Every application of the first law must begin with the identification of an appropriate control volume and its control surface, to which an analysis is subsequently applied. The first step is to indicate the control surface by drawing a dashed line. The second step is to decide whether to perform the analysis for a time interval ⌬t (Equation 1.12b) or on a rate basis (Equation 1.12c). This choice depends on the objective of the solution and on how information is given in the problem. The next step is to identify the energy terms that are relevant in the problem you are solving. To develop your confidence in taking this last step, the remainder of this section is devoted to clarifying the following energy terms: • Stored thermal and mechanical energy, Est. • Thermal and mechanical energy generation, Eg. • Thermal and mechanical energy transport across the control surfaces, that is, the inflow and outflow terms, Ein and Eout. In the statement of the first law (Equation 1.12a), the total energy, E tot, consists of kinetic energy (KE ⫽ 1⁄2mV 2, where m and V are mass and velocity, respectively), potential energy (PE ⫽ mgz, where g is the gravitational acceleration and z is the vertical coordinate), and internal energy (U). Mechanical energy is defined as the sum of kinetic and potential energy. Most often in heat transfer problems, the changes in kinetic and potential energy are small and can be neglected. The internal energy consists of a sensible component, which accounts for the translational, rotational, and/or vibrational motion of the atoms/molecules comprising the matter; a latent component, which relates to intermolecular forces influencing phase change between solid, liquid, and vapor states; a chemical component, which accounts for energy stored in the chemical bonds between atoms; and a nuclear component, which accounts for the binding forces in the nucleus. For the study of heat transfer, we focus attention on the sensible and latent components of the internal energy (Usens and Ulat, respectively), which are together referred to as thermal energy, Ut. The sensible energy is the portion that we associate mainly with changes in temperature (although it can also depend on pressure). The latent energy is the component we associate with changes in phase. For example, if the material in the control volume changes from solid to liquid (melting) or from liquid to vapor (vaporization, evaporation, boiling), the latent energy increases. Conversely, if the phase change is from vapor to liquid (condensation) or from liquid to solid (solidification, freezing), the latent energy decreases. Obviously, if no phase change is occurring, there is no change in latent energy, and this term can be neglected. Based on this discussion, the stored thermal and mechanical energy is given by Est ⫽ KE ⫹ PE ⫹ Ut, where Ut ⫽ Usens ⫹ Ulat. In many problems, the only relevant energy term will be the sensible energy, that is, Est ⫽ Usens. The energy generation term is associated with conversion from some other form of internal energy (chemical, electrical, electromagnetic, or nuclear) to thermal or mechanical energy. It is a volumetric phenomenon. That is, it occurs within the control volume and is generally proportional to the magnitude of this volume. For example, an exothermic chemical reaction may be occurring, converting chemical energy to thermal energy. The net effect is an increase in the thermal energy of the matter within the control volume. Another source of thermal energy is the conversion from electrical energy that occurs due to resistance heating when an electric current is passed through a conductor. That is, if an electric current I passes through a resistance R in the control volume, electrical energy is dissipated at a rate I2R, which corresponds to the rate at which thermal energy is generated (released)

16

Chapter 1

䊏

Introduction

within the volume. In all applications of interest in this text, if chemical, electrical, or nuclear effects exist, they are treated as sources (or sinks, which correspond to negative sources) of thermal or mechanical energy and hence are included in the generation terms of Equations 1.12b,c. The inflow and outflow terms are surface phenomena. That is, they are associated exclusively with processes occurring at the control surface and are generally proportional to the surface area. As discussed previously, the energy inflow and outflow terms include heat transfer (which can be by conduction, convection, and/or radiation) and work interactions occurring at the system boundaries (e.g., due to displacement of a boundary, a rotating shaft, and/or electromagnetic effects). For cases in which mass crosses the control volume boundary (e.g., for situations involving fluid flow), the inflow and outflow terms also include energy (thermal and mechanical) that is advected (carried) by mass entering and leaving the . control volume. For instance, if the mass flow rate entering through the boundary is m , then . the rate at which thermal and mechanical energy enters with the flow is m (ut ⫹ 1⁄2V 2 ⫹ gz), where ut is the thermal energy per unit mass. When the first law is applied to a control volume with fluid crossing its boundary, it is customary to divide the work term into two contributions. The first contribution, termed flow work, is associated with work done by pressure forces moving fluid through the boundary. For a unit mass, the amount of work is equivalent to the product of the pressure ˙ is traditionally used for the rate at and the specific volume of the fluid ( pv). The symbol W which the remaining work (not including flow work) is perfomed. If operation is under steady-state conditions (dEst /dt ⫽ 0) and if there is no thermal or mechanical energy generation, Equation 1.12c reduces to the following form of the steady-flow energy equation (see Figure 1.8), which will be familiar if you have taken a thermodynamics course: ˙ ⫽0 m˙ (ut ⫹ pv ⫹ 1⁄2 V 2 ⫹ gz)in ⫺ m˙ (ut ⫹ pv ⫹ 1⁄2 V 2 ⫹ gz)out ⫹ q ⫺ W

(1.12d)

Terms within the parentheses are expressed for a unit mass of fluid at the inflow and outflow locations. When multiplied by the mass flow rate m˙ , they yield the rate at which the corresponding form of the energy (thermal, flow work, kinetic, and potential) enters or leaves the control volume. The sum of thermal energy and flow work per unit mass may be replaced by the enthalpy per unit mass, i ⫽ ut ⫹ pv. In most open system applications of interest in this text, changes in latent energy between the inflow and outflow conditions of Equation 1.12d may be neglected, so the thermal energy reduces to only the sensible component. If the fluid is approximated as an ideal gas with constant specific heats, the difference in enthalpies (per unit mass) between the inlet and outlet flows may then be expressed as (iin ⫺ iout) ⫽ cp(Tin ⫺ Tout), where cp is

q zout

(ut , pv, V)in

zin

(ut , pv, V)out

•

W

Reference height

FIGURE 1.8 Conservation of energy for a steady-flow, open system.

1.3

䊏

17

Relationship to Thermodynamics

the specific heat at constant pressure and Tin and Tout are the inlet and outlet temperatures, respectively. If the fluid is an incompressible liquid, its specific heats at constant pressure and volume are equal, cp ⫽ cv c, and for Equation 1.12d the change in sensible energy (per unit mass) reduces to (ut,in ⫺ ut,out) ⫽ c(Tin ⫺ Tout). Unless the pressure drop is extremely large, the difference in flow work terms, (pv)in ⫺ (pv)out, is negligible for a liquid. Having already assumed steady-state conditions, no changes in latent energy, and no thermal or mechanical energy generation, there are at least four cases in which further assumptions can be made to reduce Equation 1.12d to the simplified steady-flow thermal energy equation: q ⫽ m˙ cp(Tout ⫺ Tin)

(1.12e)

The right-hand side of Equation 1.12e represents the net rate of outflow of enthalpy (thermal energy plus flow work) for an ideal gas or of thermal energy for an incompressible liquid. The first two cases for which Equation 1.12e holds can readily be verified by examining Equation 1.12d. They are: 1. An ideal gas with negligible kinetic and potential energy changes and negligible work (other than flow work). 2. An incompressible liquid with negligible kinetic and potential energy changes and negligible work, including flow work. As noted in the preceding discussion, flow work is negligible for an incompressible liquid provided the pressure variation is not too great. The second pair of cases cannot be directly derived from Equation 1.12d but require further knowledge of how mechanical energy is converted into thermal energy. These cases are: 3. An ideal gas with negligible viscous dissipation and negligible pressure variation. 4. An incompressible liquid with negligible viscous dissipation. Viscous dissipation is the conversion from mechanical energy to thermal energy associated with viscous forces acting in a fluid. It is important only in cases involving high-speed flow and/or highly viscous fluid. Since so many engineering applications satisfy one or more of the preceding four conditions, Equation 1.12e is commonly used for the analysis of heat transfer in moving fluids. It will be used in Chapter 8 in the study of convection heat transfer in internal flow. The mass flow rate m˙ of the fluid may be expressed as m˙ ⫽ VAc, where is the fluid density and Ac is the cross-sectional area of the channel through which the fluid flows. The volumetric flow rate is simply ᭙˙ ⫽ VAc ⫽ m˙ /.

EXAMPLE 1.3 The blades of a wind turbine turn a large shaft at a relatively slow speed. The rotational speed is increased by a gearbox that has an efficiency of gb ⫽ 0.93. In turn, the gearbox output shaft drives an electric generator with an efficiency of gen ⫽ 0.95. The cylindrical nacelle, which houses the gearbox, generator, and associated equipment, is of length L ⫽ 6 m and diameter D ⫽ 3 m. If the turbine produces P ⫽ 2.5 MW of electrical power, and the air and surroundings temperatures are T앝 ⫽ 25⬚C and Tsur ⫽ 20⬚C, respectively, determine the minimum possible operating temperature inside the nacelle. The emissivity of the nacelle is ⫽ 0.83,

18

Chapter 1

䊏

Introduction

and the convective heat transfer coefficient is h ⫽ 35 W/m2 䡠 K. The surface of the nacelle that is adjacent to the blade hub can be considered to be adiabatic, and solar irradiation may be neglected.

Tsur 20°C h 35 W/m2·K L6m Air

D3m

T∞ 25°C Ts ,ε 0.83 Generator, ηgen 0.95 Gearbox, ηgb 0.93 Nacelle

Hub

SOLUTION Known: Electrical power produced by a wind turbine. Gearbox and generator efficiencies, dimensions and emissivity of the nacelle, ambient and surrounding temperatures, and heat transfer coefficient. Find: Minimum possible temperature inside the enclosed nacelle. Schematic:

Tsur 20°C

Air T∞ 25°C h 35 W/m2·K

qrad qconv

L6m • Eg

D3m

Ts ε 0.83 ηgen 0.95 ηgb 0.93

Assumptions: 1. Steady-state conditions. 2. Large surroundings. 3. Surface of the nacelle that is adjacent to the hub is adiabatic.

1.3

䊏

19

Relationship to Thermodynamics

Analysis: The nacelle temperature represents the minimum possible temperature inside the nacelle, and the first law of thermodynamics may be used to determine this temperature. The first step is to perform an energy balance on the nacelle to determine the rate of heat transfer from the nacelle to the air and surroundings under steady-state conditions. This step can be accomplished using either conservation of total energy or conservation of thermal and mechanical energy; we will compare these two approaches. Conservation of Total Energy The first of the three boxed statements of the first law in Section 1.3 can be converted to a rate basis and expressed in equation form as follows: dEsttot ˙ tot ˙ tot ⫽ E in ⫺ Eout dt

(1)

˙ tot ˙ tot Under steady-state conditions, this reduces to E˙ tot in ⫺ Eout ⫽ 0. The E in term corresponds to tot the mechanical work entering the nacelle W˙ , and the E˙out term includes the electrical power output P and the rate of heat transfer leaving the nacelle q. Thus W˙ ⫺ P ⫺ q ⫽ 0

(2)

Conservation of Thermal and Mechanical Energy Alternatively, we can express conservation of thermal and mechanical energy, starting with Equation 1.12c. Under steady-state conditions, this reduces to E˙ in ⫺ E˙ out ⫹ E˙ g ⫽ 0

(3)

Here, E˙ in once again corresponds to the mechanical work W˙ . However, E˙ out now includes only the rate of heat transfer leaving the nacelle q. It does not include the electrical power, since E represents only the thermal and mechanical forms of energy. The electrical power appears in the generation term, because mechanical energy is converted to electrical energy in the generator, giving rise to a negative source of mechanical energy. That is, E˙g ⫽ ⫺P. Thus, Equation (3) becomes W˙ ⫺ q ⫺ P ⫽ 0

(4)

which is equivalent to Equation (2), as it must be. Regardless of the manner in which the first law of thermodynamics is applied, the following expression for the rate of heat transfer evolves: q ⫽ W˙ ⫺ P

(5)

The mechanical work and electrical power are related by the efficiencies of the gearbox and generator, P ⫽ W˙ gbgen

(6)

Equation (5) can therefore be written as

1 ⫺ 1 ⫽ 0.33 ⫻ 106 W q ⫽ P 1 ⫺ 1 ⫽ 2.5 ⫻ 106 W ⫻ gb gen 0.93 ⫻ 0.95

(7)

Application of the Rate Equations Heat transfer is due to convection and radiation from the exterior surface of the nacelle, governed by Equations 1.3a and 1.7, respectively. Thus

20

Chapter 1

䊏

Introduction

q ⫽ qrad ⫹ qconv⫽ A[q⬙rad ⫹ q⬙conv]

⫽ DL ⫹ D 4

[(T ⫺ T

2

4 s

4 sur)

⫹ h(Ts ⫺ T앝)] ⫽ 0.33 ⫻ 106 W

or

⫻3m⫻6m⫹

⫻ (3 m)2 4

⫻ [0.83 ⫻ 5.67 ⫻ 10⫺8 W/m2 䡠 K4 (Ts4 ⫺ (273 ⫹ 20)4)K4 ⫹ 35 W/m2 䡠 K (Ts ⫺ (273 ⫹ 25)K)] ⫽ 0.33 ⫻ 106 W The preceding equation does not have a closed-form solution, but the surface temperature can be easily determined by trial and error or by using a software package such as the Interactive Heat Transfer (IHT) software accompanying your text. Doing so yields Ts ⫽ 416 K ⫽ 143⬚C We know that the temperature inside the nacelle must be greater than the exterior surface temperature of the nacelle Ts, because the heat generated within the nacelle must be transferred from the interior of the nacelle to its surface, and from the surface to the air and surroundings. Therefore, Ts represents the minimum possible temperature inside the enclosed nacelle. 䉰

Comments: 1. The temperature inside the nacelle is very high. This would preclude, for example, performance of routine maintenance by a worker, as illustrated in the problem statement. Thermal management approaches involving fans or blowers must be employed to reduce the temperature to an acceptable level. 2. Improvements in the efficiencies of either the gearbox or the generator would not only provide more electrical power, but would also reduce the size and cost of the thermal management hardware. As such, improved efficiencies would increase revenue generated by the wind turbine and decrease both its capital and operating costs. 3. The heat transfer coefficient would not be a steady value but would vary periodically as the blades sweep past the nacelle. Therefore, the value of the heat transfer coefficient represents a time-averaged quantity.

EXAMPLE 1.4 A long conducting rod of diameter D and electrical resistance per unit length R⬘e is initially in thermal equilibrium with the ambient air and its surroundings. This equilibrium is disturbed when an electrical current I is passed through the rod. Develop an equation that could be used to compute the variation of the rod temperature with time during the passage of the current.

1.3

䊏

21

Relationship to Thermodynamics

SOLUTION Known: Temperature of a rod of prescribed diameter and electrical resistance changes with time due to passage of an electrical current. Find: Equation that governs temperature change with time for the rod. Schematic: Air

•

E out

T∞, h

Tsur

T

I

•

•

E g, E st

Diameter D

L

Assumptions: 1. At any time t, the temperature of the rod is uniform. 2. Constant properties (r, c, ⫽ a). 3. Radiation exchange between the outer surface of the rod and the surroundings is between a small surface and a large enclosure. Analysis: The first law of thermodynamics may often be used to determine an unknown temperature. In this case, there is no mechanical energy component. So relevant terms include heat transfer by convection and radiation from the surface, thermal energy generation due to ohmic heating within the conductor, and a change in thermal energy storage. Since we wish to determine the rate of change of the temperature, the first law should be applied at an instant of time. Hence, applying Equation 1.12c to a control volume of length L about the rod, it follows that E˙g ⫺ E˙out ⫽ E˙ st where thermal energy generation is due to the electric resistance heating, E˙ g ⫽ I 2R⬘e L Heating occurs uniformly within the control volume and could also be expressed in terms of a volumetric heat generation rate q˙(W/m3). The generation rate for the entire control volume is then E˙g ⫽ q˙V, where q˙ ⫽ I 2R⬘e /(D2/4). Energy outflow is due to convection and net radiation from the surface, Equations 1.3a and 1.7, respectively, 4 ) E˙out ⫽ h(DL)(T ⫺ T앝) ⫹ (DL)(T 4 ⫺ Tsur

and the change in energy storage is due to the temperature change, dU E˙st ⫽ t ⫽ d (VcT) dt dt The term E˙st is associated with the rate of change in the internal thermal energy of the rod, where and c are the mass density and the specific heat, respectively, of the rod material,

Chapter 1

䊏

Introduction

and V is the volume of the rod, V ⫽ (D2/4)L. Substituting the rate equations into the energy balance, it follows that

2 4 ) ⫽ c D L dT I 2R⬘e L ⫺ h(DL)(T ⫺ T앝) ⫺ (DL)(T 4 ⫺ T sur 4 dt

Hence 2 4 4 dT ⫽ I R⬘e ⫺ Dh(T ⫺ T앝) ⫺ D(T ⫺ Tsur) dt c(D2/4)

䉰

Comments: 1. The preceding equation could be solved for the time dependence of the rod temperature by integrating numerically. A steady-state condition would eventually be reached for which dT/dt ⫽ 0. The rod temperature is then determined by an algebraic equation of the form 4 Dh(T ⫺ T앝) ⫹ D(T 4 ⫺ Tsur ) ⫽ I 2R⬘e

2. For fixed environmental conditions (h, T앝, Tsur), as well as a rod of fixed geometry (D) and properties (, R⬘e), the steady-state temperature depends on the rate of thermal energy generation and hence on the value of the electric current. Consider an uninsulated copper wire (D ⫽ 1 mm, ⫽ 0.8, R⬘e ⫽ 0.4 ⍀/m) in a relatively large enclosure (Tsur ⫽ 300 K) through which cooling air is circulated (h ⫽ 100 W/m2 䡠 K, T앝 ⫽ 300 K). Substituting these values into the foregoing equation, the rod temperature has been computed for operating currents in the range 0 ⱕ I ⱕ 10 A, and the following results were obtained: 150 125 100

T (C)

22

75 60 50 25 0

0

2

4

5.2

6

8

10

I (amperes)

3. If a maximum operating temperature of T ⫽ 60⬚C is prescribed for safety reasons, the current should not exceed 5.2 A. At this temperature, heat transfer by radiation (0.6 W/m) is much less than heat transfer by convection (10.4 W/m). Hence, if one wished to operate at a larger current while maintaining the rod temperature within the safety limit, the convection coefficient would have to be increased by increasing the velocity of the circulating air. For h ⫽ 250 W/m2 䡠 K, the maximum allowable current could be increased to 8.1 A. 4. The IHT software is especially useful for solving equations, such as the energy balance in Comment 1, and generating the graphical results of Comment 2.

1.3

䊏

23

Relationship to Thermodynamics

EXAMPLE 1.5 A hydrogen-air Proton Exchange Membrane (PEM) fuel cell is illustrated below. It consists of an electrolytic membrane sandwiched between porous cathode and anode materials, forming a very thin, three-layer membrane electrode assembly (MEA). At the anode, protons and electrons are generated (2H2 l 4H⫹ ⫹ 4e⫺); at the cathode, the protons and electrons recombine to form water (O2 ⫹ 4e⫺ ⫹ 4H⫹ l 2H2O). The overall reaction is then 2H2 ⫹ O2 l 2H2O. The dual role of the electrolytic membrane is to transfer hydrogen ions and serve as a barrier to electron transfer, forcing the electrons to the electrical load that is external to the fuel cell. Ec

I

e

e

Tsur

e O2

H2 e

H2

H

•

Eg H2

O2

H2O e

e

Tc O2

q

q H

H2O

H2 e

e O2 H

H2O

Porous anode

Tsur

H2O O2

Porous cathode Electrolytic membrane

Air

h, T∞

The membrane must operate in a moist state in order to conduct ions. However, the presence of liquid water in the cathode material may block the oxygen from reaching the cathode reaction sites, resulting in the failure of the fuel cell. Therefore, it is critical to control the temperature of the fuel cell, Tc , so that the cathode side contains saturated water vapor. For a given set of H2 and air inlet flow rates and use of a 50 mm ⫻ 50 mm MEA, the fuel cell generates P ⫽ I 䡠 Ec ⫽ 9 W of electrical power. Saturated vapor conditions exist in the fuel cell, corresponding to Tc ⫽ Tsat ⫽ 56.4⬚C. The overall electrochemical reaction is exothermic, and the corresponding thermal generation rate of E˙g ⫽ 11.25 W must be removed from the fuel cell by convection and radiation. The ambient and surrounding

24

Chapter 1

䊏

Introduction

temperatures are T앝 ⫽ Tsur ⫽ 25⬚C, and the relationship between the cooling air velocity and the convection heat transfer coefficient h is h ⫽ 10.9 W 䡠 s0.8/m2.8 䡠 K ⫻ V 0.8 where V has units of m/s. The exterior surface of the fuel cell has an emissivity of ⫽ 0.88. Determine the value of the cooling air velocity needed to maintain steady-state operating conditions. Assume the edges of the fuel cell are well insulated.

SOLUTION Known: Ambient and surrounding temperatures, fuel cell output voltage and electrical current, heat generated by the overall electrochemical reaction, and the desired fuel cell operating temperature. Find: The required cooling air velocity V needed to maintain steady-state operation at Tc 56.4⬚C. Schematic:

W = 50 mm

H = 50 mm q

Tsur = 25C

•

Eg

Tc = 56.4C ε = 0.88

Air

T∞ = 25C h

Assumptions: 1. Steady-state conditions. 2. Negligible temperature variations within the fuel cell. 3. Fuel cell is placed in large surroundings. 4. Edges of the fuel cell are well insulated. 5. Negligible energy entering or leaving the control volume due to gas or liquid flows.

1.3

䊏

25

Relationship to Thermodynamics

Analysis: To determine the required cooling air velocity, we must first perform an energy balance on the fuel cell. Noting that there is no mechanical energy component, we see that E˙in ⫽ 0 and E˙out ⫽ E˙g. This yields qconv ⫹ qrad ⫽ E˙g ⫽ 11.25 W where 4 qrad ⫽ A(Tc4 ⫺ Tsur )

⫽ 0.88 ⫻ (2 ⫻ 0.05 m ⫻ 0.05 m) ⫻ 5.67 ⫻ 10⫺8 W/m2 䡠 K4 ⫻ (329.44 ⫺ 2984) K4 ⫽ 0.97 W Therefore, we may find qconv ⫽ 11.25 W ⫺ 0.97 W ⫽ 10.28 W ⫽ hA(Tc ⫺ T앝) ⫽ 10.9 W 䡠 s0.8/m2.8 䡠 K ⫻ V 0.8 A(Tc ⫺ T앝) which may be rearranged to yield V⫽

10.28 W 10.9 W . s0.8Ⲑm2.8 . K ⫻ (2 ⫻ 0.05 m ⫻ 0.05 m) ⫻ (56.4 ⫺ 25oC)

V ⫽ 9.4 m/s

1.25

䉰

Comments: 1. Temperature and humidity of the MEA will vary from location to location within the fuel cell. Prediction of the local conditions within the fuel cell would require a more detailed analysis. 2. The required cooling air velocity is quite high. Decreased cooling velocities could be used if heat transfer enhancement devices were added to the exterior of the fuel cell. 3. The convective heat rate is significantly greater than the radiation heat rate. 4. The chemical energy (20.25 W) of the hydrogen and oxygen is converted to electrical (9 W) and thermal (11.25 W) energy. This fuel cell operates at a conversion efficiency of (9 W)/(20.25 W) ⫻ 100 ⫽ 44%.

EXAMPLE 1.6 Large PEM fuel cells, such as those used in automotive applications, often require internal cooling using pure liquid water to maintain their temperature at a desired level (see Example 1.5). In cold climates, the cooling water must be drained from the fuel cell to an adjoining container when the automobile is turned off so that harmful freezing does not occur within the fuel cell. Consider a mass M of ice that was frozen while the automobile was not being operated. The ice is at the fusion temperature (Tf ⫽ 0⬚C) and is enclosed in a cubical container of width W on a side. The container wall is of thickness L and thermal

26

Chapter 1

䊏

Introduction

conductivity k. If the outer surface of the wall is heated to a temperature T1 > Tf to melt the ice, obtain an expression for the time needed to melt the entire mass of ice and, in turn, deliver cooling water to, and energize, the fuel cell.

SOLUTION Known: Mass and temperature of ice. Dimensions, thermal conductivity, and outer surface temperature of containing wall. Find: Expression for time needed to melt the ice. Schematic: Section A-A

A

A

T1

Ein

∆ Est

Ice-water mixture (Tf )

W

k

L

Assumptions: 1. Inner surface of wall is at Tf throughout the process. 2. Constant properties. 3. Steady-state, one-dimensional conduction through each wall. 4. Conduction area of one wall may be approximated as W 2 (L Ⰶ W). Analysis: Since we must determine the melting time tm, the first law should be applied over the time interval ⌬t ⫽ tm. Hence, applying Equation 1.12b to a control volume about the ice–water mixture, it follows that Ein ⫽ ⌬Est ⫽ ⌬Ulat where the increase in energy stored within the control volume is due exclusively to the change in latent energy associated with conversion from the solid to liquid state. Heat is transferred to the ice by means of conduction through the container wall. Since the temperature difference across the wall is assumed to remain at (T1 ⫺ Tf) throughout the melting process, the wall conduction rate is constant qcond ⫽ k(6W 2)

T1 ⫺ Tf L

and the amount of energy inflow is

Ein ⫽ k(6W 2)

T1 ⫺ Tf L

t

m

The amount of energy required to effect such a phase change per unit mass of solid is termed the latent heat of fusion hsf . Hence the increase in energy storage is ⌬Est ⫽ Mhsf

1.3

䊏

27

Relationship to Thermodynamics

By substituting into the first law expression, it follows that tm ⫽

Mhsf L 6W k(T1 ⫺ Tf) 2

䉰

Comments: 1. Several complications would arise if the ice were initially subcooled. The storage term would have to include the change in sensible (internal thermal) energy required to take the ice from the subcooled to the fusion temperature. During this process, temperature gradients would develop in the ice. 2. Consider a cavity of width W ⫽ 100 mm on a side, wall thickness L ⫽ 5 mm, and thermal conductivity k ⫽ 0.05 W/m 䡠 K. The mass of the ice in the cavity is M ⫽ s(W ⫺ 2L)3 ⫽ 920 kg/m3 ⫻ (0.100 ⫺ 0.01)3 m3 ⫽ 0.67 kg If the outer surface temperature is T1 ⫽ 30⬚C, the time required to melt the ice is tm ⫽

0.67 kg ⫻ 334,000 J/kg ⫻ 0.005 m ⫽ 12,430 s ⫽ 207 min 6(0.100 m)2 ⫻ 0.05 W/m 䡠 K (30 ⫺ 0)⬚C

The density and latent heat of fusion of the ice are s ⫽ 920 kg/m3 and hsf ⫽ 334 kJ/kg, respectively. 3. Note that the units of K and ⬚C cancel each other in the foregoing expression for tm. Such cancellation occurs frequently in heat transfer analysis and is due to both units appearing in the context of a temperature difference.

We will frequently have occasion to apply the conservation of energy requirement at the surface of a medium. In this special case, the control surfaces are located on either side of the physical boundary and enclose no mass or volume (see Figure 1.9). Accordingly, the generation and storage terms of the conservation

The Surface Energy Balance

Surroundings

Tsur q"rad T1

q"cond Fluid

q"conv T

u∞, T∞

T2 x

T∞ Control surfaces

FIGURE 1.9 The energy balance for conservation of energy at the surface of a medium.

28

Chapter 1

䊏

Introduction

expression, Equation 1.12c, are no longer relevant, and it is necessary to deal only with surface phenomena. For this case, the conservation requirement becomes E˙in ⫺ E˙out ⫽ 0

(1.13)

Even though energy generation may be occurring in the medium, the process would not affect the energy balance at the control surface. Moreover, this conservation requirement holds for both steady-state and transient conditions. In Figure 1.9, three heat transfer terms are shown for the control surface. On a unit area basis, they are conduction from the medium to the control surface (q⬙cond), convection from the surface to a fluid (q⬙conv), and net radiation exchange from the surface to the surroundings (q⬙rad). The energy balance then takes the form. q⬙cond ⫺ q⬙conv ⫺ q⬙rad ⫽ 0

(1.14)

and we can express each of the terms using the appropriate rate equations, Equations 1.2, 1.3a, and 1.7.

EXAMPLE 1.7 Humans are able to control their heat production rate and heat loss rate to maintain a nearly constant core temperature of Tc ⫽ 37⬚C under a wide range of environmental conditions. This process is called thermoregulation. From the perspective of calculating heat transfer between a human body and its surroundings, we focus on a layer of skin and fat, with its outer surface exposed to the environment and its inner surface at a temperature slightly less than the core temperature, Ti ⫽ 35⬚C ⫽ 308 K. Consider a person with a skin/fat layer of thickness L ⫽ 3 mm and effective thermal conductivity k ⫽ 0.3 W/m 䡠 K. The person has a surface area A ⫽ 1.8 m2 and is dressed in a bathing suit. The emissivity of the skin is ⫽ 0.95. 1. When the person is in still air at T앝 ⫽ 297 K, what is the skin surface temperature and rate of heat loss to the environment? Convection heat transfer to the air is characterized by a free convection coefficient of h ⫽ 2 W/m2 䡠 K. 2. When the person is in water at T앝 ⫽ 297 K, what is the skin surface temperature and heat loss rate? Heat transfer to the water is characterized by a convection coefficient of h ⫽ 200 W/m2 䡠 K.

SOLUTION Known: Inner surface temperature of a skin/fat layer of known thickness, thermal conductivity, emissivity, and surface area. Ambient conditions. Find: Skin surface temperature and heat loss rate for the person in air and the person in water.

1.3

䊏

29

Relationship to Thermodynamics

Schematic: Ti = 308 K

Skin/fat

Ts ε = 0.95

Tsur = 297 K

q"rad q"cond q"conv

T∞ = 297 K h = 2 W/m2•K (Air) h = 200 W/m2•K (Water)

k = 0.3 W/m•K L = 3 mm Air or water

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer by conduction through the skin/fat layer. 3. Thermal conductivity is uniform. 4. Radiation exchange between the skin surface and the surroundings is between a small surface and a large enclosure at the air temperature. 5. Liquid water is opaque to thermal radiation. 6. Bathing suit has no effect on heat loss from body. 7. Solar radiation is negligible. 8. Body is completely immersed in water in part 2. Analysis: 1. The skin surface temperature may be obtained by performing an energy balance at the skin surface. From Equation 1.13, E˙ in ⫺ E˙ out ⫽ 0 It follows that, on a unit area basis, q⬙cond ⫺ q⬙conv ⫺ q⬙rad ⫽ 0 or, rearranging and substituting from Equations 1.2, 1.3a, and 1.7, Ti ⫺ Ts 4 ) ⫽ h(Ts ⫺ T앝) ⫹ (T s4 ⫺ Tsur L The only unknown is Ts, but we cannot solve for it explicitly because of the fourth-power dependence of the radiation term. Therefore, we must solve the equation iteratively, which can be done by hand or by using IHT or some other equation solver. To expedite a hand solution, we write the radiation heat flux in terms of the radiation heat transfer coefficient, using Equations 1.8 and 1.9: T ⫺ Ts k i ⫽ h(Ts ⫺ T앝) ⫹ hr (Ts ⫺ Tsur) L Solving for Ts, with Tsur ⫽ T앝, we have k

kTi ⫹ (h ⫹ hr)T앝 Ts ⫽ L k ⫹ (h ⫹ h ) r L

30

Chapter 1

䊏

Introduction

We estimate hr using Equation 1.9 with a guessed value of Ts ⫽ 305 K and T앝 ⫽ 297 K, to yield hr ⫽ 5.9 W/m2 䡠 K. Then, substituting numerical values into the preceding equation, we find 0.3 W/m 䡠 K ⫻ 308 K ⫹ (2 ⫹ 5.9) W/m2 䡠 K ⫻ 297 K 3 ⫻ 10⫺3 m Ts ⫽ ⫽ 307.2 K 0.3 W/m 䡠 K ⫹ (2 ⫹ 5.9) W/m2 䡠 K 3 ⫻ 10⫺3 m With this new value of Ts, we can recalculate hr and Ts, which are unchanged. Thus the skin temperature is 307.2 K 34⬚C. 䉰 The rate of heat loss can be found by evaluating the conduction through the skin/fat layer: T ⫺ Ts (308 ⫺ 307.2) K ⫽ 146 W ⫽ 0.3 W/m 䡠 K ⫻ 1.8 m2 ⫻ qs ⫽ kA i 䉰 L 3 ⫻ 10⫺3 m 2. Since liquid water is opaque to thermal radiation, heat loss from the skin surface is by convection only. Using the previous expression with hr ⫽ 0, we find 0.3 W/m 䡠 K ⫻ 308 K ⫹ 200 W/m2 䡠 K ⫻ 297 K 3 ⫻ 10⫺3 m ⫽ 300.7 K Ts ⫽ 0.3 W/m 䡠 K ⫹ 200 W/m2 䡠 K 3 ⫻ 10⫺3 m

䉰

and qs ⫽ kA

Ti ⫺ Ts (308 ⫺ 300.7) K ⫽ 1320 W ⫽ 0.3 W/m 䡠 K ⫻ 1.8 m2 ⫻ L 3 ⫻ 10⫺3 m

䉰

Comments: 1. When using energy balances involving radiation exchange, the temperatures appearing in the radiation terms must be expressed in kelvins, and it is good practice to use kelvins in all terms to avoid confusion. 2. In part 1, heat losses due to convection and radiation are 37 W and 109 W, respectively. Thus, it would not have been reasonable to neglect radiation. Care must be taken to include radiation when the heat transfer coefficient is small (as it often is for natural convection to a gas), even if the problem statement does not give any indication of its importance. 3. A typical rate of metabolic heat generation is 100 W. If the person stayed in the water too long, the core body temperature would begin to fall. The large heat loss in water is due to the higher heat transfer coefficient, which in turn is due to the much larger thermal conductivity of water compared to air. 4. The skin temperature of 34⬚C in part 1 is comfortable, but the skin temperature of 28⬚C in part 2 is uncomfortably cold.

1.3

䊏

Relationship to Thermodynamics

31

In addition to being familiar with the transport rate equations described in Section 1.2, the heat transfer analyst must be able to work with the energy conservation requirements of Equations 1.12 and 1.13. The application of these balances is simplified if a few basic rules are followed.

Application of the Conservation Laws: Methodology

1. The appropriate control volume must be defined, with the control surfaces represented by a dashed line or lines. 2. The appropriate time basis must be identified. 3. The relevant energy processes must be identified, and each process should be shown on the control volume by an appropriately labeled arrow. 4. The conservation equation must then be written, and appropriate rate expressions must be substituted for the relevant terms in the equation. Note that the energy conservation requirement may be applied to a finite control volume or a differential (infinitesimal) control volume. In the first case, the resulting expression governs overall system behavior. In the second case, a differential equation is obtained that can be solved for conditions at each point in the system. Differential control volumes are introduced in Chapter 2, and both types of control volumes are used extensively throughout the text.

1.3.2 Relationship to the Second Law of Thermodynamics and the Efficiency of Heat Engines In this section, we are interested in the efficiency of heat engines. The discussion builds on your knowledge of thermodynamics and shows how heat transfer plays a crucial role in managing and promoting the efficiency of a broad range of energy conversion devices. Recall that a heat engine is any device that operates continuously or cyclically and that converts heat to work. Examples include internal combustion engines, power plants, and thermoelectric devices (to be discussed in Section 3.8). Improving the efficiency of heat engines is a subject of extreme importance; for example, more efficient combustion engines consume less fuel to produce a given amount of work and reduce the corresponding emissions of pollutants and carbon dioxide. More efficient thermoelectric devices can generate more electricity from waste heat. Regardless of the energy conversion device, its size, weight, and cost can all be reduced through improvements in its energy conversion efficiency. The second law of thermodynamics is often invoked when efficiency is of concern and can be expressed in a variety of different but equivalent ways. The Kelvin–Planck statement is particularly relevant to the operation of heat engines [1]. It states: It is impossible for any system to operate in a thermodynamic cycle and deliver a net amount of work to its surroundings while receiving energy by heat transfer from a single thermal reservoir.

Recall that a thermodynamic cycle is a process for which the initial and final states of the system are identical. Consequently, the energy stored in the system does not change between the initial and final states, and the first law of thermodynamics (Equation 1.12a) reduces to W ⫽ Q. A consequence of the Kelvin–Planck statement is that a heat engine must exchange heat with two (or more) reservoirs, gaining thermal energy from the higher-temperature

32

Chapter 1

䊏

Introduction

reservoir and rejecting thermal energy to the lower-temperature reservoir. Thus, converting all of the input heat to work is impossible, and W ⫽ Qin – Qout, where Qin and Qout are both defined to be positive. That is, Qin is the heat transferred from the high temperature source to the heat engine, and Qout is the heat transferred from the heat engine to the low temperature sink. The efficiency of a heat engine is defined as the fraction of heat transferred into the system that is converted to work, namely Qin ⫺ Qout Qout W ⫽ ⫽1⫺ Qin Qin Qin

(1.15)

The second law also tells us that, for a reversible process, the ratio Qout/Qin is equal to the ratio of the absolute temperatures of the respective reservoirs [1]. Thus, the efficiency of a heat engine undergoing a reversible process, called the Carnot efficiency C, is given by C ⫽ 1 ⫺

Tc Th

(1.16)

where Tc and Th are the absolute temperatures of the low- and high-temperature reservoirs, respectively. The Carnot efficiency is the maximum possible efficiency that any heat engine can achieve operating between those two temperatures. Any real heat engine, which will necessarily undergo an irreversible process, will have a lower efficiency. From our knowledge of thermodynamics, we know that, for heat transfer to take place reversibly, it must occur through an infinitesimal temperature difference between the reservoir and heat engine. However, from our newly acquired knowledge of heat transfer mechanisms, as embodied, for example, in Equations 1.2, 1.3, and 1.7, we now realize that, for heat transfer to occur, there must be a nonzero temperature difference between the reservoir and the heat engine. This reality introduces irreversibility and reduces the efficiency. With the concepts of the preceding paragraph in mind, we now consider a more realistic model of a heat engine [2–5] in which heat is transferred into the engine through a thermal resistance Rt,h , while heat is extracted from the engine through a second thermal resistance Rt,c (Figure 1.10). The subscripts h and c refer to the hot and cold sides of the heat engine, respectively. As discussed in Section 1.2.4, these thermal resistances are associated with heat transfer between the heat engine and the reservoirs across a nonzero temperature difference, by way of the mechanisms of conduction, convection, and/or radiation. For example, the resistances could represent conduction through the walls separating the heat engine from the two reservoirs. Note that the reservoir temperatures are still Th and Tc but that the temperatures seen by the heat engine are Th,i ⬍ Th and Tc,i ⬎ Tc , as shown in the diagram. The heat engine is still assumed to be internally reversible, and its efficiency is still the Carnot efficiency. However,

High-temperature side resistance

High-temperature reservoir Q

Th

in

Th,i Heat engine walls

Internally reversible heat engine

W Tc,i

Low-temperature side resistance

Qout Low-temperature reservoir

Tc

FIGURE 1.10 Internally reversible heat engine exchanging heat with high- and low-temperature reservoirs through thermal resistances.

1.3

䊏

33

Relationship to Thermodynamics

the Carnot efficiency is now based on the internal temperatures Th,i and Tc,i. Therefore, a modified efficiency that accounts for realistic (irreversible) heat transfer processes m is m ⫽ 1 ⫺

Tc,i Qout q ⫽ 1 ⫺ qout ⫽ 1 ⫺ Qin Th,i in

(1.17)

where the ratio of heat flows over a time interval, Qout /Qin, has been replaced by the corresponding ratio of heat rates, qout /qin. This replacement is based on applying energy conservation at an instant in time,1 as discussed in Section 1.3.1. Utilizing the definition of a thermal resistance, the heat transfer rates into and out of the heat engine are given by qin ⫽ (Th ⫺ Th,i)/Rt,h

(1.18a)

qout ⫽ (Tc,i ⫺ Tc)/Rt,c

(1.18b)

Equations 1.18 can be solved for the internal temperatures, to yield Th,i ⫽ Th ⫺ qin Rt,h

(1.19a)

Tc,i ⫽ Tc ⫹ qoutRt,c ⫽ Tc ⫹ qin(1 ⫺ m)Rt,c

(1.19b)

In Equation 1.19b, qout has been related to qin and m, using Equation 1.17. The more realistic, modified efficiency can then be expressed as m ⫽ 1 ⫺

Tc,i Tc ⫹ qin(1 ⫺ m)Rt,c ⫽1 ⫺ Th,i Th ⫺ qinRt,h

(1.20)

Solving for m results in m ⫽ 1 ⫺

Tc Th ⫺ qin Rtot

(1.21)

where Rtot ⫽ Rt,h ⫹ Rt,c. It is readily evident that m ⫽ C only if the thermal resistances Rt,h and Rt,c could somehow be made infinitesimally small (or if qin ⫽ 0). For realistic (nonzero) values of Rtot , m ⬍ C , and m further deteriorates as either Rtot or qin increases. As an extreme case, note that m ⫽ 0 when Th ⫽ Tc ⫹ qin Rtot , meaning that no power could be produced even though the Carnot efficiency, as expressed in Equation 1.16, is nonzero. In addition to the efficiency, another important parameter to consider is the power output of the heat engine, given by

W˙ ⫽ qinm ⫽ qin 1 ⫺

Tc Th ⫺ qin Rtot

(1.22)

It has already been noted in our discussion of Equation 1.21 that the efficiency is equal to the maximum Carnot efficiency (m ⫽ C) if qin ⫽ 0. However, under these circumstances

1

The heat engine is assumed to undergo a continuous, steady-flow process, so that all heat and work processes are occurring simultaneously, and the corresponding terms would be expressed in watts (W). For a heat engine undergoing a cyclic process with sequential heat and work processes occurring over different time intervals, we would need to introduce the time intervals for each process, and each term would be expressed in joules (J).

34

Chapter 1

䊏

Introduction

˙ is zero according to Equation 1.22. To increase W˙ , qin must be the power output W increased at the expense of decreased efficiency. In any real application, a balance must be struck between maximizing the efficiency and maximizing the power output. If provision of the heat input is inexpensive (for example, if waste heat is converted to power), a case could be made for sacrificing efficiency to maximize power output. In contrast, if fuel is expensive or emissions are detrimental (such as for a conventional fossil fuel power plant), the efficiency of the energy conversion may be as or more important than the power output. In any case, heat transfer and thermodyamic principles should be used to determine the actual efficiency and power output of a heat engine. Although we have limited our discussion of the second law to heat engines, the preceding analysis shows how the principles of thermodynamics and heat transfer can be combined to address significant problems of contemporary interest.

EXAMPLE 1.8 In a large steam power plant, the combustion of coal provides a heat rate of qin ⫽ 2500 MW at a flame temperature of Th ⫽ 1000 K. Heat is rejected from the plant to a river flowing at Tc ⫽ 300 K. Heat is transferred from the combustion products to the exterior of large tubes in the boiler by way of radiation and convection, through the boiler tubes by conduction, and then from the interior tube surface to the working fluid (water) by convection. On the cold side, heat is extracted from the power plant by condensation of steam on the exterior condenser tube surfaces, through the condenser tube walls by conduction, and from the interior of the condenser tubes to the river water by convection. Hot and cold side thermal resistances account for the combined effects of conduction, convection, and radiation and, under design conditions, they are Rt,h ⫽ 8 ⫻ 10⫺8 K/W and Rt,c ⫽ 2 ⫻ 10⫺8 K/W, respectively. 1. Determine the efficiency and power output of the power plant, accounting for heat transfer effects to and from the cold and hot reservoirs. Treat the power plant as an internally reversible heat engine. 2. Over time, coal slag will accumulate on the combustion side of the boiler tubes. This fouling process increases the hot side resistance to Rt,h ⫽ 9 ⫻ 10⫺8 K/W. Concurrently, biological matter can accumulate on the river water side of the condenser tubes, increasing the cold side resistance to Rt,c ⫽ 2.2 ⫻ 10⫺8 K/W. Find the efficiency and power output of the plant under fouled conditions.

SOLUTION Known: Source and sink temperatures and heat input rate for an internally reversible heat engine. Thermal resistances separating heat engine from source and sink under clean and fouled conditions. Find: 1. Efficiency and power output for clean conditions. 2. Efficiency and power output under fouled conditions.

1.3

䊏

35

Relationship to Thermodynamics

Schematic: Products of combustion qin 2500 MW

8

Th 1000 K

Rt,h 8 10 K/W (clean) 8 Rt,h 9 10 K/W (fouled)

Th,i Power plant

Tc,i

8

Rt,c 2 10 K/W (clean) 8 Rt,c 2.2 10 K/W (fouled)

•

W

qout Cooling water

Tc 300 K

Assumptions: 1. Steady-state conditions. 2. Power plant behaves as an internally reversible heat engine, so its efficiency is the modified efficiency. Analysis: 1. The modified efficiency of the internally reversible power plant, considering realistic heat transfer effects on the hot and cold side of the power plant, is given by Equation 1.21: m ⫽ 1 ⫺

Tc Th ⫺ qinRtot

where, for clean conditions Rtot ⫽ Rt,h ⫹ Rt,c ⫽ 8 ⫻ 10⫺8 K/W ⫹ 2 ⫻ 10⫺8 K/W ⫽ 1.0 ⫻ 10⫺7 K/W Thus m ⫽ 1 ⫺

Tc 300 K ⫽1⫺ ⫽ 0.60 ⫽ 60% 䉰 Th ⫺ qin Rtot 1000 K ⫺ 2500 ⫻ 106 W ⫻ 1.0 ⫻ 10⫺7 K/W

The power output is given by W˙ ⫽ qinm ⫽ 2500 MW ⫻ 0.60 ⫽ 1500 MW

䉰

2. Under fouled conditions, the preceding calculations are repeated to find m ⫽ 0.583 ⫽ 58.3% and W˙ ⫽ 1460 MW

䉰

Comments: 1. The actual efficiency and power output of a power plant operating between these temperatures would be much less than the foregoing values, since there would be other irreversibilities internal to the power plant. Even if these irreversibilities

36

Chapter 1

䊏

Introduction

were considered in a more comprehensive analysis, fouling effects would still reduce the plant efficiency and power output. 2. The Carnot efficiency is C ⫽ 1 ⫺ Tc /Th ⫽ 1 ⫺ 300 K/1000 K ⫽ 70%. The corresponding power output would be W˙ ⫽ qinC ⫽ 2500 MW ⫻ 0.70 ⫽ 1750 MW. Thus, if the effect of irreversible heat transfer from and to the hot and cold reservoirs, respectively, were neglected, the power output of the plant would be significantly overpredicted. 3. Fouling reduces the power output of the plant by ⌬P ⫽ 40 MW. If the plant owner sells the electricity at a price of $0.08/kW ⭈ h, the daily lost revenue associated with operating the fouled plant would be C ⫽ 40,000 kW ⫻ $0.08/kW 䡠 h ⫻ 24 h/day ⫽ $76,800/day.

1.4 Units and Dimensions The physical quantities of heat transfer are specified in terms of dimensions, which are measured in terms of units. Four basic dimensions are required for the development of heat transfer: length (L), mass (M), time (t), and temperature (T). All other physical quantities of interest may be related to these four basic dimensions. In the United States, dimensions have been customarily measured in terms of the English system of units, for which the base units are: Dimension Length (L) Mass (M) Time (t) Temperature (T)

Unit l l l l

foot (ft) pound mass (lbm) second (s) degree Fahrenheit (⬚F)

The units required to specify other physical quantities may then be inferred from this group.

In recent years, there has been a strong trend toward the global usage of a standard set of units. In 1960, the SI (Système International d’Unités) system of units was defined by the Eleventh General Conference on Weights and Measures and recommended as a worldwide standard. In response to this trend, the American Society of Mechanical Engineers (ASME) has required the use of SI units in all of its publications since 1974. For this reason and because SI units are operationally more convenient than the English system, the SI system is used for calculations of this text. However, because for some time to come, engineers might also have to work with results expressed in the English system, you should be able to convert from one system to the other. For your convenience, conversion factors are provided on the inside back cover of the text. The SI base units required for this text are summarized in Table 1.2. With regard to these units, note that 1 mol is the amount of substance that has as many atoms or molecules as there are atoms in 12 g of carbon-12 (12C); this is the gram-mole (mol). Although the mole has been recommended as the unit quantity of matter for the SI system, it is more consistent to work with the kilogram-mol (kmol, kg-mol). One kmol is simply the amount of substance that has as many atoms or molecules as there are atoms in 12 kg of 12C. As long as the use is consistent within a given problem, no difficulties arise in using either mol or kmol. The molecular weight of a substance is the mass associated with a mole or

1.4

䊏

37

Units and Dimensions

kilogram-mole. For oxygen, as an example, the molecular weight ᏹ is 16 g/mol or 16 kg/kmol. Although the SI unit of temperature is the kelvin, use of the Celsius temperature scale remains widespread. Zero on the Celsius scale (0⬚C) is equivalent to 273.15 K on the thermodynamic scale,2 in which case T (K) ⫽ T (⬚C) ⫹ 273.15 However, temperature differences are equivalent for the two scales and may be denoted as ⬚C or K. Also, although the SI unit of time is the second, other units of time (minute, hour, and day) are so common that their use with the SI system is generally accepted. The SI units comprise a coherent form of the metric system. That is, all remaining units may be derived from the base units using formulas that do not involve any numerical factors. Derived units for selected quantities are listed in Table 1.3. Note that force is measured in newtons, where a 1-N force will accelerate a 1-kg mass at 1 m/s2. Hence 1 N ⫽ 1 kg 䡠 m/s2. The unit of pressure (N/m2) is often referred to as the pascal. In the SI system, there is one unit of energy (thermal, mechanical, or electrical) called the joule (J); 1 J ⫽ 1 N 䡠 m. The unit for energy rate, or power, is then J/s, where one joule per second is equivalent to one watt (1 J/s ⫽ 1 W). Since working with extremely large or small numbers is frequently necessary, a set of standard prefixes has been introduced to simplify matters (Table 1.4). For example, 1 megawatt (MW) ⫽ 106 W, and 1 micrometer (m) ⫽ 10⫺6 m.

TABLE 1.2

SI base and supplementary units

Quantity and Symbol

Unit and Symbol

Length (L) Mass (M) Amount of substance Time (t) Electric current (I) Thermodynamic temperature (T) Plane anglea () Solid anglea ()

meter (m) kilogram (kg) mole (mol) second (s) ampere (A) kelvin (K) radian (rad) steradian (sr)

a

Supplementary unit.

TABLE 1.3

SI derived units for selected quantities

Quantity

Name and Symbol

Formula

Expression in SI Base Units

Force Pressure and stress Energy Power

newton (N) pascal (Pa) joule (J) watt (W)

m 䡠 kg/s2 N/m2 N䡠m J/s

m 䡠 kg/s2 kg/m 䡠 s2 m2 䡠 kg/s2 m2 䡠 kg/s3

2

The degree symbol is retained for designating the Celsius temperature (⬚C) to avoid confusion with the use of C for the unit of electrical charge (coulomb).

38

Chapter 1

䊏

Introduction

TABLE 1.4

Multiplying prefixes

Prefix

Abbreviation

Multiplier

femto pico nano micro milli centi hecto kilo mega giga tera peta exa

f p n m c h k M G T P E

10⫺15 10⫺12 10⫺9 10⫺6 10⫺3 10⫺2 102 103 106 109 1012 1015 1018

1.5 Analysis of Heat Transfer Problems: Methodology A major objective of this text is to prepare you to solve engineering problems that involve heat transfer processes. To this end, numerous problems are provided at the end of each chapter. In working these problems you will gain a deeper appreciation for the fundamentals of the subject, and you will gain confidence in your ability to apply these fundamentals to the solution of engineering problems. In solving problems, we advocate the use of a systematic procedure characterized by a prescribed format. We consistently employ this procedure in our examples, and we require our students to use it in their problem solutions. It consists of the following steps: 1. Known: After carefully reading the problem, state briefly and concisely what is known about the problem. Do not repeat the problem statement. 2. Find: State briefly and concisely what must be found. 3. Schematic: Draw a schematic of the physical system. If application of the conservation laws is anticipated, represent the required control surface or surfaces by dashed lines on the schematic. Identify relevant heat transfer processes by appropriately labeled arrows on the schematic. 4. Assumptions: List all pertinent simplifying assumptions. 5. Properties: Compile property values needed for subsequent calculations and identify the source from which they are obtained. 6. Analysis: Begin your analysis by applying appropriate conservation laws, and introduce rate equations as needed. Develop the analysis as completely as possible before substituting numerical values. Perform the calculations needed to obtain the desired results. 7. Comments: Discuss your results. Such a discussion may include a summary of key conclusions, a critique of the original assumptions, and an inference of trends obtained by performing additional what-if and parameter sensitivity calculations.

1.5

䊏

39

Analysis of Heat Tranfer Problems: Methodology

The importance of following steps 1 through 4 should not be underestimated. They provide a useful guide to thinking about a problem before effecting its solution. In step 7, we hope you will take the initiative to gain additional insights by performing calculations that may be computer based. The software accompanying this text provides a suitable tool for effecting such calculations.

EXAMPLE 1.9 The coating on a plate is cured by exposure to an infrared lamp providing a uniform irradiation of 2000 W/m2. It absorbs 80% of the irradiation and has an emissivity of 0.50. It is also exposed to an airflow and large surroundings for which temperatures are 20⬚C and 30⬚C, respectively. 1. If the convection coefficient between the plate and the ambient air is 15 W/m2 䡠 K, what is the cure temperature of the plate? 2. The final characteristics of the coating, including wear and durability, are known to depend on the temperature at which curing occurs. An airflow system is able to control the air velocity, and hence the convection coefficient, on the cured surface, but the process engineer needs to know how the temperature depends on the convection coefficient. Provide the desired information by computing and plotting the surface temperature as a function of h for 2 ⱕ h ⱕ 200 W/m2 䡠 K. What value of h would provide a cure temperature of 50⬚C?

SOLUTION Known: Coating with prescribed radiation properties is cured by irradiation from an infrared lamp. Heat transfer from the coating is by convection to ambient air and radiation exchange with the surroundings. Find: 1. Cure temperature for h ⫽ 15 W/m2 䡠 K. 2. Effect of airflow on the cure temperature for 2 ⱕ h ⱕ 200 W/m2 䡠 K. Value of h for which the cure temperature is 50⬚C. Schematic:

Tsur = 30°C Glamp = 2000 W/m2 T∞ = 20°C 2 ≤ h ≤ 200 W/m2•K

q"conv

Air T

Coating, α = 0.8, ε = 0.5

q"rad

α Glamp

Chapter 1

䊏

Introduction

Assumptions: 1. Steady-state conditions. 2. Negligible heat loss from back surface of plate. 3. Plate is small object in large surroundings, and coating has an absorptivity of ␣sur ⫽ ⫽ 0.5 with respect to irradiation from the surroundings. Analysis: 1. Since the process corresponds to steady-state conditions and there is no heat transfer at the back surface, the plate must be isothermal (Ts ⫽ T). Hence the desired temperature may be determined by placing a control surface about the exposed surface and applying Equation 1.13 or by placing the control surface about the entire plate and applying Equation 1.12c. Adopting the latter approach and recognizing that there is no energy generation (E˙ g ⫽ 0), Equation 1.12c reduces to E˙ in ⫺ E˙ out ⫽ 0 where E˙ st ⫽ 0 for steady-state conditions. With energy inflow due to absorption of the lamp irradiation by the coating and outflow due to convection and net radiation transfer to the surroundings, it follows that (␣G)lamp ⫺ q⬙conv ⫺ q⬙rad ⫽ 0 Substituting from Equations 1.3a and 1.7, we obtain 4 (␣G)lamp ⫺ h(T ⫺ T앝) ⫺ (T 4 ⫺ Tsur )⫽0

Substituting numerical values 0.8 ⫻ 2000 W/m2 ⫺ 15 W/m2 䡠 K (T ⫺ 293) K ⫺ 0.5 ⫻ 5.67 ⫻ 10⫺8 W/m2 䡠 K4 (T 4 ⫺ 3034) K4 ⫽ 0 and solving by trial-and-error, we obtain T ⫽ 377 K ⫽ 104⬚C

䉰

2. Solving the foregoing energy balance for selected values of h in the prescribed range and plotting the results, we obtain 240 200 160

T (C)

40

120 80 50 40 0

0

20

40 51 60 h (W/m2•K)

80

100

If a cure temperature of 50⬚C is desired, the airflow must provide a convection coefficient of h(T ⫽ 50⬚C) ⫽ 51.0 W/m2 䡠 K 䉰

1.6

䊏

Relevance of Heat Tranfer

41

Comments: 1. The coating (plate) temperature may be reduced by decreasing T앝 and Tsur, as well as by increasing the air velocity and hence the convection coefficient. 2. The relative contributions of convection and radiation to heat transfer from the plate vary greatly with h. For h ⫽ 2 W/m2 䡠 K, T ⫽ 204⬚C and radiation dominates (q⬙rad 1232 W/m2, q⬙conv 368 W/m2). Conversely, for h ⫽ 200 W/m2 䡠 K, T ⫽ 28⬚C and convection dominates (q⬙conv 1606 W/m2, q⬙rad ⫺6 W/m2). In fact, for this condition the plate temperature is slightly less than that of the surroundings and net radiation exchange is to the plate.

1.6 Relevance of Heat Transfer We will devote much time to acquiring an understanding of heat transfer effects and to developing the skills needed to predict heat transfer rates and temperatures that evolve in certain situations. What is the value of this knowledge? To what problems may it be applied? A few examples will serve to illustrate the rich breadth of applications in which heat transfer plays a critical role. The challenge of providing sufficient amounts of energy for humankind is well known. Adequate supplies of energy are needed not only to fuel industrial productivity, but also to supply safe drinking water and food for much of the world’s population and to provide the sanitation necessary to control life-threatening diseases. To appreciate the role heat transfer plays in the energy challenge, consider a flow chart that represents energy use in the United States, as shown in Figure 1.11a. Currently, about 58% of the nearly 110 EJ of energy that is consumed annually in the United States is wasted in the form of heat. Nearly 70% of the energy used to generate electricity is lost in the form of heat. The transportation sector, which relies almost exclusively on petroleumbased fuels, utilizes only 21.5% of the energy it consumes; the remaining 78.5% is released in the form of heat. Although the industrial and residential/commercial use of energy is relatively more efficient, opportunities for energy conservation abound. Creative thermal engineering, utilizing the tools of thermodynamics and heat transfer, can lead to new ways to (1) increase the efficiency by which energy is generated and converted, (2) reduce energy losses, and (3) harvest a large portion of the waste heat. As evident in Figure 1.11a, fossil fuels (petroleum, natural gas, and coal) dominate the energy portfolio in many countries, such as the United States. The combustion of fossil fuels produces massive amounts of carbon dioxide; the amount of CO2 released in the United States on an annual basis due to combustion is currently 5.99 Eg (5.99 ⫻ 1015 kg). As more CO2 is pumped into the atmosphere, mechanisms of radiation heat transfer within the atmosphere are modified, resulting in potential changes in global temperatures. In a country like the United States, electricity generation and transportation are responsible for nearly 75% of the total CO2 released into the atmosphere due to energy use (Figure 1.11b). What are some of the ways engineers are applying the principles of heat transfer to address issues of energy and environmental sustainability? The efficiency of a gas turbine engine can be significantly increased by increasing its operating temperature. Today, the temperatures of the combustion gases inside these

42

Chapter 1

䊏

Introduction

Nuclear power 8.3%

Alternative sources 6.8%

Petroleum 39.3%

Electricity generation 35.4%

68.6%

Natural gas 23.3%

Transportation 25.4%

19.9% 19.9% Waste heat 57.6%

Coal 22.9%

Industrial 21.7%

78.5%

Residential/ commercial 17.4%

Useful power 42.4%

(a)

Petroleum 43.2%

Electricity generation 40.6%

Natural gas 20.7%

Transportation 33.5%

Coal 36.1%

Industrial 16.5%

Residential/ commercial 9.4%

(b)

FIGURE 1.11 Flow charts for energy consumption and associated CO2 emissions in the United States in 2007. (a) Energy production and consumption. (b) Carbon dioxide by source of fossil fuel and end-use application. Arrow widths represent relative magnitudes of the flow streams. (Credit: U.S. Department of Energy and the Lawrence Livermore National Laboratory.)

engines far exceed the melting point of the exotic alloys used to manufacture the turbine blades and vanes. Safe operation is typically achieved by three means. First, relatively cool gases are injected through small holes at the leading edge of a turbine blade (Figure 1.12). These gases hug the blade as they are carried downstream and help insulate the blade from the hot combustion gases. Second, thin layers of a very low thermal conductivity, ceramic thermal barrier coating are applied to the blades and vanes to provide an extra layer of insulation. These coatings are produced by spraying molten ceramic powders onto the engine components using extremely high temperature sources such as plasma spray guns

1.6

䊏

43

Relevance of Heat Tranfer

(a)

(b)

FIGURE 1.12 Gas turbine blade. (a) External view showing holes for injection of cooling gases. (b) X ray view showing internal cooling passages. (Credit: Images courtesy of FarField Technology, Ltd., Christchurch, New Zealand.)

that can operate in excess of 10,000 kelvins. Third, the blades and vanes are designed with intricate, internal cooling passages, all carefully configured by the heat transfer engineer to allow the gas turbine engine to operate under such extreme conditions. Alternative sources constitute a small fraction of the energy portfolio of many nations, as illustrated in the flow chart of Figure 1.11a for the United States. The intermittent nature of the power generated by sources such as the wind and solar irradiation limits their widespread utilization, and creative ways to store excess energy for use during low-power generation periods are urgently needed. Emerging energy conversion devices such as fuel cells could be used to (1) combine excess electricity that is generated during the day (in a solar power station, for example) with liquid water to produce hydrogen, and (2) subsequently convert the stored hydrogen at night by recombining it with oxygen to produce electricity and water. Roadblocks hindering the widespread use of hydrogen fuel cells are their size, weight, and limited durability. As with the gas turbine engine, the efficiency of a fuel cell increases with temperature, but high operating temperatures and large temperature gradients can cause the delicate polymeric materials within a hydrogen fuel cell to fail. More challenging is the fact that water exists inside any hydrogen fuel cell. If this water should freeze, the polymeric materials within the fuel cell would be destroyed, and the fuel cell would cease operation. Because of the necessity to utilize very pure water in a hydrogen fuel cell, common remedies such as antifreeze cannot be used. What heat transfer mechanisms must be controlled to avoid freezing of pure water within a fuel cell located at a wind farm or solar energy station in a cold climate? How might your developing knowledge of internal forced convection, evaporation, or condensation be applied to control the operating temperatures and enhance the durability of a fuel cell, in turn promoting more widespread use of solar and wind power? Due to the information technology revolution of the last two decades, strong industrial productivity growth has brought an improved quality of life worldwide. Many information technology breakthroughs have been enabled by advances in heat transfer engineering that have ensured the precise control of temperatures of systems ranging in size from nanoscale integrated circuits, to microscale storage media including compact discs, to large data centers filled with heat-generating equipment. As electronic devices become faster and incorporate

44

Chapter 1

䊏

Introduction

greater functionality, they generate more thermal energy. Simultaneously, the devices have become smaller. Inevitably, heat fluxes (W/m2) and volumetric energy generation rates (W/m3) keep increasing, but the operating temperatures of the devices must be held to reasonably low values to ensure their reliability. For personal computers, cooling fins (also known as heat sinks) are fabricated of a high thermal conductivity material (usually aluminum) and attached to the microprocessors to reduce their operating temperatures, as shown in Figure 1.13. Small fans are used to induce forced convection over the fins. The cumulative energy that is consumed worldwide, just to (1) power the small fans that provide the airflow over the fins and (2) manufacture the heat sinks for personal computers, is estimated to be over 109 kW 䡠 h per year [6]. How might your knowledge of conduction, convection, and radiation be used to, for example, eliminate the fan and minimize the size of the heat sink? Further improvements in microprocessor technology are currently limited by our ability to cool these tiny devices. Policy makers have voiced concern about our ability to continually reduce the cost of computing and, in turn as a society, continue the growth in productivity that has marked the last 30 years, specifically citing the need to enhance heat transfer in electronics cooling [7]. How might your knowledge of heat transfer help ensure continued industrial productivity into the future? Heat transfer is important not only in engineered systems but also in nature. Temperature regulates and triggers biological responses in all living systems and ultimately marks the boundary between sickness and health. Two common examples include hypothermia, which results from excessive cooling of the human body, and heat stroke, which is triggered in warm, humid environments. Both are deadly, and both are associated with core temperatures of the body exceeding physiological limits. Both are directly linked to the convection, radiation, and evaporation processes occurring at the surface of the body, the transport of heat within the body, and the metabolic energy generated volumetrically within the body. Recent advances in biomedical engineering, such as laser surgery, have been enabled by successfully applying fundamental heat transfer principles [8, 9]. While increased temperatures resulting from contact with hot objects may cause thermal burns, beneficial hyperthermal treatments are used to purposely destroy, for example, cancerous lesions. In a

Exploded view

FIGURE 1.13 A finned heat sink and fan assembly (left) and microprocessor (right).

1.7

䊏

45

Summary

Keratin Epidermal layer Epidermis Basal cell layer

Sebaceous gland Sensory receptor

Dermis

Sweat gland Nerve fiber Hair follicle

Subcutaneous layer Vein Artery

FIGURE 1.14 Morphology of human skin.

similar manner, very low temperatures might induce frostbite, but purposeful localized freezing can selectively destroy diseased tissue during cryosurgery. Many medical therapies and devices therefore operate by destructively heating or cooling diseased tissue, while leaving the surrounding healthy tissue unaffected. The ability to design many medical devices and to develop the appropriate protocol for their use hinges on the engineer’s ability to predict and control the distribution of temperatures during thermal treatment and the distribution of chemical species in chemotherapies. The treatment of mammalian tissue is made complicated by its morphology, as shown in Figure 1.14. The flow of blood within the venular and capillary structure of a thermally treated area affects heat transfer through advection processes. Larger veins and arteries, which commonly exist in pairs throughout the body, carry blood at different temperatures and advect thermal energy at different rates. Therefore, the veins and arteries exist in a counterflow heat exchange arrangement with warm, arteriolar blood exchanging thermal energy with the cooler, venular blood through the intervening solid tissue. Networks of smaller capillaries can also affect local temperatures by perfusing blood through the treated area. In subsequent chapters, example and homework problems will deal with the analysis of these and many other thermal systems.

1.7 Summary Although much of the material of this chapter will be discussed in greater detail, you should now have a reasonable overview of heat transfer. You should be aware of the

46 TABLE 1.5

Mode Conduction

Convection

Radiation

Chapter 1

䊏

Introduction

Summary of heat transfer processes

Mechanism(s) Diffusion of energy due to random molecular motion Diffusion of energy due to random molecular motion plus energy transfer due to bulk motion (advection) Energy transfer by electromagnetic waves

Rate Equation q⬙x (W/m2) ⫽ ⫺k

dT dx

q⬙(W/m2) ⫽ h(Ts ⫺ T앝)

4 q⬙(W/m2) ⫽ (Ts4 ⫺ Tsur ) or q (W) ⫽ hr A(Ts ⫺ Tsur)

Equation Number

Transport Property or Coefficient

(1.1)

k (W/m 䡠 K)

(1.3a)

h (W/m2 䡠 K)

(1.7) (1.8)

hr (W/m2 䡠 K)

several modes of transfer and their physical origins. You will be devoting much time to acquiring the tools needed to calculate heat transfer phenomena. However, before you can use these tools effectively, you must have the intuition to determine what is happening physically. Specifically, given a physical situation, you must be able to identify the relevant transport phenomena; the importance of developing this facility must not be underestimated. The example and problems at the end of this chapter will launch you on the road to developing this intuition. You should also appreciate the significance of the rate equations and feel comfortable in using them to compute transport rates. These equations, summarized in Table 1.5, should be committed to memory. You must also recognize the importance of the conservation laws and the need to carefully identify control volumes. With the rate equations, the conservation laws may be used to solve numerous heat transfer problems. Lastly, you should have begun to acquire an appreciation for the terminology and physical concepts that underpin the subject of heat transfer. Test your understanding of the important terms and concepts introduced in this chapter by addressing the following questions: • What are the physical mechanisms associated with heat transfer by conduction, convection, and radiation? • What is the driving potential for heat transfer? What are analogs to this potential and to heat transfer itself for the transport of electric charge? • What is the difference between a heat flux and a heat rate? What are their units? • What is a temperature gradient? What are its units? What is the relationship of heat flow to a temperature gradient? • What is the thermal conductivity? What are its units? What role does it play in heat transfer? • What is Fourier’s law? Can you write the equation from memory? • If heat transfer by conduction through a medium occurs under steady-state conditions, will the temperature at a particular instant vary with location in the medium? Will the temperature at a particular location vary with time?

1.7

䊏

Summary

47

• What is the difference between natural convection and forced convection? • What conditions are necessary for the development of a hydrodynamic boundary layer? A thermal boundary layer? What varies across a hydrodynamic boundary layer? Across a thermal boundary layer? • If convection heat transfer for flow of a liquid or a vapor is not characterized by liquid/vapor phase change, what is the nature of the energy being transferred? What is it if there is such a phase change? • What is Newton’s law of cooling? Can you write the equation from memory? • What role is played by the convection heat transfer coefficient in Newton’s law of cooling? What are its units? • What effect does convection heat transfer from or to a surface have on the solid bounded by the surface? • What is predicted by the Stefan–Boltzmann law, and what unit of temperature must be used with the law? Can you write the equation from memory? • What is the emissivity, and what role does it play in characterizing radiation transfer at a surface? • What is irradiation? What are its units? • What two outcomes characterize the response of an opaque surface to incident radiation? Which outcome affects the thermal energy of the medium bounded by the surface and how? What property characterizes this outcome? • What conditions are associated with use of the radiation heat transfer coefficient? • Can you write the equation used to express net radiation exchange between a small isothermal surface and a large isothermal enclosure? • Consider the surface of a solid that is at an elevated temperature and exposed to cooler surroundings. By what mode(s) is heat transferred from the surface if (1) it is in intimate (perfect) contact with another solid, (2) it is exposed to the flow of a liquid, (3) it is exposed to the flow of a gas, and (4) it is in an evacuated chamber? • What is the inherent difference between the application of conservation of energy over a time interval and at an instant of time? • What is thermal energy storage? How does it differ from thermal energy generation? What role do the terms play in a surface energy balance?

EXAMPLE 1.10 A closed container filled with hot coffee is in a room whose air and walls are at a fixed temperature. Identify all heat transfer processes that contribute to the cooling of the coffee. Comment on features that would contribute to a superior container design.

SOLUTION Known: Hot coffee is separated from its cooler surroundings by a plastic flask, an air space, and a plastic cover. Find: Relevant heat transfer processes.

48

Chapter 1

䊏

Introduction

Schematic:

q8

q5

Hot coffee

q1

q2

q6 q3

Coffee Cover

Plastic flask

q7

q4

Air space

Room air Cover

Surroundings

Air space Plastic flask

Pathways for energy transfer from the coffee are as follows: q1: free convection from the coffee to the flask. q2: conduction through the flask. q3: free convection from the flask to the air. q4: free convection from the air to the cover. q5: net radiation exchange between the outer surface of the flask and the inner surface of the cover. q6: conduction through the cover. q7: free convection from the cover to the room air. q8: net radiation exchange between the outer surface of the cover and the surroundings.

Comments: Design improvements are associated with (1) use of aluminized (lowemissivity) surfaces for the flask and cover to reduce net radiation, and (2) evacuating the air space or using a filler material to retard free convection.

References 1. Moran, M. J., and H. N. Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, Hoboken, NJ, 2004. 2. Curzon, F. L., and B. Ahlborn, American J. Physics, 43, 22, 1975. 3. Novikov, I. I., J. Nuclear Energy II, 7, 125, 1958. 4. Callen, H. B., Thermodynamics and an Introduction to Thermostatistics, Wiley, Hoboken, NJ, 1985. 5. Bejan, A., American J. Physics, 64, 1054, 1996.

6. Bar-Cohen, A., and I. Madhusudan, IEEE Trans. Components and Packaging Tech., 25, 584, 2002. 7. Miller, R., Business Week, November 11, 2004. 8. Diller, K. R., and T. P. Ryan, J. Heat Transfer, 120, 810, 1998. 9. Datta, A.K., Biological and Bioenvironmental Heat and Mass Transfer, Marcel Dekker, New York, 2002.

䊏

49

Problems

Problems Conduction 1.1 The thermal conductivity of a sheet of rigid, extruded insulation is reported to be k ⫽ 0.029 W/m 䡠 K. The measured temperature difference across a 20-mm-thick sheet of the material is T1 ⫺ T2 ⫽ 10⬚C. (a) What is the heat flux through a 2 m ⫻ 2 m sheet of the insulation? (b) What is the rate of heat transfer through the sheet of insulation? 1.2 The heat flux that is applied to the left face of a plane wall is q⬙ ⫽ 20 W/m2. The wall is of thickness L ⫽ 10 mm and of thermal conductivity k ⫽ 12 W/m 䡠 K. If the surface temperatures of the wall are measured to be 50⬚C on the left side and 30⬚C on the right side, do steady-state conditions exist? 1.3 A concrete wall, which has a surface area of 20 m2 and is 0.30 m thick, separates conditioned room air from ambient air. The temperature of the inner surface of the wall is maintained at 25⬚C, and the thermal conductivity of the concrete is 1 W/m 䡠 K. (a) Determine the heat loss through the wall for outer surface temperatures ranging from ⫺15⬚C to 38⬚C, which correspond to winter and summer extremes, respectively. Display your results graphically. (b) On your graph, also plot the heat loss as a function of the outer surface temperature for wall materials having thermal conductivities of 0.75 and 1.25 W/m 䡠 K. Explain the family of curves you have obtained. 1.4 The concrete slab of a basement is 11 m long, 8 m wide, and 0.20 m thick. During the winter, temperatures are nominally 17⬚C and 10⬚C at the top and bottom surfaces, respectively. If the concrete has a thermal conductivity of 1.4 W/m 䡠 K, what is the rate of heat loss through the slab? If the basement is heated by a gas furnace operating at an efficiency of f ⫽ 0.90 and natural gas is priced at Cg ⫽ $0.02/MJ, what is the daily cost of the heat loss? 1.5 Consider Figure 1.3. The heat flux in the x-direction is q⬙x ⫽ 10 W/m2, the thermal conductivity and wall thickness are k ⫽ 2.3 W/m 䡠 K and L ⫽ 20 mm, respectively, and steady-state conditions exist. Determine the value of the temperature gradient in units of K/m. What is the value of the temperature gradient in units of ⬚C/m? 1.6 The heat flux through a wood slab 50 mm thick, whose inner and outer surface temperatures are 40 and 20⬚C, respectively, has been determined to be 40 W/m2. What is the thermal conductivity of the wood?

1.7 The inner and outer surface temperatures of a glass window 5 mm thick are 15 and 5⬚C. What is the heat loss through a 1 m ⫻ 3 m window? The thermal conductivity of glass is 1.4 W/m 䡠 K. 1.8 A thermodynamic analysis of a proposed Brayton cycle gas turbine yields P ⫽ 5 MW of net power production. The compressor, at an average temperature of Tc ⫽ 400⬚C, is driven by the turbine at an average temperature of Th ⫽ 1000⬚C by way of an L ⫽ 1-m-long, d ⫽ 70-mmdiameter shaft of thermal conductivity k ⫽ 40 W/m 䡠 K. Combustion chamber

Turbine

Compressor

d

Tc Shaft

Th P

m• in

L • m out

(a) Compare the steady-state conduction rate through the shaft connecting the hot turbine to the warm compressor to the net power predicted by the thermodynamics-based analysis. (b) A research team proposes to scale down the gas turbine of part (a), keeping all dimensions in the same proportions. The team assumes that the same hot and cold temperatures exist as in part (a) and that the net power output of the gas turbine is proportional to the overall volume of the device. Plot the ratio of the conduction through the shaft to the net power output of the turbine over the range 0.005 m ⱕ L ⱕ 1 m. Is a scaled-down device with L ⫽ 0.005 m feasible? 1.9 A glass window of width W ⫽ 1 m and height H ⫽ 2 m is 5 mm thick and has a thermal conductivity of kg ⫽ 1.4 W/m 䡠 K. If the inner and outer surface temperatures of the glass are 15⬚C and ⫺20⬚C, respectively, on a cold winter day, what is the rate of heat loss through the glass? To reduce heat loss through windows, it is customary to use a double pane construction in which adjoining panes are separated by an air space. If the spacing is 10 mm and the glass surfaces in contact with the air have temperatures of 10⬚C and ⫺15⬚C, what is the rate of heat loss from a 1 m ⫻ 2 m window? The thermal conductivity of air is ka ⫽ 0.024 W/m 䡠 K. 1.10 A freezer compartment consists of a cubical cavity that is 2 m on a side. Assume the bottom to be perfectly

50

Chapter 1

䊏

Introduction

insulated. What is the minimum thickness of styrofoam insulation (k ⫽ 0.030 W/m 䡠 K) that must be applied to the top and side walls to ensure a heat load of less than 500 W, when the inner and outer surfaces are ⫺10 and 35⬚C? 1.11 The heat flux that is applied to one face of a plane wall is q⬙ ⫽ 20 W/m2. The opposite face is exposed to air at temperature 30⬚C, with a convection heat transfer coefficient of 20 W/m2 䡠 K. The surface temperature of the wall exposed to air is measured and found to be 50⬚C. Do steady-state conditions exist? If not, is the temperature of the wall increasing or decreasing with time? 1.12 An inexpensive food and beverage container is fabricated from 25-mm-thick polystyrene (k ⫽ 0.023 W/m 䡠 K) and has interior dimensions of 0.8 m ⫻ 0.6 m ⫻ 0.6 m. Under conditions for which an inner surface temperature of approximately 2⬚C is maintained by an ice-water mixture and an outer surface temperature of 20⬚C is maintained by the ambient, what is the heat flux through the container wall? Assuming negligible heat gain through the 0.8 m ⫻ 0.6 m base of the cooler, what is the total heat load for the prescribed conditions? 1.13 What is the thickness required of a masonry wall having thermal conductivity 0.75 W/m 䡠 K if the heat rate is to be 80% of the heat rate through a composite structural wall having a thermal conductivity of 0.25 W/m 䡠 K and a thickness of 100 mm? Both walls are subjected to the same surface temperature difference. 1.14 A wall is made from an inhomogeneous (nonuniform) material for which the thermal conductivity varies through the thickness according to k ⫽ ax ⫹ b, where a and b are constants. The heat flux is known to be constant. Determine expressions for the temperature gradient and the temperature distribution when the surface at x ⫽ 0 is at temperature T1. 1.15 The 5-mm-thick bottom of a 200-mm-diameter pan may be made from aluminum (k ⫽ 240 W/m 䡠 K) or copper (k ⫽ 390 W/m 䡠 K). When used to boil water, the surface of the bottom exposed to the water is nominally at 110⬚C. If heat is transferred from the stove to the pan at a rate of 600 W, what is the temperature of the surface in contact with the stove for each of the two materials? 1.16 A square silicon chip (k ⫽ 150 W/m 䡠 K) is of width w ⫽ 5 mm on a side and of thickness t ⫽ 1 mm. The chip is mounted in a substrate such that its side and back surfaces are insulated, while the front surface is exposed to a coolant. If 4 W are being dissipated in circuits mounted to the back surface of the chip, what is the steady-state temperature difference between back and front surfaces?

Coolant w Chip

Circuits

t

Convection 1.17 For a boiling process such as shown in Figure 1.5c, the ambient temperature T앝 in Newton’s law of cooling is replaced by the saturation temperature of the fluid Tsat. Consider a situation where the heat flux from the hot plate is q⬙ ⫽ 20 ⫻ 105 W/m2. If the fluid is water at atmospheric pressure and the convection heat transfer coefficient is hw ⫽ 20 ⫻ 103 W/m2 䡠 K, determine the upper surface temperature of the plate, Ts,w. In an effort to minimize the surface temperature, a technician proposes replacing the water with a dielectric fluid whose saturation temperature is Tsat,d ⫽ 52⬚C. If the heat transfer coefficient associated with the dielectric fluid is hd ⫽ 3 ⫻ 103 W/m2 䡠 K, will the technician’s plan work? 1.18 You’ve experienced convection cooling if you’ve ever extended your hand out the window of a moving vehicle or into a flowing water stream. With the surface of your hand at a temperature of 30⬚C, determine the convection heat flux for (a) a vehicle speed of 35 km/h in air at ⫺5⬚C with a convection coefficient of 40 W/m2 䡠 K and (b) a velocity of 0.2 m/s in a water stream at 10⬚C with a convection coefficient of 900 W/m2 䡠 K. Which condition would feel colder? Contrast these results with a heat loss of approximately 30 W/m2 under normal room conditions. 1.19 Air at 40⬚C flows over a long, 25-mm-diameter cylinder with an embedded electrical heater. In a series of tests, measurements were made of the power per unit length, P⬘, required to maintain the cylinder surface temperature at 300⬚C for different free stream velocities V of the air. The results are as follows: Air velocity, V (m/s) Power, P⬘ (W/m)

1 450

2 658

4 983

8 1507

12 1963

(a) Determine the convection coefficient for each velocity, and display your results graphically. (b) Assuming the dependence of the convection coefficient on the velocity to be of the form h ⫽ CV n, determine the parameters C and n from the results of part (a).

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51

Problems

1.20 A wall has inner and outer surface temperatures of 16 and 6⬚C, respectively. The interior and exterior air temperatures are 20 and 5⬚C, respectively. The inner and outer convection heat transfer coefficients are 5 and 20 W/m2 䡠 K, respectively. Calculate the heat flux from the interior air to the wall, from the wall to the exterior air, and from the wall to the interior air. Is the wall under steady-state conditions? 1.21 An electric resistance heater is embedded in a long cylinder of diameter 30 mm. When water with a temperature of 25⬚C and velocity of 1 m/s flows crosswise over the cylinder, the power per unit length required to maintain the surface at a uniform temperature of 90⬚C is 28 kW/m. When air, also at 25⬚C, but with a velocity of 10 m/s is flowing, the power per unit length required to maintain the same surface temperature is 400 W/m. Calculate and compare the convection coefficients for the flows of water and air. 1.22 The free convection heat transfer coefficient on a thin hot vertical plate suspended in still air can be determined from observations of the change in plate temperature with time as it cools. Assuming the plate is isothermal and radiation exchange with its surroundings is negligible, evaluate the convection coefficient at the instant of time when the plate temperature is 225⬚C and the change in plate temperature with time (dT/dt) is ⫺0.022 K/s. The ambient air temperature is 25⬚C and the plate measures 0.3 ⫻ 0.3 m with a mass of 3.75 kg and a specific heat of 2770 J/kg 䡠 K. 1.23 A transmission case measures W ⫽ 0.30 m on a side and receives a power input of Pi ⫽ 150 hp from the engine.

1.24 A cartridge electrical heater is shaped as a cylinder of length L ⫽ 200 mm and outer diameter D ⫽ 20 mm. Under normal operating conditions, the heater dissipates 2 kW while submerged in a water flow that is at 20⬚C and provides a convection heat transfer coefficient of h ⫽ 5000 W/m2 䡠 K. Neglecting heat transfer from the ends of the heater, determine its surface temperature Ts. If the water flow is inadvertently terminated while the heater continues to operate, the heater surface is exposed to air that is also at 20⬚C but for which h ⫽ 50 W/m2 䡠 K. What is the corresponding surface temperature? What are the consequences of such an event? 1.25 A common procedure for measuring the velocity of an airstream involves the insertion of an electrically heated wire (called a hot-wire anemometer) into the airflow, with the axis of the wire oriented perpendicular to the flow direction. The electrical energy dissipated in the wire is assumed to be transferred to the air by forced convection. Hence, for a prescribed electrical power, the temperature of the wire depends on the convection coefficient, which, in turn, depends on the velocity of the air. Consider a wire of length L ⫽ 20 mm and diameter D ⫽ 0.5 mm, for which a calibration of the form V ⫽ 6.25 ⫻ 10⫺5 h2 has been determined. The velocity V and the convection coefficient h have units of m/s and W/m2 䡠 K, respectively. In an application involving air at a temperature of T앝 ⫽ 25⬚C, the surface temperature of the anemometer is maintained at Ts ⫽ 75⬚C with a voltage drop of 5 V and an electric current of 0.1 A. What is the velocity of the air? 1.26 A square isothermal chip is of width w ⫽ 5 mm on a side and is mounted in a substrate such that its side and back surfaces are well insulated; the front surface is exposed to the flow of a coolant at T앝 ⫽ 15⬚C. From reliability considerations, the chip temperature must not exceed T ⫽ 85⬚C. Coolant

T∞, h w

Transmission case, η, Ts

Air

Chip

T∞, h Pi

W

If the transmission efficiency is ⫽ 0.93 and airflow over the case corresponds to T앝 ⫽ 30⬚C and h ⫽ 200 W/m2 䡠 K, what is the surface temperature of the transmission?

If the coolant is air and the corresponding convection coefficient is h ⫽ 200 W/m2 䡠 K, what is the maximum allowable chip power? If the coolant is a dielectric liquid for which h ⫽ 3000 W/m2 䡠 K, what is the maximum allowable power? 1.27 The temperature controller for a clothes dryer consists of a bimetallic switch mounted on an electrical heater attached to a wall-mounted insulation pad.

52

Chapter 1

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Introduction

Dryer wall

Pe

Insulation pad Air T∞, h

Tset = 70°C

Electrical heater Bimetallic switch

The switch is set to open at 70⬚C, the maximum dryer air temperature. To operate the dryer at a lower air temperature, sufficient power is supplied to the heater such that the switch reaches 70⬚C (Tset) when the air temperature T is less than Tset. If the convection heat transfer coefficient between the air and the exposed switch surface of 30 mm2 is 25 W/m2 䡠 K, how much heater power Pe is required when the desired dryer air temperature is T앝 ⫽ 50⬚C?

Radiation 1.28 An overhead 25-m-long, uninsulated industrial steam pipe of 100-mm diameter is routed through a building whose walls and air are at 25⬚C. Pressurized steam maintains a pipe surface temperature of 150⬚C, and the coefficient associated with natural convection is h ⫽ 10 W/m2 䡠 K. The surface emissivity is ⫽ 0.8. (a) What is the rate of heat loss from the steam line? (b) If the steam is generated in a gas-fired boiler operating at an efficiency of f ⫽ 0.90 and natural gas is priced at Cg ⫽ $0.02 per MJ, what is the annual cost of heat loss from the line? 1.29 Under conditions for which the same room temperature is maintained by a heating or cooling system, it is not uncommon for a person to feel chilled in the winter but comfortable in the summer. Provide a plausible explanation for this situation (with supporting calculations) by considering a room whose air temperature is maintained at 20⬚C throughout the year, while the walls of the room are nominally at 27⬚C and 14⬚C in the summer and winter, respectively. The exposed surface of a person in the room may be assumed to be at a temperature of 32⬚C throughout the year and to have an emissivity of 0.90. The coefficient associated with heat transfer by natural convection between the person and the room air is approximately 2 W/m2 䡠 K.

range 40 ⱕ T ⱕ 85⬚C, what is the range of acceptable power dissipation for the package? Display your results graphically, showing also the effect of variations in the emissivity by considering values of 0.20 and 0.30. 1.32 Consider the conditions of Problem 1.22. However, now the plate is in a vacuum with a surrounding temperature of 25⬚C. What is the emissivity of the plate? What is the rate at which radiation is emitted by the surface? 1.33 If Ts Tsur in Equation 1.9, the radiation heat transfer coefficient may be approximated as hr,a ⫽ 4T 3 where T (Ts ⫹ Tsur)/2. We wish to assess the validity of this approximation by comparing values of hr and hr,a for the following conditions. In each case, represent your results graphically and comment on the validity of the approximation. (a) Consider a surface of either polished aluminum ( ⫽ 0.05) or black paint ( ⫽ 0.9), whose temperature may exceed that of the surroundings (Tsur ⫽ 25⬚C) by 10 to 100°C. Also compare your results with values of the coefficient associated with free convection in air (T앝 ⫽ Tsur), where h(W/m2 䡠 K) ⫽ 0.98 ⌬T 1/3. (b) Consider initial conditions associated with placing a workpiece at Ts ⫽ 25⬚C in a large furnace whose wall temperature may be varied over the range 100 ⱕ Tsur ⱕ 1000⬚C. According to the surface finish or coating, its emissivity may assume values of 0.05, 0.2, and 0.9. For each emissivity, plot the relative error, (hr ⫺ hr,a )/hr , as a function of the furnace temperature. 1.34 A vacuum system, as used in sputtering electrically conducting thin films on microcircuits, is comprised of a baseplate maintained by an electrical heater at 300 K and a shroud within the enclosure maintained at 77 K by a liquid-nitrogen coolant loop. The circular baseplate, insulated on the lower side, is 0.3 m in diameter and has an emissivity of 0.25.

Vacuum enclosure

1.30 A spherical interplanetary probe of 0.5-m diameter contains electronics that dissipate 150 W. If the probe surface has an emissivity of 0.8 and the probe does not receive radiation from other surfaces, as, for example, from the sun, what is its surface temperature? 1.31 An instrumentation package has a spherical outer surface of diameter D ⫽ 100 mm and emissivity ⫽ 0.25. The package is placed in a large space simulation chamber whose walls are maintained at 77 K. If operation of the electronic components is restricted to the temperature

Liquid-nitrogen filled shroud

LN2

Electrical heater Baseplate

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53

Problems

(a) How much electrical power must be provided to the baseplate heater? (b) At what rate must liquid nitrogen be supplied to the shroud if its heat of vaporization is 125 kJ/kg? (c) To reduce the liquid nitrogen consumption, it is proposed to bond a thin sheet of aluminum foil ( ⫽ 0.09) to the baseplate. Will this have the desired effect?

Relationship to Thermodynamics 1.35 An electrical resistor is connected to a battery, as shown schematically. After a brief transient, the resistor assumes a nearly uniform, steady-state temperature of 95⬚C, while the battery and lead wires remain at the ambient temperature of 25⬚C. Neglect the electrical resistance of the lead wires. I = 6A

Resistor

Battery

1.37 Consider the tube and inlet conditions of Problem 1.36. Heat transfer at a rate of q ⫽ 3.89 MW is delivered to the tube. For an exit pressure of p ⫽ 8 bar, determine (a) the temperature of the water at the outlet as well as the change in (b) combined thermal and flow work, (c) mechanical energy, and (d) total energy of the water from the inlet to the outlet of the tube. Hint: As a first estimate, neglect the change in mechanical energy in solving part (a). Relevant properties may be obtained from a thermodynamics text. 1.38 An internally reversible refrigerator has a modified coefficient of performance accounting for realistic heat transfer processes of COPm ⫽

where qin is the refrigerator cooling rate, qout is the heat ˙ is the power input. Show that COPm rejection rate, and W can be expressed in terms of the reservoir temperatures Tc and Th, the cold and hot thermal resistances Rt,c and Rt,h, and qin, as

Air

V = 24 V

COPm ⫽

T• = 25C

Lead wire

(c) Neglecting radiation from the resistor, what is the convection coefficient? 1.36 Pressurized water (pin ⫽ 10 bar, Tin ⫽ 110⬚C) enters the bottom of an L ⫽ 10-m-long vertical tube of diameter D ⫽ 100 mm at a mass flow rate of m˙ ⫽ 1.5 kg/s. The tube is located inside a combustion chamber, resulting in heat transfer to the tube. Superheated steam exits the top of the tube at pout ⫽ 7 bar, Tout ⫽ 600⬚C. Determine the change in the rate at which the following quantities enter and exit the tube: (a) the combined thermal and flow work, (b) the mechanical energy, and (c) the total energy of the water. Also, (d) determine the heat transfer rate, q. Hint: Relevant properties may be obtained from a thermodynamics text.

Tc ⫺ qin Rtot Th ⫺ Tc ⫹ qin Rtot

where Rtot ⫽ Rt,c ⫹ Rt,h. Also, show that the power input may be expressed as Th ⫺ Tc ⫹ qin Rtot W˙ ⫽ qin Tc ⫺ qin Rtot

(a) Consider the resistor as a system about which a control surface is placed and Equation 1.12c is applied. Determine the corresponding values of E˙ in(W), E˙ g(W), E˙ out(W), and E˙ st(W). If a control surface is placed about the entire system, what are the values of E˙ in, E˙ g, E˙ out, and E˙ st? (b) If electrical energy is dissipated uniformly within the resistor, which is a cylinder of diameter D ⫽ 60 mm and length L ⫽ 250 mm, what is the volumetric heat generation rate, q˙ (W/m3)?

Tc,i qin qin ⫽q ⫺q ⫽ out in T h,i ⫺ Tc,i W˙

High-temperature reservoir Q

Th

out

Th,i W

Internally reversible refrigerator

Tc,i Qin Low-temperature reservoir

High-temperature side resistance Low-temperature side resistance

Tc

1.39 A household refrigerator operates with cold- and hot-temperature reservoirs of Tc ⫽ 5⬚C and Th ⫽ 25⬚C, respectively. When new, the cold and hot side resistances are Rc,n ⫽ 0.05 K/W and Rh,n ⫽ 0.04 K/W, respectively. Over time, dust accumulates on the refrigerator’s condenser coil, which is located behind the refrigerator, increasing the hot side resistance to Rh,d ⫽ 0.1 K/W. It is desired to have a refrigerator cooling rate of qin ⫽ 750 W. Using the results of Problem 1.38, determine the modified coefficient of performance and the required power input ˙ under (a) clean and (b) dusty coil conditions. W

54

Chapter 1

䊏

Introduction exposed surface is h ⫽ 8 W/m2 䡠 K, and the surface is characterized by an emissivity of s ⫽ 0.9. The solid silicon powder is at Tsi,i ⫽ 298 K, and the solid silicon sheet exits the chamber at Tsi,o ⫽ 420 K. Both the surroundings and ambient temperatures are T앝 ⫽ Tsur ⫽ 298 K.

Energy Balance and Multimode Effects 1.40 Chips of width L ⫽ 15 mm on a side are mounted to a substrate that is installed in an enclosure whose walls and air are maintained at a temperature of Tsur ⫽ 25⬚C. The chips have an emissivity of ⫽ 0.60 and a maximum allowable temperature of Ts ⫽ 85⬚C.

Solid silicon powder

Enclosure, Tsur

Vsi Ts,o

Tsur

Ts, εs

Substrate

Air T∞, h

tsi

Molten silicon String

Pelec

Chip (Ts, ε)

Solid silicon sheet

Solid silicon sheet

H

Air T∞, h

Vsi

• •

•

Molten silicon Crucible D

L

(a) If heat is rejected from the chips by radiation and natural convection, what is the maximum operating power of each chip? The convection coefficient depends on the chip-to-air temperature difference and may be approximated as h ⫽ C(Ts ⫺ T앝)1/4, where C ⫽ 4.2 W/m2 䡠 K5/4. (b) If a fan is used to maintain airflow through the enclosure and heat transfer is by forced convection, with h ⫽ 250 W/m2 䡠 K, what is the maximum operating power? 1.41 Consider the transmission case of Problem 1.23, but now allow for radiation exchange with the ground/ chassis, which may be approximated as large surroundings at Tsur ⫽ 30⬚C. If the emissivity of the case is ⫽ 0.80, what is the surface temperature? 1.42 One method for growing thin silicon sheets for photovoltaic solar panels is to pass two thin strings of high melting temperature material upward through a bath of molten silicon. The silicon solidifies on the strings near the surface of the molten pool, and the solid silicon sheet is pulled slowly upward out of the pool. The silicon is replenished by supplying the molten pool with solid silicon powder. Consider a silicon sheet that is Wsi ⫽ 85 mm wide and tsi ⫽ 150 m thick that is pulled at a velocity of Vsi ⫽ 20 mm/min. The silicon is melted by supplying electric power to the cylindrical growth chamber of height H ⫽ 350 mm and diameter D ⫽ 300 mm. The exposed surfaces of the growth chamber are at Ts ⫽ 320 K, the corresponding convection coefficient at the

(a) Determine the electric power, Pelec, needed to operate the system at steady state. (b) If the photovoltaic panel absorbs a time-averaged solar flux of q⬙sol ⫽ 180 W/m2 and the panel has a conversion efficiency (the ratio of solar power absorbed to electric power produced) of ⫽ 0.20, how long must the solar panel be operated to produce enough electric energy to offset the electric energy that was consumed in its manufacture? 1.43 Heat is transferred by radiation and convection between the inner surface of the nacelle of the wind turbine of Example 1.3 and the outer surfaces of the gearbox and generator. The convection heat flux associated with the gearbox and the generator may be described by q⬙conv,gb ⫽ h(Tgb ⫺ T앝) and q⬙conv,gen ⫽ h(Tgen ⫺ T앝), respectively, where the ambient temperature T앝 Ts (which is the nacelle temperature) and h ⫽ 40 W/m2 䡠 K. The outer surfaces of both the gearbox and the generator are characterized by an emissivity of ⫽ 0.9. If the surface areas of the gearbox and generator are Agb ⫽ 6 m2 and Agen ⫽ 4 m2, respectively, determine their surface temperatures. 1.44 Radioactive wastes are packed in a long, thin-walled cylindrical container. The wastes generate thermal energy nonuniformly according to the relation q˙ ⫽ q˙o[1 ⫺ (r/ro)2], where q˙ is the local rate of energy generation per unit volume, q˙o is a constant, and ro is the radius of the container. Steady-state conditions are maintained by submerging the container in a liquid that is at T앝 and provides a uniform convection coefficient h.

䊏

55

Problems

estimate the magnitudes of kinetic and potential energy changes. Assume the blood’s properties are similar to those of water.

ro

T∞, h q• = q•o [1 – (r/ro)2]

Obtain an expression for the total rate at which energy is generated in a unit length of the container. Use this result to obtain an expression for the temperature Ts of the container wall. 1.45 An aluminum plate 4 mm thick is mounted in a horizontal position, and its bottom surface is well insulated. A special, thin coating is applied to the top surface such that it absorbs 80% of any incident solar radiation, while having an emissivity of 0.25. The density and specific heat c of aluminum are known to be 2700 kg/m3 and 900 J/kg 䡠 K, respectively. (a) Consider conditions for which the plate is at a temperature of 25⬚C and its top surface is suddenly exposed to ambient air at T앝 ⫽ 20⬚C and to solar radiation that provides an incident flux of 900 W/m2. The convection heat transfer coefficient between the surface and the air is h ⫽ 20 W/m2 䡠 K. What is the initial rate of change of the plate temperature? (b) What will be the equilibrium temperature of the plate when steady-state conditions are reached? (c) The surface radiative properties depend on the specific nature of the applied coating. Compute and plot the steady-state temperature as a function of the emissivity for 0.05 ⱕ ⱕ 1, with all other conditions remaining as prescribed. Repeat your calculations for values of ␣S ⫽ 0.5 and 1.0, and plot the results with those obtained for ␣S ⫽ 0.8. If the intent is to maximize the plate temperature, what is the most desirable combination of the plate emissivity and its absorptivity to solar radiation? 1.46 A blood warmer is to be used during the transfusion of blood to a patient. This device is to heat blood taken from the blood bank at 10⬚C to 37⬚C at a flow rate of 200 ml/min. The blood passes through tubing of length 2 m, with a rectangular cross section 6.4 mm ⫻ 1.6 mm At what rate must heat be added to the blood to accomplish the required temperature increase? If the fluid originates from a large tank with nearly zero velocity and flows vertically downward for its 2-m length,

1.47 Consider a carton of milk that is refrigerated at a temperature of Tm ⫽ 5⬚C. The kitchen temperature on a hot summer day is T앝 ⫽ 30⬚C. If the four sides of the carton are of height and width L ⫽ 200 mm and w ⫽ 100 mm, respectively, determine the heat transferred to the milk carton as it sits on the kitchen counter for durations of t ⫽ 10 s, 60 s, and 300 s before it is returned to the refrigerator. The convection coefficient associated with natural convection on the sides of the carton is h ⫽ 10 W/m2 䡠 K. The surface emissivity is 0.90. Assume the milk carton temperature remains at 5⬚C during the process. Your parents have taught you the importance of refrigerating certain foods from the food safety perspective. Comment on the importance of quickly returning the milk carton to the refrigerator from an energy conservation point of view. 1.48 The energy consumption associated with a home water heater has two components: (i) the energy that must be supplied to bring the temperature of groundwater to the heater storage temperature, as it is introduced to replace hot water that has been used; (ii) the energy needed to compensate for heat losses incurred while the water is stored at the prescribed temperature. In this problem, we will evaluate the first of these components for a family of four, whose daily hot water consumption is approximately 100 gal. If groundwater is available at 15⬚C, what is the annual energy consumption associated with heating the water to a storage temperature of 55⬚C? For a unit electrical power cost of $0.18/kW 䡠 h, what is the annual cost associated with supplying hot water by means of (a) electric resistance heating or (b) a heat pump having a COP of 3. 1.49 Liquid oxygen, which has a boiling point of 90 K and a latent heat of vaporization of 214 kJ/kg, is stored in a spherical container whose outer surface is of 500-mm diameter and at a temperature of ⫺10⬚C. The container is housed in a laboratory whose air and walls are at 25⬚C. (a) If the surface emissivity is 0.20 and the heat transfer coefficient associated with free convection at the outer surface of the container is 10 W/m2 䡠 K, what is the rate, in kg/s, at which oxygen vapor must be vented from the system? (b) Moisture in the ambient air will result in frost formation on the container, causing the surface emissivity to increase. Assuming the surface temperature and convection coefficient to remain at ⫺10⬚C and

56

Chapter 1

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Introduction

10 W/m2 䡠 K, respectively, compute the oxygen evaporation rate (kg/s) as a function of surface emissivity over the range 0.2 ⱕ ⱕ 0.94. 1.50 The emissivity of galvanized steel sheet, a common roofing material, is ⫽ 0.13 at temperatures around 300 K, while its absorptivity for solar irradiation is ␣S ⫽ 0.65. Would the neighborhood cat be comfortable walking on a roof constructed of the material on a day when GS ⫽ 750 W/m2, T앝 ⫽ 16⬚C, and h ⫽ 7 W/m2 䡠 K? Assume the bottom surface of the steel is insulated. 1.51 Three electric resistance heaters of length L ⫽ 250 mm and diameter D ⫽ 25 mm are submerged in a 10-gal tank of water, which is initially at 295 K. The water may be assumed to have a density and specific heat of ⫽ 990 kg/m3 and c ⫽ 4180 J/kg 䡠 K. (a) If the heaters are activated, each dissipating q1 ⫽ 500 W, estimate the time required to bring the water to a temperature of 335 K. (b) If the natural convection coefficient is given by an expression of the form h ⫽ 370 (Ts ⫺ T)1/3, where Ts and T are temperatures of the heater surface and water, respectively, what is the temperature of each heater shortly after activation and just before deactivation? Units of h and (Ts ⫺ T) are W/m2 ⭈ K and K, respectively. (c) If the heaters are inadvertently activated when the tank is empty, the natural convection coefficient associated with heat transfer to the ambient air at T앝 ⫽ 300 K may be approximated as h ⫽ 0.70 (Ts ⫺ T앝)1/3. If the temperature of the tank walls is also 300 K and the emissivity of the heater surface is ⫽ 0.85, what is the surface temperature of each heater under steady-state conditions? 1.52 A hair dryer may be idealized as a circular duct through which a small fan draws ambient air and within which the air is heated as it flows over a coiled electric resistance wire.

(a) If a dryer is designed to operate with an electric power consumption of Pelec ⫽ 500 W and to heat air from an ambient temperature of Ti ⫽ 20⬚C to a discharge temperature of To ⫽ 45⬚C, at what volu˙ should the fan operate? Heat loss metric flow rate ᭙ from the casing to the ambient air and the surroundings may be neglected. If the duct has a diameter of D ⫽ 70 mm, what is the discharge velocity Vo of the air? The density and specific heat of the air may be approximated as ⫽ 1.10 kg/m3 and cp ⫽ 1007 J/kg 䡠 K, respectively. (b) Consider a dryer duct length of L ⫽ 150 mm and a surface emissivity of ⫽ 0.8. If the coefficient associated with heat transfer by natural convection from the casing to the ambient air is h ⫽ 4 W/m2 䡠 K and the temperature of the air and the surroundings is T앝 ⫽ Tsur ⫽ 20⬚C, confirm that the heat loss from the casing is, in fact, negligible. The casing may be assumed to have an average surface temperature of Ts ⫽ 40⬚C. 1.53 In one stage of an annealing process, 304 stainless steel sheet is taken from 300 K to 1250 K as it passes through an electrically heated oven at a speed of Vs ⫽ 10 mm/s. The sheet thickness and width are ts ⫽ 8 mm and Ws ⫽ 2 m, respectively, while the height, width, and length of the oven are Ho ⫽ 2 m, Wo ⫽ 2.4 m, and Lo ⫽ 25 m, respectively. The top and four sides of the oven are exposed to ambient air and large surroundings, each at 300 K, and the corresponding surface temperature, convection coefficient, and emissivity are Ts ⫽ 350 K, h ⫽ 10 W/m2 䡠 K, and s ⫽ 0.8. The bottom surface of the oven is also at 350 K and rests on a 0.5-m-thick concrete pad whose base is at 300 K. Estimate the required electric power input, Pelec, to the oven. Tsur Pelec

Air

T∞, h

Ts, εs Lo Steel sheet

ts Vs

Surroundings, Tsur

Ts

Air T∞, h Electric resistor

Discharge

To, Vo

Concrete pad

Fan •

Inlet, ∀, Ti

D Pelec

Tb

Dryer, Ts, ε

1.54 Convection ovens operate on the principle of inducing forced convection inside the oven chamber with a fan. A small cake is to be baked in an oven when the convection feature is disabled. For this situation, the free convection coefficient associated with the cake and its

䊏

57

Problems

pan is hfr ⫽ 3 W/m2 䡠 K. The oven air and wall are at temperatures T앝 ⫽ Tsur ⫽ 180⬚C. Determine the heat flux delivered to the cake pan and cake batter when they are initially inserted into the oven and are at a temperature of Ti ⫽ 24⬚C. If the convection feature is activated, the forced convection heat transfer coefficient is hfo ⫽ 27 W/m2 䡠 K. What is the heat flux at the batter or pan surface when the oven is operated in the convection mode? Assume a value of 0.97 for the emissivity of the cake batter and pan. 1.55 Annealing, an important step in semiconductor materials processing, can be accomplished by rapidly heating the silicon wafer to a high temperature for a short period of time. The schematic shows a method involving the use of a hot plate operating at an elevated temperature Th. The wafer, initially at a temperature of Tw,i, is suddenly positioned at a gap separation distance L from the hot plate. The purpose of the analysis is to compare the heat fluxes by conduction through the gas within the gap and by radiation exchange between the hot plate and the cool wafer. The initial time rate of change in the temperature of the wafer, (dTw /dt)i, is also of interest. Approximating the surfaces of the hot plate and the wafer as blackbodies and assuming their diameter D to be much larger than the spacing L, the radiative heat flux may be expressed as q⬙rad ⫽ (Th4 ⫺ Tw4). The silicon wafer has a thickness of d ⫽ 0.78 mm, a density of 2700 kg/m3, and a specific heat of 875 J/kg 䡠 K. The thermal conductivity of the gas in the gap is 0.0436 W/m 䡠 K. D Hot plate, Th Stagnant gas, k Silicon wafer, Tw, i Gap, L

d

Positioner motion

(a) For Th ⫽ 600⬚C and Tw,i ⫽ 20⬚C, calculate the radiative heat flux and the heat flux by conduction across a gap distance of L ⫽ 0.2 mm. Also determine the value of (dTw /dt)i, resulting from each of the heating modes. (b) For gap distances of 0.2, 0.5, and 1.0 mm, determine the heat fluxes and temperature-time change as a function of the hot plate temperature for 300 ⱕ Th ⱕ 1300⬚C. Display your results graphically. Comment on the relative importance of the two heat

transfer modes and the effect of the gap distance on the heating process. Under what conditions could a wafer be heated to 900⬚C in less than 10 s? 1.56 In the thermal processing of semiconductor materials, annealing is accomplished by heating a silicon wafer according to a temperature-time recipe and then maintaining a fixed elevated temperature for a prescribed period of time. For the process tool arrangement shown as follows, the wafer is in an evacuated chamber whose walls are maintained at 27⬚C and within which heating lamps maintain a radiant flux q⬙s at its upper surface. The wafer is 0.78 mm thick, has a thermal conductivity of 30 W/m 䡠 K, and an emissivity that equals its absorptivity to the radiant flux ( ⫽ ␣l ⫽ 0.65). For q⬙s ⫽ 3.0 ⫻ 105 W/m2, the temperature on its lower surface is measured by a radiation thermometer and found to have a value of Tw,l ⫽ 997⬚C.

Heating lamps

Tsur = 27°C

qs'' = 3 × 105 W/m2 Wafer, k, ε , αl

L = 0.78 mm

Tw, l = 997°C

To avoid warping the wafer and inducing slip planes in the crystal structure, the temperature difference across the thickness of the wafer must be less than 2⬚C. Is this condition being met? 1.57 A furnace for processing semiconductor materials is formed by a silicon carbide chamber that is zone-heated on the top section and cooled on the lower section. With the elevator in the lowest position, a robot arm inserts the silicon wafer on the mounting pins. In a production operation, the wafer is rapidly moved toward the hot zone to achieve the temperature-time history required for the process recipe. In this position, the top and bottom surfaces of the wafer exchange radiation with the hot and cool zones, respectively, of the chamber. The zone temperatures are Th ⫽ 1500 K and Tc ⫽ 330 K, and the emissivity and thickness of the wafer are ⫽ 0.65 and d ⫽ 0.78 mm, respectively. With the ambient gas at T앝 ⫽ 700 K, convection coefficients at the upper and lower surfaces of the wafer are 8 and 4 W/m2 䡠 K, respectively. The silicon wafer has a density of 2700 kg/m3 and a specific heat of 875 J/kg 䡠 K.

58

Chapter 1

䊏

Introduction

Lstack SiC chamber

Gas, T•

Estack

e

Heating zone Wafer, Tw, ε hu

hl

Mounting pin holder

Hot zone, Th = 1500 K Cool zone, Tc = 330 K

Elevator Water channel

Bipolar plate

Hydrogen flow channel

Airflow channel

Membrane

(a) For an initial condition corresponding to a wafer temperature of Tw,i ⫽ 300 K and the position of the wafer shown schematically, determine the corresponding time rate of change of the wafer temperature, (dTw /dt)i. (b) Determine the steady-state temperature reached by the wafer if it remains in this position. How significant is convection heat transfer for this situation? Sketch how you would expect the wafer temperature to vary as a function of vertical distance. 1.58 Single fuel cells such as the one of Example 1.5 can be scaled up by arranging them into a fuel cell stack. A stack consists of multiple electrolytic membranes that are sandwiched between electrically conducting bipolar plates. Air and hydrogen are fed to each membrane through flow channels within each bipolar plate, as shown in the sketch. With this stack arrangement, the individual fuel cells are connected in series, electrically, producing a stack voltage of Estack ⫽ N ⫻ Ec, where Ec is the voltage produced across each membrane and N is the number of membranes in the stack. The electrical current is the same for each membrane. The cell voltage, Ec, as well as the cell efficiency, increases with temperature (the air and hydrogen fed to the stack are humidified to allow operation at temperatures greater than in Example 1.5), but the membranes will fail at temperatures exceeding T 85⬚C. Consider L ⫻ w membranes, where L ⫽ w ⫽ 100 mm, of thickness tm ⫽ 0.43 mm, that each produce Ec ⫽ 0.6 V at I ⫽ 60 A, and E˙ c,g ⫽ 45 W of thermal energy when operating at T ⫽ 80⬚C. The external surfaces of the stack are exposed to air at T앝 ⫽ 25⬚C and surroundings at Tsur ⫽ 30⬚C, with ⫽ 0.88 and h ⫽ 150 W/m2 䡠 K.

(a) Find the electrical power produced by a stack that is Lstack ⫽ 200 mm long, for bipolar plate thickness in the range 1 mm ⬍ tbp ⬍ 10 mm. Determine the total thermal energy generated by the stack. (b) Calculate the surface temperature and explain whether the stack needs to be internally heated or cooled to operate at the optimal internal temperature of 80⬚C for various bipolar plate thicknesses. (c) Identify how the internal stack operating temperature might be lowered or raised for a given bipolar plate thickness, and discuss design changes that would promote a more uniform temperature distribution within the stack. How would changes in the external air and surroundings temperature affect your answer? Which membrane in the stack is most likely to fail due to high operating temperature? 1.59 Consider the wind turbine of Example 1.3. To reduce the nacelle temperature to Ts ⫽ 30⬚C, the nacelle is vented and a fan is installed to force ambient air into and out of the nacelle enclosure. What is the minimum mass flow rate of air required if the air temperature increases to the nacelle surface temperature before exiting the nacelle? The specific heat of air is 1007 J/kg 䡠 K. 1.60 Consider the conducting rod of Example 1.4 under steady-state conditions. As suggested in Comment 3, the temperature of the rod may be controlled by varying the speed of airflow over the rod, which, in turn, alters the convection heat transfer coefficient. To consider the effect of the convection coefficient, generate plots of T versus I for values of h ⫽ 50, 100, and 250 W/m2 䡠 K. Would variations in the surface emissivity have a significant effect on the rod temperature?

䊏

59

Problems

1.61 A long bus bar (cylindrical rod used for making electrical connections) of diameter D is installed in a large conduit having a surface temperature of 30⬚C and in which the ambient air temperature is T앝 ⫽ 30⬚C. The electrical resistivity, e(⍀ 䡠 m), of the bar material is a function of temperature, e,o ⫽ e [1 ⫹ ␣ (T ⫺ To)], where e,o ⫽ 0.0171 ⍀ 䡠 m, To ⫽ 25⬚C, and ␣ ⫽ 0.00396 K⫺1. The bar experiences free convection in the ambient air, and the convection coefficient depends on the bar diameter, as well as on the difference between the surface and ambient temperatures. The governing relation is of the form, h ⫽ CD⫺0.25 (T ⫺ T앝)0.25, where C ⫽ 1.21 W 䡠 m⫺1.75 䡠 K⫺1.25. The emissivity of the bar surface is ⫽ 0.85. (a) Recognizing that the electrical resistance per unit length of the bar is R⬘e ⫽ e /Ac, where Ac is its cross-sectional area, calculate the current-carrying capacity of a 20-mm-diameter bus bar if its temperature is not to exceed 65⬚C. Compare the relative importance of heat transfer by free convection and radiation exchange. (b) To assess the trade-off between current-carrying capacity, operating temperature, and bar diameter, for diameters of 10, 20, and 40 mm, plot the bar temperature T as a function of current for the range 100 ⱕ I ⱕ 5000 A. Also plot the ratio of the heat transfer by convection to the total heat transfer. 1.62 A small sphere of reference-grade iron with a specific heat of 447 J/kg 䡠 K and a mass of 0.515 kg is suddenly immersed in a water–ice mixture. Fine thermocouple wires suspend the sphere, and the temperature is observed to change from 15 to 14⬚C in 6.35 s. The experiment is repeated with a metallic sphere of the same diameter, but of unknown composition with a mass of 1.263 kg. If the same observed temperature change occurs in 4.59 s, what is the specific heat of the unknown material? 1.63 A 50 mm ⫻ 45 mm ⫻ 20 mm cell phone charger has a surface temperature of Ts ⫽ 33⬚C when plugged into an electrical wall outlet but not in use. The surface of the charger is of emissivity ⫽ 0.92 and is subject to a free convection heat transfer coefficient of h ⫽ 4.5 W/m2 䡠 K. The room air and wall temperatures are T앝 ⫽ 22⬚C and Tsur ⫽ 20⬚C, respectively. If electricity costs C ⫽ $0.18/kW 䡠 h, determine the daily cost of leaving the charger plugged in when not in use.

Tsur w 20 mm

L 50 mm

Wall Charger

Air T∞, h

1.64 A spherical, stainless steel (AISI 302) canister is used to store reacting chemicals that provide for a uniform heat flux q⬙i to its inner surface. The canister is suddenly submerged in a liquid bath of temperature T앝 ⬍ Ti, where Ti is the initial temperature of the canister wall. Canister

Reacting chemicals

ro = 0.6 m Ti = 500 K

3 ρ = 8055 kg/m c = 510 J/kg•K

p

T∞ = 300 K h = 500 W/m2•K

q"i Bath

ri = 0.5 m

(a) Assuming negligible temperature gradients in the canister wall and a constant heat flux q⬙i , develop an equation that governs the variation of the wall temperature with time during the transient process. What is the initial rate of change of the wall temperature if q⬙i ⫽ 105 W/m2? (b) What is the steady-state temperature of the wall? (c) The convection coefficient depends on the velocity associated with fluid flow over the canister and whether the wall temperature is large enough to induce boiling in the liquid. Compute and plot the steady-state temperature as a function of h for the range 100 ⱕ h ⱕ 10,000 W/m2 䡠 K. Is there a value of h below which operation would be unacceptable? 1.65 A freezer compartment is covered with a 2-mm-thick layer of frost at the time it malfunctions. If the compartment is in ambient air at 20⬚C and a coefficient of h ⫽ 2 W/m2 䡠 K characterizes heat transfer by natural convection from the exposed surface of the layer, estimate the time required to completely melt the frost. The frost may be assumed to have a mass density of 700 kg/m3 and a latent heat of fusion of 334 kJ/kg.

60

Chapter 1

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Introduction

1.66 A vertical slab of Wood’s metal is joined to a substrate on one surface and is melted as it is uniformly irradiated by a laser source on the opposite surface. The metal is initially at its fusion temperature of Tf ⫽ 72⬚C, and the melt runs off by gravity as soon as it is formed. The absorptivity of the metal to the laser radiation is ␣1 ⫽ 0.4, and its latent heat of fusion is hsf ⫽ 33 kJ/kg. (a) Neglecting heat transfer from the irradiated surface by convection or radiation exchange with the surroundings, determine the instantaneous rate of melting in kg/s 䡠 m2 if the laser irradiation is 5 kW/m2. How much material is removed if irradiation is maintained for a period of 2 s? (b) Allowing for convection to ambient air, with T앝 ⫽ 20⬚C and h ⫽ 15 W/m2 䡠 K, and radiation exchange with large surroundings ( ⫽ 0.4, Tsur ⫽ 20⬚C), determine the instantaneous rate of melting during irradiation. 1.67 A photovoltaic panel of dimension 2 m ⫻ 4 m is installed on the roof of a home. The panel is irradiated with a solar flux of GS ⫽ 700 W/m2, oriented normal to the top panel surface. The absorptivity of the panel to the solar irradiation is ␣S ⫽ 0.83, and the efficiency of conversion of the absorbed flux to electrical power is ⫽ P/␣S GS A ⫽ 0.553 ⫺ 0.001 K⫺1Tp, where Tp is the panel temperature expressed in kelvins and A is the solar panel area. Determine the electrical power generated for (a) a still summer day, in which Tsur ⫽ T앝 ⫽ 35⬚C, h ⫽ 10 W/m2 䡠 K, and (b) a breezy winter day, for which Tsur ⫽ T앝 ⫽ ⫺15⬚C, h ⫽ 30 W/m2 䡠 K. The panel emissivity is ⫽ 0.90.

GS

Air T∞ , h

Tsur

Bank of infrared radiant heaters Gas-fired furnace Carton

Conveyor

The chief engineer of your plant will approve the purchase of the heaters if they can reduce the water content by 10% of the total mass. Would you recommend the purchase? Assume the heat of vaporization of water is hfg ⫽ 2400 kJ/kg. 1.69 Electronic power devices are mounted to a heat sink having an exposed surface area of 0.045 m2 and an emissivity of 0.80. When the devices dissipate a total power of 20 W and the air and surroundings are at 27⬚C, the average sink temperature is 42⬚C. What average temperature will the heat sink reach when the devices dissipate 30 W for the same environmental condition? Power device

Tsur = 27°C

Heat sink, Ts A s, ε

Air

T∞ = 27°C

1.70 A computer consists of an array of five printed circuit boards (PCBs), each dissipating Pb ⫽ 20 W of power. Cooling of the electronic components on a board is provided by the forced flow of air, equally distributed in passages formed by adjoining boards, and the convection coefficient associated with heat transfer from the components to the air is approximately h ⫽ 200 W/m2 䡠 K. Air enters the computer console at a temperature of Ti ⫽ 20⬚C, and flow is driven by a fan whose power consumption is Pf ⫽ 25 W. •

Outlet air ∀, To

P

Photovoltaic panel, Tp

1.68 Following the hot vacuum forming of a paper-pulp mixture, the product, an egg carton, is transported on a conveyor for 18 s toward the entrance of a gas-fired oven where it is dried to a desired final water content. Very little water evaporates during the travel time. So, to increase the productivity of the line, it is proposed that a bank of infrared radiation heaters, which provide a uniform radiant flux of 5000 W/m2, be installed over the conveyor. The carton has an exposed area of 0.0625 m2 and a mass of 0.220 kg, 75% of which is water after the forming process.

PCB, Pb

•

Inlet air ∀, Ti

Fan, Pf

䊏

61

Problems

(a) If the temperature rise of the airflow, (To ⫺ Ti), is not to exceed 15⬚C, what is the minimum allowable volu˙ of the air? The density and specific metric flow rate ᭙ heat of the air may be approximated as ⫽ 1.161 kg/m3 and cp ⫽ 1007 J/kg 䡠 K, respectively. (b) The component that is most susceptible to thermal failure dissipates 1 W/cm2 of surface area. To minimize the potential for thermal failure, where should the component be installed on a PCB? What is its surface temperature at this location? 1.71 Consider a surface-mount type transistor on a circuit board whose temperature is maintained at 35⬚C. Air at 20⬚C flows over the upper surface of dimensions 4 mm ⫻ 8 mm with a convection coefficient of 50 W/m2 䡠 K. Three wire leads, each of cross section 1 mm ⫻ 0.25 mm and length 4 mm, conduct heat from the case to the circuit board. The gap between the case and the board is 0.2 mm. Air

Transistor case Wire lead Circuit board

Gap

(a) Assuming the case is isothermal and neglecting radiation, estimate the case temperature when 150 mW is dissipated by the transistor and (i) stagnant air or (ii) a conductive paste fills the gap. The thermal conductivities of the wire leads, air, and conductive paste are 25, 0.0263, and 0.12 W/m 䡠 K, respectively. (b) Using the conductive paste to fill the gap, we wish to determine the extent to which increased heat dissipation may be accommodated, subject to the constraint that the case temperature not exceed 40⬚C. Options include increasing the air speed to achieve a larger convection coefficient h and/or changing the lead wire material to one of larger thermal conductivity. Independently considering leads fabricated from materials with thermal conductivities of 200 and 400 W/m 䡠 K, compute and plot the maximum allowable heat dissipation for variations in h over the range 50 ⱕ h ⱕ 250 W/m2 䡠 K. 1.72 The roof of a car in a parking lot absorbs a solar radiant flux of 800 W/m2, and the underside is perfectly insulated. The convection coefficient between the roof and the ambient air is 12 W/m2 䡠 K. (a) Neglecting radiation exchange with the surroundings, calculate the temperature of the roof under steadystate conditions if the ambient air temperature is 20⬚C.

(b) For the same ambient air temperature, calculate the temperature of the roof if its surface emissivity is 0.8. (c) The convection coefficient depends on airflow conditions over the roof, increasing with increasing air speed. Compute and plot the roof temperature as a function of h for 2 ⱕ h ⱕ 200 W/m2 䡠 K. 1.73 Consider the conditions of Problem 1.22, but the surroundings temperature is 25⬚C and radiation exchange with the surroundings is not negligible. If the convection coefficient is 6.4 W/m2 䡠 K and the emissivity of the plate is ⫽ 0.42, determine the time rate of change of the plate temperature, dT/dt, when the plate temperature is 225⬚C. Evaluate the heat loss by convection and the heat loss by radiation. 1.74 Most of the energy we consume as food is converted to thermal energy in the process of performing all our bodily functions and is ultimately lost as heat from our bodies. Consider a person who consumes 2100 kcal per day (note that what are commonly referred to as food calories are actually kilocalories), of which 2000 kcal is converted to thermal energy. (The remaining 100 kcal is used to do work on the environment.) The person has a surface area of 1.8 m2 and is dressed in a bathing suit. (a) The person is in a room at 20⬚C, with a convection heat transfer coefficient of 3 W/m2 䡠 K. At this air temperature, the person is not perspiring much. Estimate the person’s average skin temperature. (b) If the temperature of the environment were 33⬚C, what rate of perspiration would be needed to maintain a comfortable skin temperature of 33⬚C? 1.75 Consider Problem 1.1. (a) If the exposed cold surface of the insulation is at T2 ⫽ 20⬚C, what is the value of the convection heat transfer coefficient on the cold side of the insulation if the surroundings temperature is Tsur ⫽ 320 K, the ambient temperature is T앝 ⫽ 5⬚C, and the emissivity is ⫽ 0.95? Express your results in units of W/m2 䡠 K and W/m2 䡠 ⬚C. (b) Using the convective heat transfer coefficient you calculated in part (a), determine the surface temperature, T2, as the emissivity of the surface is varied over the range 0.05 ⱕ ⱕ 0.95. The hot wall temperature of the insulation remains fixed at T1 ⫽ 30⬚C. Display your results graphically. 1.76 The wall of an oven used to cure plastic parts is of thickness L ⫽ 0.05 m and is exposed to large surroundings and air at its outer surface. The air and the surroundings are at 300 K. (a) If the temperature of the outer surface is 400 K and its convection coefficient and emissivity are

62

Chapter 1

䊏

Introduction

h ⫽ 20 W/m2 䡠 K and ⫽ 0.8, respectively, what is the temperature of the inner surface if the wall has a thermal conductivity of k ⫽ 0.7 W/m2 䡠 K? (b) Consider conditions for which the temperature of the inner surface is maintained at 600 K, while the air and large surroundings to which the outer surface is exposed are maintained at 300 K. Explore the effects of variations in k, h, and on (i) the temperature of the outer surface, (ii) the heat flux through the wall, and (iii) the heat fluxes associated with convection and radiation heat transfer from the outer surface. Specifically, compute and plot the foregoing dependent variables for parametric variations about baseline conditions of k ⫽ 10 W/m 䡠 K, h ⫽ 20 W/m2 䡠 K, and ⫽ 0.5. The suggested ranges of the independent variables are 0.1 ⱕ k ⱕ 400 W/m 䡠 K, 2 ⱕ h ⱕ 200 W/m2 䡠 K, and 0.05 ⱕ ⱕ 1. Discuss the physical implications of your results. Under what conditions will the temperature of the outer surface be less than 45⬚C, which is a reasonable upper limit to avoid burn injuries if contact is made? 1.77 An experiment to determine the convection coefficient associated with airflow over the surface of a thick stainless steel casting involves the insertion of thermocouples into the casting at distances of 10 and 20 mm from the surface along a hypothetical line normal to the surface. The steel has a thermal conductivity of 15 W/m 䡠 K. If the thermocouples measure temperatures of 50 and 40⬚C in the steel when the air temperature is 100⬚C, what is the convection coefficient? 1.78 A thin electrical heating element provides a uniform heat flux q⬙o to the outer surface of a duct through which airflows. The duct wall has a thickness of 10 mm and a thermal conductivity of 20 W/m 䡠 K. Air

Duct

Air

Ti Duct wall

To Electrical heater Insulation

(a) At a particular location, the air temperature is 30⬚C and the convection heat transfer coefficient between the air and inner surface of the duct is 100 W/m2 䡠 K. What heat flux q⬙o is required to maintain the inner surface of the duct at Ti ⫽ 85⬚C?

(b) For the conditions of part (a), what is the temperature (To ) of the duct surface next to the heater? (c) With Ti ⫽ 85⬚C, compute and plot q⬙o and To as a function of the air-side convection coefficient h for the range 10 ⱕ h ⱕ 200 W/m2 䡠 K. Briefly discuss your results. 1.79 A rectangular forced air heating duct is suspended from the ceiling of a basement whose air and walls are at a temperature of T앝 ⫽ Tsur ⫽ 5⬚C. The duct is 15 m long, and its cross section is 350 mm ⫻ 200 mm. (a) For an uninsulated duct whose average surface temperature is 50⬚C, estimate the rate of heat loss from the duct. The surface emissivity and convection coefficient are approximately 0.5 and 4 W/m2 䡠 K, respectively. (b) If heated air enters the duct at 58⬚C and a velocity of 4 m/s and the heat loss corresponds to the result of part (a), what is the outlet temperature? The density and specific heat of the air may be assumed to be ⫽ 1.10 kg/m3 and c ⫽ 1008 J/kg 䡠 K, respectively. 1.80 Consider the steam pipe of Example 1.2. The facilities manager wants you to recommend methods for reducing the heat loss to the room, and two options are proposed. The first option would restrict air movement around the outer surface of the pipe and thereby reduce the convection coefficient by a factor of two. The second option would coat the outer surface of the pipe with a low emissivity ( ⫽ 0.4) paint. (a) Which of the foregoing options would you recommend? (b) To prepare for a presentation of your recommendation to management, generate a graph of the heat loss q⬘ as a function of the convection coefficient for 2 ⱕ h ⱕ 20 W/m 2 䡠 K and emissivities of 0.2, 0.4, and 0.8. Comment on the relative efficacy of reducing heat losses associated with convection and radiation. 1.81 During its manufacture, plate glass at 600⬚C is cooled by passing air over its surface such that the convection heat transfer coefficient is h ⫽ 5 W/m2 䡠 K. To prevent cracking, it is known that the temperature gradient must not exceed 15⬚C/mm at any point in the glass during the cooling process. If the thermal conductivity of the glass is 1.4 W/m 䡠 K and its surface emissivity is 0.8, what is the lowest temperature of the air that can initially be used for the cooling? Assume that the temperature of the air equals that of the surroundings. 1.82 The curing process of Example 1.9 involves exposure of the plate to irradiation from an infrared lamp and attendant cooling by convection and radiation exchange

䊏

63

Problems

with the surroundings. Alternatively, in lieu of the lamp, heating may be achieved by inserting the plate in an oven whose walls (the surroundings) are maintained at an elevated temperature. (a) Consider conditions for which the oven walls are at 200⬚C, airflow over the plate is characterized by T앝 ⫽ 20⬚C and h ⫽ 15 W/m2 䡠 K, and the coating has an emissivity of ⫽ 0.5. What is the temperature of the plate? (b) For ambient air temperatures of 20, 40, and 60⬚C, determine the plate temperature as a function of the oven wall temperature over the range from 150 to 250⬚C. Plot your results, and identify conditions for which acceptable curing temperatures between 100 and 110⬚C may be maintained.

1.85 A solar flux of 700 W/m2 is incident on a flat-plate solar collector used to heat water. The area of the collector is 3 m2, and 90% of the solar radiation passes through the cover glass and is absorbed by the absorber plate. The remaining 10% is reflected away from the collector. Water flows through the tube passages on the back side of the absorber plate and is heated from an inlet temperature Ti to an outlet temperature To. The cover glass, operating at a temperature of 30⬚C, has an emissivity of 0.94 and experiences radiation exchange with the sky at ⫺10⬚C. The convection coefficient between the cover glass and the ambient air at 25⬚C is 10 W/m2 䡠 K. GS Cover glass

1.83 The diameter and surface emissivity of an electrically heated plate are D ⫽ 300 mm and ⫽ 0.80, respectively. (a) Estimate the power needed to maintain a surface temperature of 200⬚C in a room for which the air and the walls are at 25⬚C. The coefficient characterizing heat transfer by natural convection depends on the surface temperature and, in units of W/m2 䡠 K, may be approximated by an expression of the form h ⫽ 0.80(Ts ⫺ T앝)1/3. (b) Assess the effect of surface temperature on the power requirement, as well as on the relative contributions of convection and radiation to heat transfer from the surface. 1.84 Bus bars proposed for use in a power transmission station have a rectangular cross section of height H ⫽ 600 mm and width W ⫽ 200 mm. The electrical resistivity, e(⍀ 䡠 m), of the bar material is a function of temperature, e ⫽ e,o[1 ⫹ ␣(T ⫺ To)], where e,o ⫽ 0.0828 ⍀ 䡠 m, To ⫽ 25⬚C, and ␣ ⫽ 0.0040 K⫺1. The emissivity of the bar’s painted surface is 0.8, and the temperature of the surroundings is 30⬚C. The convection coefficient between the bar and the ambient air at 30⬚C is 10 W/m2 䡠 K. (a) Assuming the bar has a uniform temperature T, calculate the steady-state temperature when a current of 60,000 A passes through the bar. (b) Compute and plot the steady-state temperature of the bar as a function of the convection coefficient for 10 ⱕ h ⱕ 100 W/m2 䡠 K. What minimum convection coefficient is required to maintain a safe-operating temperature below 120⬚C? Will increasing the emissivity significantly affect this result?

Air space Absorber plate Water tubing Insulation

(a) Perform an overall energy balance on the collector to obtain an expression for the rate at which useful heat is collected per unit area of the collector, q⬙u. Determine the value of q⬙u. (b) Calculate the temperature rise of the water, To ⫺ Ti, if the flow rate is 0.01 kg/s. Assume the specific heat of the water to be 4179 J/kg 䡠 K. (c) The collector efficiency is defined as the ratio of the useful heat collected to the rate at which solar energy is incident on the collector. What is the value of ?

Process Identification 1.86 In analyzing the performance of a thermal system, the engineer must be able to identify the relevant heat transfer processes. Only then can the system behavior be properly quantified. For the following systems, identify the pertinent processes, designating them by appropriately labeled arrows on a sketch of the system. Answer additional questions that appear in the problem statement. (a) Identify the heat transfer processes that determine the temperature of an asphalt pavement on a summer day. Write an energy balance for the surface of the pavement.

64

Chapter 1

䊏

Introduction

(b) Microwave radiation is known to be transmitted by plastics, glass, and ceramics but to be absorbed by materials having polar molecules such as water. Water molecules exposed to microwave radiation align and reverse alignment with the microwave radiation at frequencies up to 109 s⫺1, causing heat to be generated. Contrast cooking in a microwave oven with cooking in a conventional radiant or convection oven. In each case, what is the physical mechanism responsible for heating the food? Which oven has the greater energy utilization efficiency? Why? Microwave heating is being considered for drying clothes. How would the operation of a microwave clothes dryer differ from a conventional dryer? Which is likely to have the greater energy utilization efficiency? Why? (c) To prevent freezing of the liquid water inside the fuel cell of an automobile, the water is drained to an onboard storage tank when the automobile is not in use. (The water is transferred from the tank back to the fuel cell when the automobile is turned on.) Consider a fuel cell–powered automobile that is parked outside on a very cold evening with T앝 ⫽ ⫺20⬚C. The storage tank is initially empty at Ti,t ⫽ ⫺20⬚C, when liquid water, at atmospheric pressure and temperature Ti,w ⫽ 50⬚C, is introduced into the tank. The tank has a wall thickness tt and is blanketed with insulation of thickness tins. Identify the heat transfer processes that will promote freezing of the water. Will the likelihood of freezing change as the insulation thickness is modified? Will the likelihood of freezing depend on the tank wall’s thickness and material? Would freezing of the water be more likely if plastic (low thermal conductivity) or stainless steel (moderate thermal conductivity) tubing is used to transfer the water to and from the tank? Is there an optimal tank shape that would minimize the probability of the water freezing? Would freezing be more likely or less likely to occur if a thin sheet of aluminum foil (high thermal conductivity, low emissivity) is applied to the outside of the insulation? To fuel cell Transfer tubing

Tsur Water

tt tins h, T∞

(d) Your grandmother is concerned about reducing her winter heating bills. Her strategy is to loosely fit rigid polystyrene sheets of insulation over her double-pane windows right after the first freezing weather arrives in the autumn. Identify the relevant heat transfer processes on a cold winter night when the foamed insulation sheet is placed (i) on the inner surface and (ii) on the outer surface of her window. To avoid condensation damage, which configuration is preferred? Condensation on the window pane does not occur when the foamed insulation is not in place.

Cold, dry night air

Warm, moist room air

Exterior pane Air gap Interior pane Insulation

Insulation on inner surface

Cold, dry night air

Warm, moist room air

Exterior pane Air gap Interior pane Insulation

Insulation on outer surface

(e) There is considerable interest in developing building materials with improved insulating qualities. The development of such materials would do much to enhance energy conservation by reducing space heating requirements. It has been suggested that superior structural and insulating qualities could be obtained by using the composite shown. The material consists of a honeycomb, with cells of square cross section, sandwiched between solid slabs. The cells are filled with air, and the slabs, as well as the honeycomb matrix, are fabricated from plastics of low thermal conductivity. For heat transfer normal to the slabs, identify all heat transfer processes pertinent to the performance of the composite. Suggest ways in which this performance could be enhanced.

䊏

65

Problems

Surface slabs

(h) A thermocouple junction is used to measure the temperature of a solid material. The junction is inserted into a small circular hole and is held in place by epoxy. Identify the heat transfer processes associated with the junction. Will the junction sense a temperature less than, equal to, or greater than the solid temperature? How will the thermal conductivity of the epoxy affect the junction temperature? Hot solid

Cellular air spaces

(f) A thermocouple junction (bead) is used to measure the temperature of a hot gas stream flowing through a channel by inserting the junction into the mainstream of the gas. The surface of the channel is cooled such that its temperature is well below that of the gas. Identify the heat transfer processes associated with the junction surface. Will the junction sense a temperature that is less than, equal to, or greater than the gas temperature? A radiation shield is a small, openended tube that encloses the thermocouple junction, yet allows for passage of the gas through the tube. How does use of such a shield improve the accuracy of the temperature measurement? Cool channel Shield Hot gases

Thermocouple bead

(g) A double-glazed, glass fire screen is inserted between a wood-burning fireplace and the interior of a room. The screen consists of two vertical glass plates that are separated by a space through which room air may flow (the space is open at the top and bottom). Identify the heat transfer processes associated with the fire screen. Air channel Glass plate

Air

Thermocouple bead

Cool gases

Epoxy

1.87 In considering the following problems involving heat transfer in the natural environment (outdoors), recognize that solar radiation is comprised of long and short wavelength components. If this radiation is incident on a semitransparent medium, such as water or glass, two things will happen to the nonreflected portion of the radiation. The long wavelength component will be absorbed at the surface of the medium, whereas the short wavelength component will be transmitted by the surface. (a) The number of panes in a window can strongly influence the heat loss from a heated room to the outside ambient air. Compare the single- and double-paned units shown by identifying relevant heat transfer processes for each case.

Double pane Ambient air

Room air Single pane

(b) In a typical flat-plate solar collector, energy is collected by a working fluid that is circulated through tubes that are in good contact with the back face of an absorber plate. The back face is insulated from

66

Chapter 1

䊏

Introduction

the surroundings, and the absorber plate receives solar radiation on its front face, which is typically covered by one or more transparent plates. Identify the relevant heat transfer processes, first for the absorber plate with no cover plate and then for the absorber plate with a single cover plate. (c) The solar energy collector design shown in the schematic has been used for agricultural applications. Air is blown through a long duct whose cross section is in the form of an equilateral triangle. One side of the triangle is comprised of a double-paned, semitransparent cover; the other two sides are constructed from aluminum sheets painted flat black on the inside and covered on the outside with a layer of styrofoam insulation. During sunny periods, air entering the system is heated for delivery to either a greenhouse, grain drying unit, or storage system.

Identify all heat transfer processes associated with the cover plates, the absorber plate(s), and the air. (d) Evacuated-tube solar collectors are capable of improved performance relative to flat-plate collectors. The design consists of an inner tube enclosed in an outer tube that is transparent to solar radiation. The annular space between the tubes is evacuated. The outer, opaque surface of the inner tube absorbs solar radiation, and a working fluid is passed through the tube to collect the solar energy. The collector design generally consists of a row of such tubes arranged in front of a reflecting panel. Identify all heat transfer processes relevant to the performance of this device.

Solar radiation Evacuated tubes Reflecting panel

Doublepaned cover

Working fluid

Styrofoam

Transparent outer tube

Absorber plates Evacuated space

Inner tube

C H A P T E R

Introduction to Conduction

2

68

Chapter 2

䊏

Introduction to Conduction

R

ecall that conduction is the transport of energy in a medium due to a temperature gradient, and the physical mechanism is one of random atomic or molecular activity. In Chapter 1 we learned that conduction heat transfer is governed by Fourier’s law and that use of the law to determine the heat flux depends on knowledge of the manner in which temperature varies within the medium (the temperature distribution). By way of introduction, we restricted our attention to simplified conditions (one-dimensional, steady-state conduction in a plane wall). However, Fourier’s law is applicable to transient, multidimensional conduction in complex geometries. The objectives of this chapter are twofold. First, we wish to develop a deeper understanding of Fourier’s law. What are its origins? What form does it take for different geometries? How does its proportionality constant (the thermal conductivity) depend on the physical nature of the medium? Our second objective is to develop, from basic principles, the general equation, termed the heat equation, which governs the temperature distribution in a medium. The solution to this equation provides knowledge of the temperature distribution, which may then be used with Fourier’s law to determine the heat flux.

2.1 The Conduction Rate Equation Although the conduction rate equation, Fourier’s law, was introduced in Section 1.2, it is now appropriate to consider its origin. Fourier’s law is phenomenological; that is, it is developed from observed phenomena rather than being derived from first principles. Hence, we view the rate equation as a generalization based on much experimental evidence. For example, consider the steady-state conduction experiment of Figure 2.1. A cylindrical rod of known material is insulated on its lateral surface, while its end faces are maintained at different temperatures, with T1 T2. The temperature difference causes conduction heat transfer in the positive x-direction. We are able to measure the heat transfer rate qx, and we seek to determine how qx depends on the following variables: T, the temperature difference; x, the rod length; and A, the cross-sectional area. We might imagine first holding T and x constant and varying A. If we do so, we find that qx is directly proportional to A. Similarly, holding T and A constant, we observe that qx varies inversely with x. Finally, holding A and x constant, we find that qx is directly proportional to T. The collective effect is then qx A T x In changing the material (e.g., from a metal to a plastic), we would find that this proportionality remains valid. However, we would also find that, for equal values of A, x, and T, ∆T = T1 – T2

A, T1

T2

qx

x

∆x

FIGURE 2.1 Steady-state heat conduction experiment.

2.1

䊏

69

The Conduction Rate Equation

the value of qx would be smaller for the plastic than for the metal. This suggests that the proportionality may be converted to an equality by introducing a coefficient that is a measure of the material behavior. Hence, we write qx kA T x where k, the thermal conductivity (W/m 䡠 K) is an important property of the material. Evaluating this expression in the limit as x l 0, we obtain for the heat rate qx kA dT dx

(2.1)

or for the heat flux qx

qx k dT A dx

(2.2)

Recall that the minus sign is necessary because heat is always transferred in the direction of decreasing temperature. Fourier’s law, as written in Equation 2.2, implies that the heat flux is a directional quantity. In particular, the direction of qx is normal to the cross-sectional area A. Or, more generally, the direction of heat flow will always be normal to a surface of constant temperature, called an isothermal surface. Figure 2.2 illustrates the direction of heat flow qx in a plane wall for which the temperature gradient dT/dx is negative. From Equation 2.2, it follows that qx is positive. Note that the isothermal surfaces are planes normal to the x-direction. Recognizing that the heat flux is a vector quantity, we can write a more general statement of the conduction rate equation (Fourier’s law) as follows:

冢 ⭸T⭸x j ⭸T⭸y k ⭸T⭸z 冣

q kT k i

(2.3)

where is the three-dimensional del operator and T(x, y, z) is the scalar temperature field. It is implicit in Equation 2.3 that the heat flux vector is in a direction perpendicular to the isothermal surfaces. An alternative form of Fourier’s law is therefore q qn n k

⭸T n ⭸n

T(x)

T1 q''x T2 x

FIGURE 2.2 The relationship between coordinate system, heat flow direction, and temperature gradient in one dimension.

(2.4)

70

Chapter 2

䊏

Introduction to Conduction

qy''

qn''

qx'' y

n Isotherm

x

FIGURE 2.3 The heat flux vector normal to an isotherm in a two-dimensional coordinate system.

where qn is the heat flux in a direction n, which is normal to an isotherm, and n is the unit normal vector in that direction. This is illustrated for the two-dimensional case in Figure 2.3. The heat transfer is sustained by a temperature gradient along n. Note also that the heat flux vector can be resolved into components such that, in Cartesian coordinates, the general expression for q is q iqx jqy kqz

(2.5)

where, from Equation 2.3, it follows that qx k

⭸T ⭸x

qy k

⭸T ⭸y

qz k

⭸T ⭸z

(2.6)

Each of these expressions relates the heat flux across a surface to the temperature gradient in a direction perpendicular to the surface. It is also implicit in Equation 2.3 that the medium in which the conduction occurs is isotropic. For such a medium, the value of the thermal conductivity is independent of the coordinate direction. Fourier’s law is the cornerstone of conduction heat transfer, and its key features are summarized as follows. It is not an expression that may be derived from first principles; it is instead a generalization based on experimental evidence. It is an expression that defines an important material property, the thermal conductivity. In addition, Fourier’s law is a vector expression indicating that the heat flux is normal to an isotherm and in the direction of decreasing temperature. Finally, note that Fourier’s law applies for all matter, regardless of its state (solid, liquid, or gas).

2.2 The Thermal Properties of Matter To use Fourier’s law, the thermal conductivity of the material must be known. This property, which is referred to as a transport property, provides an indication of the rate at which energy is transferred by the diffusion process. It depends on the physical structure of matter, atomic and molecular, which is related to the state of the matter. In this section we consider various forms of matter, identifying important aspects of their behavior and presenting typical property values.

2.2.1

Thermal Conductivity

From Fourier’s law, Equation 2.6, the thermal conductivity associated with conduction in the x-direction is defined as qx kx ⬅ (⭸T/⭸x)

2.2

䊏

71

The Thermal Properties of Matter

Similar definitions are associated with thermal conductivities in the y- and z-directions (ky, kz), but for an isotropic material the thermal conductivity is independent of the direction of transfer, kx ky kz ⬅ k. From the foregoing equation, it follows that, for a prescribed temperature gradient, the conduction heat flux increases with increasing thermal conductivity. In general, the thermal conductivity of a solid is larger than that of a liquid, which is larger than that of a gas. As illustrated in Figure 2.4, the thermal conductivity of a solid may be more than four orders of magnitude larger than that of a gas. This trend is due largely to differences in intermolecular spacing for the two states. In the modern view of materials, a solid may be comprised of free electrons and atoms bound in a periodic arrangement called the lattice. Accordingly, transport of thermal energy may be due to two effects: the migration of free electrons and lattice vibrational waves. When viewed as a particle-like phenomenon, the lattice vibration quanta are termed phonons. In pure metals, the electron contribution to conduction heat transfer dominates, whereas in nonconductors and semiconductors, the phonon contribution is dominant. Kinetic theory yields the following expression for the thermal conductivity [1]:

The Solid State

k 1 C c mfp 3

(2.7)

For conducting materials such as metals, C ⬅ Ce is the electron specific heat per unit volume, c is the mean electron velocity, and mfp ⬅ e is the electron mean free path, which is defined as the average distance traveled by an electron before it collides with either an imperfection in the material or with a phonon. In nonconducting solids, C ⬅ Cph is the phonon specific heat, c is the average speed of sound, and mfp ⬅ ph is the phonon mean free path, which again is determined by collisions with imperfections or other phonons. In all cases, the thermal conductivity increases as the mean free path of the energy carriers (electrons or phonons) is increased.

Zinc Silver PURE METALS Nickel Aluminum ALLOYS Plastics Ice Oxides NONMETALLIC SOLIDS Foams Fibers INSULATION SYSTEMS Oils Water Mercury LIQUIDS Carbon Hydrogen dioxide GASES

0.01

0.1

1 10 Thermal conductivity (W/m•K)

100

1000

FIGURE 2.4 Range of thermal conductivity for various states of matter at normal temperatures and pressure.

Chapter 2

䊏

Introduction to Conduction

When electrons and phonons carry thermal energy leading to conduction heat transfer in a solid, the thermal conductivity may be expressed as k ke kph

(2.8)

To a first approximation, ke is inversely proportional to the electrical resistivity, e. For pure metals, which are of low e, ke is much larger than kph. In contrast, for alloys, which are of substantially larger e, the contribution of kph to k is no longer negligible. For nonmetallic solids, k is determined primarily by kph, which increases as the frequency of interactions between the atoms and the lattice decreases. The regularity of the lattice arrangement has an important effect on kph, with crystalline (well-ordered) materials like quartz having a higher thermal conductivity than amorphous materials like glass. In fact, for crystalline, nonmetallic solids such as diamond and beryllium oxide, kph can be quite large, exceeding values of k associated with good conductors, such as aluminum. The temperature dependence of k is shown in Figure 2.5 for representative metallic and nonmetallic solids. Values for selected materials of technical importance are also provided in Table A.1 (metallic solids) and Tables A.2 and A.3 (nonmetallic solids). More detailed treatments of thermal conductivity are available in the literature [2]. In the preceding discussion, the bulk thermal conductivity is described, and the thermal conductivity values listed in Tables A.1 through A.3 are appropriate for use when the physical dimensions of the material of interest are relatively large. This is the case in many commonplace engineering problems. However, in several

The Solid State: Micro- and Nanoscale Effects

500 400

Silver Copper

300

Gold Aluminum Aluminum alloy 2024 Tungsten

200

100 Thermal conductivity (W/m•K)

72

Platinum 50 Iron

20

Stainless steel, AISI 304

10

Aluminum oxide

5 Pyroceram

2 Fused quartz 1 100

300

500 1000 Temperature (K)

2000

4000

FIGURE 2.5 The temperature dependence of the thermal conductivity of selected solids.

2.2

䊏

73

The Thermal Properties of Matter

areas of technology, such as microelectronics, the material’s characteristic dimensions can be on the order of micrometers or nanometers, in which case care must be taken to account for the possible modifications of k that can occur as the physical dimensions become small. Cross sections of films of the same material having thicknesses L1 and L2 are shown in Figure 2.6. Electrons or phonons that are associated with conduction of thermal energy are also shown qualitatively. Note that the physical boundaries of the film act to scatter the energy carriers and redirect their propagation. For large L/mfp1 (Figure 2.6a), the effect of the boundaries on reducing the average energy carrier path length is minor, and conduction heat transfer occurs as described for bulk materials. However, as the film becomes thinner, the physical boundaries of the material can decrease the average net distance traveled by the energy carriers, as shown in Figure 2.6b. Moreover, electrons and phonons moving in the thin x-direction (representing conduction in the x-direction) are affected by the boundaries to a more significant degree than energy carriers moving in the y-direction. As such, for films characterized by small L/mfp, we find that kx ky k, where k is the bulk thermal conductivity of the film material. For L/mfp 1, the predicted values of kx and ky may be estimated to within 20% from the following expression [1]: mfp kx 1 k 3L ky k

1

(2.9a)

2mfp 3L

(2.9b)

Equations 2.9a, b reveal that the values of kx and ky are within approximately 5% of the bulk thermal conductivity if L/mfp 7 (for kx ) and L/mfp 4.5 (for ky). Values of the mean free path as well as critical film thicknesses below which microscale effects must be considered, Lcrit, are included in Table 2.1 for several materials at T ⬇ 300 K. For films with mfp L Lcrit, kx and ky are reduced from the bulk value as indicated in Equations 2.9a,b.

y L1

L2 < L1

(a)

(b)

x

FIGURE 2.6 Electron or phonon trajectories in (a) a relatively thick film and (b) a relatively thin film with boundary effects.

The quantity mfp/L is a dimensionless parameter known as the Knudsen number. Large Knudsen numbers (small L/mfp) suggest potentially significant nano- or microscale effects. 1

Chapter 2

䊏

Introduction to Conduction

No general guidelines exist for predicting values of the thermal conductivities for L/mfp 1. Note that, in solids, the value of mfp decreases as the temperature increases. In addition to scattering from physical boundaries, as in the case of Figure 2.6b, energy carriers may be redirected by chemical dopants embedded within a material or by grain boundaries that separate individual clusters of material in otherwise homogeneous matter. Nanostructured materials are chemically identical to their conventional counterparts but are processed to provide very small grain sizes. This feature impacts heat transfer by increasing the scattering and reflection of energy carriers at grain boundaries. Measured values of the thermal conductivity of a bulk, nanostructured yttria-stabilized zirconia material are shown in Figure 2.7. This particular ceramic is widely used for insulation purposes in high-temperature combustion devices. Conduction is dominated by phonon transfer, and the mean free path of the phonon energy carriers is, from Table 2.1, mfp 25 nm at 300 K. As the grain sizes are reduced to characteristic dimensions less than 25 nm (and more grain boundaries are introduced in the material per unit volume), significant reduction of the thermal conductivity occurs. Extrapolation of the results of Figure 2.7 to higher temperatures is not recommended, since the mean free path decreases with increasing temperature (mfp ⬇ 4 nm at T ⬇ 1525 K ) and grains of the material may coalesce, merge, and enlarge at elevated temperatures. Therefore, L/mfp becomes larger at high temperatures, and

TABLE 2.1 Mean free path and critical film thickness for various materials at T 艐 300 K [3,4] Material Aluminum oxide Diamond (IIa) Gallium arsenide Gold Silicon Silicon dioxide Yttria-stabilized zirconia

mfp (nm)

Lcrit, x (nm)

Lcrit,y (nm)

5.08 315 23 31 43 0.6 25

36 2200 160 220 290 4 170

22 1400 100 140 180 3 110

2.5 L = 98 nm 2 Thermal conductivity (W/m•K)

74

L = 55 nm L = 32 nm

1.5 L = 23 nm

1

L = 10 nm

0.5 λmfp (T = 300 K) = 25 nm 0

0

100

200

300

Temperature (K)

400

500

FIGURE 2.7 Measured thermal conductivity of yttria-stabilized zirconia as a function of temperature and mean grain size, L [3].

2.2

75

The Thermal Properties of Matter

䊏

reduction of k due to nanoscale effects becomes less pronounced. Research on heat transfer in nanostructured materials continues to reveal novel ways engineers can manipulate the nanostructure to reduce or increase thermal conductivity [5]. Potentially important consequences include applications such as gas turbine engine technology [6], microelectronics [7], and renewable energy [8]. The Fluid State The fluid state includes both liquids and gases. Because the intermolecular spacing is much larger and the motion of the molecules is more random for the fluid state than for the solid state, thermal energy transport is less effective. The thermal conductivity of gases and liquids is therefore generally smaller than that of solids. The effect of temperature, pressure, and chemical species on the thermal conductivity of a gas may be explained in terms of the kinetic theory of gases [9]. From this theory it is known that the thermal conductivity is directly proportional to the density of the gas, the mean molecular speed c, and the mean free path mfp, which is the average distance traveled by an energy carrier (a molecule) before experiencing a collision.

k 艐 1 cv c mfp 3

(2.10)

For an ideal gas, the mean free path may be expressed as mfp

k BT 兹2d 2p

(2.11)

where kB is Boltzmann’s constant, kB 1.381 1023 J/K, d is the diameter of the gas molecule, representative values of which are included in Figure 2.8, and p is the pressure. 0.3 Hydrogen

= 2.016, d ⫽ 0.274

Thermal conductivity (W/m•K)

Helium 4.003, 0.219

0.2

Water (steam, 1 atm) 18.02, 0.458

0.1

Carbon dioxide 44.01, 0.464

Air 28.97, 0.372

0

0

200

400 600 Temperature (K)

800

1000

FIGURE 2.8 The temperature dependence of the thermal conductivity of selected gases at normal pressures. Molecular diameters (d) are in nm [10]. Molecular weights (ᏹ) of the gases are also shown.

Chapter 2

䊏

Introduction to Conduction

As expected, the mean free path is small for high pressure or low temperature, which causes densely packed molecules. The mean free path also depends on the diameter of the molecule, with larger molecules more likely to experience collisions than small molecules; in the limiting case of an infinitesimally small molecule, the molecules cannot collide, resulting in an infinite mean free path. The mean molecular speed, c, can be determined from the kinetic theory of gases, and Equation 2.10 may ultimately be expressed as k

9␥ 5 cv 4 d 2

冪 ᏺ

ᏹkBT

(2.12)

where the parameter ␥ is the ratio of specific heats, ␥ ⬅ cp /cv, and ᏺ is Avogadro’s number, ᏺ 6.022 1023 molecules per mol. Equation 2.12 can be used to estimate the thermal conductivity of gas, although more accurate models have been developed [10]. It is important to note that the thermal conductivity is independent of pressure except in extreme cases as, for example, when conditions approach that of a perfect vacuum. Therefore, the assumption that k is independent of gas pressure for large volumes of gas is appropriate for the pressures of interest in this text. Accordingly, although the values of k presented in Table A.4 pertain to atmospheric pressure or the saturation pressure corresponding to the prescribed temperature, they may be used over a much wider pressure range. Molecular conditions associated with the liquid state are more difficult to describe, and physical mechanisms for explaining the thermal conductivity are not well understood [11]. The thermal conductivity of nonmetallic liquids generally decreases with increasing temperature. As shown in Figure 2.9, water, glycerine, and engine oil are notable exceptions. The thermal conductivity of liquids is usually insensitive to pressure except near the critical point. Also, thermal conductivity generally decreases with increasing molecular weight. Values of

0.8

Water

0.6 Thermal conductivity (W/m•K)

76

Ammonia

0.4 Glycerine

0.2 Engine oil Freon 12 0 200

300

400 Temperature (K)

500

FIGURE 2.9 The temperature dependence of the thermal conductivity of selected nonmetallic liquids under saturated conditions.

2.2

䊏

The Thermal Properties of Matter

77

the thermal conductivity are often tabulated as a function of temperature for the saturated state of the liquid. Tables A.5 and A.6 present such data for several common liquids. Liquid metals are commonly used in high heat flux applications, such as occur in nuclear power plants. The thermal conductivity of such liquids is given in Table A.7. Note that the values are much larger than those of the nonmetallic liquids [12]. The Fluid State: Micro- and Nanoscale Effects As for the solid state, the bulk thermal conductivity of a fluid may be modified when the characteristic dimension of the system becomes small, in particular for small values of L/mfp. Similar to the situation of a thin solid film shown in Figure 2.6b, the molecular mean free path is restricted when a fluid is constrained by a small physical dimension, affecting conduction across a thin fluid layer. Mixtures of fluids and solids can also be formulated to tailor the transport properties of the resulting suspension. For example, nanofluids are base liquids that are seeded with nanometer-sized solid particles. Their very small size allows the solid particles to remain suspended within the base liquid for a long time. From the heat transfer perspective, a nanofluid exploits the high thermal conductivity that is characteristic of most solids, as is evident in Figure 2.5, to boost the relatively low thermal conductivity of base liquids, typical values of which are shown in Figure 2.9. Typical nanofluids involve liquid water seeded with nominally spherical nanoparticles of Al2O3 or CuO. Insulation Systems Thermal insulations consist of low thermal conductivity materials combined to achieve an even lower system thermal conductivity. In conventional fiber-, powder-, and flake-type insulations, the solid material is finely dispersed throughout an air space. Such systems are characterized by an effective thermal conductivity, which depends on the thermal conductivity and surface radiative properties of the solid material, as well as the nature and volumetric fraction of the air or void space. A special parameter of the system is its bulk density (solid mass/total volume), which depends strongly on the manner in which the material is packed. If small voids or hollow spaces are formed by bonding or fusing portions of the solid material, a rigid matrix is created. When these spaces are sealed from each other, the system is referred to as a cellular insulation. Examples of such rigid insulations are foamed systems, particularly those made from plastic and glass materials. Reflective insulations are composed of multilayered, parallel, thin sheets or foils of high reflectivity, which are spaced to reflect radiant energy back to its source. The spacing between the foils is designed to restrict the motion of air, and in high-performance insulations, the space is evacuated. In all types of insulation, evacuation of the air in the void space will reduce the effective thermal conductivity of the system. Heat transfer through any of these insulation systems may include several modes: conduction through the solid materials; conduction or convection through the air in the void spaces; and radiation exchange between the surfaces of the solid matrix. The effective thermal conductivity accounts for all of these processes, and values for selected insulation systems are summarized in Table A.3. Additional background information and data are available in the literature [13, 14]. As with thin films, micro- and nanoscale effects can influence the effective thermal conductivity of insulating materials. The value of k for a nanostructured silica aerogel material that is composed of approximately 5% by volume solid material and 95% by volume air that is trapped within pores of L ⬇ 20 nm is shown in Figure 2.10. Note that at T ⬇ 300 K, the mean free path for air at atmospheric pressure is approximately 80 nm. As the gas pressure is reduced, mfp would increase for an unconfined gas, but the molecular

Chapter 2

䊏

Introduction to Conduction

0.014 Effective thermal conductivity (W/m•K)

78

0.012 0.01 0.008 0.006 0.004 0.002 0 10ⴚ3

10ⴚ2

10ⴚ1

100

Pressure (atm)

FIGURE 2.10 Measured thermal conductivity of carbon-doped silica aerogel as a function of pressure at T 艐 300 K [15].

motion of the trapped air is restricted by the walls of the small pores and k is reduced to extremely small values relative to the thermal conductivities of conventional matter reported in Figure 2.4.

2.2.2

Other Relevant Properties

In our analysis of heat transfer problems, it will be necessary to use several properties of matter. These properties are generally referred to as thermophysical properties and include two distinct categories, transport and thermodynamic properties. The transport properties include the diffusion rate coefficients such as k, the thermal conductivity (for heat transfer), and , the kinematic viscosity (for momentum transfer). Thermodynamic properties, on the other hand, pertain to the equilibrium state of a system. Density () and specific heat (cp) are two such properties used extensively in thermodynamic analysis. The product cp (J/m3 䡠 K), commonly termed the volumetric heat capacity, measures the ability of a material to store thermal energy. Because substances of large density are typically characterized by small specific heats, many solids and liquids, which are very good energy storage media, have comparable heat capacities (cp 1 MJ/m3 䡠 K). Because of their very small densities, however, gases are poorly suited for thermal energy storage (cp ⬇ 1 kJ/m3 䡠 K). Densities and specific heats are provided in the tables of Appendix A for a wide range of solids, liquids, and gases. In heat transfer analysis, the ratio of the thermal conductivity to the heat capacity is an important property termed the thermal diffusivity ␣, which has units of m2/s: k ␣ rc

p

It measures the ability of a material to conduct thermal energy relative to its ability to store thermal energy. Materials of large ␣ will respond quickly to changes in their thermal environment, whereas materials of small ␣ will respond more sluggishly, taking longer to reach a new equilibrium condition. The accuracy of engineering calculations depends on the accuracy with which the thermophysical properties are known [16–18]. Numerous examples could be cited of flaws

2.2

䊏

79

The Thermal Properties of Matter

in equipment and process design or failure to meet performance specifications that were attributable to misinformation associated with the selection of key property values used in the initial system analysis. Selection of reliable property data is an integral part of any careful engineering analysis. The casual use of data that have not been well characterized or evaluated, as may be found in some literature or handbooks, is to be avoided. Recommended data values for many thermophysical properties can be obtained from Reference 19. This reference, available in most institutional libraries, was prepared by the Thermophysical Properties Research Center (TPRC) at Purdue University.

EXAMPLE 2.1 The thermal diffusivity ␣ is the controlling transport property for transient conduction. Using appropriate values of k, , and cp from Appendix A, calculate ␣ for the following materials at the prescribed temperatures: pure aluminum, 300 and 700 K; silicon carbide, 1000 K; paraffin, 300 K.

SOLUTION Known: Definition of the thermal diffusivity ␣. Find: Numerical values of ␣ for selected materials and temperatures. Properties: Table A.1, pure aluminum (300 K):

冧

2702 kg/m3 k 237 W/m 䡠 K cp 903 J/kg 䡠 K ␣ c 3 p 2702 kg/m 903 J/kg 䡠 K k 237 W/m 䡠 K 97.1 106 m2/s

䉰

Table A.1, pure aluminum (700 K): 2702 kg/m3 cp 1090 J/kg 䡠 K k 225 W/m 䡠 K

at 300 K at 700 K (by linear interpolation) at 700 K (by linear interpolation)

Hence

␣ kc p

225 W/m 䡠 K 76 106 m2/s 2702 kg/m3 1090 J/kg 䡠 K

䉰

Table A.2, silicon carbide (1000 K):

3160 kg/m3 cp 1195 J/kg 䡠 K k 87 W/m 䡠 K

冧

at 300 K 87 W/m 䡠 K at 1000 K ␣ 3160 kg/m3 1195 J/kg 䡠 K at 1000 K 23 106 m2/s

䉰

80

Chapter 2

䊏

Introduction to Conduction

Table A.3, paraffin (300 K):

冧

900 kg/m3 0.24 W/m 䡠 K cp 2890 J/kg 䡠 K ␣ kc p 900 kg/m3 2890 J/kg 䡠 K k 0.24 W/m 䡠 K 9.2 108 m2/s

䉰

Comments: 1. Note the temperature dependence of the thermophysical properties of aluminum and silicon carbide. For example, for silicon carbide, ␣(1000 K) ⬇ 0.1 ␣(300 K); hence properties of this material have a strong temperature dependence. 2. The physical interpretation of ␣ is that it provides a measure of heat transport (k) relative to energy storage (cp). In general, metallic solids have higher ␣, whereas nonmetallics (e.g., paraffin) have lower values of ␣. 3. Linear interpolation of property values is generally acceptable for engineering calculations. 4. Use of the low-temperature (300 K) density at higher temperatures ignores thermal expansion effects but is also acceptable for engineering calculations. 5. The IHT software provides a library of thermophysical properties for selected solids, liquids, and gases that can be accessed from the toolbar button, Properties. See Example 2.1 in IHT.

EXAMPLE 2.2 The bulk thermal conductivity of a nanofluid containing uniformly dispersed, noncontacting spherical nanoparticles may be approximated by knf

kp 2kbf 2(kp kbf)

冤 k 2k p

bf

(kp kbf)

冥k

bf

where is the volume fraction of the nanoparticles, and kbf, kp, and knf are the thermal conductivities of the base fluid, particle, and nanofluid, respectively. Likewise, the dynamic viscosity may be approximated as [20] nf bf (1 2.5) Determine the values of knf, nf, cp,nf, nf, and ␣nf for a mixture of water and Al2O3 nanoparticles at a temperature of T 300 K and a particle volume fraction of 0.05. The thermophysical properties of the particle are kp 36.0 W/m 䡠 K, p 3970 kg/m3, and cp,p 0.765 kJ/kg 䡠 K.

SOLUTION Known: Expressions for the bulk thermal conductivity and viscosity of a nanofluid with spherical nanoparticles. Nanoparticle properties.

2.2

䊏

81

The Thermal Properties of Matter

Find: Values of the nanofluid thermal conductivity, density, specific heat, dynamic viscosity, and thermal diffusivity. Schematic: Water Nanoparticle kp ⫽ 36.0 W/m·K ρp ⫽ 3970 kg/m3 cp,p ⫽ 0.765 kJ/kg·K

Assumptions: 1. Constant properties. 2. Density and specific heat are not affected by nanoscale phenomena. 3. Isothermal conditions. Properties: Table A.6 (T 300 K): Water; kbf 0.613 W/m K, bf 997 kg/m3, cp,bf 4.179 kJ/kg K, bf 855 106 N s/m2. Analysis: From the problem statement, knf

kp 2kbf 2(kp kbf)

冤 k 2k p

bf

(kp kbf)

冥k

bf

䡠 K 2 0.613 W/m 䡠 K 2 0.05(36.0 0.613) W/m 䡠 K 冤36.036.0W/m W/m 䡠 K 2 0.613 W/m 䡠 K 0.05(36.0 0.613) W/m 䡠 K 冥 0.613 W/m 䡠 K

0.705 W/m K

䉰

Consider the control volume shown in the schematic to be of total volume V. Then the conservation of mass principle yields nfV bfV(1 ) pV or, after dividing by the volume V, nf 997 kg/m3 (1 0.05) 3970 kg/m3 0.05 1146 kg/m3

䉰

Similarly, the conservation of energy principle yields, nfVcp,nf T bfV(1 )cp,bf T pVcp,p T Dividing by the volume V, temperature T, and nanofluid density nf yields cp,nf

bf cp,bf (1 ) pcp,p nf

997 kg/m3 4.179 kJ/kg 䡠 K (1 0.05) 3970 kg/m3 0.765 kJ/kg 䡠 K (0.05) 1146 kg/m3 3.587 kJ/kg 䡠 K 䉰

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From the problem statement, the dynamic viscosity of the nanofluid is nf 855 106 N 䡠 s/m2 (1 2.5 0.05) 962 106 N 䡠 s/m2 The nanofluid’s thermal diffusivity is k 0.705 W/m 䡠 K 171 109 m2/s ␣nf cnf 3 nf p,nf 1146 kg/m 3587 J/kg 䡠 K

䉰

䉰

Comments: 1. Ratios of the properties of the nanofluid to the properties of water are as follows. knf 0.705 W/m 䡠 K 1.150 kbf 0.613 W/m 䡠 K cp,nf 3587 J/kg 䡠 K cp,bf 4179 J/kg 䡠 K 0.858

nf 1146 kg/m3 bf 997 kg/m3 1.149 nf 962 106 N 䡠 s/m2 bf 855 106 N 䡠 s/m2 1.130

␣nf 171 109 m2 /s ␣bf 147 109 m2/s 1.166 The relatively large thermal conductivity and thermal diffusivity of the nanofluid enhance heat transfer rates in some applications. However, all of the thermophysical properties are affected by the addition of the nanoparticles, and, as will become evident in Chapters 6 through 9, properties such as the viscosity and specific heat are adversely affected. This condition can degrade thermal performance when the use of nanofluids involves convection heat transfer. 2. The expression for the nanofluid’s thermal conductivity (and viscosity) is limited to dilute mixtures of noncontacting, spherical particles. In some cases, the particles do not remain separated but can agglomerate into long chains, providing effective paths for heat conduction through the fluid and larger bulk thermal conductivities. Hence, the expression for the thermal conductivity represents the minimum possible enhancement of the thermal conductivity by spherical nanoparticles. An expression for the maximum possible isotropic thermal conductivity of a nanofluid, corresponding to agglomeration of the spherical particles, is available [21], as are expressions for dilute suspensions of nonspherical particles [22]. Note that these expressions can also be applied to nanostructured composite materials consisting of a particulate phase interspersed within a host binding medium, as will be discussed in more detail in Chapter 3. 3. The nanofluid’s density and specific heat are determined by applying the principles of mass and energy conservation, respectively. As such, these properties do not depend on the manner in which the nanoparticles are dispersed within the base liquid.

2.3 The Heat Diffusion Equation A major objective in a conduction analysis is to determine the temperature field in a medium resulting from conditions imposed on its boundaries. That is, we wish to know the temperature distribution, which represents how temperature varies with position in the medium. Once this distribution is known, the conduction heat flux at any point in the medium or on its surface may be computed from Fourier’s law. Other important

2.3

䊏

83

The Heat Diffusion Equation

quantities of interest may also be determined. For a solid, knowledge of the temperature distribution could be used to ascertain structural integrity through determination of thermal stresses, expansions, and deflections. The temperature distribution could also be used to optimize the thickness of an insulating material or to determine the compatibility of special coatings or adhesives used with the material. We now consider the manner in which the temperature distribution can be determined. The approach follows the methodology described in Section 1.3.1 of applying the energy conservation requirement. In this case, we define a differential control volume, identify the relevant energy transfer processes, and introduce the appropriate rate equations. The result is a differential equation whose solution, for prescribed boundary conditions, provides the temperature distribution in the medium. Consider a homogeneous medium within which there is no bulk motion (advection) and the temperature distribution T(x, y, z) is expressed in Cartesian coordinates. Following the methodology of applying conservation of energy (Section 1.3.1), we first define an infinitesimally small (differential) control volume, dx 䡠 dy 䡠 dz, as shown in Figure 2.11. Choosing to formulate the first law at an instant of time, the second step is to consider the energy processes that are relevant to this control volume. In the absence of motion (or with uniform motion), there are no changes in mechanical energy and no work being done on the system. Only thermal forms of energy need be considered. Specifically, if there are temperature gradients, conduction heat transfer will occur across each of the control surfaces. The conduction heat rates perpendicular to each of the control surfaces at the x-, y-, and z-coordinate locations are indicated by the terms qx, qy, and qz, respectively. The conduction heat rates at the opposite surfaces can then be expressed as a Taylor series expansion where, neglecting higher-order terms, ⭸qx dx ⭸x ⭸qy qy dy qy dy ⭸y qx dx qx

qz dz qz

(2.13a) (2.13b)

⭸qz dz ⭸z

(2.13c)

T(x, y, z)

qz + dz qy + dy

dz •

Eg

qx

•

qx + dx

E st

z y x

dy

qy dx qz

FIGURE 2.11 Differential control volume, dx dy dz, for conduction analysis in Cartesian coordinates.

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Introduction to Conduction

In words, Equation 2.13a simply states that the x-component of the heat transfer rate at x dx is equal to the value of this component at x plus the amount by which it changes with respect to x times dx. Within the medium there may also be an energy source term associated with the rate of thermal energy generation. This term is represented as E˙ g q˙ dx dy dz

(2.14)

where q˙ is the rate at which energy is generated per unit volume of the medium (W/m3). In addition, changes may occur in the amount of the internal thermal energy stored by the material in the control volume. If the material is not experiencing a change in phase, latent energy effects are not pertinent, and the energy storage term may be expressed as ⭸T E˙ st cp dx dy dz (2.15) ⭸t where cp ⭸T/⭸t is the time rate of change of the sensible (thermal) energy of the medium per unit volume. Once again it is important to note that the terms E˙ g and E˙ st represent different physical processes. The energy generation term E˙ g is a manifestation of some energy conversion process involving thermal energy on one hand and some other form of energy, such as chemical, electrical, or nuclear, on the other. The term is positive (a source) if thermal energy is being generated in the material at the expense of some other energy form; it is negative (a sink) if thermal energy is being consumed. In contrast, the energy storage term E˙ st refers to the rate of change of thermal energy stored by the matter. The last step in the methodology outlined in Section 1.3.1 is to express conservation of energy using the foregoing rate equations. On a rate basis, the general form of the conservation of energy requirement is E˙in E˙g E˙out E˙st

(1.12c)

Hence, recognizing that the conduction rates constitute the energy inflow E˙ in and outflow E˙ out, and substituting Equations 2.14 and 2.15, we obtain ⭸T qx qy qz q˙ dx dy dz qxdx qydy qzdz cp dx dy dz (2.16) ⭸t Substituting from Equations 2.13, it follows that

⭸qy ⭸qz ⭸qx ⭸T dx dy dz q˙ dx dy dz cp dx dy dz ⭸x ⭸y ⭸z ⭸t

(2.17)

The conduction heat rates in an isotropic material may be evaluated from Fourier’s law, ⭸T ⭸x ⭸T qy k dx dz ⭸y ⭸T qz k dx dy ⭸z

qx k dy dz

(2.18a) (2.18b) (2.18c)

where each heat flux component of Equation 2.6 has been multiplied by the appropriate control surface (differential) area to obtain the heat transfer rate. Substituting

2.3

䊏

85

The Heat Diffusion Equation

Equations 2.18 into Equation 2.17 and dividing out the dimensions of the control volume (dx dy dz), we obtain

冢 冣

冢 冣

冢 冣

⭸ ⭸T ⭸T ⭸T ⭸T ⭸ ⭸ k k k q˙ cp ⭸x ⭸x ⭸y ⭸y ⭸z ⭸z ⭸t

(2.19)

Equation 2.19 is the general form, in Cartesian coordinates, of the heat diffusion equation. This equation, often referred to as the heat equation, provides the basic tool for heat conduction analysis. From its solution, we can obtain the temperature distribution T(x, y, z) as a function of time. The apparent complexity of this expression should not obscure the fact that it describes an important physical condition, that is, conservation of energy. You should have a clear understanding of the physical significance of each term appearing in the equation. For example, the term ⭸(k⭸T/⭸x)/⭸x is related to the net conduction heat flux into the control volume for the x-coordinate direction. That is, multiplying by dx,

冢 冣

⭸ ⭸T k dx qx qxdx ⭸x ⭸x

(2.20)

with similar expressions applying for the fluxes in the y- and z-directions. In words, the heat equation, Equation 2.19, therefore states that at any point in the medium the net rate of energy transfer by conduction into a unit volume plus the volumetric rate of thermal energy generation must equal the rate of change of thermal energy stored within the volume. It is often possible to work with simplified versions of Equation 2.19. For example, if the thermal conductivity is constant, the heat equation is ⭸2T ⭸2T ⭸2T q˙ 1 ⭸T ⭸x2 ⭸y2 ⭸z2 k ␣ ⭸t

(2.21)

where ␣ k/cp is the thermal diffusivity. Additional simplifications of the general form of the heat equation are often possible. For example, under steady-state conditions, there can be no change in the amount of energy storage; hence Equation 2.19 reduces to

冢 冣

冢 冣

冢 冣

⭸ ⭸T ⭸T ⭸T ⭸ ⭸ k k k q˙ 0 ⭸x ⭸x ⭸y ⭸y ⭸z ⭸z

(2.22)

Moreover, if the heat transfer is one-dimensional (e.g., in the x-direction) and there is no energy generation, Equation 2.22 reduces to

冢 冣

d k dT 0 dx dx

(2.23)

The important implication of this result is that, under steady-state, one-dimensional conditions with no energy generation, the heat flux is a constant in the direction of transfer (dqx /dx 0). The heat equation may also be expressed in cylindrical and spherical coordinates. The differential control volumes for these two coordinate systems are shown in Figures 2.12 and 2.13.

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Chapter 2

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Introduction to Conduction

qz + dz

rdφ

qr

qφ + dφ dz z r T(r,φ ,z)

qφ qr + dr

dr

y

r

φ

x qz

FIGURE 2.12 Differential control volume, dr 䡠 r d 䡠 dz, for conduction analysis in cylindrical coordinates (r, , z).

When the del operator of Equation 2.3 is expressed in cylindrical coordinates, the general form of the heat flux vector and hence of Fourier’s law is

Cylindrical Coordinates

⭸T ⭸T k 冣 冢 ⭸T⭸r j 1r ⭸ ⭸z

q kT k i

(2.24)

where ⭸T ⭸r

qr k

⭸T q kr ⭸

qz k

⭸T ⭸z

qθ + dθ r sin θ dφ qr

qφ + dφ rdθ

qφ

z θ

r φ

T(r, φ , θ)

qr + dr

dr

y qθ

x

FIGURE 2.13 Differential control volume, dr 䡠 r sin d 䡠 r d, for conduction analysis in spherical coordinates (r, , ).

(2.25)

2.3

䊏

87

The Heat Diffusion Equation

are heat flux components in the radial, circumferential, and axial directions, respectively. Applying an energy balance to the differential control volume of Figure 2.12, the following general form of the heat equation is obtained:

冢 冣

冢 冣

冢 冣

1 ⭸ kr ⭸T 1 ⭸ k ⭸T ⭸ k ⭸T q˙ c ⭸T p r ⭸r ⭸r ⭸z ⭸z ⭸t r 2 ⭸ ⭸ Spherical Coordinates

(2.26)

In spherical coordinates, the general form of the heat flux vector

and Fourier’s law is 1 ⭸T 冢 ⭸T⭸r j 1r ⭸T⭸ k r sin ⭸冣

q kT k i

(2.27)

where qr k

⭸T ⭸r

⭸T q kr ⭸

q

k ⭸T r sin ⭸

(2.28)

are heat flux components in the radial, polar, and azimuthal directions, respectively. Applying an energy balance to the differential control volume of Figure 2.13, the following general form of the heat equation is obtained:

冢

冣

冢 冣

⭸ ⭸T 1 1 ⭸ kr 2 ⭸T k 2 ⭸r 2 2 ⭸ ⭸r ⭸ r r sin

冢

冣

⭸ ⭸T ⭸T 1 k sin q˙ cp ⭸t ⭸ r 2 sin ⭸

(2.29)

You should attempt to derive Equation 2.26 or 2.29 to gain experience in applying conservation principles to differential control volumes (see Problems 2.35 and 2.36). Note that the temperature gradient in Fourier’s law must have units of K/m. Hence, when evaluating the gradient for an angular coordinate, it must be expressed in terms of the differential change in arc length. For example, the heat flux component in the circumferential direction of a cylindrical coordinate system is q (k/r)(⭸T/⭸), not q k(⭸T/⭸).

EXAMPLE 2.3 The temperature distribution across a wall 1 m thick at a certain instant of time is given as T(x) a bx cx2 where T is in degrees Celsius and x is in meters, while a 900 C, b 300 C/m, and . c 50 C/m2. A uniform heat generation, q 1000 W/m3, is present in the wall of area 2 3 10 m having the properties 1600 kg/m , k 40 W/m 䡠 K, and cp 4 kJ/kg 䡠 K.

88

Chapter 2

䊏

Introduction to Conduction

1. Determine the rate of heat transfer entering the wall (x 0) and leaving the wall (x 1 m). 2. Determine the rate of change of energy storage in the wall. 3. Determine the time rate of temperature change at x 0, 0.25, and 0.5 m.

SOLUTION Known: Temperature distribution T(x) at an instant of time t in a one-dimensional wall with uniform heat generation. Find: 1. Heat rates entering, qin (x 0), and leaving, qout (x 1 m), the wall. 2. Rate of change of energy storage in the wall, E˙ st. 3. Time rate of temperature change at x 0, 0.25, and 0.5 m. Schematic: q• = 1000 W/m3 k = 40 W/m•K ρ = 1600 kg/m3 cp = 4 kJ/kg•K

A = 10 m2

T(x) = a + bx + cx2 •

Eg •

E st qin

qout

L=1m x

Assumptions: 1. One-dimensional conduction in the x-direction. 2. Isotropic medium with constant properties. . 3. Uniform internal heat generation, q (W/m3). Analysis: 1. Recall that once the temperature distribution is known for a medium, it is a simple matter to determine the conduction heat transfer rate at any point in the medium or at its surfaces by using Fourier’s law. Hence the desired heat rates may be determined by using the prescribed temperature distribution with Equation 2.1. Accordingly, qin qx(0) kA

⭸T 兩 kA(b 2cx)x0 ⭸x x0

qin bkA 300 C/m 40 W/m 䡠 K 10 m2 120 kW

䉰

2.3

䊏

89

The Heat Diffusion Equation

Similarly, qout qx(L) kA

⭸T 兩 kA(b 2cx)xL ⭸x xL

qout (b 2cL)kA [300 C/m 2(50 C/m2) 1 m] 40 W/m 䡠 K 10 m2 160 kW

䉰

2. The rate of change of energy storage in the wall E˙ st may be determined by applying an overall energy balance to the wall. Using Equation 1.12c for a control volume about the wall, E˙ in E˙ g E˙ out E˙ st where E˙ g q˙AL, it follows that E˙ st E˙ in E˙ g E˙ out qin q˙AL qout E˙ st 120 kW 1000 W/m3 10 m2 1 m 160 kW E˙ st 30 kW

䉰

3. The time rate of change of the temperature at any point in the medium may be determined from the heat equation, Equation 2.21, rewritten as ⭸T k ⭸2T q˙ c cp ⭸t p ⭸x2 From the prescribed temperature distribution, it follows that

冢 冣

⭸2T ⭸ ⭸T 2 ⭸x ⭸x ⭸x

⭸ (b 2cx) 2c 2(50 C/m2) 100 C/m2 ⭸x

Note that this derivative is independent of position in the medium. Hence the time rate of temperature change is also independent of position and is given by ⭸T 40 W/m 䡠 K (100 C/m2) ⭸t 1600 kg/m3 4 kJ/kg 䡠 K

1000 W/m3 1600 kg/m3 4 kJ/kg 䡠 K

⭸T 6.25 104 C/s 1.56 104 C/s ⭸t 4.69 104 C/s

䉰

90

Chapter 2

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Introduction to Conduction

Comments: 1. From this result, it is evident that the temperature at every point within the wall is decreasing with time. 2. Fourier’s law can always be used to compute the conduction heat rate from knowledge of the temperature distribution, even for unsteady conditions with internal heat generation.

Microscale Effects For most practical situations, the heat diffusion equations generated in this text may be used with confidence. However, these equations are based on Fourier’s law, which does not account for the finite speed at which thermal information is propagated within the medium by the various energy carriers. The consequences of the finite propagation speed may be neglected if the heat transfer events of interest occur over a sufficiently long time scale, t, such that mfp 1 (2.30) ct The heat diffusion equations of this text are likewise invalid for problems where boundary scattering must be explicitly considered. For example, the temperature distribution within the thin film of Figure 2.6b cannot be determined by applying the foregoing heat diffusion equations. Additional discussion of micro- and nanoscale heat transfer applications and analysis methods is available in the literature [1, 5, 10, 23].

2.4 Boundary and Initial Conditions To determine the temperature distribution in a medium, it is necessary to solve the appropriate form of the heat equation. However, such a solution depends on the physical conditions existing at the boundaries of the medium and, if the situation is time dependent, on conditions existing in the medium at some initial time. With regard to the boundary conditions, there are several common possibilities that are simply expressed in mathematical form. Because the heat equation is second order in the spatial coordinates, two boundary conditions must be expressed for each coordinate needed to describe the system. Because the equation is first order in time, however, only one condition, termed the initial condition, must be specified. Three kinds of boundary conditions commonly encountered in heat transfer are summarized in Table 2.2. The conditions are specified at the surface x 0 for a one-dimensional system. Heat transfer is in the positive x-direction with the temperature distribution, which may be time dependent, designated as T(x, t). The first condition corresponds to a situation for which the surface is maintained at a fixed temperature Ts. It is commonly termed a Dirichlet condition, or a boundary condition of the first kind. It is closely approximated, for example, when the surface is in contact with a melting solid or a boiling liquid. In both cases, there is heat transfer at the surface, while the surface remains at the temperature of the phase change process. The second condition corresponds to the existence of a fixed or constant heat flux qs at the surface. This heat flux is related to the temperature gradient at the surface by Fourier’s law, Equation 2.6, which may be expressed as qx (0) k

⭸T 兩 qs ⭸x x0

2.4

䊏

91

Boundary and Initial Conditions

TABLE 2.2 Boundary conditions for the heat diffusion equation at the surface (x 0) 1.

Ts

Constant surface temperature T(0, t) Ts

(2.31)

T(x, t) x

2.

Constant surface heat flux (a) Finite heat flux ⭸T k 兩 qs ⭸x x0

qs'' T(x, t)

(2.32) x

(b) Adiabatic or insulated surface ⭸T 兩 0 ⭸x x0

T(x, t)

(2.33) x

3.

Convection surface condition ⭸T k 兩 h[T앝 T(0, t)] ⭸x x0

T(0, t)

(2.34)

T∞, h x

T(x, t)

It is termed a Neumann condition, or a boundary condition of the second kind, and may be realized by bonding a thin film electric heater to the surface. A special case of this condition corresponds to the perfectly insulated, or adiabatic, surface for which ⭸T/⭸x冷 x0 0. The boundary condition of the third kind corresponds to the existence of convection heating (or cooling) at the surface and is obtained from the surface energy balance discussed in Section 1.3.1.

EXAMPLE 2.4 A long copper bar of rectangular cross section, whose width w is much greater than its thickness L, is maintained in contact with a heat sink at its lower surface, and the temperature throughout the bar is approximately equal to that of the sink, To. Suddenly, an electric current is passed through the bar and an airstream of temperature T앝 is passed over the top surface, while the bottom surface continues to be maintained at To. Obtain the differential equation and the boundary and initial conditions that could be solved to determine the temperature as a function of position and time in the bar.

SOLUTION Known: Copper bar initially in thermal equilibrium with a heat sink is suddenly heated by passage of an electric current.

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Introduction to Conduction

Find: Differential equation and boundary and initial conditions needed to determine temperature as a function of position and time within the bar. Schematic: Copper bar (k, α) T(x, y, z, t) T(x, t)

y

Air

T∞, h

x

Air

w

T∞, h

L

T(L, t) •

q

z

L

I

Heat sink

To

x

To = T(0, t)

Assumptions: 1. Since the bar is long and w L, end and side effects are negligible and heat transfer within the bar is primarily one dimensional in the x-direction. 2. Uniform volumetric heat generation, q˙. 3. Constant properties. Analysis: The temperature distribution is governed by the heat equation (Equation 2.19), which, for the one-dimensional and constant property conditions of the present problem, reduces to ⭸2T q˙ 1 ⭸T ⭸x2 k ␣ ⭸t

(1)

䉰

where the temperature is a function of position and time, T(x, t). Since this differential equation is second order in the spatial coordinate x and first order in time t, there must be two boundary conditions for the x-direction and one condition, termed the initial condition, for time. The boundary condition at the bottom surface corresponds to case 1 of Table 2.2. In particular, since the temperature of this surface is maintained at a value, To, which is fixed with time, it follows that T(0, t) To

(2)

䉰

The convection surface condition, case 3 of Table 2.2, is appropriate for the top surface. Hence k

⭸T 兩 h[T(L, t) T앝] ⭸x xL

(3)

䉰

The initial condition is inferred from recognition that, before the change in conditions, the bar is at a uniform temperature To. Hence T(x, 0) To

(4)

䉰

2.4

䊏

93

Boundary and Initial Conditions

. If To, T앝, q, and h are known, Equations 1 through 4 may be solved to obtain the time-varying temperature distribution T(x, t) following imposition of the electric current.

Comments: 1. The heat sink at x 0 could be maintained by exposing the surface to an ice bath or by attaching it to a cold plate. A cold plate contains coolant channels machined in a solid of large thermal conductivity (usually copper). By circulating a liquid (usually water) through the channels, the plate and hence the surface to which it is attached may be maintained at a nearly uniform temperature. 2. The temperature of the top surface T(L, t) will change with time. This temperature is an unknown and may be obtained after finding T(x, t). 3. We may use our physical intuition to sketch temperature distributions in the bar at selected times from the beginning to the end of the transient process. If we assume that T앝 To and that the electric current is sufficiently large to heat the bar to temperatures in excess of T⬁, the following distributions would correspond to the initial condition (t 0), the final (steady-state) condition (t l 앝), and two intermediate times.

T(x, t)

T(x, ∞), Steady-state condition T∞

T∞

b

a T(x, 0), Initial condition

To L

0 Distance, x

Note how the distributions comply with the initial and boundary conditions. What is a special feature of the distribution labeled (b)? 4. Our intuition may also be used to infer the manner in which the heat flux varies with time at the surfaces (x 0, L) of the bar. On qx t coordinates, the transient variations are as follows. +

q"x (x, t)

q"x (L, t)

0

q"x (0, t)

– 0

Time, t

Convince yourself that the foregoing variations are consistent with the temperature distributions of Comment 3. For t l 앝, how are qx (0) and qx (L) related to the volumetric rate of energy generation?

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Chapter 2

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Introduction to Conduction

2.5 Summary Despite the relative brevity of this chapter, its importance must not be underestimated. Understanding the conduction rate equation, Fourier’s law, is essential. You must be cognizant of the importance of thermophysical properties; over time, you will develop a sense of the magnitudes of the properties of many real materials. Likewise, you must recognize that the heat equation is derived by applying the conservation of energy principle to a differential control volume and that it is used to determine temperature distributions within matter. From knowledge of the distribution, Fourier’s law can be used to determine the corresponding conduction heat rates. A firm grasp of the various types of thermal boundary conditions that are used in conjunction with the heat equation is vital. Indeed, Chapter 2 is the foundation on which Chapters 3 through 5 are based, and you are encouraged to revisit this chapter often. You may test your understanding of various concepts by addressing the following questions. • In the general formulation of Fourier’s law (applicable to any geometry), what are the vector and scalar quantities? Why is there a minus sign on the right-hand side of the equation? • What is an isothermal surface? What can be said about the heat flux at any location on this surface? • What form does Fourier’s law take for each of the orthogonal directions of Cartesian, cylindrical, and spherical coordinate systems? In each case, what are the units of the temperature gradient? Can you write each equation from memory? • An important property of matter is defined by Fourier’s law. What is it? What is its physical significance? What are its units? • What is an isotropic material? • Why is the thermal conductivity of a solid generally larger than that of a liquid? Why is the thermal conductivity of a liquid larger than that of a gas? • Why is the thermal conductivity of an electrically conducting solid generally larger than that of a nonconductor? Why are materials such as beryllium oxide, diamond, and silicon carbide (see Table A.2) exceptions to this rule? • Is the effective thermal conductivity of an insulation system a true manifestation of the efficacy with which heat is transferred through the system by conduction alone? • Why does the thermal conductivity of a gas increase with increasing temperature? Why is it approximately independent of pressure? • What is the physical significance of the thermal diffusivity? How is it defined and what are its units? • What is the physical significance of each term appearing in the heat equation? • Cite some examples of thermal energy generation. If the rate at which thermal energy is generated per unit volume, q˙, varies with location in a medium of volume V, how can the rate of energy generation for the entire medium, E˙ g, be determined from knowledge of q˙(x, y, z)? • For a chemically reacting medium, what kind of reaction provides a source of thermal energy (q˙ 0)? What kind of reaction provides a sink for thermal energy (q˙ 0)? • To solve the heat equation for the temperature distribution in a medium, boundary conditions must be prescribed at the surfaces of the medium. What physical conditions are commonly suitable for this purpose?

䊏

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Problems

References 1. Flik, M. I., B.-I. Choi, and K. E. Goodson, J. Heat Transfer, 114, 666, 1992. 2. Klemens, P. G., “Theory of the Thermal Conductivity of Solids,” in R. P. Tye, Ed., Thermal Conductivity, Vol. 1, Academic Press, London, 1969. 3. Yang, H.-S., G.-R. Bai, L. J. Thompson, and J. A. Eastman, Acta Materialia, 50, 2309, 2002. 4. Chen, G., J. Heat Transfer, 118, 539, 1996. 5. Carey, V. P., G. Chen, C. Grigoropoulos, M. Kaviany, and A. Majumdar, Nano. and Micro. Thermophys. Engng. 12, 1, 2008. 6. Padture, N. P., M. Gell, and E. H. Jordan, Science, 296, 280, 2002. 7. Schelling, P. K., L. Shi, and K. E. Goodson, Mat. Today, 8, 30, 2005. 8. Baxter, J., Z. Bian, G. Chen, D. Danielson, M. S. Dresselhaus, A. G. Federov, T. S. Fisher, C. W. Jones, E. Maginn, W. Kortshagen, A. Manthiram, A. Nozik, D. R. Rolison, T. Sands, L. Shi, D. Sholl, and Y. Wu, Energy and Environ. Sci., 2, 559, 2009. 9. Vincenti, W. G., and C. H. Kruger Jr., Introduction to Physical Gas Dynamics, Wiley, New York, 1986. 10. Zhang, Z. M., Nano/Microscale Heat Transfer, McGrawHill, New York, 2007. 11. McLaughlin, E., “Theory of the Thermal Conductivity of Fluids,” in R. P. Tye, Ed., Thermal Conductivity, Vol. 2, Academic Press, London, 1969. 12. Foust, O. J., Ed., “Sodium Chemistry and Physical Properties,” in Sodium-NaK Engineering Handbook, Vol. 1, Gordon & Breach, New York, 1972.

13. Mallory, J. F., Thermal Insulation, Reinhold Book Corp., New York, 1969. 14. American Society of Heating, Refrigeration and Air Conditioning Engineers, Handbook of Fundamentals, Chapters 23–25 and 31, ASHRAE, New York, 2001. 15. Zeng, S. Q., A. Hunt, and R. Greif, J. Heat Transfer, 117, 1055, 1995. 16. Sengers, J. V., and M. Klein, Eds., The Technical Importance of Accurate Thermophysical Property Information, National Bureau of Standards Technical Note No. 590, 1980. 17. Najjar, M. S., K. J. Bell, and R. N. Maddox, Heat Transfer Eng., 2, 27, 1981. 18. Hanley, H. J. M., and M. E. Baltatu, Mech. Eng., 105, 68, 1983. 19. Touloukian, Y. S., and C. Y. Ho, Eds., Thermophysical Properties of Matter, The TPRC Data Series (13 volumes on thermophysical properties: thermal conductivity, specific heat, thermal radiative, thermal diffusivity, and thermal linear expansion), Plenum Press, New York, 1970 through 1977. 20. Chow, T. S., Phys. Rev. E, 48, 1977, 1993. 21. Keblinski, P., R. Prasher, and J. Eapen, J. Nanopart. Res., 10, 1089, 2008. 22. Hamilton, R. L., and O. K. Crosser, I&EC Fundam. 1, 187, 1962. 23. Cahill, D. G., W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar, H. J. Maris, R. Merlin, and S. R. Phillpot, App. Phys. Rev., 93, 793, 2003.

Problems Fourier’s Law 2.1 Assume steady-state, one-dimensional heat conduction through the axisymmetric shape shown below. T1

T2

2.2 Assume steady-state, one-dimensional conduction in the axisymmetric object below, which is insulated around its perimeter. T1

T2 T1 > T2

T1 > T2 x x

L

Assuming constant properties and no internal heat generation, sketch the temperature distribution on T x coordinates. Briefly explain the shape of your curve.

L

If the properties remain constant and no internal heat generation occurs, sketch the heat flux distribution, qx (x), and the temperature distribution, T(x). Explain the shapes of your curves. How do your curves depend on the thermal conductivity of the material?

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2.3 A hot water pipe with outside radius r1 has a temperature T1. A thick insulation, applied to reduce the heat loss, has an outer radius r2 and temperature T2. On T r coordinates, sketch the temperature distribution in the insulation for one-dimensional, steady-state heat transfer with constant properties. Give a brief explanation, justifying the shape of your curve.

T1, A1

r

x

2.4 A spherical shell with inner radius r1 and outer radius r2 has surface temperatures T1 and T2, respectively, where T1 T2. Sketch the temperature distribution on T r coordinates assuming steady-state, one-dimensional conduction with constant properties. Briefly justify the shape of your curve. 2.5 Assume steady-state, one-dimensional heat conduction through the symmetric shape shown.

T2 < T1 A2 > A1

The thermal conductivity of the solid depends on temperature according to the relation k k0 aT, where a is a positive constant, and the sides of the cone are well insulated. Do the following quantities increase, decrease, or remain the same with increasing x: the heat transfer rate qx , the heat flux qx , the thermal conductivity k, and the temperature gradient dT/dx? 2.8 To determine the effect of the temperature dependence of the thermal conductivity on the temperature distribution in a solid, consider a material for which this dependence may be represented as

qx

k ko aT x

Assuming that there is no internal heat generation, derive an expression for the thermal conductivity k(x) for these conditions: A(x) (1 x), T(x) 300 (1 2x x3), and q 6000 W, where A is in square meters, T in kelvins, and x in meters. 2.6 A composite rod consists of two different materials, A and B, each of length 0.5L. T1

T2

T1 < T 2

A

x

0.5 L

B

L

The thermal conductivity of Material A is half that of Material B, that is, kA/kB 0.5. Sketch the steady-state temperature and heat flux distributions, T(x) and qx , respectively. Assume constant properties and no internal heat generation in either material. 2.7 A solid, truncated cone serves as a support for a system that maintains the top (truncated) face of the cone at a temperature T1, while the base of the cone is at a temperature T2 T1.

where ko is a positive constant and a is a coefficient that may be positive or negative. Sketch the steady-state temperature distribution associated with heat transfer in a plane wall for three cases corresponding to a 0, a 0, and a 0. 2.9 A young engineer is asked to design a thermal protection barrier for a sensitive electronic device that might be exposed to irradiation from a high-powered infrared laser. Having learned as a student that a low thermal conductivity material provides good insulating characteristics, the engineer specifies use of a nanostructured aerogel, characterized by a thermal conductivity of ka 0.005 W/m 䡠 K, for the protective barrier. The engineer’s boss questions the wisdom of selecting the aerogel because it has a low thermal conductivity. Consider the sudden laser irradiation of (a) pure aluminum, (b) glass, and (c) aerogel. The laser provides irradiation of G 10 106 W/m2. The absorptivities of the materials are ␣ 0.2, 0.9, and 0.8 for the aluminum, glass, and aerogel, respectively, and the initial temperature of the barrier is Ti 300 K. Explain why the boss is concerned. Hint: All materials experience thermal expansion (or contraction), and local stresses that develop within a material are, to a first approximation, proportional to the local temperature gradient. 2.10 A one-dimensional plane wall of thickness 2L 100 mm experiences uniform thermal energy generation of q˙ 1000 W/m3 and is convectively cooled at x 50 mm by an ambient fluid characterized by T앝 20 C. If the steady-state temperature distribution

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within the wall is T(x) a(L2 x2) b where a 10 C/m2 and b 30 C, what is the thermal conductivity of the wall? What is the value of the convection heat transfer coefficient, h?

Insulation 1m

k = 10 W/m•K y

2.11 Consider steady-state conditions for one-dimensional conduction in a plane wall having a thermal conductivity k 50 W/m 䡠 K and a thickness L 0.25 m, with no internal heat generation.

T1 L

Determine the heat flux and the unknown quantity for each case and sketch the temperature distribution, indicating the direction of the heat flux. Case

T1(ⴗC)

T2(ⴗC)

1 2 3 4 5

50 30 70

20 10 Ao

160 80 200

T(x)

qx(x)

T(x) T2

T2

T1

T1 x (a)

2.16 Steady-state, one-dimensional conduction occurs in a rod of constant thermal conductivity k and variable crosssectional area Ax(x) Aoeax, where Ao and a are constants. The lateral surface of the rod is well insulated. Ax(x) = Aoeax

2.12 Consider a plane wall 100 mm thick and of thermal conductivity 100 W/m 䡠 K. Steady-state conditions are known to exist with T1 400 K and T2 600 K. Determine the heat flux qx and the temperature gradient dT/dx for the coordinate systems shown.

T2

x A, TA = 0°C

dT/dx (K/m)

40 30

T(x)

2m

2.15 Consider the geometry of Problem 2.14 for the case where the thermal conductivity varies with temperature as k ko aT, where ko 10 W/m 䡠 K, a 103 W/m 䡠 K2, and T is in kelvins. The gradient at surface B is ⭸T/⭸x 30 K/m. What is ⭸T/⭸y at surface A?

T2

x

B, TB = 100°C

x

L

(a) Write an expression for the conduction heat rate, qx(x). Use this expression to determine the temperature distribution T(x) and qualitatively sketch the distribution for T(0) T(L). (b) Now consider conditions for which thermal energy is generated in the rod at a volumetric rate q˙ q˙o exp(ax), where q˙o is a constant. Obtain an expression for qx(x) when the left face (x 0) is well insulated.

T1

x

x (b)

(c)

2.13 A cylinder of radius ro, length L, and thermal conductivity k is immersed in a fluid of convection coefficient h and unknown temperature T앝. At a certain instant the temperature distribution in the cylinder is T(r) a br2, where a and b are constants. Obtain expressions for the heat transfer rate at ro and the fluid temperature. 2.14 In the two-dimensional body illustrated, the gradient at surface A is found to be ⭸T/⭸y 30 K/m. What are ⭸T/⭸y and ⭸T/⭸x at surface B?

Thermophysical Properties 2.17 An apparatus for measuring thermal conductivity employs an electrical heater sandwiched between two identical samples of diameter 30 mm and length 60 mm, which are pressed between plates maintained at a uniform temperature To 77 C by a circulating fluid. A conducting grease is placed between all the surfaces to ensure good thermal contact. Differential thermocouples are imbedded in the samples with a spacing of 15 mm. The lateral sides of the samples are insulated to ensure onedimensional heat transfer through the samples.

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Sample Heater leads

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Introduction to Conduction

Plate, To

(a) Explain why the apparatus of Problem 2.17 cannot be used to obtain an accurate measurement of the aerogel’s thermal conductivity.

∆T1

(b) The engineer designs a new apparatus for which an electric heater of diameter D 150 mm is sandwiched between two thin plates of aluminum. The steady-state temperatures of the 5-mm-thick aluminum plates, T1 and T2, are measured with thermocouples. Aerogel sheets of thickness t 5 mm are placed outside the aluminum plates, while a coolant with an inlet temperature of Tc,i 25 C maintains the exterior surfaces of the aerogel at a low temperature. The circular aerogel sheets are formed so that they encase the heater and aluminum sheets, providing insulation to minimize radial heat losses. At steady state, T1 T2 55 C, and the heater draws 125 mA at 10 V. Determine the value of the aerogel thermal conductivity ka.

Insulation ∆T2

Sample

Plate, To

(a) With two samples of SS316 in the apparatus, the heater draws 0.353 A at 100 V, and the differential thermocouples indicate T1 T2 25.0 C. What is the thermal conductivity of the stainless steel sample material? What is the average temperature of the samples? Compare your result with the thermal conductivity value reported for this material in Table A.1. (b) By mistake, an Armco iron sample is placed in the lower position of the apparatus with one of the SS316 samples from part (a) in the upper portion. For this situation, the heater draws 0.601 A at 100 V, and the differential thermocouples indicate T1 T2 15.0 C. What are the thermal conductivity and average temperature of the Armco iron sample? (c) What is the advantage in constructing the apparatus with two identical samples sandwiching the heater rather than with a single heater–sample combination? When would heat leakage out of the lateral surfaces of the samples become significant? Under what conditions would you expect T1 T2 ? 2.18 An engineer desires to measure the thermal conductivity of an aerogel material. It is expected that the aerogel will have an extremely small thermal conductivity. Heater leads

Tc,i

Coolant in

(c) Calculate the temperature difference across the thickness of the 5-mm-thick aluminum plates. Comment on whether it is important to know the axial locations at which the temperatures of the aluminum plates are measured. (d) If liquid water is used as the coolant with a total flow rate of m˙ 1 kg/min (0.5 kg/min for each of the two streams), calculate the outlet temperature of the water, Tc,o. 2.19 Consider a 300 mm 300 mm window in an aircraft. For a temperature difference of 80 C from the inner to the outer surface of the window, calculate the heat loss through L 10-mm-thick polycarbonate, soda lime glass, and aerogel windows, respectively. The thermal conductivities of the aerogel and polycarbonate are kag 0.014 W/m 䡠 K and kpc 0.21 W/m 䡠 K, respectively. Evaluate the thermal conductivity of the soda lime glass at 300 K. If the aircraft has 130 windows and the cost to heat the cabin air is $1/kW 䡠 h, compare the costs associated with the heat loss through the windows for an 8-hour intercontinental flight. 2.20 Consider a small but known volume of metal that has a large thermal conductivity.

t

Aerogel sample

D

Heater x Aluminum plate T 2 T1

(a) Since the thermal conductivity is large, spatial temperature gradients that develop within the metal in response to mild heating are small. Neglecting spatial temperature gradients, derive a differential equation that could be solved for the temperature of the metal versus time T(t) if the metal is subjected to a fixed surface heat rate q supplied by an electric heater. (b) A student proposes to identify the unknown metal by comparing measured and predicted thermal

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99

Problems

responses. Once a match is made, relevant thermophysical properties might be determined, and, in turn, the metal may be identified by comparison to published property data. Will this approach work? Consider aluminum, gold, and silver as the candidate metals.

Sample 1, D, L, ρ

To(t)

Heater leads Sample 2, D, L, ρ

2.21 Use IHT to perform the following tasks. (a) Graph the thermal conductivity of pure copper, 2024 aluminum, and AISI 302 stainless steel over the temperature range 300 T 600 K. Include all data on a single graph, and comment on the trends you observe. (b) Graph the thermal conductivity of helium and air over the temperature range 300 T 800 K. Include the data on a single graph, and comment on the trends you observe. (c) Graph the kinematic viscosity of engine oil, ethylene glycol, and liquid water over the temperature range 300 T 360 K. Include all data on a single graph, and comment on the trends you observe. (d) Graph the thermal conductivity of a water-Al2O3 nanofluid at T 300 K over the volume fraction range 0 0.08. See Example 2.2. 2.22 Calculate the thermal conductivity of air, hydrogen, and carbon dioxide at 300 K, assuming ideal gas behavior. Compare your calculated values to values from Table A.4. 2.23 A method for determining the thermal conductivity k and the specific heat cp of a material is illustrated in the sketch. Initially the two identical samples of diameter D 60 mm and thickness L 10 mm and the thin heater are at a uniform temperature of Ti 23.00 C, while surrounded by an insulating powder. Suddenly the heater is energized to provide a uniform heat flux qo on each of the sample interfaces, and the heat flux is maintained constant for a period of time, to. A short time after sudden heating is initiated, the temperature at this interface To is related to the heat flux as

冢ct k冣

1/ 2

To(t) Ti 2qo

p

For a particular test run, the electrical heater dissipates 15.0 W for a period of to 120 s, and the temperature at the interface is To(30 s) 24.57 C after 30 s of heating. A long time after the heater is deenergized, t t0, the samples reach the uniform temperature of To(앝) 33.50 C. The density of the sample materials, determined by measurement of volume and mass, is 3965 kg/m3.

Determine the specific heat and thermal conductivity of the test material. By looking at values of the thermophysical properties in Table A.1 or A.2, identify the test sample material. 2.24 Compare and contrast the heat capacity cp of common brick, plain carbon steel, engine oil, water, and soil. Which material provides the greatest amount of thermal energy storage per unit volume? Which material would you expect to have the lowest cost per unit heat capacity? Evaluate properties at 300 K. 2.25 A cylindrical rod of stainless steel is insulated on its exterior surface except for the ends. The steady-state temperature distribution is T(x) a bx/L, where a 305 K and b 10 K. The diameter and length of the rod are D 20 mm and L 100 mm, respectively. Determine the heat flux along the rod, qx . Hint: The mass of the rod is M 0.248 kg.

The Heat Equation 2.26 At a given instant of time, the temperature distribution within an infinite homogeneous body is given by the function T(x, y, z) x2 2y2 z2 xy 2yz Assuming constant properties and no internal heat generation, determine the regions where the temperature changes with time. 2.27 A pan is used to boil water by placing it on a stove, from which heat is transferred at a fixed rate qo. There are two stages to the process. In Stage 1, the water is taken from its initial (room) temperature Ti to the boiling point, as heat is transferred from the pan by natural convection. During this stage, a constant value of the convection coefficient h may be assumed, while the bulk temperature of the water increases with time, T앝 T앝(t). In Stage 2, the water has come to a boil, and its temperature remains at a fixed value, T앝 Tb, as heating continues. Consider a pan bottom of thickness L and diameter D, with a coordinate system corresponding to x 0 and x L for the surfaces in contact with the stove and water, respectively. (a) Write the form of the heat equation and the boundary/ initial conditions that determine the variation of

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Chapter 2

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Introduction to Conduction

temperature with position and time, T(x, t), in the pan bottom during Stage 1. Express your result in terms of the parameters qo, D, L, h, and T앝, as well as appropriate properties of the pan material. (b) During Stage 2, the surface of the pan in contact with the water is at a fixed temperature, T(L, t) TL Tb. Write the form of the heat equation and boundary conditions that determine the temperature distribution T(x) in the pan bottom. Express your result in terms of the parameters qo, D, L, and TL, as well as appropriate properties of the pan material. 2.28 Uniform internal heat generation at q˙ 5 107 W/m3 is occurring in a cylindrical nuclear reactor fuel rod of 50-mm diameter, and under steady-state conditions the temperature distribution is of the form T(r) a br2, where T is in degrees Celsius and r is in meters, while a 800 C and b 4.167 105 C/m2. The fuel rod properties are k 30 W/m 䡠 K, 1100 kg/m3, and cp 800 J/kg K. (a) What is the rate of heat transfer per unit length of the rod at r 0 (the centerline) and at r 25 mm (the surface)? (b) If the reactor power level is suddenly increased to . q2 108 W/m3, what is the initial time rate of temperature change at r 0 and r 25 mm? 2.29 Consider a one-dimensional plane wall with constant properties and uniform internal generation q˙. The left face is insulated, and the right face is held at a uniform temperature.

thickness 50 mm is observed to be T( C) a bx2, where a 200 C, b 2000 C/m2, and x is in meters. (a) What is the heat generation rate q˙ in the wall? (b) Determine the heat fluxes at the two wall faces. In what manner are these heat fluxes related to the heat generation rate? 2.31 The temperature distribution across a wall 0.3 m thick at a certain instant of time is T(x) a bx cx2, where T is in degrees Celsius and x is in meters, a 200 C, b 200 C/m, and c 30 C/m2. The wall has a thermal conductivity of 1 W/m 䡠 K. (a) On a unit surface area basis, determine the rate of heat transfer into and out of the wall and the rate of change of energy stored by the wall. (b) If the cold surface is exposed to a fluid at 100 C, what is the convection coefficient? 2.32 A plane wall of thickness 2L 40 mm and thermal conductivity k 5 W/m 䡠 K experiences uniform volumetric . heat generation at a rate q, while convection heat transfer occurs at both of its surfaces (x L, L), each of which is exposed to a fluid of temperature T앝 20 C. Under steady-state conditions, the temperature distribution in the wall is of the form T(x) a bx cx2 where a 82.0 C, b 210 C/m, c 2 104 C/m2, and x is in meters. The origin of the x-coordinate is at the midplane of the wall. (a) Sketch the temperature distribution and identify significant physical features. (b) What is the volumetric rate of heat generation q˙ in the wall?

ξ Tc

•

q

(c) Determine the surface heat fluxes, qx(L) and qx(L). How are these fluxes related to the heat generation rate? (d) What are the convection coefficients for the surfaces at x L and x L?

x

(e) Obtain an expression for the heat flux distribution qx(x). Is the heat flux zero at any location? Explain any significant features of the distribution.

(a) Using the appropriate form of the heat equation, derive an expression for the x-dependence of the steady-state heat flux q(x).

(f) If the source of the heat generation is suddenly deactivated (q˙ 0), what is the rate of change of energy stored in the wall at this instant?

(b) Using a finite volume spanning the range 0 x , derive an expression for q() and compare the expression to your result for part (a).

(g) What temperature will the wall eventually reach with q˙ 0? How much energy must be removed by the fluid per unit area of the wall (J/m2) to reach this state? The density and specific heat of the wall material are 2600 kg/m3 and 800 J/kg 䡠 K, respectively.

2.30 The steady-state temperature distribution in a onedimensional wall of thermal conductivity 50 W/m 䡠 K and

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Problems

2.33 Temperature distributions within a series of onedimensional plane walls at an initial time, at steady state, and at several intermediate times are as shown. t→∞

t⫽0

t→∞

t⫽0 x

L

x (b)

(a)

x

L

t→∞

t→∞

t⫽0

t⫽0

L

x

(c)

L

(d)

For each case, write the appropriate form of the heat diffusion equation. Also write the equations for the initial condition and the boundary conditions that are applied at x 0 and x L. If volumetric generation occurs, it is uniform throughout the wall. The properties are constant. 2.34 One-dimensional, steady-state conduction with uniform internal energy generation occurs in a plane wall with a thickness of 50 mm and a constant thermal conductivity of 5 W/m 䡠 K. For these conditions, the temperature distribution has the form T(x) a bx cx2. The surface at x 0 has a temperature of T(0) ⬅ To 120 C and experiences convection with a fluid for which T앝 20 C and h 500 W/m2 䡠 K. The surface at x L is well insulated.

(a) Applying an overall energy balance to the wall, calculate the volumetric energy generation rate q˙. (b) Determine the coefficients a, b, and c by applying the boundary conditions to the prescribed temperature distribution. Use the results to calculate and plot the temperature distribution. (c) Consider conditions for which the convection coefficient is halved, but the volumetric energy generation rate remains unchanged. Determine the new values of a, b, and c, and use the results to plot the temperature distribution. Hint: recognize that T(0) is no longer 120 C. (d) Under conditions for which the volumetric energy generation rate is doubled, and the convection coefficient remains unchanged (h 500 W/m2 䡠 K), determine the new values of a, b, and c and plot the corresponding temperature distribution. Referring to the results of parts (b), (c), and (d) as Cases 1, 2, and 3, respectively, compare the temperature distributions for the three cases and discuss the effects of h and q˙ on the distributions. 2.35 Derive the heat diffusion equation, Equation 2.26, for cylindrical coordinates beginning with the differential control volume shown in Figure 2.12. 2.36 Derive the heat diffusion equation, Equation 2.29, for spherical coordinates beginning with the differential control volume shown in Figure 2.13. 2.37 The steady-state temperature distribution in a semitransparent material of thermal conductivity k and thickness L exposed to laser irradiation is of the form T(x)

A ax e Bx C ka2

where A, a, B, and C are known constants. For this situation, radiation absorption in the material is manifested by a distributed heat generation term, q˙(x). Laser irradiation

x To = 120°C

T(x)

L Semitransparent medium, T(x)

T∞ = 20°C h = 500 W/m2•K •

q , k = 5 W/m•K

Fluid

x

L = 50 mm

(a) Obtain expressions for the conduction heat fluxes at the front and rear surfaces. (b) Derive an expression for q˙(x). (c) Derive an expression for the rate at which radiation is absorbed in the entire material, per unit surface

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Chapter 2

䊏

Introduction to Conduction (b) With the temperature at x 0 and the fluid temperature fixed at T(0) 0 C and T앝 20 C, respectively, compute and plot the temperature at x L, T(L), as a function of h for 10 h 100 W/m2 䡠 K. Briefly explain your results.

area. Express your result in terms of the known constants for the temperature distribution, the thermal conductivity of the material, and its thickness. 2.38 One-dimensional, steady-state conduction with no energy generation is occurring in a cylindrical shell of inner radius r1 and outer radius r2. Under what condition is the linear temperature distribution shown possible? T(r) T(r1)

T(r2) r1

r

r2

2.39 One-dimensional, steady-state conduction with no energy generation is occurring in a spherical shell of inner radius r1 and outer radius r2. Under what condition is the linear temperature distribution shown in Problem 2.38 possible? 2.40 The steady-state temperature distribution in a onedimensional wall of thermal conductivity k and thickness L is of the form T ax3 bx2 cx d. Derive expressions for the heat generation rate per unit volume in the wall and the heat fluxes at the two wall faces (x 0, L). 2.41 One-dimensional, steady-state conduction with no energy generation is occurring in a plane wall of constant thermal conductivity. 120

Ambient air T∞, h

GS

Ts

L Coal, k, q•

x

(a) Write the steady-state form of the heat diffusion equation for the layer of coal. Verify that this equation is satisfied by a temperature distribution of the form T(x) Ts

冢

q˙L2 x2 1 2 2k L

冣

From this distribution, what can you say about conditions at the bottom surface (x 0)? Sketch the temperature distribution and label key features.

100 80

T(⬚C)

2.42 A plane layer of coal of thickness L 1 m experiences uniform volumetric generation at a rate of q˙ 20 W/m3 due to slow oxidation of the coal particles. Averaged over a daily period, the top surface of the layer transfers heat by convection to ambient air for which h 5 W/m2 䡠 K and T앝 25 C, while receiving solar irradiation in the amount GS 400 W/m2. Irradiation from the atmosphere may be neglected. The solar absorptivity and emissivity of the surface are each ␣S 0.95.

(b) Obtain an expression for the rate of heat transfer by conduction per unit area at x L. Applying an energy balance to a control surface about the top surface of the layer, obtain an expression for Ts. Evaluate Ts and T(0) for the prescribed conditions.

60 40 20 0

x •

q = 0, k = 4.5 W/m•K T∞ = 20°C h = 30 W/m2•K 0.18 m Air

(a) Is the prescribed temperature distribution possible? Briefly explain your reasoning.

(c) Daily average values of GS and h depend on a number of factors, such as time of year, cloud cover, and wind conditions. For h 5 W/m2 䡠 K, compute and plot TS and T(0) as a function of GS for 50 GS 500 W/m2. For GS 400 W/m2, compute and plot TS and T(0) as a function of h for 5 h 50 W/m2 䡠 K. 2.43 The cylindrical system illustrated has negligible variation of temperature in the r- and z-directions. Assume

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Problems

that r ro ri is small compared to ri, and denote the length in the z-direction, normal to the page, as L. Insulation

φ

2.48 Passage of an electric current through a long conducting rod of radius ri and thermal conductivity kr results in uniform volumetric heating at a rate of q˙. The conducting rod is wrapped in an electrically nonconducting cladding material of outer radius ro and thermal conductivity kc, and convection cooling is provided by an adjoining fluid.

ri r o

T2

T1

(a) Beginning with a properly defined control volume and considering energy generation and storage effects, derive the differential equation that prescribes the variation in temperature with the angular coordinate . Compare your result with Equation 2.26. (b) For steady-state conditions with no internal heat generation and constant properties, determine the temperature distribution T() in terms of the constants T1, T2, ri, and ro. Is this distribution linear in ? (c) For the conditions of part (b) write the expression for the heat rate q. 2.44 Beginning with a differential control volume in the form of a cylindrical shell, derive the heat diffusion equation for a one-dimensional, cylindrical, radial coordinate system with internal heat generation. Compare your result with Equation 2.26. 2.45 Beginning with a differential control volume in the form of a spherical shell, derive the heat diffusion equation for a one-dimensional, spherical, radial coordinate system with internal heat generation. Compare your result with Equation 2.29. 2.46 A steam pipe is wrapped with insulation of inner and outer radii ri and ro, respectively. At a particular instant the temperature distribution in the insulation is known to be of the form

冢冣

r T(r) C1 ln r C2 o Are conditions steady-state or transient? How do the heat flux and heat rate vary with radius? 2.47 For a long circular tube of inner and outer radii r1 and r2, respectively, uniform temperatures T1 and T2 are maintained at the inner and outer surfaces, while thermal energy generation is occurring within the tube wall (r1 r r2). Consider steady-state conditions for which T1 T2. Is it possible to maintain a linear radial temperature distribution in the wall? If so, what special conditions must exist?

Conducting rod, q•, kr

ri

T∞, h ro Cladding, kc

For steady-state conditions, write appropriate forms of the heat equations for the rod and cladding. Express appropriate boundary conditions for the solution of these equations. 2.49 Two-dimensional, steady-state conduction occurs in a hollow cylindrical solid of thermal conductivity k 16 W/m 䡠 K, outer radius r o 1 m and overall length 2zo 5 m, where the origin of the coordinate system is located at the midpoint of the center line. The inner surface of the cylinder is insulated, and the temperature distribution within the cylinder has the form T(r, z) a br2 clnr dz2, where a 20 C, b 150 C/m2, c 12 C, d 300 C/m2 and r and z are in meters. (a) Determine the inner radius ri of the cylinder. (b) Obtain an expression for the volumetric rate of heat generation, q˙(W/m3). (c) Determine the axial distribution of the heat flux at the outer surface, qr(ro, z). What is the heat rate at the outer surface? Is it into or out of the cylinder? (d) Determine the radial distribution of the heat flux at the end faces of the cylinder, qr (r, zo) and qr (r, zo). What are the corresponding heat rates? Are they into or out of the cylinder? (e) Verify that your results are consistent with an overall energy balance on the cylinder. 2.50 An electric cable of radius r1 and thermal conductivity kc is enclosed by an insulating sleeve whose outer surface is of radius r2 and experiences convection heat transfer and radiation exchange with the adjoining air and large surroundings, respectively. When electric

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Introduction to Conduction

current passes through the cable, thermal energy is generated within the cable at a volumetric rate q˙. Tsur

2.51 A spherical shell of inner and outer radii ri and ro, respectively, contains heat-dissipating components, and at a particular instant the temperature distribution in the shell is known to be of the form C1 T(r) r C2

Electrical cable Insulation

Ts, 1 r1

Ambient air T∞, h

r2 Ts, 2

(a) Write the steady-state forms of the heat diffusion equation for the insulation and the cable. Verify that these equations are satisfied by the following temperature distributions: Insulation: T(r) Ts,2 (Ts,1 Ts,2) Cable: T(r) Ts,1

冢

q˙ r 21 r2 1 2 4kc r1

ln(r/r2) ln(r1/r2)

冣

Sketch the temperature distribution, T(r), in the cable and the sleeve, labeling key features. (b) Applying Fourier’s law, show that the rate of conduction heat transfer per unit length through the sleeve may be expressed as qr

Are conditions steady-state or transient? How do the heat flux and heat rate vary with radius? 2.52 A chemically reacting mixture is stored in a thin-walled spherical container of radius r1 200 mm, and the exothermic reaction generates heat at a uniform, but temperaturedependent volumetric rate of q˙ q˙o exp(A/To), where q˙o 5000 W/m3, A 75 K, and To is the mixture temperature in kelvins. The vessel is enclosed by an insulating material of outer radius r2, thermal conductivity k, and emissivity . The outer surface of the insulation experiences convection heat transfer and net radiation exchange with the adjoining air and large surroundings, respectively.

Tsur Chemical reaction, q• (To) Ambient air

T∞, h Insulation, k, ε

2pks(Ts,1 Ts,2) ln(r2/r1)

Applying an energy balance to a control surface placed around the cable, obtain an alternative expression for qr , expressing your result in terms of q˙ and r1. (c) Applying an energy balance to a control surface placed around the outer surface of the sleeve, obtain an expression from which Ts,2 may be determined as a function of q˙, r1, h, T앝, , and Tsur. (d) Consider conditions for which 250 A are passing through a cable having an electric resistance per unit length of Re 0.005 /m, a radius of r1 15 mm, and a thermal conductivity of kc 200 W/m 䡠 K. For ks 15 W/m 䡠 K, r2 15.5 mm, h 25 W/m2 K, 0.9, T앝 25 C, and Tsur 35 C, evaluate the surface temperatures, Ts,1 and Ts,2, as well as the temperature To at the centerline of the cable. (e) With all other conditions remaining the same, compute and plot To, Ts,1, and Ts,2 as a function of r2 for 15.5 r2 20 mm.

r1

r2

(a) Write the steady-state form of the heat diffusion equation for the insulation. Verify that this equation is satisfied by the temperature distribution T(r) Ts,1 (Ts,1 Ts,2)

冤11(r(r /r/r))冥 1

1

2

Sketch the temperature distribution, T(r), labeling key features. (b) Applying Fourier’s law, show that the rate of heat transfer by conduction through the insulation may be expressed as qr

4k(Ts,1 Ts,2) (1/r1) (1/r2)

Applying an energy balance to a control surface about the container, obtain an alternative expression for qr, expressing your result in terms of q˙ and r1.

䊏

105

Problems

(c) With the system operating as described in part (b), the surface x L also experiences a sudden loss of coolant. This dangerous situation goes undetected for 15 min, at which time the power to the heater is deactivated. Assuming no heat losses from the surfaces of the plates, what is the eventual (t l 앝), uniform, steady-state temperature distribution in the plates? Show this distribution as Case 3 on your sketch, and explain its key features. Hint: Apply the conservation of energy requirement on a time-interval basis, Eq. 1.12b, for the initial and final conditions corresponding to Case 2 and Case 3, respectively.

(c) Applying an energy balance to a control surface placed around the outer surface of the insulation, obtain an expression from which Ts,2 may be determined as a function of q˙, r1, h, T앝, , and Tsur. (d) The process engineer wishes to maintain a reactor temperature of To T(r1) 95 C under conditions for which k 0.05 W/m 䡠 K, r2 208 mm, h 5 W/m2 䡠 K, 0.9, T앝 25 C, and Tsur 35 C. What is the actual reactor temperature and the outer surface temperature Ts,2 of the insulation? (e) Compute and plot the variation of Ts,2 with r2 for 201 r2 210 mm. The engineer is concerned about potential burn injuries to personnel who may come into contact with the exposed surface of the insulation. Is increasing the insulation thickness a practical solution to maintaining Ts,2 45 C? What other parameter could be varied to reduce Ts,2?

Graphical Representations 2.53 A thin electrical heater dissipating 4000 W/m2 is sandwiched between two 25-mm-thick plates whose exposed surfaces experience convection with a fluid for which T앝 20 C and h 400 W/m2 䡠 K. The thermophysical properties of the plate material are 2500 kg/m3, c 700 J/kg 䡠 K, and k 5 W/m 䡠 K. Electric heater, q"o

(d) On T t coordinates, sketch the temperature history at the plate locations x 0, L during the transient period between the distributions for Cases 2 and 3. Where and when will the temperature in the system achieve a maximum value? 2.54 The one-dimensional system of mass M with constant properties and no internal heat generation shown in the figure is initially at a uniform temperature Ti. The electrical heater is suddenly energized, providing a uniform heat flux qo at the surface x 0. The boundaries at x L and elsewhere are perfectly insulated. Insulation

L x

System, mass M Electrical heater

ρ , c, k Fluid

Fluid

T∞, h

T∞, h

–L

0

+L

x

(a) On T x coordinates, sketch the steady-state temperature distribution for L x L. Calculate values of the temperatures at the surfaces, x L, and the midpoint, x 0. Label this distribution as Case 1, and explain its salient features. (b) Consider conditions for which there is a loss of coolant and existence of a nearly adiabatic condition on the x L surface. On the T x coordinates used for part (a), sketch the corresponding steady-state temperature distribution and indicate the temperatures at x 0, L. Label the distribution as Case 2, and explain its key features.

(a) Write the differential equation, and identify the boundary and initial conditions that could be used to determine the temperature as a function of position and time in the system. (b) On T x coordinates, sketch the temperature distributions for the initial condition (t 0) and for several times after the heater is energized. Will a steady-state temperature distribution ever be reached? (c) On qx t coordinates, sketch the heat flux qx (x, t) at the planes x 0, x L/2, and x L as a function of time. (d) After a period of time te has elapsed, the heater power is switched off. Assuming that the insulation is perfect, the system will eventually reach a final uniform temperature Tf. Derive an expression that can be used to determine Tf as a function of the parameters qo , te, Ti, and the system characteristics M, cp, and As (the heater surface area). 2.55 Consider a one-dimensional plane wall of thickness 2L. The surface at x L is subjected to convective conditions characterized by T앝,1, h1, while the surface

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Introduction to Conduction

at x L is subjected to conditions T앝,2, h2. The initial temperature of the wall is To (T앝,1 T앝,2)/2 where T앝,1 T앝,2.

To h1, T∞,1

h2, T∞,2

T∞,1 T∞,2 2L

x

(a) Write the differential equation, and identify the boundary and initial conditions that could be used to determine the temperature distribution T(x, t) as a function of position and time. (b) On T x coordinates, sketch the temperature distributions for the initial condition, the steady-state condition, and for two intermediate times for the case h1 h2. (c) On qx t coordinates, sketch the heat flux qx (x, t) at the planes x 0, L, and L. (d) The value of h1 is now doubled with all other conditions being identical as in parts (a) through (c). On T x coordinates drawn to the same scale as used in part (b), sketch the temperature distributions for the initial condition, the steady-state condition, and for two intermediate times. Compare the sketch to that of part (b). (e) Using the doubled value of h1, sketch the heat flux qx(x, t) at the planes x 0, L, and L on the same plot you prepared for part (c). Compare the two responses. 2.56 A large plate of thickness 2L is at a uniform temperature of Ti 200 C, when it is suddenly quenched by dipping it in a liquid bath of temperature T앝 20 C. Heat transfer to the liquid is characterized by the convection coefficient h. (a) If x 0 corresponds to the midplane of the wall, on T x coordinates, sketch the temperature distributions for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and two intermediate times. (b) On qx t coordinates, sketch the variation with time of the heat flux at x L.

(c) If h 100 W/m2 䡠 K, what is the heat flux at x L and t 0? If the wall has a thermal conductivity of k 50 W/m 䡠 K what is the corresponding temperature gradient at x L? (d) Consider a plate of thickness 2L 20 mm with a density of 2770 kg/m3 and a specific heat cp 875 J/kg 䡠 K. By performing an energy balance on the plate, determine the amount of energy per unit surface area of the plate (J/m2) that is transferred to the bath over the time required to reach steady-state conditions. (e) From other considerations, it is known that, during the quenching process, the heat flux at x L and x L decays exponentially with time according to the relation, qx A exp(Bt), where t is in seconds, A 1.80 104 W/m2, and B 4.126 103 s1. Use this information to determine the energy per unit surface area of the plate that is transferred to the fluid during the quenching process. 2.57 The plane wall with constant properties and no internal heat generation shown in the figure is initially at a uniform temperature Ti. Suddenly the surface at x L is heated by a fluid at T앝 having a convection heat transfer coefficient h. The boundary at x 0 is perfectly insulated.

T∞, h Insulation

x

L

(a) Write the differential equation, and identify the boundary and initial conditions that could be used to determine the temperature as a function of position and time in the wall. (b) On T x coordinates, sketch the temperature distributions for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and two intermediate times. (c) On qx t coordinates, sketch the heat flux at the locations x 0, x L. That is, show qualitatively how qx (0, t) and qx (L, t) vary with time. (d) Write an expression for the total energy transferred to the wall per unit volume of the wall (J/m3). 2.58 Consider the steady-state temperature distributions within a composite wall composed of Material A and Material B for the two cases shown. There is no

䊏

107

Problems

internal generation, and the conduction process is onedimensional. T(x)

T(x)

(b) On qx x coordinates, sketch the heat flux corresponding to the four temperature distributions of part (a). (c) On qx t coordinates, sketch the heat flux at the locations x 0 and x L. That is, show qualitatively how qx (0, t) and qx (L, t) vary with time. (d) Derive an expression for the steady-state temperature at the heater surface, T(0, 앝), in terms of qo , T앝, k, h, and L.

LA

LB

kA

LA kB

LB

kA

x

kB x

Case 1

Case 2

Answer the following questions for each case. Which material has the higher thermal conductivity? Does the thermal conductivity vary significantly with temperature? If so, how? Describe the heat flux distribution qx(x) through the composite wall. If the thickness and thermal conductivity of each material were both doubled and the boundary temperatures remained the same, what would be the effect on the heat flux distribution? Case 1. Linear temperature distributions exist in both materials, as shown. Case 2. Nonlinear temperature distributions exist in both materials, as shown. 2.59 A plane wall has constant properties, no internal heat generation, and is initially at a uniform temperature Ti. Suddenly, the surface at x L is heated by a fluid at T앝 having a convection coefficient h. At the same instant, the electrical heater is energized, providing a constant heat flux qo at x 0.

T∞, h

Heater

2.60 A plane wall with constant properties is initially at a uniform temperature To. Suddenly, the surface at x L is exposed to a convection process with a fluid at T앝 (To) having a convection coefficient h. Also, suddenly the wall experiences a uniform internal volumetric heating q˙ that is sufficiently large to induce a maximum steadystate temperature within the wall, which exceeds that of the fluid. The boundary at x 0 remains at To.

k, q• (t ≥ 0) To T∞, h

L x

(a) On T x coordinates, sketch the temperature distributions for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and for two intermediate times. Show also the distribution for the special condition when there is no heat flow at the x L boundary. (b) On qx t coordinates, sketch the heat flux for the locations x 0 and x L, that is, qx(0, t) and qx(L, t), respectively. 2.61 Consider the conditions associated with Problem 2.60, but now with a convection process for which T앝 To.

Insulation

L x

(a) On T x coordinates, sketch the temperature distributions for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and for two intermediate times.

(a) On T x coordinates, sketch the temperature distributions for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and for two intermediate times. Identify key features of the distributions, especially the location of the maximum temperature and the temperature gradient at x L. (b) On qx t coordinates, sketch the heat flux for the locations x 0 and x L, that is, qx(0, t) and qx(L, t), respectively. Identify key features of the flux histories.

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䊏

2.62 Consider the steady-state temperature distribution within a composite wall composed of Materials A and B.

determine a relationship between the temperature gradient dT/dr and the local radius r, for r1 r r2. (c) On T r coordinates, sketch the temperature distribution over the range 0 r r2.

T(x)

LA

LB

kA

kB x

The conduction process is one-dimensional. Within which material does uniform volumetric generation occur? What is the boundary condition at x LA? How would the temperature distribution change if the thermal conductivity of Material A were doubled? How would the temperature distribution change if the thermal conductivity of Material B were doubled? Does a contact resistance exist at the interface between the two materials? Sketch the heat flux distribution qx(x) through the composite wall. 2.63 A spherical particle of radius r1 experiences uniform ther. mal generation at a rate of q. The particle is encapsulated by a spherical shell of outside radius r2 that is cooled by ambient air. The thermal conductivities of the particle and shell are k1 and k2, respectively, where k1 2k2.

2.64 A long cylindrical rod, initially at a uniform temperature Ti, is suddenly immersed in a large container of liquid at T앝 Ti. Sketch the temperature distribution within the rod, T(r), at the initial time, at steady state, and at two intermediate times. On the same graph, carefully sketch the temperature distributions that would occur at the same times within a second rod that is the same size as the first rod. The densities and specific heats of the two rods are identical, but the thermal conductivity of the second rod is very large. Which rod will approach steady-state conditions sooner? Write the appropriate boundary conditions that would be applied at r 0 and r D/2 for either rod. 2.65 A plane wall of thickness L 0.1 m experiences uniform . volumetric heating at a rate q. One surface of the wall (x 0) is insulated, and the other surface is exposed to a fluid at T앝 20 C, with convection heat transfer characterized by h 1000 W/m2 䡠 K. Initially, the temperature distribution in the wall is T(x, 0) a bx2, where a 300 C, b 1.0 104 C/m2, and x is in meters. Suddenly, the volumetric heat generation is deactivated . (q 0 for t 0), while convection heat transfer continues to occur at x L. The properties of the wall are 7000 kg/m3, cp 450 J/kg 䡠 K, and k 90 W/m 䡠 K. •

k, ρ , cp, q (t < – 0)

Chemical reaction •

q T∞, h

x r1

Ambient air T∞, h

r2 Control volume B Control volume A

(a) By applying the conservation of energy principle to spherical control volume A, which is placed at an arbitrary location within the sphere, determine a relationship between the temperature gradient dT/dr and the local radius r, for 0 r r1. (b) By applying the conservation of energy principle to spherical control volume B, which is placed at an arbitrary location within the spherical shell,

L

(a) Determine the magnitude of the volumetric energy . generation rate q associated with the initial condition (t 0). (b) On T x coordinates, sketch the temperature distribution for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and two intermediate conditions. (c) On qx t coordinates, sketch the variation with time of the heat flux at the boundary exposed to the convection process, qx(L, t). Calculate the corresponding value of the heat flux at t 0, qx(L, 0). (d) Calculate the amount of energy removed from the wall per unit area (J/m2) by the fluid stream

䊏

109

Problems

as the wall cools from its initial to steady-state condition. 2.66 A plane wall that is insulated on one side (x 0) is initially at a uniform temperature Ti, when its exposed surface at x L is suddenly raised to a temperature Ts. (a) Verify that the following equation satisfies the heat equation and boundary conditions:

冢

冣 冢 冣

T(x, t) Ts 2 ␣t x C1 exp cos Ti Ts 4 L2 2L

where C1 is a constant and ␣ is the thermal diffusivity. (b) Obtain expressions for the heat flux at x 0 and x L. (c) Sketch the temperature distribution T(x) at t 0, at t l 앝, and at an intermediate time. Sketch the variation with time of the heat flux at x L, qL(t). (d) What effect does ␣ have on the thermal response of the material to a change in surface temperature? 2.67 A composite one-dimensional plane wall is of overall thickness 2L. Material A spans the domain L x 0 and experiences an exothermic chemical reaction leading . to a uniform volumetric generation rate of qA. Material B spans the domain 0 x L and undergoes an endothermic chemical reaction corresponding to a uniform . . volumetric generation rate of qB qA. The surfaces at x L are insulated. Sketch the steady-state temperature and heat flux distributions T(x) and qx(x), respectively, over the domain L x L for kA kB, kA 0.5kB, and kA 2kB. Point out the important features of the distributions you have drawn. If q˙B 2q˙A, can you sketch the steady-state temperature distribution? 2.68 Typically, air is heated in a hair dryer by blowing it across a coiled wire through which an electric current is passed. Thermal energy is generated by electric resistance heating within the wire and is transferred by convection from the surface of the wire to the air. Consider conditions for which the wire is initially at room temperature, Ti, and resistance heating is concurrently initiated with airflow at t 0.

Coiled wire (ro, L, k, ρ , cp)

•

q

Airflow

Air

Pelec

T∞, h ro

r

(a) For a wire radius ro, an air temperature T앝, and a convection coefficient h, write the form of the heat equation and the boundary/initial conditions that govern the transient thermal response, T(r, t), of the wire. (b) If the length and radius of the wire are 500 mm and 1 mm, respectively, what is the volumetric rate of thermal energy generation for a power consumption of Pelec 500 W? What is the convection heat flux under steady-state conditions? (c) On T r coordinates, sketch the temperature distributions for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and for two intermediate times. (d) On qr t coordinates, sketch the variation of the heat flux with time for locations at r 0 and r ro. 2.69 The steady-state temperature distribution in a composite plane wall of three different materials, each of constant thermal conductivity, is shown. 1

2

3

4

T A

B

C

q"2

q"3

q"4 x

(a) Comment on the relative magnitudes of q2 and q3 , and of q3 and q4 . (b) Comment on the relative magnitudes of kA and kB, and of kB and kC. (c) Sketch the heat flux as a function of x.

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C H A P T E R

One-Dimensional, Steady-State Conduction

3

112

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

I

n this chapter we treat situations for which heat is transferred by diffusion under onedimensional, steady-state conditions. The term one-dimensional refers to the fact that only one coordinate is needed to describe the spatial variation of the dependent variables. Hence, in a one-dimensional system, temperature gradients exist along only a single coordinate direction, and heat transfer occurs exclusively in that direction. The system is characterized by steady-state conditions if the temperature at each point is independent of time. Despite their inherent simplicity, one-dimensional, steady-state models may be used to accurately represent numerous engineering systems. We begin our consideration of one-dimensional, steady-state conduction by discussing heat transfer with no internal generation of thermal energy (Sections 3.1 through 3.4). The objective is to determine expressions for the temperature distribution and heat transfer rate in common (planar, cylindrical, and spherical) geometries. For such geometries, an additional objective is to introduce the concept of thermal resistance and to show how thermal circuits may be used to model heat flow, much as electrical circuits are used for current flow. The effect of internal heat generation is treated in Section 3.5, and again our objective is to obtain expressions for determining temperature distributions and heat transfer rates. In Section 3.6, we consider the special case of one-dimensional, steady-state conduction for extended surfaces. In their most common form, these surfaces are termed fins and are used to enhance heat transfer by convection to an adjoining fluid. In addition to determining related temperature distributions and heat rates, our objective is to introduce performance parameters that may be used to determine their efficacy. Finally, in Sections 3.7 through 3.9 we apply heat transfer and thermal resistance concepts to the human body, including the effects of metabolic heat generation and perfusion; to thermoelectric power generation driven by the Seebeck effect; and to micro- and nanoscale conduction in thin gas layers and thin solid films.

3.1 The Plane Wall For one-dimensional conduction in a plane wall, temperature is a function of the x-coordinate only and heat is transferred exclusively in this direction. In Figure 3.1a, a plane wall separates two fluids of different temperatures. Heat transfer occurs by convection from the hot fluid at T앝,1 to one surface of the wall at Ts,1, by conduction through the wall, and by convection from the other surface of the wall at Ts,2 to the cold fluid at T앝,2. We begin by considering conditions within the wall. We first determine the temperature distribution, from which we can then obtain the conduction heat transfer rate.

3.1.1

Temperature Distribution

The temperature distribution in the wall can be determined by solving the heat equation with the proper boundary conditions. For steady-state conditions with no distributed source or sink of energy within the wall, the appropriate form of the heat equation is Equation 2.23

冢 冣

d k dT 0 dx dx

(3.1)

3.1

䊏

113

The Plane Wall

T∞,1 Ts,1

Ts,2 T∞,2 qx Hot fluid

T∞,1, h1 x

x=L Cold fluid

T∞,2, h2

(a)

T∞,1 qx

Ts,1

T∞,2

Ts,2

1 ____

L ____

h1A

kA

FIGURE 3.1 Heat transfer through a plane wall. (a) Temperature distribution. (b) Equivalent thermal circuit.

1 ____

h2A

(b)

Hence, from Equation 2.2, it follows that, for one-dimensional, steady-state conduction in a plane wall with no heat generation, the heat flux is a constant, independent of x. If the thermal conductivity of the wall material is assumed to be constant, the equation may be integrated twice to obtain the general solution T(x) C1x C2

(3.2)

To obtain the constants of integration, C1 and C2, boundary conditions must be introduced. We choose to apply conditions of the first kind at x 0 and x L, in which case T(0) Ts,1

and

T(L) Ts,2

Applying the condition at x 0 to the general solution, it follows that Ts,1 C2 Similarly, at x L, Ts,2 C1L C2 C1L Ts,1 in which case Ts,2 Ts,1 C1 L Substituting into the general solution, the temperature distribution is then T(x) (Ts,2 Ts,1) x Ts,1 L

(3.3)

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Chapter 3

䊏

One-Dimensional, Steady-State Conduction

From this result it is evident that, for one-dimensional, steady-state conduction in a plane wall with no heat generation and constant thermal conductivity, the temperature varies linearly with x. Now that we have the temperature distribution, we may use Fourier’s law, Equation 2.1, to determine the conduction heat transfer rate. That is, qx kA dT kA (Ts,1 Ts,2) dx L

(3.4)

Note that A is the area of the wall normal to the direction of heat transfer and, for the plane wall, it is a constant independent of x. The heat flux is then qx

qx k (T Ts,2) A L s,1

(3.5)

Equations 3.4 and 3.5 indicate that both the heat rate qx and heat flux qx are constants, independent of x. In the foregoing paragraphs we have used the standard approach to solving conduction problems. That is, the general solution for the temperature distribution is first obtained by solving the appropriate form of the heat equation. The boundary conditions are then applied to obtain the particular solution, which is used with Fourier’s law to determine the heat transfer rate. Note that we have opted to prescribe surface temperatures at x 0 and x L as boundary conditions, even though it is the fluid temperatures, not the surface temperatures, that are typically known. However, since adjoining fluid and surface temperatures are easily related through a surface energy balance (see Section 1.3.1), it is a simple matter to express Equations 3.3 through 3.5 in terms of fluid, rather than surface, temperatures. Alternatively, equivalent results could be obtained directly by using the surface energy balances as boundary conditions of the third kind in evaluating the constants of Equation 3.2 (see Problem 3.1).

3.1.2

Thermal Resistance

At this point we note that, for the special case of one-dimensional heat transfer with no internal energy generation and with constant properties, a very important concept is suggested by Equation 3.4. In particular, an analogy exists between the diffusion of heat and electrical charge. Just as an electrical resistance is associated with the conduction of electricity, a thermal resistance may be associated with the conduction of heat. Defining resistance as the ratio of a driving potential to the corresponding transfer rate, it follows from Equation 3.4 that the thermal resistance for conduction in a plane wall is Rt,cond ⬅

Ts,1 Ts,2 L qx kA

(3.6)

Similarly, for electrical conduction in the same system, Ohm’s law provides an electrical resistance of the form Re

Es,1 Es,2 L I A

(3.7)

3.1

䊏

115

The Plane Wall

The analogy between Equations 3.6 and 3.7 is obvious. A thermal resistance may also be associated with heat transfer by convection at a surface. From Newton’s law of cooling, q hA(Ts T앝)

(3.8)

The thermal resistance for convection is then Rt,conv ⬅

Ts T앝 1 q hA

(3.9)

Circuit representations provide a useful tool for both conceptualizing and quantifying heat transfer problems. The equivalent thermal circuit for the plane wall with convection surface conditions is shown in Figure 3.1b. The heat transfer rate may be determined from separate consideration of each element in the network. Since qx is constant throughout the network, it follows that qx

T앝,1 Ts,1 Ts,1 Ts,2 Ts,2 T앝,2 1/h1A L/kA 1/h2A

(3.10)

In terms of the overall temperature difference, T앝,1 T앝,2, and the total thermal resistance, Rtot, the heat transfer rate may also be expressed as qx

T앝,1 T앝,2 Rtot

(3.11)

Because the conduction and convection resistances are in series and may be summed, it follows that Rtot 1 L 1 h1A kA h2A

(3.12)

Radiation exchange between the surface and surroundings may also be important if the convection heat transfer coefficient is small (as it often is for natural convection in a gas). A thermal resistance for radiation may be defined by reference to Equation 1.8: Rt,rad

Ts Tsur 1 qrad hr A

(3.13)

For radiation between a surface and large surroundings, hr is determined from Equation 1.9. Surface radiation and convection resistances act in parallel, and if T앝 Tsur, they may be combined to obtain a single, effective surface resistance.

3.1.3

The Composite Wall

Equivalent thermal circuits may also be used for more complex systems, such as composite walls. Such walls may involve any number of series and parallel thermal resistances due to layers of different materials. Consider the series composite wall of Figure 3.2. The onedimensional heat transfer rate for this system may be expressed as qx

T앝,1 T앝,4 Rt

(3.14)

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Chapter 3

䊏

One-Dimensional, Steady-State Conduction

T∞,1 Ts,1

T2

T3 Ts,4

Hot fluid

LA

LB

LC

kA

kB

kC

A

B

C

T∞,4

T∞,1, h1

x

qx

Cold fluid

1 ____

LA ____

LB ____

LC ____

1 ____

h1A

kA A

kB A

kC A

h4 A

T∞,1

Ts,1

T2

T3

Ts,4

T∞,4, h4

T∞,4

FIGURE 3.2 Equivalent thermal circuit for a series composite wall.

where T앝,1 T앝,4 is the overall temperature difference, and the summation includes all thermal resistances. Hence qx

T앝,1 T앝,4 [(1/h1A) (LA /kAA) (LB /kBA) (LC /kC A) (1/h4A)]

(3.15)

Alternatively, the heat transfer rate can be related to the temperature difference and resistance associated with each element. For example, qx

T앝,1 Ts,1 Ts,1 T2 T T3 … 2 (1/h1A) (LA/kAA) (LB /kBA)

(3.16)

With composite systems, it is often convenient to work with an overall heat transfer coefficient U, which is defined by an expression analogous to Newton’s law of cooling. Accordingly, qx ⬅ UA T

(3.17)

where T is the overall temperature difference. The overall heat transfer coefficient is related to the total thermal resistance, and from Equations 3.14 and 3.17 we see that UA 1/Rtot. Hence, for the composite wall of Figure 3.2, 1 U 1 Rtot A [(1/h1) (LA /kA) (LB /kB) (LC /kC) (1/h4)]

(3.18)

In general, we may write Rtot

1 兺R T q UA t

(3.19)

3.1

䊏

117

The Plane Wall

LE

LF = LG

Area, A

LH

kF

F

T1

T2 kE

kG

kH

E

G

H

x LF ________ kF(A/2) LE ____

LH ____ kHA

kEA qx

LG ________ kG(A/2)

T1

T2

(a)

qx

LE ________ kE(A/2)

LF ________ kF(A/2)

LH ________ kH(A/2)

T1 L E ________ kE(A/2)

LG ________ kG(A/2)

LH ________ kH(A/2)

T2

FIGURE 3.3 Equivalent thermal circuits for a series–parallel composite wall.

(b)

Composite walls may also be characterized by series–parallel configurations, such as that shown in Figure 3.3. Although the heat flow is now multidimensional, it is often reasonable to assume one-dimensional conditions. Subject to this assumption, two different thermal circuits may be used. For case (a) it is presumed that surfaces normal to the x-direction are isothermal, whereas for case (b) it is assumed that surfaces parallel to the x-direction are adiabatic. Different results are obtained for Rtot, and the corresponding values of q bracket the actual heat transfer rate. These differences increase with increasing 冨kF kG冨, as multidimensional effects become more significant.

3.1.4

Contact Resistance

Although neglected until now, it is important to recognize that, in composite systems, the temperature drop across the interface between materials may be appreciable. This temperature change is attributed to what is known as the thermal contact resistance, Rt,c. The effect is shown in Figure 3.4, and for a unit area of the interface, the resistance is defined as Rt,c

TA TB qx

(3.20)

The existence of a finite contact resistance is due principally to surface roughness effects. Contact spots are interspersed with gaps that are, in most instances, air filled. Heat transfer is therefore due to conduction across the actual contact area and to conduction and/or radiation across the gaps. The contact resistance may be viewed as two parallel resistances: that due to

118

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

q"x qc"ontact q"x

TA ∆T

A

B

TB

T

qg"ap A

B

FIGURE 3.4 Temperature drop due to thermal contact resistance.

x

the contact spots and that due to the gaps. The contact area is typically small, and, especially for rough surfaces, the major contribution to the resistance is made by the gaps. For solids whose thermal conductivities exceed that of the interfacial fluid, the contact resistance may be reduced by increasing the area of the contact spots. Such an increase may be effected by increasing the joint pressure and/or by reducing the roughness of the mating surfaces. The contact resistance may also be reduced by selecting an interfacial fluid of large thermal conductivity. In this respect, no fluid (an evacuated interface) eliminates conduction across the gap, thereby increasing the contact resistance. Likewise, if the characteristic gap width L becomes small (as, for example, in the case of very smooth surfaces in contact), L/mfp can approach values for which the thermal conductivity of the interfacial gas is reduced by microscale effects, as discussed in Section 2.2. Although theories have been developed for predicting Rt,c, the most reliable results are those that have been obtained experimentally. The effect of loading on metallic interfaces can be seen in Table 3.1a, which presents an approximate range of thermal resistances under vacuum conditions. The effect of interfacial fluid on the thermal resistance of an aluminum interface is shown in Table 3.1b. Contrary to the results of Table 3.1, many applications involve contact between dissimilar solids and/or a wide range of possible interstitial (filler) materials (Table 3.2). Any interstitial substance that fills the gap between contacting surfaces and whose thermal conductivity exceeds that of air will decrease the contact resistance. Two classes of materials that are well suited for this purpose are soft metals and thermal greases. The metals, which include

TABLE 3.1 Thermal contact resistance for (a) metallic interfaces under vacuum conditions and (b) aluminum interface (10-m surface roughness, 105 N/m2) with different interfacial fluids [1] Thermal Resistance, Rⴖt, c ⫻ 104 (m2 䡠 K/W) (a) Vacuum Interface Contact pressure 100 kN/m2 Stainless steel 6–25 Copper 1–10 Magnesium 1.5–3.5 Aluminum 1.5–5.0

2

10,000 kN/m 0.7–4.0 0.1–0.5 0.2–0.4 0.2–0.4

(b) Interfacial Fluid Air 2.75 Helium 1.05 Hydrogen 0.720 Silicone oil 0.525 Glycerine 0.265

3.1

䊏

119

The Plane Wall

TABLE 3.2

Thermal resistance of representative solid/solid interfaces

Interface Silicon chip/lapped aluminum in air (27–500 kN/m2) Aluminum/aluminum with indium foil filler (⬃100 kN/m2) Stainless/stainless with indium foil filler (⬃3500 kN/m2) Aluminum/aluminum with metallic (Pb) coating Aluminum/aluminum with Dow Corning 340 grease (⬃100 kN/m2) Stainless/stainless with Dow Corning 340 grease (⬃3500 kN/m2) Silicon chip/aluminum with 0.02-mm epoxy Brass/brass with 15-m tin solder

Rⴖt,c ⫻ 104 (m2 䡠 K/W)

Source

0.3–0.6

[2]

⬃0.07

[1, 3]

⬃0.04

[1, 3]

0.01–0.1

[4]

⬃0.07

[1, 3]

⬃0.04

[1, 3]

0.2–0.9

[5]

0.025–0.14

[6]

indium, lead, tin, and silver, may be inserted as a thin foil or applied as a thin coating to one of the parent materials. Silicon-based thermal greases are attractive on the basis of their ability to completely fill the interstices with a material whose thermal conductivity is as much as 50 times that of air. Unlike the foregoing interfaces, which are not permanent, many interfaces involve permanently bonded joints. The joint could be formed from an epoxy, a soft solder rich in lead, or a hard solder such as a gold/tin alloy. Due to interface resistances between the parent and bonding materials, the actual thermal resistance of the joint exceeds the theoretical value (L/k) computed from the thickness L and thermal conductivity k of the joint material. The thermal resistance of epoxied and soldered joints is also adversely affected by voids and cracks, which may form during manufacture or as a result of thermal cycling during normal operation. Comprehensive reviews of thermal contact resistance results and models are provided by Snaith et al. [3], Madhusudana and Fletcher [7], and Yovanovich [8].

3.1.5

Porous Media

In many applications, heat transfer occurs within porous media that are combinations of a stationary solid and a fluid. When the fluid is either a gas or a liquid, the resulting porous medium is said to be saturated. In contrast, all three phases coexist in an unsaturated porous medium. Examples of porous media include beds of powder with a fluid occupying the interstitial regions between individual granules, as well as the insulation systems and nanofluids of Section 2.2.1. A saturated porous medium that consists of a stationary solid phase through which a fluid flows is referred to as a packed bed and is discussed in Section 7.8. Consider a saturated porous medium that is subjected to surface temperatures T1 at x 0 and T2 at x L, as shown in Figure 3.5a. After steady-state conditions are reached and if T1 T2, the heat rate may be expressed as qx

keff A (T1 T2) L

(3.21)

120

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

where keff is an effective thermal conductivity. Equation 3.21 is valid if fluid motion, as well as radiation heat transfer within the medium, are negligible. The effective thermal conductivity varies with the porosity or void fraction of the medium which is defined as the volume of fluid relative to the total volume (solid and fluid). In addition, keff depends on the thermal conductivities of each of the phases and, in this discussion, it is assumed that ks kf. The detailed solid phase geometry, for example the size distribution and packing arrangement of individual powder particles, also affects the value of keff. Contact resistances that might evolve at interfaces between adjacent solid particles can impact the value of keff. As discussed in Section 2.2.1, nanoscale phenomena might also influence the effective thermal conductivity. Hence, prediction of keff can be difficult and, in general, requires detailed knowledge of parameters that might not be readily available. Despite the complexity of the situation, the value of the effective thermal conductivity may be bracketed by considering the composite walls of Figures 3.5b and 3.5c. In Figure 3.5b, the medium is modeled as an equivalent, series composite wall consisting of a fluid region of length L and a solid region of length (1 – )L. Applying Equations 3.17 and 3.18 to this model for which there is no convection (h1 h2 0) and only two conduction terms, it follows that qx

A T (1 )L /ks L /kf

(3.22)

Equating this result to Equation 3.21, we then obtain keff,min

1 (1 )/ks /kf

(3.23)

Alternatively, the medium of Figure 3.5a could be described by the equivalent, parallel composite wall consisting of a fluid region of width w and a solid region of width (1 – )w, as shown in Figure 3.5c. Combining Equation 3.21 with an expression for the equivalent resistance of two resistors in parallel gives keff,max kf (1 )ks

(3.24)

L

(1 − )L

L

L

Area A , ks, kf, keff

T1

Area A ks

T1

kf

Area A T1

ks

qx.s

A w

x

x

T1

L

T2

qx

T1

keff A (a)

qx.f

kf

x

q

T2

qx

T2

qx

(b)

(1 − )L ks A

(1 − )w

T2

w

x

L kf A

T2

qx T1

(c)

L ks(1 − )A

T2

L kf A

FIGURE 3.5 A porous medium. (a) The medium and its properties. (b) Series thermal resistance representation. (c) Parallel resistance representation.

3.1

䊏

121

The Plane Wall

While Equations 3.23 and 3.24 provide the minimum and maximum possible values of keff, more accurate expressions have been derived for specific composite systems within which nanoscale effects are negligible. Maxwell [9] derived an expression for the effective electrical conductivity of a solid matrix interspersed with uniformly distributed, noncontacting spherical inclusions. Noting the analogy between Equations 3.6 and 3.7, Maxwell’s result may be used to determine the effective thermal conductivity of a saturated porous medium consisting of an interconnected solid phase within which a dilute distribution of spherical fluid regions exists, resulting in an expression of the form [10] keff

kf 2ks 2(ks kf)

冤 k 2k (k k ) 冥k

s

f

s

s

(3.25)

f

Equation 3.25 is valid for relatively small porosities ( 0.25) as shown schematically in Figure 3.5a [11]. It is equivalent to the expression introduced in Example 2.2 for a fluid that contains a dilute mixture of solid particles, but with reversal of the fluid and solid. When analyzing conduction within porous media, it is important to consider the potential directional dependence of the effective thermal conductivity. For example, the media represented in Figure 3.5b or Figure 3.5c would not be characterized by isotropic properties, since the effective thermal conductivity in the x-direction is clearly different from values of keff in the vertical direction. Hence, although Equations 3.23 and 3.24 can be used to bracket the actual value of the effective thermal conductivity, they will generally overpredict the possible range of keff for isotropic media. For isotropic media, expressions have been developed to determine the minimum and maximum possible effective thermal conductivities based solely on knowledge of the porosity and the thermal conductivities of the solid and fluid. Specifically, the maximum possible value of keff in an isotropic porous medium is given by Equation 3.25, which corresponds to an interconnected, high thermal conductivity solid phase. The minimum possible value of keff for an isotropic medium corresponds to the case where the fluid phase forms long, randomly oriented fingers within the medium [12]. Additional information regarding conduction in saturated porous media is available [13].

EXAMPLE 3.1 In Example 1.7, we calculated the heat loss rate from a human body in air and water environments. Now we consider the same conditions except that the surroundings (air or water) are at 10 C. To reduce the heat loss rate, the person wears special sporting gear (snow suit and wet suit) made from a nanostructured silica aerogel insulation with an extremely low thermal conductivity of 0.014 W/m K. The emissivity of the outer surface of the snow and wet suits is 0.95. What thickness of aerogel insulation is needed to reduce the heat loss rate to 100 W (a typical metabolic heat generation rate) in air and water? What are the resulting skin temperatures?

SOLUTION Known: Inner surface temperature of a skin/fat layer of known thickness, thermal conductivity, and surface area. Thermal conductivity and emissivity of snow and wet suits. Ambient conditions.

122

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Find: Insulation thickness needed to reduce heat loss rate to 100 W and corresponding skin temperature. Schematic: Ti = 35°C

Ts

ε = 0.95

Tsur = 10°C

Insulation

Skin/fat

kins = 0.014 W/m•K

ksf = 0.3 W/m•K

T∞ = 10°C h = 2 W/m2•K (Air) h = 200 W/m2•K (Water) Lins

Lsf = 3 mm

Air or water

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer by conduction through the skin/fat and insulation layers. 3. Contact resistance is negligible. 4. Thermal conductivities are uniform. 5. Radiation exchange between the skin surface and the surroundings is between a small surface and a large enclosure at the air temperature. 6. Liquid water is opaque to thermal radiation. 7. Solar radiation is negligible. 8. Body is completely immersed in water in part 2. Analysis: The thermal circuit can be constructed by recognizing that resistance to heat flow is associated with conduction through the skin/fat and insulation layers and convection and radiation at the outer surface. Accordingly, the circuit and the resistances are of the following form (with hr 0 for water): 1 ____

Lsf ____ ksf A q

Ti

Lins kins A

hr A Tsur Tsur = T∞

Ts

T∞ 1 ____

hA

The total thermal resistance needed to achieve the desired heat loss rate is found from Equation 3.19, Rtot

Ti T앝 (35 10) K 0.25 K/ W q 100 W

The total thermal resistance between the inside of the skin/fat layer and the cold surroundings includes conduction resistances for the skin/fat and insulation layers and an

3.1

123

The Plane Wall

䊏

effective resistance associated with convection and radiation, which act in parallel. Hence, Rtot

冢

Lsf L 1 1 ins ksf A kins A 1/hA 1/hr A

冣

1

1 A

冢Lk

sf

sf

L ins 1 kins h hr

冣

This equation can be solved for the insulation thickness.

Air The radiation heat transfer coefficient is approximated as having the same value as in Example 1.7: hr = 5.9 W/m2 䡠 K.

冤

Lins kins ARtot

冥

Lsf 1 ksf h hr

冤

冥

3 1 0.014 W/m 䡠 K 1.8 m2 0.25 K/W 3 10 m 0.3 W/m 䡠 K (2 5.9) W/m2 䡠 K

0.0044 m 4.4 mm

䉰

Water

冤

Lins kins ARtot

冥

Lsf 1 ksf h

冤

冥

3 1 0.014 W/m 䡠 K 1.8 m2 0.25 K/W 3 10 m 0.3 W/m 䡠 K 200 W/m2 䡠 K

䉰

0.0061 m 6.1 mm

These required thicknesses of insulation material can easily be incorporated into the snow and wet suits. The skin temperature can be calculated by considering conduction through the skin/fat layer: q

ksf A(Ti Ts) Lsf

or solving for Ts, Ts Ti

3 qLsf 35 C 100 W 3 10 m2 34.4 C ksfA 0.3 W/m 䡠 K 1.8 m

䉰

The skin temperature is the same in both cases because the heat loss rate and skin/fat properties are the same.

Comments: 1. The nanostructured silica aerogel is a porous material that is only about 5% solid. Its thermal conductivity is less than the thermal conductivity of the gas that fills its pores. As explained in Section 2.2, the reason for this seemingly impossible result is that the pore size is only around 20 nm, which reduces the mean free path of the gas and hence decreases its thermal conductivity.

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

2. By reducing the heat loss rate to 100 W, a person could remain in the cold environments indefinitely without becoming chilled. The skin temperature of 34.4 C would feel comfortable. 3. In the water case, the thermal resistance of the insulation dominates and all other resistances can be neglected. 4. The convection heat transfer coefficient associated with the air depends on the wind conditions, and it can vary over a broad range. As it changes, so will the outer surface temperature of the insulation layer. Since the radiation heat transfer coefficient depends on this temperature, it will also vary. We can perform a more complete analysis that takes this into account. The radiation heat transfer coefficient is given by Equation 1.9: 2 2 hr (Ts,o Tsur)(Ts,o Tsur )

(1)

Here Ts,o is the outer surface temperature of the insulation layer, which can be calculated from

冤kL A kL A冥

Ts,o Ti q

sf

ins

sf

(2)

ins

Since this depends on the insulation thickness, we also need the previous equation for Lins:

冢

Lins kins ARtot

Lsf 1 ksf h hr

冣

(3)

With all other values known, these three equations can be solved for the required insulation thickness. Using all the values from above, these equations have been solved for values of h in the range 0 h 100 W/m2 K, and the results are represented graphically. 7

6

Lins (mm)

124

5

4

3

0

10

20

30

40

50

60

70

80

90

100

h (W/m2•K)

Increasing h reduces the corresponding convection resistance, which then requires additional insulation to maintain the heat transfer rate at 100 W. Once the heat transfer coefficient exceeds approximately 60 W/m2 䡠 K, the convection resistance is negligible and further increases in h have little effect on the required insulation thickness.

3.1

䊏

125

The Plane Wall

The outer surface temperature and radiation heat transfer coefficient can also be calculated. As h increases from 0 to 100 W/m2 䡠 K, Ts,o decreases from 294 to 284 K, while hr decreases from 5.2 to 4.9 W/m2 䡠 K. The initial estimate of hr 5.9 W/m2 䡠 K was not highly accurate. Using this more complete model of the radiation heat transfer, with h 2 W/m2 䡠 K, the radiation heat transfer coefficient is 5.1 W/m2 K, and the required insulation thickness is 4.2 mm, close to the value calculated in the first part of the problem. 5. See Example 3.1 in IHT. This problem can also be solved using the thermal resistance network builder, Models/Resistance Networks, available in IHT.

EXAMPLE 3.2 A thin silicon chip and an 8-mm-thick aluminum substrate are separated by a 0.02-mm-thick epoxy joint. The chip and substrate are each 10 mm on a side, and their exposed surfaces are cooled by air, which is at a temperature of 25 C and provides a convection coefficient of 100 W/m2 䡠 K. If the chip dissipates 104 W/m2 under normal conditions, will it operate below a maximum allowable temperature of 85 C?

SOLUTION Known: Dimensions, heat dissipation, and maximum allowable temperature of a silicon chip. Thickness of aluminum substrate and epoxy joint. Convection conditions at exposed chip and substrate surfaces. Find:

Whether maximum allowable temperature is exceeded.

Schematic: Air

Silicon chip

q1"

T∞ = 25°C h = 100 W/m2•K q1"

q2" L = 8 mm

Aluminum substrate

T∞

h qc"

q"c Epoxy joint (0.02 mm)

_1_

Insulation

Tc R"t,c _L_

k _1_

h Air

T∞ = 25°C h = 100 W/m2•K

T∞ q2"

Assumptions: 1. Steady-state conditions. 2. One-dimensional conduction (negligible heat transfer from sides of composite). 3. Negligible chip thermal resistance (an isothermal chip). 4. Constant properties. 5. Negligible radiation exchange with surroundings.

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Properties: Table A.1, pure aluminum (T ⬃ 350 K): k 239 W/m 䡠 K. Analysis: Heat dissipated in the chip is transferred to the air directly from the exposed surface and indirectly through the joint and substrate. Performing an energy balance on a control surface about the chip, it follows that, on the basis of a unit surface area, qc q1 q2 or qc

Tc T앝 Tc T앝 (L/k) (1/h) (1/h) Rt,c

To conservatively estimate Tc, the maximum possible value of Rt,c 0.9 104 m2 䡠 K/W is obtained from Table 3.2. Hence

冤

Tc T앝 qc h

1 Rt,c (L/k) (1/h)

冥

1

or Tc 25 C 104 W/m2

冤

100

冥

1 (0.9 0.33 100) 104

1

m2 䡠 K/W

Tc 25 C 50.3 C 75.3 C

䉰

Hence the chip will operate below its maximum allowable temperature.

Comments: 1. The joint and substrate thermal resistances are much less than the convection resistance. The joint resistance would have to increase to the unrealistically large value of 50 104 m2 䡠 K/W, before the maximum allowable chip temperature would be exceeded. 2. The allowable power dissipation may be increased by increasing the convection coefficients, either by increasing the air velocity and/or by replacing the air with a more effective heat transfer fluid. Exploring this option for 100 h 2000 W/m2 䡠 K with Tc 85 C, the following results are obtained. 2.5

Tc = 85°C

2.0

q"c × 10–5 (W/m2)

126

1.5 1.0 0.5 0

0

500

1000

1500

2000

h (W/m2•K)

As h l 앝, q2 l 0 and virtually all of the chip power is transferred directly to the fluid stream.

3.1

䊏

127

The Plane Wall

3. As calculated, the difference between the air temperature (T앝 25 C) and the chip temperature (Tc 75.3 C) is 50.3 K. Keep in mind that this is a temperature difference and therefore is the same as 50.3 C. 4. Consider conditions for which airflow over the chip (upper) or substrate (lower) surface ceases due to a blockage in the air supply channel. If heat transfer from either surface is negligible, what are the resulting chip temperatures for qc 104 W/m2? [Answer, 126 C or 125 C]

EXAMPLE 3.3 A photovoltaic panel consists of (top to bottom) a 3-mm-thick ceria-doped glass (kg 1.4 W/m 䡠 K), a 0.1-mm-thick optical grade adhesive (ka 145 W/m 䡠 K), a very thin layer of silicon within which solar energy is converted to electrical energy, a 0.1-mm-thick solder layer (ksdr 50 W/m 䡠 K), and a 2-mm-thick aluminum nitride substrate (kan 120 W/m 䡠 K). The solar-to-electrical conversion efficiency within the silicon layer decreases with increasing silicon temperature, Tsi, and is described by the expression a – bTsi, where a 0.553 and b 0.001 K1. The temperature T is expressed in kelvins over the range 300 K Tsi 525 K. Of the incident solar irradiation, G 700 W/m2, 7% is reflected from the top surface of the glass, 10% is absorbed at the top surface of the glass, and 83% is transmitted to and absorbed within the silicon layer. Part of the solar irradiation absorbed in the silicon is converted to thermal energy, and the remainder is converted to electrical energy. The glass has an emissivity of 0.90, and the bottom as well as the sides of the panel are insulated. Determine the electric power P produced by an L 1-m-long, w 0.1-m-wide solar panel for conditions characterized by h 35 W/m2 䡠 K and T앝 Tsur 20 C.

Air

T∞ = 20°C h = 35 W/m2 •K

Tsur = 20°C G = 700 W/m2

Glass Adhesive Electric power to grid, P

Solder

Silicon layer

Substrate

Lg = 3 mm La = 0.1 mm Lan = 2 mm

Lsdr = 0.1 mm

L=1m

SOLUTION Known: Dimensions and materials of a photovoltaic solar panel. Material properties, solar irradiation, convection coefficient and ambient temperature, emissivity of top panel surface and surroundings temperature. Partitioning of the solar irradiation, and expression for the solar-to-electrical conversion efficiency. Find:

Electric power produced by the photovoltaic panel.

128

Chapter 3

One-Dimensional, Steady-State Conduction

䊏

Schematic: qrad

qconv Tsur

Solar irradiation G = 700 W/m2

Air

T∞ = 20°C h = 35 W/m2·K

Tsur = 20°C

1 hLw Tg,top Lg kg Lw Tg,bot La ka Lw 0.83ηGLw

0.10GLw

0.07G (reflected) 0.10G (absorbed at surface)

Glass

T

1 hr Lw

Lg = 3 mm

Adhesive Silicon layer

0.83GLw

0.83G (absorbed in silicon) La = 0.1 mm

Tsi

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer. 3. Constant properties. 4. Negligible thermal contact resistances. 5. Negligible temperature differences within the silicon layer. Analysis: Recognize that there is no heat transfer to the bottom insulated surface of the solar panel. Hence, the solder layer and aluminum nitride substrate do not affect the solution, and all of the solar energy absorbed by the panel must ultimately leave the panel in the form of radiation and convection heat transfer from the top surface of the glass, and electric power to the grid, P 0.83 GLw. Performing an energy balance on the node associated with the silicon layer yields 0.83 GLw 0.83 GLw

Tsi Tg,top Lg La kaLw kg Lw

Substituting the expression for the solar-to-electrical conversion efficiency and simplifying leads to 0.83 G(1 a bTsi)

Tsi Tg,top La L g ka kg

(1)

Performing a second energy balance on the node associated with the top surface of the glass gives 4 4 Tsur ) 0.83 GLw(1 ) 0.1 GLw hLw(Tg,top T앝) Lw(Tg,top

Substituting the expression for the solar-to-electrical conversion efficiency into the preceding equation and simplifying provides 4 4 Tsur ) 0.83 G(1 a bTsi) 0.1 G h(Tg,top T앝) (Tg,top

(2)

3.1

䊏

The Plane Wall

129

Finally, substituting known values into Equations 1 and 2 and solving simultaneously yields Tsi 307 K 34 C, providing a solar-to-electrical conversion efficiency of 0.553 – 0.001 K1 307 K 0.247. Hence, the power produced by the photovoltaic panel is P 0.83 GLw 0.247 0.83 700 W/m2 1 m 0.1 m 14.3 W

䉰

Comments: 1. The correct application of the conservation of energy requirement is crucial to determining the silicon temperature and the electric power. Note that solar energy is converted to both thermal and electrical energy, and the thermal circuit is used to quantify only the thermal energy transfer. 2. Because of the thermally insulated boundary condition, it is not necessary to include the solder or substrate layers in the analysis. This is because there is no conduction through these materials and, from Fourier’s law, there can be no temperature gradients within these materials. At steady state, Tsdr Tan Tsi. 3. As the convection coefficient increases, the temperature of the silicon decreases. This leads to a higher solar-to-electrical conversion efficiency and increased electric power output. Similarly, higher silicon temperatures and less power production are associated with smaller convection coefficients. For example, P 13.6 W and 14.6 W for h 15 W/m2 䡠 K and 55 W/m2 䡠 K, respectively. 4. The cost of a photovoltaic system can be reduced significantly by concentrating the solar energy onto the relatively expensive photovoltaic panel using inexpensive focusing mirrors or lenses. However, good thermal management then becomes even more important. For example, if the irradiation supplied to the panel were increased to G 7,000 W/m2 through concentration, the conversion efficiency drops to 0.160 as the silicon temperature increases to Tsi 119 C, even for h 55 W/m2 䡠 K. A key to reducing the cost of photovoltaic power generation is developing innovative cooling technologies for use in concentrating photovoltaic systems. 5. The simultaneous solution of Equations 1 and 2 may be achieved by using IHT, another commercial code, or a handheld calculator. A trial-and-error solution could also be obtained, but with considerable effort. Equations 1 and 2 could be combined to write a single transcendental expression for the silicon temperature, but the equation must still be solved numerically or by trial-and-error.

EXAMPLE 3.4 The thermal conductivity of a D 14-nm-diameter carbon nanotube is measured with an instrument that is fabricated of a wafer of silicon nitride at a temperature of T앝 300 K. The 20-m-long nanotube rests on two 0.5-m-thick, 10 m 10 m square islands that are separated by a distance s 5 m. A thin layer of platinum is used as an electrical resistor on the heated island (at temperature Th) to dissipate q 11.3 W of electrical power. On the sensing island, a similar layer of platinum is used to determine its temperature, Ts. The platinum’s electrical resistance, R(Ts) E/I, is found by measuring the voltage drop and electrical current across the platinum layer. The temperature of the sensing island, Ts, is then determined from the relationship of the platinum electrical resistance to its temperature.

130

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Each island is suspended by two Lsn 250-m-long silicon nitride beams that are wsn 3 m wide and tsn 0.5 m thick. A platinum line of width wpt 1 m and thickness tpt 0.2 m is deposited within each silicon nitride beam to power the heated island or to detect the voltage drop associated with the determination of Ts. The entire experiment is performed in a vacuum with Tsur 300 K and at steady state, Ts 308.4 K. Estimate the thermal conductivity of the carbon nanotube.

SOLUTION Known: Dimensions, heat dissipated at the heated island, and temperatures of the sensing island and surrounding silicon nitride wafer. Find:

The thermal conductivity of the carbon nanotube.

Schematic:

Tsur = 300 K

Carbon nanotube

D = 14 nm

Sensing island

Heated island

s = 5 µm

Sensing island Ts = 308.4 K Heated island

Th

s = 5 µm Lsn = 250 µm

10 µm 10 µm

tpt = 0.2 µm wpt = 1 µm

tsn = 0.5 µm wsn = 3 µm Silicon nitride block

T∞ = 300 K

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer. 3. The heated and sensing islands are isothermal. 4. Radiation exchange between the surfaces and the surroundings is negligible. 5. Negligible convection losses.

3.1

䊏

131

The Plane Wall

6. Ohmic heating in the platinum signal lines is negligible. 7. Constant properties. 8. Negligible contact resistance between the nanotube and the islands.

Properties: Table A.1, platinum (325 K, assumed): kpt 71.6 W/m 䡠 K. Table A.2, silicon nitride (325 K, assumed): ksn 15.5 W/m 䡠 K. Analysis: Energy that is dissipated at the heated island is transferred to the silicon nitride block through the support beams of the heated island, the carbon nanotube, and subsequently through the support beams of the sensing island. Therefore, the thermal circuit may be constructed as follows qh /2

qs /2

T∞

T∞

Rt,sup

q

Rt,sup

Th

s kcn Acn

Rt,sup

Ts Rt,sup

T∞

T∞

qh /2

qs /2

where each supporting beam provides a thermal resistance Rt,sup that is composed of a resistance due to the silicon nitride (sn) in parallel with a resistance due to the platinum (pt) line. The cross-sectional areas of the materials in the support beams are Apt wpttpt (1 106 m) (0.2 106 m) 2 1013 m2 Asn wsntsn Apt (3 106 m) (0.5 106 m) 2 1013 m2 1.3 1012 m2 while the cross-sectional area of the carbon nanotube is Acn D2/4 (14 109 m)2/4 1.54 1016 m2 The thermal resistance of each support is

冤 L k LA 冥 冤71.6 W/m 䡠 K 2 10 250 10 m

Rt,sup

kpt Apt pt

sn

sn

1

sn

6

13

冥

m2 15.5 W/m 䡠 K 1.3 1012 m2 250 106 m

7.25 106 K/W The combined heat loss through both sensing island supports is qs 2(Ts T )/Rt,sup 2 (308.4 K 300 K)/(7.25 106 K/W) 2.32 106 W 2.32 W

1

132

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

It follows that qh q qs 11.3 W 2.32 W 8.98 W and Th attains a value of Th T 1 qh Rt,sup 300 K 2

8.98 106 W 7.25 106 K/W 332.6 K 2

For the portion of the thermal circuit connecting Th and Ts, qs

Th Ts s/(kcn Acn)

from which kcn

qss 2.32 106 W 5 106 m Acn(Th Ts) 1.54 1016 m2 (332.6 K 308.4 K)

kcn 3113 W/m 䡠 K

䉰

Comments: 1. The measured thermal conductivity is extremely large, as evident by comparing its value to the thermal conductivities of pure metals shown in Figure 2.4. Carbon nanotubes might be used to dope otherwise low thermal conductivity materials to improve heat transfer. 2. Contact resistances between the carbon nanotube and the heated and sensing islands were neglected because little is known about such resistances at the nanoscale. However, if a contact resistance were included in the analysis, the measured thermal conductivity of the carbon nanotube would be even higher than the predicted value. 3. The significance of radiation heat transfer may be estimated by approximating the heated island as a blackbody radiating to Tsur from both its top and bottom surfaces. Hence, qrad,b ⬇ 5.67 108 W/m2 䡠 K4 2 (10 106 m)2 (332.64 3004)K4 4.7 108 W 0.047 W, and radiation is negligible.

3.2 An Alternative Conduction Analysis The conduction analysis of Section 3.1 was performed using the standard approach. That is, the heat equation was solved to obtain the temperature distribution, Equation 3.3, and Fourier’s law was then applied to obtain the heat transfer rate, Equation 3.4. However, an alternative approach may be used for the conditions presently of interest. Considering conduction in the system of Figure 3.6, we recognize that, for steady-state conditions with no heat generation and no heat loss from the sides, the heat transfer rate qx must be a constant independent of x. That is, for any differential element dx, qx qxdx. This condition is, of course, a consequence of the energy conservation requirement, and it must apply even if the area varies with position A(x) and the thermal conductivity varies with temperature k(T). Moreover, even though the temperature distribution may be two-dimensional, varying with x and y, it is often reasonable to neglect the y-variation and to assume a one-dimensional distribution in x. For the above conditions it is possible to work exclusively with Fourier’s law when performing a conduction analysis. In particular, since the conduction rate is a constant, the

3.2

䊏

133

An Alternative Conduction Analysis

Insulation

qx Adiabatic surface

T1

T0, A(x) z y

qx+dx

x1 x

x

qx

dx

FIGURE 3.6 System with a constant conduction heat transfer rate.

x0

rate equation may be integrated, even though neither the rate nor the temperature distribution is known. Consider Fourier’s law, Equation 2.1, which may be applied to the system of Figure 3.6. Although we may have no knowledge of the value of qx or the form of T(x), we do know that qx is a constant. Hence we may express Fourier’s law in the integral form qx

dx 冕 A(x) 冕 k(T ) dT x

T

x0

T0

(3.26)

The cross-sectional area may be a known function of x, and the material thermal conductivity may vary with temperature in a known manner. If the integration is performed from a point x0 at which the temperature T0 is known, the resulting equation provides the functional form of T(x). Moreover, if the temperature T T1 at some x x1 is also known, integration between x0 and x1 provides an expression from which qx may be computed. Note that, if the area A is uniform and k is independent of temperature, Equation 3.26 reduces to qx x k T A

(3.27)

where x x1 x0 and T T1 – T0. We frequently elect to solve diffusion problems by working with integrated forms of the diffusion rate equations. However, the limiting conditions for which this may be done should be firmly fixed in our minds: steady-state and one-dimensional transfer with no heat generation.

EXAMPLE 3.5 The diagram shows a conical section fabricated from pyroceram. It is of circular cross section with the diameter D ax, where a 0.25. The small end is at x1 50 mm and the large end at x2 250 mm. The end temperatures are T1 400 K and T2 600 K, while the lateral surface is well insulated. T2 T1

x1 x2

x

134

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

1. Derive an expression for the temperature distribution T(x) in symbolic form, assuming one-dimensional conditions. Sketch the temperature distribution. 2. Calculate the heat rate qx through the cone.

SOLUTION Known: Conduction in a circular conical section having a diameter D ax, where a 0.25. Find: 1. Temperature distribution T(x). 2. Heat transfer rate qx. Schematic: T2 = 600 K T1 = 400 K qx

x1 = 0.05 m x2 = 0.25 m x Pyroceram

Assumptions: 1. Steady-state conditions. 2. One-dimensional conduction in the x-direction. 3. No internal heat generation. 4. Constant properties. Properties: Table A.2, pyroceram (500 K): k 3.46 W/m 䡠 K. Analysis: 1. Since heat conduction occurs under steady-state, one-dimensional conditions with no internal heat generation, the heat transfer rate qx is a constant independent of x. Accordingly, Fourier’s law, Equation 2.1, may be used to determine the temperature distribution qx kA dT dx where A D2/4 a2x2/4. Separating variables, 4qxdx kdT a2x2 Integrating from x1 to any x within the cone, and recalling that qx and k are constants, it follows that 4qx a2

冕 dxx k冕 dT x

x1

T

2

T1

3.2

䊏

135

An Alternative Conduction Analysis

Hence

冢

冣

4qx 1x x1 k(T T1) 1 a2 or solving for T T(x) T1

冢

4qx 1 1 a2k x1 x

冣

Although qx is a constant, it is as yet an unknown. However, it may be determined by evaluating the above expression at x x2, where T(x2) T2. Hence T2 T1

冢

4qx 1 1 a2k x1 x2

冣

and solving for qx qx

a2k(T1 T2) 4[(1/x1) (1/x2)]

Substituting for qx into the expression for T(x), the temperature distribution becomes T(x) T1 (T1 T2)

(1/x) (1/x ) 冤(1/x ) (1/x )冥 1

1

䉰

2

From this result, temperature may be calculated as a function of x and the distribution is as shown.

T(x)

T2

T1 x2

x1 x

Note that, since dT/dx – 4qx/ka2x2 from Fourier’s law, it follows that the temperature gradient and heat flux decrease with increasing x. 2. Substituting numerical values into the foregoing result for the heat transfer rate, it follows that qx

(0.25)2 3.46 W/m 䡠 K (400 600) K 2.12 W 4 (1/0.05 m 1/0.25 m)

䉰

Comments: When the parameter a increases, the cross-sectional area changes more rapidly with distance, causing the one-dimensional assumption to become less appropriate.

136

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

3.3 Radial Systems Cylindrical and spherical systems often experience temperature gradients in the radial direction only and may therefore be treated as one-dimensional. Moreover, under steady-state conditions with no heat generation, such systems may be analyzed by using the standard method, which begins with the appropriate form of the heat equation, or the alternative method, which begins with the appropriate form of Fourier’s law. In this section, the cylindrical system is analyzed by means of the standard method and the spherical system by means of the alternative method.

3.3.1

The Cylinder

A common example is the hollow cylinder whose inner and outer surfaces are exposed to fluids at different temperatures (Figure 3.7). For steady-state conditions with no heat generation, the appropriate form of the heat equation, Equation 2.26, is

冢

冣

1 d kr dT 0 r dr dr

(3.28)

where, for the moment, k is treated as a variable. The physical significance of this result becomes evident if we also consider the appropriate form of Fourier’s law. The rate at which energy is conducted across any cylindrical surface in the solid may be expressed as qr kA dT k(2rL) dT dr dr

(3.29)

where A 2rL is the area normal to the direction of heat transfer. Since Equation 3.28 dictates that the quantity kr(dT/dr) is independent of r, it follows from Equation 3.29 that the conduction heat transfer rate qr (not the heat flux qr ) is a constant in the radial direction. Hot fluid T∞,1, h1 Cold fluid T∞,2, h2

Ts,1 r

Ts,2 Ts,1

r1

r2 r

r1

L r2 qr

Ts,2

FIGURE 3.7

T∞,1

Ts,1

________ 1

h12 π r1L

Ts,2 In( r2/r1) ________ 2 π kL

Hollow cylinder with convective surface conditions.

T∞,2

________ 1

h22 π r2L

3.3

䊏

137

Radial Systems

We may determine the temperature distribution in the cylinder by solving Equation 3.28 and applying appropriate boundary conditions. Assuming the value of k to be constant, Equation 3.28 may be integrated twice to obtain the general solution T(r) C1 ln r C2

(3.30)

To obtain the constants of integration C1 and C2, we introduce the following boundary conditions: T(r1) Ts,1

and

T(r2) Ts,2

Applying these conditions to the general solution, we then obtain Ts,1 C1 ln r1 C2

and

Ts,2 C1 ln r2 C2

Solving for C1 and C2 and substituting into the general solution, we then obtain T(r)

冢冣

Ts,1 Ts,2 ln rr Ts,2 2 ln (r1 /r2)

(3.31)

Note that the temperature distribution associated with radial conduction through a cylindrical wall is logarithmic, not linear, as it is for the plane wall under the same conditions. The logarithmic distribution is sketched in the inset of Figure 3.7. If the temperature distribution, Equation 3.31, is now used with Fourier’s law, Equation 3.29, we obtain the following expression for the heat transfer rate: qr

2Lk(Ts,1 Ts,2) ln (r2 /r1)

(3.32)

From this result it is evident that, for radial conduction in a cylindrical wall, the thermal resistance is of the form Rt,cond

ln (r2 /r1) 2Lk

(3.33)

This resistance is shown in the series circuit of Figure 3.7. Note that since the value of qr is independent of r, the foregoing result could have been obtained by using the alternative method, that is, by integrating Equation 3.29. Consider now the composite system of Figure 3.8. Recalling how we treated the composite plane wall and neglecting the interfacial contact resistances, the heat transfer rate may be expressed as qr

T앝,1 T앝,4 ln (r2 /r1) ln (r3 /r2) ln (r4 /r3) 1 1 2r1Lh1 2kAL 2kBL 2kCL 2r4Lh4

(3.34)

The foregoing result may also be expressed in terms of an overall heat transfer coefficient. That is, qr

T앝,1 T앝,4 UA(T앝,1 T앝,4) Rtot

(3.35)

138

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Ts,4

T∞,4, h4

T∞,1, h1

T3 T2 Ts,1

r1

r2

r3 r4

L

T∞,1, h1

A

B

C

T∞,4, h4 T∞,1

Ts,1 T2 T3 Ts,4

qr

T∞,1

Ts,1 In(r2/r1) _________ 2π kAL

1 __________ h12 π r1L

FIGURE 3.8

T2

T3

In(r3/r2) _________ 2π kBL

Ts,4 In(r4/r3) _________ 2π kCL

T∞,4 T∞,4 1 __________ h42 π r4L

Temperature distribution for a composite cylindrical wall.

If U is defined in terms of the inside area, A1 2r1L, Equations 3.34 and 3.35 may be equated to yield U1

1 r r r 1 1 ln 2 1 ln r3 r1 ln r4 r1 1 h1 kA r1 kB r2 kC r3 r4 h4

(3.36)

This definition is arbitrary, and the overall coefficient may also be defined in terms of A4 or any of the intermediate areas. Note that U1A1 U2A2 U3A3 U4A4 (Rt)1

(3.37)

and the specific forms of U2, U3, and U4 may be inferred from Equations 3.34 and 3.35.

EXAMPLE 3.6 The possible existence of an optimum insulation thickness for radial systems is suggested by the presence of competing effects associated with an increase in this thickness. In particular, although the conduction resistance increases with the addition of insulation, the convection resistance decreases due to increasing outer surface area. Hence there may exist an insulation thickness that minimizes heat loss by maximizing the total resistance to heat transfer. Resolve this issue by considering the following system.

3.3

䊏

139

Radial Systems

1. A thin-walled copper tube of radius ri is used to transport a low-temperature refrigerant and is at a temperature Ti that is less than that of the ambient air at T앝 around the tube. Is there an optimum thickness associated with application of insulation to the tube? 2. Confirm the above result by computing the total thermal resistance per unit length of tube for a 10-mm-diameter tube having the following insulation thicknesses: 0, 2, 5, 10, 20, and 40 mm. The insulation is composed of cellular glass, and the outer surface convection coefficient is 5 W/m2 䡠 K.

SOLUTION Known: Radius ri and temperature Ti of a thin-walled copper tube to be insulated from the ambient air. Find: 1. Whether there exists an optimum insulation thickness that minimizes the heat transfer rate. 2. Thermal resistance associated with using cellular glass insulation of varying thickness. Schematic: T∞ h = 5 W/m2•K r ri Air

Ti Insulation, k

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer in the radial (cylindrical) direction. 3. Negligible tube wall thermal resistance. 4. Constant properties for insulation. 5. Negligible radiation exchange between insulation outer surface and surroundings. Properties: Table A.3, cellular glass (285 K, assumed): k 0.055 W/m 䡠 K. Analysis: 1. The resistance to heat transfer between the refrigerant and the air is dominated by conduction in the insulation and convection in the air. The thermal circuit is therefore q'

Ti

T∞ In(r/ri) ________ 2π k

1 _______ 2 π rh

where the conduction and convection resistances per unit length follow from Equations 3.33 and 3.9, respectively. The total thermal resistance per unit length of tube is then Rtot

ln (r/ri ) 1 2k 2rh

140

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

where the rate of heat transfer per unit length of tube is q

T앝 Ti Rtot

An optimum insulation thickness would be associated with the value of r that minimized q or maximized Rtot. Such a value could be obtained from the requirement that dRtot 0 dr Hence 1 1 0 2kr 2r 2h or rk h To determine whether the foregoing result maximizes or minimizes the total resistance, the second derivative must be evaluated. Hence d 2Rtot 1 2 13 dr 2 2kr r h or, at r k/h,

冢

冣

d 2Rtot 1 1 2 1 1 0 dr 2 (k /h) k 2k 2k 3/h2 Since this result is always positive, it follows that r k/h is the insulation radius for which the total resistance is a minimum, not a maximum. Hence an optimum insulation thickness does not exist. From the above result it makes more sense to think in terms of a critical insulation radius rcr ⬅ k h which maximizes heat transfer, that is, below which q increases with increasing r and above which q decreases with increasing r. 2. With h 5 W/m2 䡠 K and k 0.055 W/m 䡠 K, the critical radius is 䡠 K 0.011 m rcr 0.055 W/m 2 5 W/m 䡠 K Hence rcr ri and heat transfer will increase with the addition of insulation up to a thickness of rcr ri (0.011 0.005) m 0.006 m

3.3

䊏

141

Radial Systems

The thermal resistances corresponding to the prescribed insulation thicknesses may be calculated and are plotted as follows: 8

R'tot

R't (m•K/W)

6

R'cond

4

R'conv 2

0

0

6

10

20

30

40

50

r – ri (mm)

Comments: 1. The effect of the critical radius is revealed by the fact that, even for 20 mm of insulation, the total resistance is not as large as the value for no insulation. 2. If ri rcr, as it is in this case, the total resistance decreases and the heat rate therefore increases with the addition of insulation. This trend continues until the outer radius of the insulation corresponds to the critical radius. The trend is desirable for electrical current flow through a wire, since the addition of electrical insulation would aid in transferring heat dissipated in the wire to the surroundings. Conversely, if ri rcr, any addition of insulation would increase the total resistance and therefore decrease the heat loss. This behavior would be desirable for steam flow through a pipe, where insulation is added to reduce heat loss to the surroundings. 3. For radial systems, the problem of reducing the total resistance through the application of insulation exists only for small diameter wires or tubes and for small convection coefficients, such that rcr ri. For a typical insulation (k ⬇ 0.03 W/m 䡠 K) and free convection in air (h ⬇ 10 W/m2 䡠 K), rcr (k/h) ⬇ 0.003 m. Such a small value tells us that, normally, ri rcr and we need not be concerned with the effects of a critical radius. 4. The existence of a critical radius requires that the heat transfer area change in the direction of transfer, as for radial conduction in a cylinder (or a sphere). In a plane wall the area perpendicular to the direction of heat flow is constant and there is no critical insulation thickness (the total resistance always increases with increasing insulation thickness).

3.3.2

The Sphere

Now consider applying the alternative method to analyzing conduction in the hollow sphere of Figure 3.9. For the differential control volume of the figure, energy conservation requires that qr qrdr for steady-state, one-dimensional conditions with no heat generation. The appropriate form of Fourier’s law is qr kA dT k(4r 2) dT dr dr 2 where A 4r is the area normal to the direction of heat transfer.

(3.38)

142

Chapter 3

r1

䊏

One-Dimensional, Steady-State Conduction

r

qr

r2

qr + dr

Ts, 2 Ts, 1

dr

FIGURE 3.9 Conduction in a spherical shell.

Acknowledging that qr is a constant, independent of r, Equation 3.38 may be expressed in the integral form qr 4

冕

r2

r1

冕

dr r2

Ts,2

k(T) dT

(3.39)

Ts,1

Assuming constant k, we then obtain qr

4k(Ts,1 Ts,2) (1/r1) (1/r2)

(3.40)

Remembering that the thermal resistance is defined as the temperature difference divided by the heat transfer rate, we obtain

冢

Rt,cond 1 r1 r1 2 4k 1

冣

(3.41)

Note that the temperature distribution and Equations 3.40 and 3.41 could have been obtained by using the standard approach, which begins with the appropriate form of the heat equation. Spherical composites may be treated in much the same way as composite walls and cylinders, where appropriate forms of the total resistance and overall heat transfer coefficient may be determined.

3.4 Summary of One-Dimensional Conduction Results Many important problems are characterized by one-dimensional, steady-state conduction in plane, cylindrical, or spherical walls without thermal energy generation. Key results for these three geometries are summarized in Table 3.3, where T refers to the temperature difference, Ts,1 Ts,2, between the inner and outer surfaces identified in Figures 3.1, 3.7, and 3.9. In each case, beginning with the heat equation, you should be able to derive the corresponding expressions for the temperature distribution, heat flux, heat rate, and thermal resistance.

3.5 Conduction with Thermal Energy Generation In the preceding section we considered conduction problems for which the temperature distribution in a medium was determined solely by conditions at the boundaries of the medium. We now want to consider the additional effect on the temperature distribution of processes that may be occurring within the medium. In particular, we wish to consider situations for which thermal energy is being generated due to conversion from some other energy form.

3.5

䊏

143

Conduction with Thermal Energy Generation

TABLE 3.3 One-dimensional, steady-state solutions to the heat equation with no generation

Heat equation Temperature distribution

Plane Wall

Cylindrical Walla

Spherical Walla

d 2T 0 dx2

dT 1 d r dr r dr 0

冢 冣

1 d 2 dT r 0 dr r 2 dr

Ts,1 T

Ts, 2 T

T L

Heat flux (q⬙)

k

Heat rate (q)

kA

Thermal resistance (Rt,cond)

x L

T L

L kA

ln (r/r2) ln (r1/r2)

Ts,1

冢 冣 1 (r /r) T 冤 1 (r /r )冥 1

1

2

k T r ln (r2 /r1)

k T r 2[(1/r1) (1/r2)]

2Lk T ln (r2 /r1)

4k T (1/r1) (1/r2)

ln (r2 /r1) 2Lk

(1/r1) (1/r2) 4 k

The critical radius of insulation is rcr k/h for the cylinder and rcr 2k/h for the sphere.

a

A common thermal energy generation process involves the conversion from electrical to thermal energy in a current-carrying medium (Ohmic, or resistance, or Joule heating). The rate at which energy is generated by passing a current I through a medium of electrical resistance Re is E˙g I 2Re

(3.42)

If this power generation (W) occurs uniformly throughout the medium of volume V, the volumetric generation rate (W/m3) is then q˙ ⬅

E˙ g V

I 2Re V

(3.43)

Energy generation may also occur as a result of the deceleration and absorption of neutrons in the fuel element of a nuclear reactor or exothermic chemical reactions occurring within a medium. Endothermic reactions would, of course, have the inverse effect (a thermal energy sink) of converting thermal energy to chemical bonding energy. Finally, a conversion from electromagnetic to thermal energy may occur due to the absorption of radiation within the medium. The process occurs, for example, when gamma rays are absorbed in external nuclear reactor components (cladding, thermal shields, pressure vessels, etc.) or when visible radiation is absorbed in a semitransparent medium. Remember not to confuse energy generation with energy storage (Section 1.3.1).

3.5.1

The Plane Wall

Consider the plane wall of Figure 3.10a, in which there is uniform energy generation per unit volume (q˙ is constant) and the surfaces are maintained at Ts,1 and Ts,2. For constant thermal conductivity k, the appropriate form of the heat equation, Equation 2.22, is d 2T q˙ 0 dx2 k

(3.44)

144

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

The general solution is T

q˙ 2 x C1x C2 2k

(3.45)

where C1 and C2 are the constants of integration. For the prescribed boundary conditions, T(L) Ts,1

T(L) Ts,2

and

The constants may be evaluated and are of the form C1

Ts,2 Ts,1 2L

q˙ 2 Ts,1 Ts,2 L 2k 2

C2

and

in which case the temperature distribution is T(x)

冢

冣

2 Ts,2 Ts,1 x Ts,1 Ts,2 q˙L2 1 x2 2k 2 L 2 L

(3.46)

The heat flux at any point in the wall may, of course, be determined by using Equation 3.46 with Fourier’s law. Note, however, that with generation the heat flux is no longer independent of x. The preceding result simplifies when both surfaces are maintained at a common temperature, Ts,1 Ts,2 ⬅ Ts. The temperature distribution is then symmetrical about the midplane, Figure 3.10b, and is given by T(x)

冢

冣

2 q˙L2 1 x 2 Ts 2k L

(3.47)

x –L

x +L

–L

q•

+L

T0

T(x)

Ts,1

q• T(x)

Ts

Ts

T∞,1,h1

T∞ ,h (a)

q"conv

q"cond

Ts,2

T∞, h

T∞,2,h2

(b )

q• T0 T(x) Ts q"conv

q"cond

T∞, h (c)

FIGURE 3.10 Conduction in a plane wall with uniform heat generation. (a) Asymmetrical boundary conditions. (b) Symmetrical boundary conditions. (c) Adiabatic surface at midplane.

3.5

䊏

145

Conduction with Thermal Energy Generation

The maximum temperature exists at the midplane T(0) ⬅ T0

q˙L2 Ts 2k

(3.48)

in which case the temperature distribution, Equation 3.47, may be expressed as

冢冣

T(x) T0 x Ts T0 L

2

(3.49)

It is important to note that at the plane of symmetry in Figure 3.10b, the temperature gradient is zero, (dT/dx)x0 0. Accordingly, there is no heat transfer across this plane, and it may be represented by the adiabatic surface shown in Figure 3.10c. One implication of this result is that Equation 3.47 also applies to plane walls that are perfectly insulated on one side (x 0) and maintained at a fixed temperature Ts on the other side (x L). To use the foregoing results, the surface temperature(s) Ts must be known. However, a common situation is one for which it is the temperature of an adjoining fluid, T앝, and not Ts, which is known. It then becomes necessary to relate Ts to T앝. This relation may be developed by applying a surface energy balance. Consider the surface at x L for the symmetrical plane wall (Figure 3.10b) or the insulated plane wall (Figure 3.10c). Neglecting radiation and substituting the appropriate rate equations, the energy balance given by Equation 1.13 reduces to k dT dx

冏

xL

h(Ts T앝)

(3.50)

Substituting from Equation 3.47 to obtain the temperature gradient at x L, it follows that Ts T앝

q˙L h

(3.51)

Hence Ts may be computed from knowledge of T앝, q˙ , L, and h. Equation 3.51 may also be obtained by applying an overall energy balance to the plane wall of Figure 3.10b or 3.10c. For example, relative to a control surface about the wall of Figure 3.10c, the rate at which energy is generated within the wall must be balanced by the rate at which energy leaves via convection at the boundary. Equation 1.12c reduces to E˙g E˙out

(3.52)

q˙L h(Ts T앝)

(3.53)

or, for a unit surface area,

Solving for Ts, Equation 3.51 is obtained. Equation 3.51 may be combined with Equation 3.47 to eliminate Ts from the temperature distribution, which is then expressed in terms of the known quantities q˙, L, k, h, and T앝. The same result may be obtained directly by using Equation 3.50 as a boundary condition to evaluate the constants of integration appearing in Equation 3.45.

EXAMPLE 3.7 A plane wall is a composite of two materials, A and B. The wall of material A has uniform heat generation q˙ 1.5 106 W/m3, kA 75 W/m K, and thickness LA 50 mm. The

146

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

wall material B has no generation with kB 150 W/m 䡠 K and thickness LB 20 mm. The inner surface of material A is well insulated, while the outer surface of material B is cooled by a water stream with T앝 30 C and h 1000 W/m2 䡠 K. 1. Sketch the temperature distribution that exists in the composite under steady-state conditions. 2. Determine the temperature T0 of the insulated surface and the temperature T2 of the cooled surface.

SOLUTION Known: Plane wall of material A with internal heat generation is insulated on one side and bounded by a second wall of material B, which is without heat generation and is subjected to convection cooling. Find: 1. Sketch of steady-state temperature distribution in the composite. 2. Inner and outer surface temperatures of the composite. Schematic: T0

T1

T2 T∞ = 30°C h = 1000 W/m2•K

Insulation

q•A = 1.5 × 106 W/m3 kA = 75 W/m•K

q"

A

LA = 50 mm x

B

LB = 20 mm

Water

kB = 150 W/m•K q• B = 0

Assumptions: 1. Steady-state conditions. 2. One-dimensional conduction in x-direction. 3. Negligible contact resistance between walls. 4. Inner surface of A adiabatic. 5. Constant properties for materials A and B. Analysis: 1. From the prescribed physical conditions, the temperature distribution in the composite is known to have the following features, as shown: (a) Parabolic in material A. (b) Zero slope at insulated boundary. (c) Linear in material B. (d) Slope change kB/kA 2 at interface.

3.5

䊏

147

Conduction with Thermal Energy Generation

The temperature distribution in the water is characterized by (e) Large gradients near the surface. b

a

T(x)

T0

d

c

T1 T2

e A

T∞

B

LA

0

LA + LB x

2. The outer surface temperature T2 may be obtained by performing an energy balance on a control volume about material B. Since there is no generation in this material, it follows that, for steady-state conditions and a unit surface area, the heat flux into the material at x LA must equal the heat flux from the material due to convection at x LA LB. Hence q h(T2 T )

(1)

The heat flux q may be determined by performing a second energy balance on a control volume about material A. In particular, since the surface at x 0 is adiabatic, there is no inflow and the rate at which energy is generated must equal the outflow. Accordingly, for a unit surface area, q˙LA q

(2)

Combining Equations 1 and 2, the outer surface temperature is T2 T

q˙LA h

T2 30 C 1.5 10 W/m 2 0.05 m 105 C 1000 W/m 䡠 K 6

3

䉰

From Equation 3.48 the temperature at the insulated surface is T0

q˙L2A T1 2kA

(3)

where T1 may be obtained from the following thermal circuit: q''

T1

T2 Rcond, '' B

T∞ Rconv ''

That is, T1 T (Rcond,B Rconv) q where the resistances for a unit surface area are Rcond, B

LB kB

Rconv 1 h

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Hence, T1 30 C

m 1 冢1500.02 W/m 䡠 K 1000 W/m 䡠 K冣 2

1.5 106 W/m3 0.05 m T1 30 C 85 C 115 C Substituting into Equation 3, 1.5 106 W/m3 (0.05 m)2 115 C 2 75 W/m 䡠 K T0 25oC 115oC 140oC

T0

䉰

Comments: 1. Material A, having heat generation, cannot be represented by a thermal circuit element. 2. Since the resistance to heat transfer by convection is significantly larger than that due to conduction in material B, Rconv/Rcond 7.5, the surface-to-fluid temperature difference is much larger than the temperature drop across material B, (T2 – T앝)/(T1 – T2) 7.5. This result is consistent with the temperature distribution plotted in part 1. 3. The surface and interface temperatures (T0, T1, and T2) depend on the generation rate q˙, the thermal conductivities kA and kB, and the convection coefficient h. Each material will have a maximum allowable operating temperature, which must not be exceeded if thermal failure of the system is to be avoided. We explore the effect of one of these parameters by computing and plotting temperature distributions for values of h 200 and 1000 W/m2 䡠 K, which would be representative of air and liquid cooling, respectively. 450

440

h = 200 W/m2•K

T (°C)

430

420

410

400 0

10

20

30

40

50

60

70

60

70

x (mm) 150

140

h = 1000 W/m2•K

130

T (°C)

148

120

110

100 0

10

20

30

40

x (mm)

50

3.5

Conduction with Thermal Energy Generation

䊏

149

For h 200 W/m2 䡠 K, there is a significant increase in temperature throughout the system and, depending on the selection of materials, thermal failure could be a problem. Note the slight discontinuity in the temperature gradient, dT/dx, at x 50 mm. What is the physical basis for this discontinuity? We have assumed negligible contact resistance at this location. What would be the effect of such a resistance on the temperature distribution throughout the system? Sketch a representative distribution. What would be the effect on the temperature distribution of an increase in q˙, kA, or kB? Qualitatively sketch the effect of such changes on the temperature distribution. 4. This example is solved in the Advanced section of IHT.

3.5.2

Radial Systems

Heat generation may occur in a variety of radial geometries. Consider the long, solid cylinder of Figure 3.11, which could represent a current-carrying wire or a fuel element in a nuclear reactor. For steady-state conditions, the rate at which heat is generated within the cylinder must equal the rate at which heat is convected from the surface of the cylinder to a moving fluid. This condition allows the surface temperature to be maintained at a fixed value of Ts. To determine the temperature distribution in the cylinder, we begin with the appropriate form of the heat equation. For constant thermal conductivity k, Equation 2.26 reduces to

冢 冣

1 d r dT q˙ 0 r dr dr k

(3.54)

Separating variables and assuming uniform generation, this expression may be integrated to obtain q˙ r dT r2 C1 dr 2k

(3.55)

Repeating the procedure, the general solution for the temperature distribution becomes T(r)

q˙ 2 r C1 ln r C2 4k

(3.56)

Cold fluid

T∞, h

qr

Ts

q• L

r ro

FIGURE 3.11 Conduction in a solid cylinder with uniform heat generation.

150

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

To obtain the constants of integration C1 and C2, we apply the boundary conditions dT dr

冏

0

T(r0) Ts

and

r0

The first condition results from the symmetry of the situation. That is, for the solid cylinder the centerline is a line of symmetry for the temperature distribution and the temperature gradient must be zero. Recall that similar conditions existed at the midplane of a wall having symmetrical boundary conditions (Figure 3.10b). From the symmetry condition at r 0 and Equation 3.55, it is evident that C1 0. Using the surface boundary condition at r ro with Equation 3.56, we then obtain C2 Ts

q˙ 2 r 4k o

(3.57)

冣

(3.58)

The temperature distribution is therefore T(r)

冢

2 q˙ro2 1 r 2 Ts 4k ro

Evaluating Equation 3.58 at the centerline and dividing the result into Equation 3.58, we obtain the temperature distribution in nondimensional form,

冢冣

T(r) Ts 1 rr o To Ts

2

(3.59)

where To is the centerline temperature. The heat rate at any radius in the cylinder may, of course, be evaluated by using Equation 3.58 with Fourier’s law. To relate the surface temperature, Ts, to the temperature of the cold fluid T앝, either a surface energy balance or an overall energy balance may be used. Choosing the second approach, we obtain q˙(ro2L) h(2ro L)(Ts T앝) or Ts T앝

3.5.3

q˙ro 2h

(3.60)

Tabulated Solutions

Appendix C provides a convenient and systematic procedure for treating the different combinations of surface conditions that may be applied to one-dimensional planar and radial (cylindrical and spherical) geometries with uniform thermal energy generation. From the tabulated results of this appendix, it is a simple matter to obtain distributions of the temperature, heat flux, and heat rate for boundary conditions of the second kind (a uniform surface heat flux) and the third kind (a surface heat flux that is proportional to a convection coefficient h or the overall heat transfer coefficient U). You are encouraged to become familiar with the contents of the appendix.

3.5.4

Application of Resistance Concepts

We conclude our discussion of heat generation effects with a word of caution. In particular, when such effects are present, the heat transfer rate is not a constant, independent of the

3.5

䊏

151

Conduction with Thermal Energy Generation

spatial coordinate. Accordingly, it would be incorrect to use the conduction resistance concepts and the related heat rate equations developed in Sections 3.1 and 3.3.

EXAMPLE 3.8 Consider a long solid tube, insulated at the outer radius r2 and cooled at the inner radius r1, with uniform heat generation q˙ (W/m3) within the solid. 1. Obtain the general solution for the temperature distribution in the tube. 2. In a practical application a limit would be placed on the maximum temperature that is permissible at the insulated surface (r r2). Specifying this limit as Ts,2, identify appropriate boundary conditions that could be used to determine the arbitrary constants appearing in the general solution. Determine these constants and the corresponding form of the temperature distribution. 3. Determine the heat removal rate per unit length of tube. 4. If the coolant is available at a temperature T앝, obtain an expression for the convection coefficient that would have to be maintained at the inner surface to allow for operation at prescribed values of Ts,2 and q˙ .

SOLUTION Known: Solid tube with uniform heat generation is insulated at the outer surface and cooled at the inner surface. Find: 1. General solution for the temperature distribution T(r). 2. Appropriate boundary conditions and the corresponding form of the temperature distribution. 3. Heat removal rate for specified maximum temperature. 4. Corresponding required convection coefficient at the inner surface. Schematic:

Ts,2 Ts,1

q'conv

q'cond

r1 T∞, h

r2

q•, k

Coolant

T∞, h

Insulation

152

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Assumptions: 1. Steady-state conditions. 2. One-dimensional radial conduction. 3. Constant properties. 4. Uniform volumetric heat generation. 5. Outer surface adiabatic. Analysis: 1. To determine T(r), the appropriate form of the heat equation, Equation 2.26, must be solved. For the prescribed conditions, this expression reduces to Equation 3.54, and the general solution is given by Equation 3.56. Hence, this solution applies in a cylindrical shell, as well as in a solid cylinder (Figure 3.11). 2. Two boundary conditions are needed to evaluate C1 and C2, and in this problem it is appropriate to specify both conditions at r2. Invoking the prescribed temperature limit, T(r2) Ts,2

(1)

and applying Fourier’s law, Equation 3.29, at the adiabatic outer surface dT dr

冏

0

(2)

r2

Using Equations 3.56 and 1, it follows that Ts,2

q˙ 2 r C1 ln r2 C2 4k 2

(3)

Similarly, from Equations 3.55 and 2 0

q˙ 2 r C1 2k 2

(4)

q˙ 2 r 2k 2

(5)

Hence, from Equation 4, C1 and from Equation 3 C2 Ts,2

q˙ 2 q˙ 2 r r ln r2 4k 2 2k 2

(6)

Substituting Equations 5 and 6 into the general solution, Equation 3.56, it follows that T(r) Ts,2

r q˙ 2 q˙ (r r2) r 22 ln r2 4k 2 2k

(7)

3.5

䊏

153

Conduction with Thermal Energy Generation

3. The heat removal rate may be determined by obtaining the conduction rate at r1 or by evaluating the total generation rate for the tube. From Fourier’s law qr k2r dT dr Hence, substituting from Equation 7 and evaluating the result at r1,

冢

qr(r1) k2r1

冣

q˙ q˙ r 2 r1 r2 q˙(r22 r12) 2k 2k 1

(8)

Alternatively, because the tube is insulated at r2, the rate at which heat is generated in the tube must equal the rate of removal at r1. That is, for a control volume about the tube, the energy conservation requirement, Equation 1.12c, reduces to E˙ g E˙ out 0, where E˙ g q˙(r22 r12)L and E˙out qcond L qr(r1)L. Hence qr(r1) q˙(r22 r12)

(9)

4. Applying the energy conservation requirement, Equation 1.13, to the inner surface, it follows that qcond qconv or

q˙(r22 r12) h2r1(Ts,1 T ) Hence h

q˙(r22 r12) 2r1(Ts,1 T앝)

(10)

where Ts,1 may be obtained by evaluating Equation 7 at r r1.

Comments: 1. Note that, through application of Fourier’s law in part 3, the sign on qr(r1) was found to be negative, Equation 8, implying that heat flow is in the negative r-direction. However, in applying the energy balance, we acknowledged that heat flow was out of the wall. Hence we expressed qcond as qr(r1) and we expressed qconv in terms of (Ts,1 – T앝), rather than (T앝 – Ts,1). 2. Results of the foregoing analysis may be used to determine the convection coefficient required to maintain the maximum tube temperature Ts,2 below a prescribed value. Consider a tube of thermal conductivity k 5 W/m K and inner and outer radii of r1 20 mm and r2 25 mm, respectively, with a maximum allowable temperature of Ts,2 350 C. The tube experiences heat generation at a rate of q· 5 106 W/m3, and the coolant is at a temperature of T앝 80 C. Obtaining T(r1) Ts,1 336.5 C from Equation 7 and substituting into Equation 10, the required convection coefficient is found to be h 110 W/m2 䡠 K. Using the IHT Workspace, parametric calculations may be performed to determine the effects of the convection coefficient and the generation rate on the maximum tube temperature, and results are plotted as a function of h for three values of q·.

154

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Maximum tube temperature, Ts,2 (°C)

500 •

q × 10–6 (W/m3) 7.5 5.0 2.5

400

300

200

100 20

60 100 140 Convection coefficient, h (W/m2•K)

180

For each generation rate, the minimum value of h needed to maintain Ts,2 350 C may be determined from the figure. 3. The temperature distribution, Equation 7, may also be obtained by using the results of Appendix C. Applying a surface energy balance at r r1, with q(r) q˙(r22 r21)L, (Ts,2 Ts,1) may be determined from Equation C.8 and the result substituted into Equation C.2 to eliminate Ts,1 and obtain the desired expression.

3.6 Heat Transfer from Extended Surfaces The term extended surface is commonly used to depict an important special case involving heat transfer by conduction within a solid and heat transfer by convection (and/or radiation) from the boundaries of the solid. Until now, we have considered heat transfer from the boundaries of a solid to be in the same direction as heat transfer by conduction in the solid. In contrast, for an extended surface, the direction of heat transfer from the boundaries is perpendicular to the principal direction of heat transfer in the solid. Consider a strut that connects two walls at different temperatures and across which there is fluid flow (Figure 3.12). With T1 T2, temperature gradients in the x-direction sustain heat transfer by conduction in the strut. However, with T1 T2 T앝, there is concurrent heat

T2 qx, 2 L

x

qconv

Fluid

T∞, h

T1 T1

qx, 1

T2 T(x)

T1 > T2 > T∞

0

FIGURE 3.12 Combined conduction and convection in a structural element.

3.6

䊏

155

Heat Transfer from Extended Surfaces

transfer by convection to the fluid, causing qx, and hence the magnitude of the temperature gradient, 兩dT/dx兩, to decrease with increasing x. Although there are many different situations that involve such combined conduction– convection effects, the most frequent application is one in which an extended surface is used specifically to enhance heat transfer between a solid and an adjoining fluid. Such an extended surface is termed a fin. Consider the plane wall of Figure 3.13a . If Ts is fixed, there are two ways in which the heat transfer rate may be increased. The convection coefficient h could be increased by increasing the fluid velocity, and/or the fluid temperature T앝 could be reduced. However, there are many situations for which increasing h to the maximum possible value is either insufficient to obtain the desired heat transfer rate or the associated costs are prohibitive. Such costs are related to the blower or pump power requirements needed to increase h through increased fluid motion. Moreover, the second option of reducing T앝 is often impractical. Examining Figure 3.13b , however, we see that there exists a third option. That is, the heat transfer rate may be increased by increasing the surface area across which the convection occurs. This may be done by employing fins that extend from the wall into the surrounding fluid. The thermal conductivity of the fin material can have a strong effect on the temperature distribution along the fin and therefore influences the degree to which the heat transfer rate is enhanced. Ideally, the fin material should have a large thermal conductivity to minimize temperature variations from its base to its tip. In the limit of infinite thermal conductivity, the entire fin would be at the temperature of the base surface, thereby providing the maximum possible heat transfer enhancement. Examples of fin applications are easy to find. Consider the arrangement for cooling engine heads on motorcycles and lawn mowers or for cooling electric power transformers. Consider also the tubes with attached fins used to promote heat exchange between air and the working fluid of an air conditioner. Two common finned-tube arrangements are shown in Figure 3.14. Different fin configurations are illustrated in Figure 3.15. A straight fin is any extended surface that is attached to a plane wall. It may be of uniform cross-sectional area, or its cross-sectional area may vary with the distance x from the wall. An annular fin is one that is circumferentially attached to a cylinder, and its cross section varies with radius from the wall of the cylinder. The foregoing fin types have rectangular cross sections, whose area may be expressed as a product of the fin thickness t and the width w for straight fins or the circumference 2r for annular fins. In contrast a pin fin, or spine, is an extended surface of circular cross section. Pin fins may also be of uniform or nonuniform cross section. In any

T∞, h

T∞, h

A

q = hA(Ts – T∞)

Ts, A

Ts (a)

(b)

FIGURE 3.13 Use of fins to enhance heat transfer from a plane wall. (a) Bare surface. (b) Finned surface.

156

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Liquid flow Gas flow

Liquid flow Gas flow

FIGURE 3.14 Schematic of typical finned-tube heat exchangers.

application, selection of a particular fin configuration may depend on space, weight, manufacturing, and cost considerations, as well as on the extent to which the fins reduce the surface convection coefficient and increase the pressure drop associated with flow over the fins.

3.6.1

A General Conduction Analysis

As engineers we are primarily interested in knowing the extent to which particular extended surfaces or fin arrangements could improve heat transfer from a surface to the surrounding fluid. To determine the heat transfer rate associated with a fin, we must first obtain the temperature distribution along the fin. As we have done for previous systems, we begin by performing an energy balance on an appropriate differential element. Consider the extended surface of Figure 3.16. The analysis is simplified if certain assumptions are made. We choose to assume one-dimensional conditions in the longitudinal (x-) direction, even though conduction within the fin is actually two-dimensional. The rate at which energy is convected to the fluid from any point on the fin surface must be balanced by the net rate at which energy reaches that point due to conduction in the transverse (y-, z-) direction. However, in practice the fin is thin, and temperature changes in the transverse

t w

x

r

x (a)

(b)

x (c)

(d)

FIGURE 3.15 Fin configurations. (a) Straight fin of uniform cross section. (b) Straight fin of nonuniform cross section. (c) Annular fin. (d) Pin fin.

3.6

䊏

157

Heat Transfer from Extended Surfaces

dAs

qx

dqconv Ac(x) qx+dx

dx

x

z y

FIGURE 3.16 Energy balance for an extended surface.

x

direction within the fin are small compared with the temperature difference between the fin and the environment. Hence, we may assume that the temperature is uniform across the fin thickness, that is, it is only a function of x. We will consider steady-state conditions and also assume that the thermal conductivity is constant, that radiation from the surface is negligible, that heat generation effects are absent, and that the convection heat transfer coefficient h is uniform over the surface. Applying the conservation of energy requirement, Equation 1.12c, to the differential element of Figure 3.16, we obtain qx qxdx dqconv

(3.61)

qx kAc dT dx

(3.62)

From Fourier’s law we know that

where Ac is the cross-sectional area, which may vary with x. Since the conduction heat rate at x dx may be expressed as qxdx qx

dqx dx dx

(3.63)

it follows that

冢

冣

qxdx kAc dT k d Ac dT dx dx dx dx

(3.64)

The convection heat transfer rate may be expressed as dqconv hdAs(T T )

(3.65)

where dAs is the surface area of the differential element. Substituting the foregoing rate equations into the energy balance, Equation 3.61, we obtain

冢

冣

d A dT h dAs (T T ) 0 앝 dx c dx k dx

158

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

or

冢

冣

冢

冣

d 2T 1 dAc dT 1 h dAs (T T ) 0

Ac dx dx Ac k dx dx 2

(3.66)

This result provides a general form of the energy equation for an extended surface. Its solution for appropriate boundary conditions provides the temperature distribution, which may be used with Equation 3.62 to calculate the conduction rate at any x.

3.6.2

Fins of Uniform Cross-Sectional Area

To solve Equation 3.66 it is necessary to be more specific about the geometry. We begin with the simplest case of straight rectangular and pin fins of uniform cross section (Figure 3.17). Each fin is attached to a base surface of temperature T(0) Tb and extends into a fluid of temperature T앝. For the prescribed fins, Ac is a constant and As Px, where As is the surface area measured from the base to x and P is the fin perimeter. Accordingly, with dAc /dx 0 and dAs /dx P, Equation 3.66 reduces to d 2T hP (T T ) 0 앝 dx 2 kAc

(3.67)

To simplify the form of this equation, we transform the dependent variable by defining an excess temperature as (x) ⬅ T(x) T앝

(3.68)

where, since T앝 is a constant, d/dx dT/dx. Substituting Equation 3.68 into Equation 3.67, we then obtain d 2 m2 0 dx 2

(3.69)

T∞, h qconv T ∞, h Tb

qconv

t Ac

Tb

qf

qf

D

w x L P = 2w + 2t Ac = wt (a)

x Ac L P = πD Ac = π D2/4 (b)

FIGURE 3.17 Straight fins of uniform cross section. (a) Rectangular fin. (b) Pin fin.

3.6

䊏

159

Heat Transfer from Extended Surfaces

where m2 ⬅ hP kAc

(3.70)

Equation 3.69 is a linear, homogeneous, second-order differential equation with constant coefficients. Its general solution is of the form (x) C1emx C2emx

(3.71)

By substitution it may readily be verified that Equation 3.71 is indeed a solution to Equation 3.69. To evaluate the constants C1 and C2 of Equation 3.71, it is necessary to specify appropriate boundary conditions. One such condition may be specified in terms of the temperature at the base of the fin (x 0) (0) Tb T앝 ⬅ b

(3.72)

The second condition, specified at the fin tip (x L), may correspond to one of four different physical situations. The first condition, Case A, considers convection heat transfer from the fin tip. Applying an energy balance to a control surface about this tip (Figure 3.18), we obtain hAc[T(L) T앝] kAc dT dx or h(L) k

d dx

冏

xL

冏

(3.73)

xL

That is, the rate at which energy is transferred to the fluid by convection from the tip must equal the rate at which energy reaches the tip by conduction through the fin. Substituting Equation 3.71 into Equations 3.72 and 3.73, we obtain, respectively, b C1 C2

(3.74)

and h(C1emL C2emL) km(C2emL C1emL) Solving for C1 and C2, it may be shown, after some manipulation, that cosh m(L x) (h /mk) sin h m(L x) b cosh mL (h /mk) sin h mL

(3.75)

The form of this temperature distribution is shown schematically in Figure 3.18. Note that the magnitude of the temperature gradient decreases with increasing x. This trend is a consequence of the reduction in the conduction heat transfer qx(x) with increasing x due to continuous convection losses from the fin surface. We are particularly interested in the amount of heat transferred from the entire fin. From Figure 3.18, it is evident that the fin heat transfer rate qf may be evaluated in two

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Fluid, T∞

qconv Tb dT dx x=L

–kAc __

qb = qf

hAc[T(L) – T∞]

θb θ (x)

160

0

FIGURE 3.18 Conduction and convection in a fin of uniform cross section.

L

0

x

alternative ways, both of which involve use of the temperature distribution. The simpler procedure, and the one that we will use, involves applying Fourier’s law at the fin base. That is, qf qb kAc dT dx

冏

kAc

x0

d dx

冏

(3.76)

x0

Hence, knowing the temperature distribution, (x), qf may be evaluated, giving qf 兹hPkAcb

sinh mL (h /mk) cosh mL cosh mL (h/mk) sinh mL

(3.77)

However, conservation of energy dictates that the rate at which heat is transferred by convection from the fin must equal the rate at which it is conducted through the base of the fin. Accordingly, the alternative formulation for qf is

冕 h[T(x) T ] dA q 冕 h(x) dA

qf

Af

f

s

s

(3.78)

Af

where Af is the total, including the tip, fin surface area. Substitution of Equation 3.75 into Equation 3.78 would yield Equation 3.77. The second tip condition, Case B, corresponds to the assumption that the convective heat loss from the fin tip is negligible, in which case the tip may be treated as adiabatic and d dx

冏

0

xL

Substituting from Equation 3.71 and dividing by m, we then obtain C1emL C2emL 0

(3.79)

3.6

䊏

161

Heat Transfer from Extended Surfaces

Using this expression with Equation 3.74 to solve for C1 and C2 and substituting the results into Equation 3.71, we obtain cosh m(L x) b cosh mL

(3.80)

Using this temperature distribution with Equation 3.76, the fin heat transfer rate is then qf 兹hPkAc b tanh mL

(3.81)

In the same manner, we can obtain the fin temperature distribution and heat transfer rate for Case C, where the temperature is prescribed at the fin tip. That is, the second boundary condition is (L) L, and the resulting expressions are of the form (L /b) sinh mx sinh m(L x) b sinh mL qf 兹hPkAc b

(3.82)

cosh mL L /b sinh mL

(3.83)

The very long fin, Case D, is an interesting extension of these results. In particular, as L l 앝, L l 0 and it is easily verified that emx b

(3.84)

qf 兹hPkAc b

(3.85)

The foregoing results are summarized in Table 3.4. A table of hyperbolic functions is provided in Appendix B.1.

TABLE 3.4 Case A

Temperature distribution and heat loss for fins of uniform cross section Tip Condition (x ⴝ L) Convection heat transfer: h(L) kd/dx冨x⫽L

B

Adiabatic: d/dx冨x⫽L 0

C

Prescribed temperature: (L) L

D

Infinite fin (L l 앝): (L) 0

⬅ T T앝 b (0) Tb T앝

m2 ⬅ hP/kAc M ⬅ 兹h 苶P 苶kA 苶苶 c b

Temperature Distribution /b

Fin Heat Transfer Rate qƒ

cosh m(L x) (h/mk) sinh m(L x) cosh mL (h/mk) sinh mL (3.75) cosh m(L x) cosh mL (3.80) (L/b) sinh mx sinh m(L x) sinh mL (3.82) emx

(3.84)

M

sinh mL (h/mk) cosh mL cosh mL (h/mk) sinh mL (3.77) M tanh mL (3.81) M

(cosh mL L/b) sinh mL (3.83) M

(3.85)

162

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

EXAMPLE 3.9 A very long rod 5 mm in diameter has one end maintained at 100 C. The surface of the rod is exposed to ambient air at 25 C with a convection heat transfer coefficient of 100 W/m2 䡠 K. 1. Determine the temperature distributions along rods constructed from pure copper, 2024 aluminum alloy, and type AISI 316 stainless steel. What are the corresponding heat losses from the rods? 2. Estimate how long the rods must be for the assumption of infinite length to yield an accurate estimate of the heat loss.

SOLUTION Known: A long circular rod exposed to ambient air. Find: 1. Temperature distribution and heat loss when rod is fabricated from copper, an aluminum alloy, or stainless steel. 2. How long rods must be to assume infinite length. Schematic: Air

Tb = 100°C

T∞ = 25°C h = 100 W/m2•K

k, L→∞, D = 5 mm

Assumptions: 1. Steady-state conditions. 2. One-dimensional conduction along the rod. 3. Constant properties. 4. Negligible radiation exchange with surroundings. 5. Uniform heat transfer coefficient. 6. Infinitely long rod. Properties: Table A.1, copper [T (Tb T앝)/2 62.5 C ⬇ 335 K]: k 398 W/m 䡠 K. Table A.1, 2024 aluminum (335 K): k 180 W/m 䡠 K. Table A.1, stainless steel, AISI 316 (335 K): k 14 W/m 䡠 K. Analysis: 1. Subject to the assumption of an infinitely long fin, the temperature distributions are determined from Equation 3.84, which may be expressed as T T앝 (Tb T앝)emx

3.6

䊏

163

Heat Transfer from Extended Surfaces

where m (hP/kAc)1/2 (4h/kD)1/2. Substituting for h and D, as well as for the thermal conductivities of copper, the aluminum alloy, and the stainless steel, respectively, the values of m are 14.2, 21.2, and 75.6 m1. The temperature distributions may then be computed and plotted as follows: 100 316 SS

T (°C)

80

2024 Al Cu

60

40

T∞

20

0

50

100

150

200

250

300

x (mm)

From these distributions, it is evident that there is little additional heat transfer associated with extending the length of the rod much beyond 50, 200, and 300 mm, respectively, for the stainless steel, the aluminum alloy, and the copper. From Equation 3.85, the heat loss is qf 兹hPkAc b Hence for copper,

冤

qf 100 W/m2 䡠 K 0.005 m

冥

398 W/m 䡠 K (0.005 m)2 4

1/2

(100 25) C

8.3 W

䉰

Similarly, for the aluminum alloy and stainless steel, respectively, the heat rates are qf 5.6 W and 1.6 W. 2. Since there is no heat loss from the tip of an infinitely long rod, an estimate of the validity of this approximation may be made by comparing Equations 3.81 and 3.85. To a satisfactory approximation, the expressions provide equivalent results if tanh mL 0.99 or mL 2.65. Hence a rod may be assumed to be infinitely long if L L앝 ⬅ 2.65 m 2.65

冢 冣 kAc hP

1/2

For copper, L앝 2.65

冤

冥

398 W/m 䡠 K (/4)(0.005 m)2 100 W/m2 䡠 K (0.005 m)

1/2

0.19 m

䉰

Results for the aluminum alloy and stainless steel are L앝 0.13 m and L앝 0.04 m, respectively.

164

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Comments: 1. The foregoing results suggest that the fin heat transfer rate may accurately be predicted from the infinite fin approximation if mL 2.65. However, if the infinite fin approximation is to accurately predict the temperature distribution T(x), a larger value of mL would be required. This value may be inferred from Equation 3.84 and the requirement that the tip temperature be very close to the fluid temperature. Hence, if we require that (L)/b exp(mL) 0.01, it follows that mL 4.6, in which case L앝 ⬇ 0.33, 0.23, and 0.07 m for the copper, aluminum alloy, and stainless steel, respectively. These results are consistent with the distributions plotted in part 1. 2. This example is solved in the Advanced section of IHT.

3.6.3

Fin Performance

Recall that fins are used to increase the heat transfer from a surface by increasing the effective surface area. However, the fin itself represents a conduction resistance to heat transfer from the original surface. For this reason, there is no assurance that the heat transfer rate will be increased through the use of fins. An assessment of this matter may be made by evaluating the fin effectiveness f. It is defined as the ratio of the fin heat transfer rate to the heat transfer rate that would exist without the fin. Therefore f

qf hAc,bb

(3.86)

where Ac,b is the fin cross-sectional area at the base. In any rational design the value of f should be as large as possible, and in general, the use of fins may rarely be justified unless 2. f ⬃ Subject to any one of the four tip conditions that have been considered, the effectiveness for a fin of uniform cross section may be obtained by dividing the appropriate expression for qf in Table 3.4 by hAc,bb. Although the installation of fins will alter the surface convection coefficient, this effect is commonly neglected. Hence, assuming the convection coefficient of the finned surface to be equivalent to that of the unfinned base, it follows that, for the infinite fin approximation (Case D), the result is

冢 冣

f kP hAc

1/2

(3.87)

Several important trends may be inferred from this result. Obviously, fin effectiveness is enhanced by the choice of a material of high thermal conductivity. Aluminum alloys and copper come to mind. However, although copper is superior from the standpoint of thermal conductivity, aluminum alloys are the more common choice because of additional benefits related to lower cost and weight. Fin effectiveness is also enhanced by increasing the ratio of the perimeter to the cross-sectional area. For this reason, the use of thin, but closely spaced fins, is preferred, with the proviso that the fin gap not be reduced to a value for which flow between the fins is severely impeded, thereby reducing the convection coefficient. Equation 3.87 also suggests that the use of fins can be better justified under conditions for which the convection coefficient h is small. Hence from Table 1.1 it is evident that the need for fins is stronger when the fluid is a gas rather than a liquid and when the surface heat transfer is by free convection. If fins are to be used on a surface separating a gas and a liquid, they are

3.6

䊏

165

Heat Transfer from Extended Surfaces

generally placed on the gas side, which is the side of lower convection coefficient. A common example is the tubing in an automobile radiator. Fins are applied to the outer tube surface, over which there is flow of ambient air (small h), and not to the inner surface, through which there is flow of water (large h). Note that, if f 2 is used as a criterion to justify the implementation of fins, Equation 3.87 yields the requirement that (kP/hAc) 4. Equation 3.87 provides an upper limit to f, which is reached as L approaches infinity. However, it is certainly not necessary to use very long fins to achieve near maximum heat transfer enhancement. As seen in Example 3.8, 99% of the maximum possible fin heat transfer rate is achieved for mL 2.65. Hence, it would make no sense to extend the fins beyond L 2.65/m. Fin performance may also be quantified in terms of a thermal resistance. Treating the difference between the base and fluid temperatures as the driving potential, a fin resistance may be defined as Rt,f qb

(3.88)

f

This result is extremely useful, particularly when representing a finned surface by a thermal circuit. Note that, according to the fin tip condition, an appropriate expression for qf may be obtained from Table 3.4. Dividing Equation 3.88 into the expression for the thermal resistance due to convection at the exposed base, Rt,b 1 hAc,b

(3.89)

and substituting from Equation 3.86, it follows that f

Rt,b Rt, f

(3.90)

Hence the fin effectiveness may be interpreted as a ratio of thermal resistances, and to increase f it is necessary to reduce the conduction/convection resistance of the fin. If the fin is to enhance heat transfer, its resistance must not exceed that of the exposed base. Another measure of fin thermal performance is provided by the fin efficiency f. The maximum driving potential for convection is the temperature difference between the base (x 0) and the fluid, b Tb – T앝. Hence the maximum rate at which a fin could dissipate energy is the rate that would exist if the entire fin surface were at the base temperature. However, since any fin is characterized by a finite conduction resistance, a temperature gradient must exist along the fin and the preceding condition is an idealization. A logical definition of fin efficiency is therefore qf f ⬅ q max

qf hAf b

(3.91)

where Af is the surface area of the fin. For a straight fin of uniform cross section and an adiabatic tip, Equations 3.81 and 3.91 yield hf M tanh mL tanh mL hPLb mL

(3.92)

Referring to Table B.1, this result tells us that f approaches its maximum and minimum values of 1 and 0, respectively, as L approaches 0 and 앝.

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

In lieu of the somewhat cumbersome expression for heat transfer from a straight rectangular fin with an active tip, Equation 3.77, it has been shown that approximate, yet accurate, predictions may be obtained by using the adiabatic tip result, Equation 3.81, with a corrected fin length of the form Lc L (t/2) for a rectangular fin and Lc L (D/4) for a pin fin [14]. The correction is based on assuming equivalence between heat transfer from the actual fin with tip convection and heat transfer from a longer, hypothetical fin with an adiabatic tip. Hence, with tip convection, the fin heat rate may be approximated as qf M tanh mLc

(3.93)

and the corresponding efficiency as hf

tanh mLc mLc

(3.94)

Errors associated with the approximation are negligible if (ht/k) or (hD/2k) 0.0625 [15]. If the width of a rectangular fin is much larger than its thickness, w t, the perimeter may be approximated as P 2w, and

冢 冣

mLc hP kAc

冢 冣

1/2

Lc 2h kt

1/2

Lc

Multiplying numerator and denominator by L1/2 c and introducing a corrected fin profile area, Ap Lc t, it follows that

冢 冣

mLc 2h kAp

1/2

L3/2 c

(3.95)

Hence, as shown in Figures 3.19 and 3.20, the efficiency of a rectangular fin with tip convection may be represented as a function of Lc3/2(h/kAp)1/2. 100

y ~ x2

80

y

x

Lc = L Ap = Lt /3

t/2 L

60

η f (%)

166

Lc = L + t/2 Ap = Lc t

40

L

t/2

y~x y

20

t/2

x Lc = L Ap = Lt /2

L 0

0

0.5

1.0

1.5

2.0

1/2 L3/2 c (h/kAp)

FIGURE 3.19 Efficiency of straight fins (rectangular, triangular, and parabolic profiles).

2.5

3.6

䊏

167

Heat Transfer from Extended Surfaces

100

80

η f (%)

60 1 = r2c /r1 40

2

20

L

r2c = r2 + t/2 t Lc = L + t/2 Ap = Lc t

3

5

r1 r2 0

0

0.5

1.0

1.5

2.0

2.5

1/2 L3/2 c (h/kAp)

FIGURE 3.20 Efficiency of annular fins of rectangular profile.

3.6.4

Fins of Nonuniform Cross-Sectional Area

Analysis of fin thermal behavior becomes more complex if the fin is of nonuniform cross section. For such cases the second term of Equation 3.66 must be retained, and the solutions are no longer in the form of simple exponential or hyperbolic functions. As a special case, consider the annular fin shown in the inset of Figure 3.20. Although the fin thickness is uniform (t is independent of r), the cross-sectional area, Ac 2rt, varies with r. Replacing x by r in Equation 3.66 and expressing the surface area as As 2(r 2 r 12), the general form of the fin equation reduces to d 2T 1 dT 2h (T T ) 0 앝 kt dr 2 r dr or, with m2 ⬅ 2h/kt and ⬅ T – T앝, d 2 1 d m2 0 dr 2 r dr The foregoing expression is a modified Bessel equation of order zero, and its general solution is of the form (r) C1I0(mr) C2K0(mr) where I0 and K0 are modified, zero-order Bessel functions of the first and second kinds, respectively. If the temperature at the base of the fin is prescribed, (r1) b, and an adiabatic tip is presumed, d/dr冨r 2 0, C1 and C2 may be evaluated to yield a temperature distribution of the form I (mr)K1(mr2) K0(mr)I1(mr2) 0 b I0(mr1)K1(mr2) K0(mr1)I1(mr2)

168

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

where I1(mr) d[I0(mr)]/d(mr) and K1(mr) –d[K0(mr)]/d(mr) are modified, first-order Bessel functions of the first and second kinds, respectively. The Bessel functions are tabulated in Appendix B. With the fin heat transfer rate expressed as qf kAc,b dT dr

冏

k(2r1t)

rr1

d dr

冏

rr1

it follows that qf 2kr1tbm

K1(mr1)I1(mr2) I1(mr1)K1(mr2) K0(mr1)I1(mr2) I0(mr1)K1(mr2)

from which the fin efficiency becomes f

qf h2(r22

r21)b

2r1 K1(mr1)I1(mr2) I1(mr1)K1(mr2) 2 2 K (mr )I (mr ) I (mr )K (mr ) m(r2 r1) 0 1 1 2 0 1 1 2

(3.96)

This result may be applied for an active (convecting) tip, if the tip radius r2 is replaced by a corrected radius of the form r2c r2 (t/2). Results are represented graphically in Figure 3.20. Knowledge of the thermal efficiency of a fin may be used to evaluate the fin resistance, where, from Equations 3.88 and 3.91, it follows that Rt, f

1 hAff

(3.97)

Expressions for the efficiency and surface area of several common fin geometries are summarized in Table 3.5. Although results for the fins of uniform thickness or diameter

TABLE 3.5

Efficiency of common fin shapes

Straight Fins Rectangular a Aƒ 2wLc Lc L (t/2) Ap tL

tanh mLc mLc

(3.94)

1 I1(2mL) mL I0(2mL)

(3.98)

2 [4(mL)2 1]1/2 1

(3.99)

f

t w L

Triangular a Aƒ 2w[L2 (t/2)2]1/2 Ap (t/2)L

f

t w L

Parabolica Aƒ w[C1L (L2/t)ln (t/L C1)] C1 [1 (t/L)2]1/2 Ap (t/3)L

y = (t/2)(1 – x/L)2

f

t w L x

3.6

TABLE 3.5

䊏

169

Heat Transfer from Extended Surfaces

Continued

Circular Fin Rectangular a 2 Aƒ 2 (r 2c r 12) r2c r2 (t/2) V (r 22 r 12)t

W-121

t

L r1

f C2

K1(mr1)I1(mr2c) I1(mr1)K1(mr2c) I0(mr1)K1(mr2c) K0(mr1)I1(mr2c) (2r1/m) C2 2 (r 2c r 21)

(3.96)

r2

Pin Fins Rectangular b Aƒ DLc Lc L (D/4) V (D2/4)L

tanh mLc mLc

(3.100)

2 I2(2mL) mL I1(2mL)

(3.101)

2 [4/9(mL)2 1]1/2 1

(3.102)

f

D

L

Triangular b D 2 [L (D/2)2]1/2 2 V (/12)D2L

f

Aƒ

D

L

Parabolic b Aƒ

L3 {C3C4 8D L ln [(2DC4/L) C3]}

y = (D/2)(1 – x/L)2

D

2D

f

L

C3 1 2(D/L)2 C4 [1 (D/L)2]1/2 V (/20)D2 L

x

m (2h/kt)1/2. m (4h/kD)1/2.

a b

were obtained by assuming an adiabatic tip, the effects of convection may be treated by using a corrected length (Equations 3.94 and 3.100) or radius (Equation 3.96). The triangular and parabolic fins are of nonuniform thickness that reduces to zero at the fin tip. Expressions for the profile area, Ap, or the volume, V, of a fin are also provided in Table 3.5. The volume of a straight fin is simply the product of its width and profile area, V wAp. Fin design is often motivated by a desire to minimize the fin material and/or related manufacturing costs required to achieve a prescribed cooling effectiveness. Hence, a straight triangular fin is attractive because, for equivalent heat transfer, it requires much less volume (fin material) than a rectangular profile. In this regard, heat dissipation per unit volume, (q/V)f,

170

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

is largest for a parabolic profile. However, since (q/V)f for the parabolic profile is only slightly larger than that for a triangular profile, its use can rarely be justified in view of its larger manufacturing costs. The annular fin of rectangular profile is commonly used to enhance heat transfer to or from circular tubes.

3.6.5

Overall Surface Efficiency

In contrast to the fin efficiency f, which characterizes the performance of a single fin, the overall surface efficiency o characterizes an array of fins and the base surface to which they are attached. Representative arrays are shown in Figure 3.21, where S designates the fin pitch. In each case the overall efficiency is defined as q q ho q t t max hAtb

(3.103)

where qt is the total heat rate from the surface area At associated with both the fins and the exposed portion of the base (often termed the prime surface). If there are N fins in the array, each of surface area Af , and the area of the prime surface is designated as Ab, the total surface area is At NAf Ab

(3.104)

The maximum possible heat rate would result if the entire fin surface, as well as the exposed base, were maintained at Tb. The total rate of heat transfer by convection from the fins and the prime (unfinned) surface may be expressed as qt Nf hAf b hAbb

(3.105)

where the convection coefficient h is assumed to be equivalent for the finned and prime surfaces and f is the efficiency of a single fin. Hence

冤

qt h[Nf Af (At NAf )]b hAt 1

NAf At

冥

(1 f ) b

(3.106)

r2 r1

t t S Tb

Tb

S

w T∞, h

L

(a)

FIGURE 3.21 Representative fin arrays. (a) Rectangular fins. (b) Annular fins.

(b)

3.6

䊏

171

Heat Transfer from Extended Surfaces

Substituting Equation (3.106) into (3.103), it follows that o 1

NAf At

(1 f)

(3.107)

From knowledge of o, Equation 3.103 may be used to calculate the total heat rate for a fin array. Recalling the definition of the fin thermal resistance, Equation 3.88, Equation 3.103 may be used to infer an expression for the thermal resistance of a fin array. That is, Rt,o qb t

1 hohAt

(3.108)

where Rt,o is an effective resistance that accounts for parallel heat flow paths by conduction/convection in the fins and by convection from the prime surface. Figure 3.22 illustrates the thermal circuits corresponding to the parallel paths and their representation in terms of an effective resistance. If fins are machined as an integral part of the wall from which they extend (Figure 3.22a), there is no contact resistance at their base. However, more commonly, fins are manufactured separately and are attached to the wall by a metallurgical or adhesive joint. Alternatively, the attachment may involve a press fit, for which the fins are forced into slots machined on the wall material. In such cases (Figure 3.22b), there is a thermal contact resistance Rt,c, which

(Nηf hAf)–1

qf Nqf

Tb

qb

T∞ qb

Tb

[h(At – NAf)]–1

qt

Tb

T∞, h

T∞

(ηo hAt)–1

(a)

R"t,c

(Nηf hA f)–1

R"t ,c /NAc,b

qf Tb

Nqf qb

Tb

T∞ qb [h(At – NAf)]–1

T∞, h

qt

Tb

( ηo(c)h At)–1 (b)

FIGURE 3.22 Fin array and thermal circuit. (a) Fins that are integral with the base. (b) Fins that are attached to the base.

T∞

172

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

may adversely influence overall thermal performance. An effective circuit resistance may again be obtained, where, with the contact resistance, Rt,o(c) qb t

1 ho(c)hAt

(3.109)

It is readily shown that the corresponding overall surface efficiency is ho(c) 1

NAf At

冢1 C 冣 hf

(3.110a)

1

where C1 1 f hAf (Rt,c /Ac,b)

(3.110b)

In manufacturing, care must be taken to render Rt,c Rt,f.

EXAMPLE 3.10 The engine cylinder of a motorcycle is constructed of 2024-T6 aluminum alloy and is of height H 0.15 m and outside diameter D 50 mm. Under typical operating conditions the outer surface of the cylinder is at a temperature of 500 K and is exposed to ambient air at 300 K, with a convection coefficient of 50 W/m2 䡠 K. Annular fins are integrally cast with the cylinder to increase heat transfer to the surroundings. Consider five such fins, which are of thickness t 6 mm, length L 20 mm, and equally spaced. What is the increase in heat transfer due to use of the fins?

SOLUTION Known: Operating conditions of a finned motorcycle cylinder. Find:

Increase in heat transfer associated with using fins.

Schematic: Engine cylinder cross section (2024 T6 Al alloy)

S H = 0.15 m

Tb = 500 K t = 6 mm

T∞ = 300 K h = 50 W/m2•K Air

r1 = 25 mm L = 20 mm r2 = 45 mm

Assumptions: 1. Steady-state conditions. 2. One-dimensional radial conduction in fins. 3. Constant properties.

3.6

䊏

173

Heat Transfer from Extended Surfaces

4. Negligible radiation exchange with surroundings. 5. Uniform convection coefficient over outer surface (with or without fins).

Properties: Table A.1, 2024-T6 aluminum (T 400 K): k 186 W/m 䡠 K. Analysis: With the fins in place, the heat transfer rate is given by Equation 3.106

冤

qt hAt 1

NAf At

冥

(1 f ) b

2 r 12) 2[(0.048 m)2 (0.025 m)2] 0.0105 m2 and, from Equawhere Af 2(r 2c tion 3.104, At NAƒ 2r1(H Nt) 0.0527 m2 2 (0.025 m) [0.15 m 0.03 m] 0.0716 m2. With r2c /r1 1.92, Lc 0.023 m, Ap 1.380 104 m2, we obtain 1/2 L3/2 0.15. Hence, from Figure 3.20, the fin efficiency is ƒ ⬇ 0.95. c (h/kAp) With the fins, the total heat transfer rate is then

冤

冥

2 qt 50 W/m2 䡠 K 0.0716 m2 1 0.0527 m2 (0.05) 200 K 690 W 0.0716 m Without the fins, the convection heat transfer rate would be

qwo h(2r1H)b 50 W/m2 䡠 K(2 0.025 m 0.15 m)200 K 236 W Hence

q qt qwo 454 W

䉰

Comments: 1. Although the fins significantly increase heat transfer from the cylinder, considerable improvement could still be obtained by increasing the number of fins. We assess this possibility by computing qt as a function of N, first by fixing the fin thickness at t 6 mm and increasing the number of fins by reducing the spacing between fins. Prescribing a fin clearance of 2 mm at each end of the array and a minimum fin gap of 4 mm, the maximum allowable number of fins is N H/S 0.15 m/(0.004 0.006) m 15. The parametric calculations yield the following variation of qt with N: 1600 1400

t = 6 mm qt (W)

1200 1000 800 600

5

7

9 11 Number of fins, N

13

15

The number of fins could also be increased by reducing the fin thickness. If the fin gap is fixed at (S t) 4 mm and manufacturing constraints dictate a minimum allowable fin thickness of 2 mm, up to N 25 fins may be accommodated. In this case the parametric calculations yield

Chapter 3

One-Dimensional, Steady-State Conduction

䊏

3000 2500 (S – t) = 4 mm 2000

qt (W)

174

1500 1000 500

5

10

15 Number of fins, N

25

20

The foregoing calculations are based on the assumption that h is not affected by a reduction in the fin gap. The assumption is reasonable as long as there is no interaction between boundary layers that develop on the opposing surfaces of adjoining fins. Note that, since NAf 2r1(H – Nt) for the prescribed conditions, qt increases nearly linearly with increasing N. 2. The Models/Extended Surfaces option in the Advanced section of IHT provides readyto-solve models for straight, pin, and circular fins, as well as for fin arrays. The models include the efficiency relations of Figures 3.19 and 3.20 and Table 3.5.

EXAMPLE 3.11 In Example 1.5, we saw that to generate an electrical power of P 9 W, the temperature of the PEM fuel cell had to be maintained at Tc ⬇ 56.4 C, which required removal of 11.25 W from the fuel cell and a cooling air velocity of V 9.4 m/s for T앝 25 C. To provide these convective conditions, the fuel cell is centered in a 50 mm 26 mm rectangular duct, with 10-mm gaps between the exterior of the 50 mm 50 mm 6 mm fuel cell and the top and bottom of the well-insulated duct wall. A small fan, powered by the fuel cell, is used to circulate the cooling air. Inspection of a particular fan vendor’s data sheets suggests that the ratio of the fan power consumption to the fan’s volumetric flow rate is ˙ 102 m3/s. Pf / ˙ f C 1000 W/(m3/s) for the range 104 f Duct

Duct

W

Without finned heat sink

W H

H

Lf

tc

tf

tb

Fuel cell Lc

Fuel cell

Wc a •

T∞, f Air

•

T∞, f Air

Lc Wc

With finned heat sink

3.6

䊏

175

Heat Transfer from Extended Surfaces

1. Determine the net electric power produced by the fuel cell–fan system, Pnet P Pf . 2. Consider the effect of attaching an aluminum (k 200 W/m 䡠 K) finned heat sink, of identical top and bottom sections, onto the fuel cell body. The contact joint has a thermal resistance of Rt,c 103 m2 䡠 K/W, and the base of the heat sink is of thickness tb 2 mm. Each of the N rectangular fins is of length Lf 8 mm and thickness tf 1 mm, and spans the entire length of the fuel cell, Lc 50 mm. With the heat sink in place, radiation losses are negligible and the convective heat transfer coefficient may be related to the size and geometry of a typical air channel by an expression of the form h 1.78 kair (Lf a)/(Lf 䡠 a), where a is the distance between fins. Draw an equivalent thermal circuit for part 2 and determine the total number of fins needed to reduce the fan power consumption to half of the value found in part 1.

SOLUTION Known: Dimensions of a fuel cell and finned heat sink, fuel cell operating temperature, rate of thermal energy generation, power production. Relationship between power consumed by a cooling fan and the fan airflow rate. Relationship between the convection coefficient and the air channel dimensions. Find: 1. The net power produced by the fuel cell–fan system when there is no heat sink. 2. The number of fins needed to reduce the fan power consumption found in part 1 by 50%. Schematic: Lc = 50 mm A

Fuel cell

T∞

Fan

H = 26 mm

tc = 6 mm

A

Finned heat sink

Finned heat sink Air

T∞ = 25°C, V Lf = 8 mm

a

tc = 6 mm

Fuel cell, Tc = 56.4°C

tb = 2 mm

tf = 1 mm

W = Wc = 50 mm Section A–A

H = 26 mm

176

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Assumptions: 1. Steady-state conditions. 2. Negligible heat transfer from the edges of the fuel cell, as well as from the front and back faces of the finned heat sink. 3. One-dimensional heat transfer through the heat sink. 4. Adiabatic fin tips. 5. Constant properties. 6. Negligible radiation when the heat sink is in place. – Properties: Table A.4. air (T 300 K): kair 0.0263 W/m 䡠 K, cp 1007 J/kg 䡠 K, 1.1614 kg/m3.

Analysis: 1. The volumetric flow rate of cooling air is ˙ f VAc, where Ac W (H – tc) is the crosssectional area of the flow region between the duct walls and the unfinned fuel cell. Therefore, ˙ f V[W(H tc)] 9.4 m/s [0.05 m (0.026 m 0.006 m)] 9.4 103 m3/s and Pnet P C ˙ f 9.0 W 1000 W/(m3/s) 9.4 103 m3/s 0.4 W

䉰

With this arrangement, the fan consumes more power than is generated by the fuel cell, and the system cannot produce net power. 2. To reduce the fan power consumption by 50%, the volumetric flow rate of air must be reduced to ˙ f 4.7 103 m3/s. The thermal circuit includes resistances for the contact joint, conduction through the base of the finned heat sink, and resistances for the exposed base of the finned side of the heat sink, as well as the fins. Rt,b Tc q

T∞ Rt,c

Rt,base Rt, f(N)

The thermal resistances for the contact joint and the base are /2LcWc (103 m2 䡠 K/W)/(2 0.05 m 0.05 m) 0.2 K/W Rt,c Rt,c and Rt,base tb /(2kLcWc) (0.002 m)/(2 200 W/m 䡠 K 0.05 m 0.05 m) 0.002 K/W

3.6

䊏

Heat Transfer from Extended Surfaces

177

where the factors of two account for the two sides of the heat sink assembly. For the portion of the base exposed to the cooling air, the thermal resistance is Rt,b 1/[h (2Wc Ntf )Lc ] 1/[h (2 0.05 m N 0.001 m) 0.05 m] which cannot be evaluated until the total number of fins on both sides, N, and h are determined. For a single fin, Rt, f b/qf , where, from Table 3.4 for a fin with an insulated fin tip, Rt, f (hPkAc)1/2/tanh(mLf). In our case, P 2(Lc tf) 2 (0.05 m 0.001 m) 0.102 m, Ac Lctf 0.05 m 0.001 m 0.00005 m2, and m 兹hP/kAc [h 0.102 m/(200 W/m 䡠 K 0.00005 m2)]1/2 Hence, Rt, f

(h 0.102 m 200 W/m 䡠 K 0.00005 m2)1/2 tanh(m 0.008 m)

and for N fins, Rt, f(N) Rt, f /N. As for Rt,b, Rt,f cannot be evaluated until h and N are determined. Also, h depends on a, the distance between fins, which in turn depends on N, according to a (2Wc Ntf)/N (2 0.05 m N 0.001 m)/N. Thus, specification of N will make it possible to calculate all resistances. From the thermal resistance network, the total thermal resistance is Rtot Rt,c Rt,base Requiv, where Requiv [Rt, b1 Rt, f(N)1]1. The equivalent fin resistance, Requiv, corresponding to the desired fuel cell temperature is found from the expression q

Tc T앝 Tc T앝 Rtot Rt,c Rt,base Requiv

in which case, Requiv

Tc T앝 (Rt,c Rt,base) q

(56.4 C 25 C)/11.25 W (0.2 0.002) K/W 2.59 K/W For N 22, the following values of the various parameters are obtained: a 0.0035 m, h 19.1 W/m2 䡠 K, m 13.9 m1, Rt,f(N) 2.94 K/W, Rt,b 13.5 K/W, Requiv 2.41 K/W, and Rtot 2.61 K/W, resulting in a fuel cell temperature of 54.4 C. Fuel cell temperatures associated with N 20 and N 24 fins are Tc 58.9 C and 50.7 C, respectively. The actual fuel cell temperature is closest to the desired value when N 22. Therefore, a total of 22 fins, 11 on top and 11 on the bottom, should be specified, resulting in Pnet P Pf 9.0 W 4.7 W 4.3 W

䉰

Comments: 1. The performance of the fuel cell–fan system is enhanced significantly by combining the finned heat sink with the fuel cell. Good thermal management can transform an impractical proposal into a viable concept. 2. The temperature of the cooling air increases as heat is transferred from the fuel cell. The temperature of the air leaving the finned heat sink may be calculated from an overall ˙ ). For part 1, T 25 C energy balance on the airflow, which yields To Ti q/(cp f o 3 3 3 10.28 W/(1.1614 kg/m 1007 J/kg 䡠 K 9.4 10 m /s) 25.9 C. For part 2, the

178

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

outlet air temperature is To 27.0 C. Hence, the operating temperature of the fuel cell will be slightly higher than predicted under the assumption that the cooling air temperature is constant at 25 C and will be closer to the desired value. 3. For the conditions in part 2, the convection heat transfer coefficient does not vary with the air velocity. The insensitivity of the value of h to the fluid velocity occurs frequently in cases where the flow is confined within passages of small cross-sectional area, as will be discussed in detail in Chapter 8. The fin’s influence on increasing or reducing the value of h relative to that of an unfinned surface should be taken into account in critical applications. 4. A more detailed analysis of the system would involve prediction of the pressure drop associated with the fan-induced flow of air through the gaps between the fins. 5. The adiabatic fin tip assumption is valid since the duct wall is well insulated.

3.7 The Bioheat Equation The topic of heat transfer within the human body is becoming increasingly important as new medical treatments are developed that involve extreme temperatures [16] and as we explore more adverse environments, such as the Arctic, underwater, or space. There are two main phenomena that make heat transfer in living tissues more complex than in conventional engineering materials: metabolic heat generation and the exchange of thermal energy between flowing blood and the surrounding tissue. Pennes [17] introduced a modification to the heat equation, now known as the Pennes or bioheat equation, to account for these effects. The bioheat equation is known to have limitations, but it continues to be a useful tool for understanding heat transfer in living tissues. In this section, we present a simplified version of the bioheat equation for the case of steady-state, one-dimensional heat transfer. Both the metabolic heat generation and exchange of thermal energy with the blood can be viewed as effects of thermal energy generation. Therefore, we can rewrite Equation 3.44 to account for these two heat sources as d2T q˙m q˙p (3.111) 0 k dx2 where q˙m and q˙p are the metabolic and perfusion heat source terms, respectively. The perfusion term accounts for energy exchange between the blood and the tissue and is an energy source or sink according to whether heat transfer is from or to the blood, respectively. The thermal conductivity has been assumed constant in writing Equation 3.111. Pennes proposed an expression for the perfusion term by assuming that within any small volume of tissue, the blood flowing in the small capillaries enters at an arterial temperature, Ta, and exits at the local tissue temperature, T. The rate at which heat is gained by the tissue is the rate at which heat is lost from the blood. If the perfusion rate is (m3/s of volumetric blood flow per m3 of tissue), the heat lost from the blood can be calculated from Equation 1.12e, or on a unit volume basis, q˙ p bcb(Ta T)

(3.112)

where b and cb are the blood density and specific heat, respectively. Note that b is the blood mass flow rate per unit volume of tissue.

3.7

䊏

179

The Bioheat Equation

Substituting Equation 3.112 into Equation 3.111, we find d 2T q˙m bcb(Ta T) 0 k dx 2

(3.113)

Drawing on our experience with extended surfaces, it is convenient to define an excess temperature of the form ⬅ T Ta q˙ m /bcb. Then, if we assume that Ta, q˙ m, , and the blood properties are all constant, Equation 3.113 can be rewritten as d 2 ˜ 2 m 0 (3.114) dx2 ˜ 2 bcb/k. This equation is identical in form to Equation 3.69. Depending on the where m form of the boundary conditions, it may therefore be possible to use the results of Table 3.4 to estimate the temperature distribution within the living tissue.

EXAMPLE 3.12 In Example 1.7, the temperature at the inner surface of the skin/fat layer was given as 35 C. In reality, this temperature depends on the existing heat transfer conditions, including phenomena occurring farther inside the body. Consider a region of muscle with a skin/fat layer over it. At a depth of Lm 30 mm into the muscle, the temperature can be assumed to be at the core body temperature of Tc 37 C. The muscle thermal conductivity . is km 0.5 W/m 䡠 K. The metabolic heat generation rate within the muscle is qm 700 W/m3. The perfusion rate is 0.0005 s1; the blood density and specific heat are b 1000 kg/m3 and cb 3600 J/kg 䡠 K, respectively, and the arterial blood temperature Ta is the same as the core body temperature. The thickness, emissivity, and thermal conductivity of the skin/fat layer are as given in Example 1.7; perfusion and metabolic heat generation within this layer can be neglected. We wish to predict the heat loss rate from the body and the temperature at the inner surface of the skin/fat layer for air and water environments of Example 1.7.

SOLUTION Known: Dimensions and thermal conductivities of a muscle layer and a skin/fat layer. Skin emissivity and surface area. Metabolic heat generation rate and perfusion rate within the muscle layer. Core body and arterial temperatures. Blood density and specific heat. Ambient conditions. Find:

Heat loss rate from body and temperature at inner surface of the skin/fat layer.

Schematic: Tc = 37°C

Muscle

Skin/Fat

q•m = 700 W/m3 q•p

Ti

ε = 0.95

ksf = 0.3 W/m•K

km = 0.5 W/m•K = 0.0005 sⴚ1

Lm = 30 mm x

Tsur = 297 K

Lsf = 3 mm

Air or water

T∞ = 297 K h = 2 W/m2•K (air) h = 200 W/m2•K (water)

180

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer through the muscle and skin/fat layers. 3. Metabolic heat generation rate, perfusion rate, arterial temperature, blood properties, and thermal conductivities are all uniform. 4. Radiation heat transfer coefficient is known from Example 1.7. 5. Solar irradiation is negligible. Analysis: We will combine an analysis of the muscle layer with a treatment of heat transfer through the skin/fat layer and into the environment. The rate of heat transfer through the skin/fat layer and into the environment can be expressed in terms of a total resistance, Rtot, as q

Ti T앝 Rtot

(1)

As in Example 3.1 and for exposure of the skin to the air, Rtot accounts for conduction through the skin/fat layer in series with heat transfer by convection and radiation, which act in parallel with each other. Thus, Rtot

冢

Lsf 1 1 ksf A 1/hA 1/hr A

冣

1

冢

L 1 sf 1 A ksf h hr

冣

Using the values from Example 1.7 for air, Rtot

冢

冣

1 0.003 m 1 0.076 K/W 1.8 m2 0.3 W/m 䡠 K (2 5.9) W/m2 䡠 K

For water, with hr 0 and h 200 W/m2 䡠 K, Rtot 0.0083 W/m2 䡠 K. Heat transfer in the muscle layer is governed by Equation 3.114. The boundary conditions are specified in terms of the temperatures, Tc and Ti, where Ti is, as yet, unknown. In terms of the excess temperature , the boundary conditions are then q˙ (0) Tc Ta mc c

and

b b

q˙ (Lm) Ti Ta mc i b b

Since we have two boundary conditions involving prescribed temperatures, the solution for is given by case C of Table 3.4, ˜ x sinh m ˜ (Lm x) ( / )sinh m i c ˜ Lm c sinh m The value of qf given in Table 3.4 would correspond to the heat transfer rate at x 0, but this is not of particular interest here. Rather, we seek the rate at which heat leaves the muscle and enters the skin/fat layer so that we can equate this quantity with the rate at which heat is transferred through the skin/fat layer and into the environment. Therefore, we calculate the heat transfer rate at x Lm as q

冏

km A dT dx xLm

冏

xLm

km A

d dx

冏

xLm

˜ c kmAm

˜ Lm 1 (i /c) cosh m ˜ Lm sinh m

(2)

3.7

䊏

181

The Bioheat Equation

Combining Equations 1 and 2 yields ˜ c km Am

˜ Lm 1 (i /c) cosh m T T앝 i ˜ Lm Rtot sinh m

This expression can be solved for Ti, recalling that Ti also appears in i.

Ti

冤 冢

冣

冥

q˙ ˜ Lm kmAm ˜ Rtot c Ta m cosh m ˜ Lm T앝 sinh m c b b

˜ Lm km Am ˜ Rtot cosh m ˜ Lm sinh m

where m˜ 兹bcb /km [0.0005 s1 1000 kg/m3 3600 J/kg 䡠 K/0.5 W/m 䡠 K]1/2 60 m1 sinh (m˜ Lm) sinh (60 m1 0.03 m) 2.94 and cosh (m˜ Lm) cosh (60 m1 0.03 m) 3.11 q˙ q˙ c Tc Ta mc mc b b

b b

1

0.0005 s

700 W/m3 1000 kg/m3 3600 J/kg 䡠 K

0.389 K The excess temperature can be expressed in kelvins or degrees Celsius, since it is a temperature difference. Thus, for air: {24 C 2.94 0.5 W/m 䡠 K 1.8 m2 60 m1 0.076 K/W[0.389 C (37 C 0.389 C) 3.11]} Ti 34.8 C 2.94 0.5 W/m 䡠 K 1.8 m2 60 m1 0.076 K/W 3.11

䉰

This result agrees well with the value of 35 C that was assumed for Example 1.7. Next we can find the heat loss rate: q

Ti T앝 34.8 C 24 C 142 W Rtot 0.076 K/W

䉰

Again this agrees well with the previous result. Repeating the calculation for water, we find Ti 28.2 C

䉰

q 514 W

䉰

Here the calculation of Example 1.7 was not accurate because it incorrectly assumed that the inside of the skin/fat layer would be at 35 C. Furthermore, the skin temperature in this case would be only 25.4 C based on this more complete calculation.

Comments: 1. In reality, our bodies adjust in many ways to the thermal environment. For example, if we are too cold, we will shiver, which increases our metabolic heat generation rate. If we are too warm, the perfusion rate near the skin surface will increase, locally raising the skin temperature to increase heat loss to the environment.

182

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

2. Measuring the true thermal conductivity of living tissue is very challenging, first because of the necessity of making invasive measurements in a living being, and second because it is difficult to experimentally separate the effects of heat conduction and perfusion. It is easier to measure an effective thermal conductivity that would account for the combined contributions of conduction and perfusion. However, this effective conductivity value necessarily depends on the perfusion rate, which in turn varies with the thermal environment and physical condition of the specimen. 3. The calculations can be repeated for a range of values of the perfusion rate, and the dependence of the heat loss rate on the perfusion rate is illustrated below. The effect is stronger for the case of the water environment, because the muscle temperature is lower and therefore the effect of perfusion by the warm arterial blood is more pronounced. 700 600 500

q(W)

400

Water environment

300

Air environment 200 100 0 0

0.0002

0.0004

0.0006

0.0008

0.001

(sⴚ1)

3.8 Thermoelectric Power Generation As noted in Section 1.6, approximately 60% of the energy consumed globally is wasted in the form of low-grade heat. As such, an opportunity exists to harvest this energy stream and convert some of it to useful power. One approach involves thermoelectric power generation, which operates on a fundamental principle termed the Seebeck effect that states when a temperature gradient is established within a material, a corresponding voltage gradient is induced. The Seebeck coefficient S is a material property representing the proportionality between voltage and temperature gradients and, accordingly, has units of volts/K. For a constant property material experiencing one-dimensional conduction, as illustrated in Figure 3.23a, (E1 E2) S(T1 T2)

(3.115)

Electrically conducting materials can exhibit either positive or negative values of the Seebeck coefficient, depending on how they scatter electrons. The Seebeck coefficient is very small in metals, but can be relatively large in some semiconducting materials. If the material of Figure 3.23a is installed in an electric circuit, the voltage difference induced by the Seebeck effect can drive an electric current I, and electric power can be

3.8

䊏

183

Thermoelectric Power Generation

T1, E1

q1

Thin metallic conductor

T1, E1

L

qP,1

n-type semiconductor, Sn

Thin metallic conductor

I

p-type semiconductor, Sp

I

x T2, E2 qP,2

+L T2, E2 I

q2/2

q2/2

I

Re,load (a)

(b)

FIGURE 3.23 Thermoelectric phenomena. (a) The Seebeck effect. (b) A simplified thermoelectric circuit consisting of one pair (N 1) of semiconducting pellets.

generated from waste heat that induces a temperature difference across the material. A simplified thermoelectric circuit, consisting of two pellets of semiconducting material, is shown in Figure 3.23b. By blending minute amounts of a secondary element into the pellet material, the direction of the current induced by the Seebeck effect can be manipulated. The resulting p- and n-type semiconductors, which are characterized by positive and negative Seebeck coefficients, respectively, can be arranged as shown in the figure. Heat is supplied to the top and lost from the bottom of the assembly, and thin metallic conductors connect the semiconductors to an external load represented by the electrical resistance, Re,load. Ultimately, the amount of electric power that is produced is governed by the heat transfer rates to and from the pair of semiconducting pellets shown in Figure 3.23b. In addition to inducing an electric current I, thermoelectric effects also induce the generation or absorption of heat at the interface between two dissimilar materials. This heat source or heat sink phenomenon is known as the Peltier effect, and the amount of heat absorbed qP is related to the Seebeck coefficients of the adjoining materials by an equation of the form qP I(Sp Sn)T ISp-nT

(3.116)

where the individual Seebeck coefficients in the preceding expression, Sp and Sn, correspond to the p- and n-type semiconductors, and the differential Seebeck coefficient is Sp-n ⬅ Sp – Sn. Temperature is expressed in kelvins in Equation 3.116. The heat absorption is positive (generation is negative) when the electric current flows from the n-type to the p-type semiconductor. Hence, in Figure 3.23b, Peltier heat absorption occurs at the warm interface between the semiconducting pellets and the upper, thin metallic conductor, while Peltier heat generation occurs at the cool interface between the pellets and the lower conductor. When T1 T2, the heat transfer rates to and from the device, q1 and q2, respectively, may be found by solving the appropriate form of the energy equation. For steady-state, one-dimensional conduction within the assembly of Figure 3.23b the analysis proceeds as follows. Assuming the thin metallic connectors are of relatively high thermal and electrical conductivity, Ohmic dissipation occurs exclusively within the semiconducting pellets, each of which has a cross-sectional area Ac,s. The thermal resistances of the metallic conductors are assumed to be negligible, as is heat transfer within any gas trapped between the semiconducting pellets. Recognizing that the electrical resistance of each of the two pellets may be

184

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

expressed as Re,s e,s(2L)/Ac,s where e,s is the electrical resistivity of the semiconducting material, Equation 3.43 may be used to find the uniform volumetric generation rate within each pellet q˙

I 2e,s

(3.117)

A2c,s

Assuming negligible contact resistances and identical, as well as constant, thermophysical properties in each of the two pellets (with the exception being Sp Sn), Equation C.7 may be used to write expressions for the heat conduction out of and into the semiconducting material q(x L) 2Ac,s

冤

q(x L) 2Ac,s

冥

I 2e,s L ks (T1 T2) 2 2L Ac,s

冤

冥

I 2e,sL ks (T1 T2) 2 2L Ac,s

(3.118a)

(3.118b)

The factor of 2 outside the brackets accounts for heat transfer in both pellets and, as evident, q(x L) q(x –L). Because of the Peltier effect, q1 and q2 are not equal to the heat transfer rates into and out of the pellets as expressed in Equations 3.118a,b. Incorporating Equation 3.116 in an energy balance for a control surface about the interface between the thin metallic conductor and the semiconductor material at x –L yields q1 q(x L) qP,1 q(x L) ISp-nT1

(3.119)

q2 q(x L) ISn-pT2 q(x L) ISp-nT2

(3.120)

Similarly at x L,

Combining Equations 3.118b and 3.119 yields Ac,sks I 2e,sL q1 (T1 T2) ISp-nT1 2 L Ac,s Similarly, combining Equations 3.118a and 3.120 gives Ac,sks I 2e,sL q2 (T1 T2) ISp-nT2 2 L Ac,s

(3.121)

(3.122)

From an overall energy balance on the thermoelectric device, the electric power produced by the Seebeck effect is P q1 q2 (3.123) Substituting Equations 3.121 and 3.122 into this expression yields P ISp-n(T1 T2) 4

I 2e,sL ISp-n(T1 T2) I 2 Re,tot Ac,s

(3.124)

where Re,tot 2Re,s. The voltage difference induced by the Seebeck effect is relatively small for a single pair of semiconducting pellets. To amplify the voltage difference, thermoelectric modules are fabricated, as shown schematically in Figure 3.24a where N 1 pairs of semiconducting pellets are wired in series. Thin layers of a dielectric material, usually a ceramic, sandwich the module to provide structural rigidity and electrical insulation from the surroundings. Assuming the

3.8

䊏

185

Thermoelectric Power Generation

Thin ceramic insulators q″

Thin metallic conductors

1

n

p n

p n

p

n

p n

p n

p

2L

I n=1

n=2

n=3

q″

n=N1 n=N

2

I

Re,load (a)

T∞,1 Rt,conv,1 qconv,1

ISp-n,effT1 I2Re,eff

T1 Thermoelectric module

Rt,cond,mod

ISp-n,eff(T1 T2) 2I2Re,eff I2Re,load

T2 qconv,2

ISp-n,effT2 I2Re,eff

Rt,conv,2

T∞,2 (b)

FIGURE 3.24 Thermoelectric module. (a) Cross-section of a module consisting of N semiconductor pairs. (b) Equivalent thermal circuit for a convectively heated and cooled module.

thermal resistances of the thin ceramic layers are negligible, q1, q2, and the total module electric power, PN, can be written by modifying Equations 3.121, 3.122, 3.124 as q1

1 (T T2) ISp-n,eff T1 I 2 Re,eff Rt,cond,mod 1

(3.125)

q2

1 (T T2) ISp-n,eff T2 I 2 Re,eff Rt,cond,mod 1

(3.126)

PN q1 q2 ISp-n,eff (T1 T2) 2 I 2 Re,eff

(3.127)

where Sp-n,eff NSp-n, and Re,eff NRe,s are the effective Seebeck coefficient and the total internal electrical resistance of the module while Rt,cond,mod L/NAsks is the conduction resistance associated with the module’s p-n semiconductor matrix. An equivalent thermal circuit for a convectively heated and cooled thermoelectric module is shown in Figure 3.24b. If heating or cooling were to be applied by radiation or conduction, the resistance network outside of the thermoelectric module portion of the circuit would be modified accordingly.

186

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Returning to the single thermoelectric circuit of Figure 3.23b, the efficiency is defined as TE ⬅ P/q1. From Equations 3.121 and 3.124, it can be seen that efficiency depends on the electrical current in a complex manner. However, the efficiency can be maximized by adjusting the current through changes in the load resistance. The resulting maximum efficiency is given as [18]

冢

TE 1

冣

T2 兹1 ZT 1 T1 兹1 ZT T /T 2 1

(3.128)

where T (T1 T2)/2, S ⬅ Sp Sn , and 2 Z S (3.129) e,s ks – – Since the efficiency increases with increasing ZT , ZT may be seen as a dimensionless – figure of merit associated with thermoelectric generation [19]. As ZT l 앝, TE l (1 T2 /T1) (1 Tc /Th) ⬅ C where C is the Carnot efficiency. As discussed in Section 1.3.2, the Carnot efficiency and, in turn, the thermoelectric efficiency cannot be determined until the appropriate hot and cold temperatures are calculated from a heat transfer analysis. – Because ZT is defined in terms of interrelated electrical and thermal conductivities, extensive research is being conducted to tailor the properties of the semiconducting pellets, primarily by manipulating the nanostructure of the material so as to independently control phonon and electron motion and, in turn, the thermal and electrical conductivities of the mater– ial. Currently, ZT values of approximately unity at room temperature are readily achieved. Finally, we note that thermoelectric modules can be operated in reverse; supplying electric power to the module allows one to control the heat transfer rates to or from the outer ceramic surfaces. Such thermoelectric chillers or thermoelectric heaters are used in a wide variety of applications. A comprehensive discussion of one-dimensional, steady-state heat transfer modeling associated with thermoelectric heating and cooling modules is available [20].

EXAMPLE 3.13 An array of M 48 thermoelectric modules is installed on the exhaust of a sports car. Each module has an effective Seebeck coefficient of Sp-n,eff 0.1435 V/K, and an internal electrical resistance of Re,eff 4 . In addition, each module is of width and length W 54 mm and contains N 100 pairs of semiconducting pellets. Each pellet has an overall length of 2L 5 mm and cross-sectional area Ac,s 1.2 105 m2 and is characterized by a thermal conductivity of ks 1.2 W/m 䡠 K. The hot side of each module is exposed to exhaust gases at T앝,1 550 C with h1 40 W/m2 䡠 K, while the opposite side of each module is cooled by pressurized water at T앝,2 105 C with h2 500 W/m2 䡠 K. If the modules are wired in series, and the load resistance is Re,load 400 , what is the electric power harvested from the hot exhaust gases? Pressurized water T∞,2 105°C h2 500 W/m2 • K 2L 5 mm

Exhaust gas T∞,1 550°C h1 40 W/m2 • K

W M 48 thermoelectric modules

W 54 mm N 100 pellet pairs

3.8

䊏

187

Thermoelectric Power Generation

SOLUTION Known: Thermoelectric module properties and dimensions, number of semiconductor pairs in each module, and number of modules in the array. Temperature of exhaust gas and pressurized water, as well as convection coefficients at the hot and cold module surfaces. Modules are wired in series, and the electrical resistance of the load is known. Find:

Power produced by the module array.

Schematic: 2L = 5 mm

Pressurized water T∞,2 = 105°C

h2 = 500

W/m2 • K

W = 54 mm I

h1 = 40 W/m2 • K

Exhaust gas

Re,load = 400 Ω

T∞,1 = 550°C I Pressurized water T∞,2 = 105°C

M = 48 Thermoelectric modules N = 100 semiconductor pairs per module

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer. 3. Constant properties. 4. Negligible electrical and thermal contact resistances. 5. Negligible radiation exchange and negligible heat transfer within the gas inside the modules. 6. Negligible conduction resistance posed by the metallic contacts and ceramic insulators of the modules. Analysis: We begin by analyzing a single module. The conduction resistance of each module’s semiconductor array is Rt,cond,mod

L 2.5 103 m 1.736 K/W NAc,s ks 100 1.2 105 m2 1.2 W/m 䡠 K

From Equation 3.125, q1

1 Rt,cond,mod

(T1 T2) ISp-n,eff T1 I 2 Re,eff

(T1 T2) 1.736 K/W

I 0.1435 V/K T1 I 2 4

(1)

while from Equation 3.126, q2

1 Rt,cond,mod

(T1 T2) ISp-n,eff T2 I 2 Re,eff

I 0.1435 V/K T2 I 2 4

(T1 T2) 1.736 K/W (2)

188

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

At the hot surface, Newton’s law of cooling may be written as q1 h1W 2(T앝,1 T1) 40 W/m2 䡠 K (0.054 m)2 [(550 273) K T1]

(3)

whereas at the cool surface, q2 h2W 2(T2 T앝,2) 500 W/m2 䡠 K (0.054 m)2 [T2 (105 273) K]

(4)

Four equations have been written that include five unknowns, q1, q2, T1, T2, and I. An additional equation is obtained from the electrical circuit. With the modules wired in series, the total electric power produced by all M 48 modules is equal to the electric power dissipated in the load resistance. Equation 3.127 yields Ptot MPN M[ISp-n,eff(T1 T2) 2I 2Re,eff] 48[I 0.1435 V/K (T1 T2) 2I 2 4 ] (5)

Since the electric power produced by the thermoelectric module is dissipated in the electrical load, it follows that Ptot I 2Rload I 2 400 Equations 1 through 6 may be solved simultaneously, yielding Ptot 46.9 W.

(6) 䉰

Comments: 1. Equations 1 through 5 can be readily written by inspecting the equivalent thermal circuit of Figure 3.24b. 2. The module surface temperatures are T1 173 C and T2 134 C, respectively. If these surface temperatures were specified in the problem statement, the electric power could be obtained directly from Equations 5 and 6. In any practical design of a thermoelectric generator, however, a heat transfer analysis must be conducted to determine the power generated. 3. Power generation is very sensitive to the convection heat transfer resistances. For h1 h2 l 앝, Ptot 5900 W. To reduce the thermal resistance between the module and fluid streams, finned heat sinks are often used to increase the temperature difference across the modules and, in turn, increase their power output. Good thermal management and design are crucial to maximizing the power generation. 4. Harvesting the thermal energy contained in the exhaust with thermoelectrics can eliminate the need for an alternator, resulting in an increase in the net power produced by the engine, a reduction in the automobile’s weight, and an increase in gas mileage of up to 10%. 5. Thermoelectric modules, operating in the heating mode, can be embedded in car seats and powered by thermoelectric exhaust harvesters, reducing energy costs associated with heating the entire passenger cabin. The seat modules can also be operated in the cooling mode, potentially eliminating the need for vapor compression air conditioning. Common refrigerants, such as R134a, are harmful greenhouse gases, and are emitted into the atmosphere by leakage through seals and connections, and by catastrophic leaks due to collisions. Replacing automobile vapor compression air conditioners with personalized thermoelectric seat coolers can eliminate the equivalent of 45 million metric tons of CO2 released into the atmosphere every year in the United States alone.

3.9

䊏

189

Micro- and Nanoscale Conduction

3.9 Micro- and Nanoscale Conduction We conclude the discussion of one-dimensional, steady-state conduction by considering situations for which the physical dimensions are on the order of, or smaller than, the mean free path of the energy carriers, leading to potentially important nano- or microscale effects.

Conduction Through Thin Gas Layers

3.9.1

Figure 3.25 shows instantaneous trajectories of gas molecules between two isothermal, solid surfaces separated by a distance L. As discussed in Section 1.2.1, even in the absence of bulk fluid motion individual molecules continually impinge on the two solid boundaries that are held at uniform surface temperatures Ts,1 and Ts,2, respectively. The molecules also collide with each other, exchanging energy within the gaseous medium. When the thickness of the gas layer is large, L L1 (Figure 3.25a), a particular gas molecule will collide more frequently with other gas molecules than with either of the solid boundaries. Alternatively, for a very thin gas layer, L L2 L1 (Figure 3.25b), the probability of a molecule striking either of the solid boundaries is high relative to the likelihood of it colliding with another molecule. The energy content of a gas molecule is associated with its translational, rotational, and vibrational kinetic energies. It is this molecular-scale kinetic energy that ultimately defines the temperature of the gas, and collisions between individual molecules determine the value of the thermal conductivity, as discussed in Section 2.2.1. However, the manner in which a gas molecule is reflected or scattered from the solid walls also affects its level of kinetic energy and, in turn, its temperature. Hence, wall–molecule collisions can become important in determining the heat rate, qx, as L/mfp becomes small. The collision with and subsequent scattering of an individual gas molecule from a solid wall can be described by a thermal accommodation coefficient, ␣t, ␣t

Ti Tsc Ti Ts

(3.130)

where Ti is the effective molecule temperature just prior to striking the solid surface, Tsc is the temperature of the molecule immediately after it is scattered or reflected by the surface, and Ts is the surface temperature. When the temperature of the scattered molecule is identical to the wall temperature, ␣t 1. Alternatively, if Tsc Ti, the molecule’s kinetic energy and temperature are unaffected by a collision with the wall and ␣t 0.

Ts,1

Ts,1

x

x L1 (a)

qx

qx

Ts,2

Ts,2

x

x L2 (b)

FIGURE 3.25 Molecule trajectories in (a) a relatively thick gas layer and (b) a relatively thin gas layer. Molecules collide with each other, and with the two solid walls.

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For one-dimensional conduction within an ideal gas contained between two surfaces held at temperatures Ts,1 and Ts,2 Ts,1, the heat rate through the gas layer may be expressed as [21] q

Ts,1 Ts,2 (Rt,mm Rt,ms)

(3.131)

where, at the molecular level, the thermal resistances are associated with molecule–molecule and molecule-surface collisions Rt,mm L kA

and

Rt,ms

mfp 2 ␣t ␣t kA

冤

冥冤9␥␥ 15冥

(3.132a,b)

In the preceding expression, ␥ ⬅ cp /cv is the specific heat ratio of the ideal gas. The two solids are assumed to be the same material with equal values of ␣t, and the temperature difference is assumed to be small relative to the cold wall, (Ts,1 – Ts,2)/Ts,2 1. Equations 3.132a,b may be combined to yield Rt,ms mfp 2 ␣t ␣t Rt,mm L

冤

冥冤9␥␥ 15冥

from which it is evident that Rt,ms may be neglected if L/mfp is large and ␣t 0. In this case, Equation 3.131 reduces to Equation 3.6. However, Rt,ms can be significant if L/mfp is small. From Equation 2.11 the mean free path increases as the gas pressure is decreased. Hence, Rt,ms increases with decreasing gas pressure, and the heat rate can be pressure dependent when L/mfp is small. Values of ␣t for specific gas and surface combinations range from 0.87 to 0.97 for air–aluminum and air–steel, but can be less than 0.02 when helium interacts with clean metallic surfaces [21]. Equations 3.131, 3.132a,b may be applied to situations for which L/mfp 0.1. For air at atmospheric pressure, this corresponds to L 10 nm.

3.9.2

Conduction Through Thin Solid Films

One-dimensional conduction across or along thin solid films was discussed in Section 2.2.1 in terms of the thermal conductivities kx and ky. The heat transfer rate across a thin solid film may be approximated by combining Equation 2.9a with Equation 3.5, yielding qx

k[1 mfp /(3L)]A kx A (Ts,1 Ts,2) (Ts,1 Ts,2) L L

(3.133)

When L/mfp is large, Equation (3.133) reduces to Equation 3.4. Many alternative expressions for kx are available and are discussed in the literature [21].

3.10 Summary Despite its inherent mathematical simplicity, one-dimensional, steady-state heat transfer occurs in numerous engineering applications. Although one-dimensional, steady-state conditions may not apply exactly, the assumptions may often be made to obtain results of reasonable accuracy. You should therefore be thoroughly familiar with the means by which such

3.10

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Summary

191

problems are treated. In particular, you should be comfortable with the use of equivalent thermal circuits and with the expressions for the conduction resistances that pertain to each of the three common geometries. You should also be familiar with how the heat equation and Fourier’s law may be used to obtain temperature distributions and the corresponding fluxes. The implications of an internally distributed source of energy should also be clearly understood. In addition, you should appreciate the important role that extended surfaces can play in the design of thermal systems and should have the facility to effect design and performance calculations for such surfaces. Finally, you should understand how the preceding concepts can be applied to analyze heat transfer in the human body, thermoelectric power generation, and micro- and nanoscale conduction. You may test your understanding of this chapter’s key concepts by addressing the following questions. • Under what conditions may it be said that the heat flux is a constant, independent of the direction of heat flow? For each of these conditions, use physical considerations to convince yourself that the heat flux would not be independent of direction if the condition were not satisfied. • For one-dimensional, steady-state conduction in a cylindrical or spherical shell without heat generation, is the radial heat flux independent of radius? Is the radial heat rate independent of radius? • For one-dimensional, steady-state conduction without heat generation, what is the shape of the temperature distribution in a plane wall? In a cylindrical shell? In a spherical shell? • What is the thermal resistance? How is it defined? What are its units? • For conduction across a plane wall, can you write the expression for the thermal resistance from memory? Similarly, can you write expressions for the thermal resistance associated with conduction across cylindrical and spherical shells? From memory, can you express the thermal resistances associated with convection from a surface and net radiation exchange between the surface and large surroundings? • What is the physical basis for existence of a critical insulation radius? How do the thermal conductivity and the convection coefficient affect its value? • How is the conduction resistance of a solid affected by its thermal conductivity? How is the convection resistance at a surface affected by the convection coefficient? How is the radiation resistance affected by the surface emissivity? • If heat is transferred from a surface by convection and radiation, how are the corresponding thermal resistances represented in a circuit? • Consider steady-state conduction through a plane wall separating fluids of different temperatures, T앝,i and T앝,o, adjoining the inner and outer surfaces, respectively. If the convection coefficient at the outer surface is five times larger than that at the inner surface, ho 5hi, what can you say about relative proximity of the corresponding surface temperatures, Ts,o and Ts,i, to their adjoining fluid temperatures? • Can a thermal conduction resistance be applied to a solid cylinder or sphere? • What is a contact resistance? How is it defined? What are its units for an interface of prescribed area? What are they for a unit area? • How is the contact resistance affected by the roughness of adjoining surfaces? • If the air in the contact region between two surfaces is replaced by helium, how is the thermal contact resistance affected? How is it affected if the region is evacuated? • What is the overall heat transfer coefficient? How is it defined, and how is it related to the total thermal resistance? What are its units? • In a solid circular cylinder experiencing uniform volumetric heating and convection heat transfer from its surface, how does the heat flux vary with radius? How does the heat rate vary with radius?

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• In a solid circular sphere experiencing uniform volumetric heating and convection heat transfer from its surface, how does the heat flux vary with radius? How does the heat rate vary with radius? • Is it possible to achieve steady-state conditions in a solid cylinder or sphere that is experiencing heat generation and whose surface is perfectly insulated? Explain. • Can a material experiencing heat generation be represented by a thermal resistance and included in a circuit analysis? If so, why? If not, why not? • What is the physical mechanism associated with cooking in a microwave oven? How do conditions differ from a conventional (convection or radiant) oven? • If radiation is incident on the surface of a semitransparent medium and is absorbed as it propagates through the medium, will the corresponding volumetric rate of heat generation q˙ be distributed uniformly in the medium? If not, how will q˙ vary with distance from the surface? • In what way is a plane wall that is of thickness 2L and experiences uniform volumetric heating and equivalent convection conditions at both surfaces similar to a plane wall that is of thickness L and experiences the same volumetric heating and convection conditions at one surface but whose opposite surface is well insulated? • What purpose is served by attaching fins to a surface? • In the derivation of the general form of the energy equation for an extended surface, why is the assumption of one-dimensional conduction an approximation? Under what conditions is it a good approximation? • Consider a straight fin of uniform cross section (Figure 3.15a). For an x-location in the fin, sketch the temperature distribution in the transverse (y-) direction, placing the origin of the coordinate at the midplane of the fin (t/2 y t/2). What is the form of a surface energy balance applied at the location (x, t/2)? • What is the fin effectiveness? What is its range of possible values? Under what conditions are fins most effective? • What is the fin efficiency? What is its range of possible values? Under what conditions will the efficiency be large? • What is the fin resistance? What are its units? • How are the effectiveness, efficiency, and thermal resistance of a fin affected if its thermal conductivity is increased? If the convection coefficient is increased? If the length of the fin is increased? If the thickness (or diameter) of the fin is increased? • Heat is transferred from hot water flowing through a tube to air flowing over the tube. To enhance the rate of heat transfer, should fins be installed on the tube interior or exterior surface? • A fin may be manufactured as an integral part of a surface by using a casting or extrusion process, or it may be separately brazed or adhered to the surface. From thermal considerations, which option is preferred? • Describe the physical origins of the two heat source terms in the bioheat equation. Under what conditions is the perfusion term a heat sink? • How do heat sinks increase the electric power generated by a thermoelectric device? • Under what conditions do thermal resistances associated with molecule–wall interactions become important?

䊏

Problems

193

References 1. Fried, E., “Thermal Conduction Contribution to Heat Transfer at Contacts,” in R. P. Tye, Ed., Thermal Conductivity, Vol. 2, Academic Press, London, 1969. 2. Eid, J. C., and V. W. Antonetti, “Small Scale Thermal Contact Resistance of Aluminum Against Silicon,” in C. L. Tien, V. P. Carey, and J. K. Ferrel, Eds., Heat Transfer—1986, Vol. 2, Hemisphere, New York, 1986, pp. 659–664. 3. Snaith, B., P. W. O’Callaghan, and S. D. Probert, Appl. Energy, 16, 175, 1984. 4. Yovanovich, M. M., “Theory and Application of Constriction and Spreading Resistance Concepts for Microelectronic Thermal Management,” Presented at the International Symposium on Cooling Technology for Electronic Equipment, Honolulu, 1987. 5. Peterson, G. P., and L. S. Fletcher, “Thermal Contact Resistance of Silicon Chip Bonding Materials,” Proceedings of the International Symposium on Cooling Technology for Electronic Equipment, Honolulu, 1987, pp. 438–448. 6. Yovanovich, M. M., and M. Tuarze, AIAA J. Spacecraft Rockets, 6, 1013, 1969. 7. Madhusudana, C. V., and L. S. Fletcher, AIAA J., 24, 510, 1986. 8. Yovanovich, M. M., “Recent Developments in Thermal Contact, Gap and Joint Conductance Theories and Experiment,” in C. L. Tien, V. P. Carey, and J. K. Ferrel, Eds., Heat Transfer—1986, Vol. 1, Hemisphere, New York, 1986, pp. 35–45.

9. Maxwell, J. C., A Treatise on Electricity and Magnetism, 3rd ed., Oxford University Press, Oxford, 1892. 10. Hamilton, R. L., and O. K. Crosser, I&EC Fund. 1, 187, 1962. 11. Jeffrey, D. J., Proc. Roy. Soc. A, 335, 355, 1973. 12. Hashin Z., and S. Shtrikman, J. Appl. Phys., 33, 3125, 1962. 13. Aichlmayr, H. T., and F. A. Kulacki, “The Effective Thermal Conductivity of Saturated Porous Media,” in J. P. Hartnett, A. Bar-Cohen, and Y. I Cho, Eds., Advances in Heat Transfer, Vol. 39, Academic Press, London, 2006. 14. Harper, D. R., and W. B. Brown, “Mathematical Equations for Heat Conduction in the Fins of Air Cooled Engines,” NACA Report No. 158, 1922. 15. Schneider, P. J., Conduction Heat Transfer, AddisonWesley, Reading, MA, 1957. 16. Diller, K. R., and T. P. Ryan, J. Heat Transfer, 120, 810, 1998. 17. Pennes, H. H., J. Applied Physiology, 85, 5, 1998. 18. Goldsmid, H. J., “Conversion Efficiency and Figure-ofMerit,” in D. M. Rowe, Ed., CRC Handbook of Thermoelectrics, Chap. 3, CRC Press, Boca Raton, 1995. 19. Majumdar, A., Science, 303, 777, 2004. 20. Hodes, M., IEEE Trans. Com. Pack. Tech., 28, 218, 2005. 21. Zhang, Z. M., Nano/Microscale Heat Transfer, McGrawHill, New York, 2007.

Problems Plane and Composite Walls 3.1 Consider the plane wall of Figure 3.1, separating hot and cold fluids at temperatures T앝,1 and T앝,2, respectively. Using surface energy balances as boundary conditions at x 0 and x L (see Equation 2.34), obtain the temperature distribution within the wall and the heat flux in terms of T앝,1, T앝,2, h1, h2, k, and L. 3.2 A new building to be located in a cold climate is being designed with a basement that has an L 200-mm-thick wall. Inner and outer basement wall temperatures are Ti 20 C and To 0 C, respectively. The architect can specify the wall material to be either aerated concrete block with kac 0.15 W/m 䡠 K, or stone mix concrete. To reduce the conduction heat flux through the stone mix wall to a level equivalent to that of the aerated concrete wall, what thickness of extruded polystyrene sheet must be applied onto the inner surface of the stone mix con-

crete wall? Floor dimensions of the basement are 20 m 30 m, and the expected rental rate is $50/m2/ month. What is the yearly cost, in terms of lost rental income, if the stone mix concrete wall with polystyrene insulation is specified? 3.3 The rear window of an automobile is defogged by passing warm air over its inner surface. (a) If the warm air is at T앝,i 40 C and the corresponding convection coefficient is hi 30 W/m2 䡠 K, what are the inner and outer surface temperatures of 4-mm-thick window glass, if the outside ambient air temperature is T앝,o 10 C and the associated convection coefficient is ho 65 W/m2 䡠 K? (b) In practice T앝,o and ho vary according to weather conditions and car speed. For values of ho 2, 65, and 100 W/m2 䡠 K, compute and plot the inner and outer surface temperatures as a function of T앝,o for –30 T앝,o 0 C.

194

Chapter 3

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One-Dimensional, Steady-State Conduction

3.4 The rear window of an automobile is defogged by attaching a thin, transparent, film-type heating element to its inner surface. By electrically heating this element, a uniform heat flux may be established at the inner surface.

(a) Show the thermal circuit representing the steady-state heat transfer situation. Be sure to label all elements, nodes, and heat rates. Leave in symbolic form. (b) Assume the following conditions: T앝 20 C, h 50 W/m2 䡠 K, and T1 30 C. Calculate the heat flux q0 that is required to maintain the bonded surface at T0 60 C.

(a) For 4-mm-thick window glass, determine the electrical power required per unit window area to maintain an inner surface temperature of 15 C when the interior air temperature and convection coefficient are T앝,i 25 C and hi 10 W/m2 䡠 K, while the exterior (ambient) air temperature and convection coefficient are T앝,o 10 C and ho 65 W/m2 䡠 K.

(c) Compute and plot the required heat flux as a function of the film thickness for 0 Lƒ 1 mm. (d) If the film is not transparent and all of the radiant heat flux is absorbed at its upper surface, determine the heat flux required to achieve bonding. Plot your results as a function of Lƒ for 0 Lƒ 1 mm.

(b) In practice T앝,o and ho vary according to weather conditions and car speed. For values of ho 2, 20, 65, and 100 W/m2 䡠 K, determine and plot the electrical power requirement as a function of T앝,o for 30 T앝,o 0 C. From your results, what can you conclude about the need for heater operation at low values of ho? How is this conclusion affected by the value of T앝,o? If h V n, where V is the vehicle speed and n is a positive exponent, how does the vehicle speed affect the need for heater operation?

3.7 The walls of a refrigerator are typically constructed by sandwiching a layer of insulation between sheet metal panels. Consider a wall made from fiberglass insulation of thermal conductivity ki 0.046 W/m 䡠 K and thickness Li 50 mm and steel panels, each of thermal conductivity kp 60 W/m 䡠 K and thickness Lp 3 mm. If the wall separates refrigerated air at T앝, i 4 C from ambient air at T앝,o 25 C, what is the heat gain per unit surface area? Coefficients associated with natural convection at the inner and outer surfaces may be approximated as hi ho 5 W/m2 䡠 K.

3.5 A dormitory at a large university, built 50 years ago, has exterior walls constructed of Ls 25-mm-thick sheathing with a thermal conductivity of ks 0.1 W/m 䡠 K. To reduce heat losses in the winter, the university decides to encapsulate the entire dormitory by applying an Li 25-mm-thick layer of extruded insulation characterized by ki 0.029 W/m 䡠 K to the exterior of the original sheathing. The extruded insulation is, in turn, covered with an Lg 5-mm-thick architectural glass with kg 1.4 W/m 䡠 K. Determine the heat flux through the original and retrofitted walls when the interior and exterior air temperatures are T앝,i 22 C and T앝,o 20 C, respectively. The inner and outer convection heat transfer coefficients are hi 5 W/m2 䡠 K and ho 25 W/m2 䡠 K, respectively. 3.6 In a manufacturing process, a transparent film is being bonded to a substrate as shown in the sketch. To cure the bond at a temperature T0, a radiant source is used to provide a heat flux q0 (W/m2), all of which is absorbed at the bonded surface. The back of the substrate is maintained at T1 while the free surface of the film is exposed to air at T앝 and a convection heat transfer coefficient h. Air

q0"

T∞, h Lf

Film

Ls

Substrate

Bond, T0

T1

Lf = 0.25 mm kf = 0.025 W/m•K Ls = 1.0 mm ks = 0.05 W/m•K

3.8 A t 10-mm-thick horizontal layer of water has a top surface temperature of Tc 4 C and a bottom surface temperature of Th 2 C. Determine the location of the solid–liquid interface at steady state. 3.9 A technique for measuring convection heat transfer coefficients involves bonding one surface of a thin metallic foil to an insulating material and exposing the other surface to the fluid flow conditions of interest. T∞, h Foil ( P"elec, Ts)

L

Foam Insulation (k)

Tb

By passing an electric current through the foil, heat is dissipated uniformly within the foil and the corresponding flux, Pelec, may be inferred from related voltage and current measurements. If the insulation thickness L and thermal conductivity k are known and the fluid, foil, and insulation temperatures (T앝, Ts, Tb) are measured, the convection coefficient may be determined. Consider conditions for which T앝 Tb 25 C, Pelec 2000 W/m2, L 10 mm, and k 0.040 W/m 䡠 K.

䊏

195

Problems

(a) With water flow over the surface, the foil temperature measurement yields Ts 27 C. Determine the convection coefficient. What error would be incurred by assuming all of the dissipated power to be transferred to the water by convection? (b) If, instead, air flows over the surface and the temperature measurement yields Ts 125 C, what is the convection coefficient? The foil has an emissivity of 0.15 and is exposed to large surroundings at 25 C. What error would be incurred by assuming all of the dissipated power to be transferred to the air by convection? (c) Typically, heat flux gages are operated at a fixed temperature (Ts), in which case the power dissipation provides a direct measure of the convection coefficient. For Ts 27 C, plot Pelec as a function of ho for 10 ho 1000 W/m2 䡠 K. What effect does ho have on the error associated with neglecting conduction through the insulation? 3.10 The wind chill, which is experienced on a cold, windy day, is related to increased heat transfer from exposed human skin to the surrounding atmosphere. Consider a layer of fatty tissue that is 3 mm thick and whose interior surface is maintained at a temperature of 36 C. On a calm day the convection heat transfer coefficient at the outer surface is 25 W/m2 䡠 K, but with 30 km/h winds it reaches 65 W/m2 䡠 K. In both cases the ambient air temperature is 15 C. (a) What is the ratio of the heat loss per unit area from the skin for the calm day to that for the windy day? (b) What will be the skin outer surface temperature for the calm day? For the windy day? (c) What temperature would the air have to assume on the calm day to produce the same heat loss occurring with the air temperature at 15 C on the windy day? 3.11 Determine the thermal conductivity of the carbon nanotube of Example 3.4 when the heating island temperature is measured to be Th 332.6 K, without evaluating the thermal resistances of the supports. The conditions are the same as in the example. 3.12 A thermopane window consists of two pieces of glass 7 mm thick that enclose an air space 7 mm thick. The window separates room air at 20 C from outside ambient air at 10 C. The convection coefficient associated with the inner (room-side) surface is 10 W/m2 䡠 K. (a) If the convection coefficient associated with the outer (ambient) air is ho 80 W/m2 䡠 K, what is the heat loss through a window that is 0.8 m long by 0.5 m wide? Neglect radiation, and assume the air enclosed between the panes to be stagnant. (b) Compute and plot the effect of ho on the heat loss for 10 ho 100 W/m2 䡠 K. Repeat this calculation for a

triple-pane construction in which a third pane and a second air space of equivalent thickness are added. 3.13 A house has a composite wall of wood, fiberglass insulation, and plaster board, as indicated in the sketch. On a cold winter day, the convection heat transfer coefficients are ho 60 W/m2 䡠 K and hi 30 W/m2 䡠 K. The total wall surface area is 350 m2. Glass fiber blanket (28 kg/m3), kb Plaster board, kp

Plywood siding, ks

Inside

Outside

hi, T∞, i = 20°C

ho, T∞, o = –15°C

10 mm

100 mm

Lp

20 mm

Ls

Lb

(a) Determine a symbolic expression for the total thermal resistance of the wall, including inside and outside convection effects for the prescribed conditions. (b) Determine the total heat loss through the wall. (c) If the wind were blowing violently, raising ho to 300 W/m2 䡠 K, determine the percentage increase in the heat loss. (d) What is the controlling resistance that determines the amount of heat flow through the wall? 3.14 Consider the composite wall of Problem 3.13 under conditions for which the inside air is still characterized by T앝,i 20 C and hi 30 W/m2 䡠 K. However, use the more realistic conditions for which the outside air is characterized by a diurnal (time) varying temperature of the form T앝,o(K) 273 5 sin

冢224 t冣

T앝,o(K) 273 11 sin

冢224 t冣

0 t 12 h 12 t 24 h

with ho 60 W/m2 䡠 K. Assuming quasi-steady conditions for which changes in energy storage within the wall may be neglected, estimate the daily heat loss through the wall if its total surface area is 200 m2. 3.15 Consider a composite wall that includes an 8-mm-thick hardwood siding, 40-mm by 130-mm hardwood studs on 0.65-m centers with glass fiber insulation (paper

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One-Dimensional, Steady-State Conduction

faced, 28 kg/m3), and a 12-mm layer of gypsum (vermiculite) wall board. Wood siding Stud 130 mm

x

Insulation Wall board

3.19 The wall of a drying oven is constructed by sandwiching an insulation material of thermal conductivity k 0.05 W/m 䡠 K between thin metal sheets. The oven air is at T앝,i 300 C, and the corresponding convection coefficient is hi 30 W/m2 䡠 K. The inner wall surface absorbs a radiant flux of qrad 100 W/m2 from hotter objects within the oven. The room air is at T앝,o 25 C, and the overall coefficient for convection and radiation from the outer surface is ho 10 W/m2 䡠 K.

40 mm

What is the thermal resistance associated with a wall that is 2.5 m high by 6.5 m wide (having 10 studs, each 2.5 m high)? Assume surfaces normal to the x-direction are isothermal. 3.16 Work Problem 3.15 assuming surfaces parallel to the x-direction are adiabatic. 3.17 Consider the oven of Problem 1.54. The walls of the oven consist of L 30-mm-thick layers of insulation characterized by kins 0.03 W/m 䡠 K that are sandwiched between two thin layers of sheet metal. The exterior surface of the oven is exposed to air at 23 C with hext 2 W/m2 䡠 K. The interior oven air temperature is 180 C. Neglecting radiation heat transfer, determine the steady-state heat flux through the oven walls when the convection mode is disabled and the free convection coefficient at the inner oven surface is hfr 3 W/m2 䡠 K. Determine the heat flux through the oven walls when the convection mode is activated, in which case the forced convection coefficient at the inner oven surface is hfo 27 W/m2 䡠 K. Does operation of the oven in its convection mode result in significantly increased heat losses from the oven to the kitchen? Would your conclusion change if radiation were included in your analysis? 3.18 The composite wall of an oven consists of three materials, two of which are of known thermal conductivity, kA 20 W/m 䡠 K and kC 50 W/m 䡠 K, and known thickness, LA 0.30 m and LC 0.15 m. The third material, B, which is sandwiched between materials A and C, is of known thickness, LB 0.15 m, but unknown thermal conductivity kB. Ts, i

kA

kB

kC

LA

LB

LC

Ts,o

Air

T∞, h

Under steady-state operating conditions, measurements reveal an outer surface temperature of Ts,o 20 C, an inner surface temperature of Ts,i 600 C, and an oven air temperature of T앝 800 C. The inside convection coefficient h is known to be 25 W/m2 䡠 K. What is the value of kB?

Absorbed radiation, q"rad

Insulation, k

To

Oven air

Room air

T∞,i, hi

T∞,o, ho

L

(a) Draw the thermal circuit for the wall and label all temperatures, heat rates, and thermal resistances. (b) What insulation thickness L is required to maintain the outer wall surface at a safe-to-touch temperature of To 40 C? 3.20 The t 4-mm-thick glass windows of an automobile have a surface area of A 2.6 m2. The outside temperature is T앝,o 32 C while the passenger compartment is maintained at T앝,i 22 C. The convection heat transfer coefficient on the exterior window surface is ho 90 W/m2 䡠 K. Determine the heat gain through the windows when the interior convection heat transfer coefficient is hi 15 W/m2 䡠 K. By controlling the airflow in the passenger compartment the interior heat transfer coefficient can be reduced to hi 5 W/m2 䡠 K without sacrificing passenger comfort. Determine the heat gain through the window for the reduced inside heat transfer coefficient. 3.21 The thermal characteristics of a small, dormitory refrigerator are determined by performing two separate experiments, each with the door closed and the refrigerator placed in ambient air at T앝 25 C. In one case, an electric heater is suspended in the refrigerator cavity, while the refrigerator is unplugged. With the heater dissipating 20 W, a steady-state temperature of 90 C is recorded within the cavity. With the heater removed and the refrigerator now in operation, the second experiment involves maintaining a steady-state cavity temperature of 5 C for a fixed time interval and recording the electrical energy required to operate the refrigerator. In such an experiment for which steady operation is maintained over a 12-hour period, the input electrical energy is 125,000 J. Determine the refrigerator’s coefficient of performance (COP). 3.22 In the design of buildings, energy conservation requirements dictate that the exterior surface area, As, be minimized. This requirement implies that, for a desired floor

䊏

197

Problems

space, there may be optimum values associated with the number of floors and horizontal dimensions of the building. Consider a design for which the total floor space, Af , and the vertical distance between floors, Hf , are prescribed.

aerogel (k 0.006 W/m 䡠 K). The temperatures of the surroundings and the ambient are Tsur 300 K and T앝 298 K, respectively. The outer surface is characterized by a convective heat transfer coefficient of h 12 W/m2 䡠 K.

(a) If the building has a square cross section of width W on a side, obtain an expression for the value of W that would minimize heat loss to the surroundings. Heat loss may be assumed to occur from the four vertical side walls and from a flat roof. Express your result in terms of Af and Hf.

(b) Calculate the outer surface temperature of the canister for the four cases (high and low thermal conductivity; high and low surface emissivity).

(b) If Af 32,768 m2 and Hf 4 m, for what values of W and Nf (the number of floors) is the heat loss minimized? If the average overall heat transfer coefficient is U 1 W/m2 䡠 K and the difference between the inside and ambient air temperatures is 25 C, what is the corresponding heat loss? What is the percentage reduction in heat loss compared with a building for Nf 2?

3.24 A firefighter’s protective clothing, referred to as a turnout coat, is typically constructed as an ensemble of three layers separated by air gaps, as shown schematically.

3.23 When raised to very high temperatures, many conventional liquid fuels dissociate into hydrogen and other components. Thus the advantage of a solid oxide fuel cell is that such a device can internally reform readily available liquid fuels into hydrogen that can then be used to produce electrical power in a manner similar to Example 1.5. Consider a portable solid oxide fuel cell, operating at a temperature of Tfc 800 C. The fuel cell is housed within a cylindrical canister of diameter D 75 mm and length L 120 mm. The outer surface of the canister is insulated with a low-thermal-conductivity material. For a particular application, it is desired that the thermal signature of the canister be small, to avoid its detection by infrared sensors. The degree to which the canister can be detected with an infrared sensor may be estimated by equating the radiation heat flux emitted from the exterior surface of the canister (Equation 1.5; Es sT 4s ) to the heat flux emitted from an equivalent black surface, (Eb T b4). If the equivalent black surface temperature Tb is near the surroundings temperature, the thermal signature of the canister is too small to be detected—the canister is indistinguishable from the surroundings. (a) Determine the required thickness of insulation to be applied to the cylindrical wall of the canister to ensure that the canister does not become highly visible to an infrared sensor (i.e., Tb Tsur 5 K). Consider cases where (i) the outer surface is covered with a very thin layer of dirt (s 0.90) and (ii) the outer surface is comprised of a very thin polished aluminum sheet (s 0.08). Calculate the required thicknesses for two types of insulating material, calcium silicate (k 0.09 W/m 䡠 K) and

(c) Calculate the heat loss from the cylindrical walls of the canister for the four cases.

Moisture barrier (mb)

Shell (s)

1 mm

Fire-side

ks, Ls

kmb Lmb

Air gap

Thermal liner (tl)

1 mm

k tl L tl

Firefighter

Air gap

Representative dimensions and thermal conductivities for the layers are as follows. Layer Shell (s) Moisture barrier (mb) Thermal liner (tl)

Thickness (mm) 0.8 0.55 3.5

k (W/m 䡠 K) 0.047 0.012 0.038

The air gaps between the layers are 1 mm thick, and heat is transferred by conduction and radiation exchange through the stagnant air. The linearized radiation coefficient for a gap may be approximated 3 as, hrad (T1 T2)(T 12 T 22) 艐 4T avg , where Tavg represents the average temperature of the surfaces comprising the gap, and the radiation flux across the gap may be expressed as qrad hrad (T1 T2). (a) Represent the turnout coat by a thermal circuit, labeling all the thermal resistances. Calculate and tabulate the thermal resistances per unit area (m2 䡠 K/W) for each of the layers, as well as for the conduction and radiation processes in the gaps. Assume that a value of Tavg 470 K may be used to approximate the radiation resistance of both gaps. Comment on the relative magnitudes of the resistances. (b) For a pre-flash-over fire environment in which firefighters often work, the typical radiant heat flux on the fire-side of the turnout coat is 0.25 W/cm2.

198

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One-Dimensional, Steady-State Conduction

What is the outer surface temperature of the turnout coat if the inner surface temperature is 66 C, a condition that would result in burn injury? 3.25 A particular thermal system involves three objects of fixed shape with conduction resistances of R1 1 K/W, R2 2 K/W and R3 4 K/W, respectively. An objective is to minimize the total thermal resistance Rtot associated with a combination of R1, R2, and R3. The chief engineer is willing to invest limited funds to specify an alternative material for just one of the three objects; the alternative material will have a thermal conductivity that is twice its nominal value. Which object (1, 2, or 3) should be fabricated of the higher thermal conductivity material to most significantly decrease Rtot? Hint: Consider two cases, one for which the three thermal resistances are arranged in series, and the second for which the three resistances are arranged in parallel.

Contact Resistance 3.26 A composite wall separates combustion gases at 2600 C from a liquid coolant at 100 C, with gas- and liquid-side convection coefficients of 50 and 1000 W/m2 䡠 K. The wall is composed of a 10-mm-thick layer of beryllium oxide on the gas side and a 20-mm-thick slab of stainless steel (AISI 304) on the liquid side. The contact resistance between the oxide and the steel is 0.05 m2 䡠 K/W. What is the heat loss per unit surface area of the composite? Sketch the temperature distribution from the gas to the liquid. 3.27 Approximately 106 discrete electrical components can be placed on a single integrated circuit (chip), with electrical heat dissipation as high as 30,000 W/m2. The chip, which is very thin, is exposed to a dielectric liquid at its outer surface, with ho 1000 W/m2 䡠 K and T앝,o 20 C, and is joined to a circuit board at its inner surface. The thermal contact resistance between the chip and the board is 104 m2 䡠 K/W, and the board thickness and thermal conductivity are Lb 5 mm and kb 1 W/m 䡠 K, respectively. The other surface of the board is exposed to ambient air for which hi 40 W/m2 䡠 K and T앝,i 20 C.

(a) Sketch the equivalent thermal circuit corresponding to steady-state conditions. In variable form, label appropriate resistances, temperatures, and heat fluxes. (b) Under steady-state conditions for which the chip heat dissipation is qc 30,000 W/m2, what is the chip temperature? (c) The maximum allowable heat flux, qc,m, is determined by the constraint that the chip temperature must not exceed 85 C. Determine qc,m for the foregoing conditions. If air is used in lieu of the dielectric liquid, the convection coefficient is reduced by approximately an order of magnitude. What is the value of qc,m for ho 100 W/m2 䡠 K? With air cooling, can significant improvements be realized by using an aluminum oxide circuit board and/or by using a conductive paste at the chip/board interface for which Rt, c 105 m2 䡠 K/W? 3.28 Two stainless steel plates 10 mm thick are subjected to a contact pressure of 1 bar under vacuum conditions for which there is an overall temperature drop of 100 C across the plates. What is the heat flux through the plates? What is the temperature drop across the contact plane? 3.29 Consider a plane composite wall that is composed of two materials of thermal conductivities kA 0.1 W/m 䡠 K and kB 0.04 W/m 䡠 K and thicknesses LA 10 mm and LB 20 mm. The contact resistance at the interface between the two materials is known to be 0.30 m2 䡠 K/W. Material A adjoins a fluid at 200 C for which h 10 W/m2 䡠 K, and material B adjoins a fluid at 40 C for which h 20 W/m2 䡠 K. (a) What is the rate of heat transfer through a wall that is 2 m high by 2.5 m wide? (b) Sketch the temperature distribution. 3.30 The performance of gas turbine engines may be improved by increasing the tolerance of the turbine blades to hot gases emerging from the combustor. One approach to achieving high operating temperatures involves application of a thermal barrier coating (TBC) to the exterior surface of a blade, while passing cooling air through the blade. Typically, the blade is made from a high-temperature superalloy, such as Inconel (k ⬇ 25 W/m 䡠 K), while a ceramic, such as zirconia (k ⬇ 1.3 W/m 䡠 K), is used as a TBC.

Coolant

T∞,o, ho

Superalloy Chip q"c, Tc Thermal contact resistance, R"t,c Board, kb

Lb

Cooling air

T∞,i, hi Hot gases

T∞,o, ho

Air

Bonding agent

T∞,i, hi

TBC

䊏

199

Problems

Consider conditions for which hot gases at T앝,o 1700 K and cooling air at T앝,i 400 K provide outer and inner surface convection coefficients of ho 1000 W/m2 䡠 K and hi 500 W/m2 䡠 K, respectively. If a 0.5-mm-thick zirconia TBC is attached to a 5-mmthick Inconel blade wall by means of a metallic bonding agent, which provides an interfacial thermal resistance of Rt,c 104 m2 䡠 K/W, can the Inconel be maintained at a temperature that is below its maximum allowable value of 1250 K? Radiation effects may be neglected, and the turbine blade may be approximated as a plane wall. Plot the temperature distribution with and without the TBC. Are there any limits to the thickness of the TBC? 3.31 A commercial grade cubical freezer, 3 m on a side, has a composite wall consisting of an exterior sheet of 6.35-mm-thick plain carbon steel, an intermediate layer of 100-mm-thick cork insulation, and an inner sheet of 6.35-mm-thick aluminum alloy (2024). Adhesive interfaces between the insulation and the metallic strips are each characterized by a thermal contact resistance of Rt,c 2.5 104 m2 䡠 K/W. What is the steady-state cooling load that must be maintained by the refrigerator under conditions for which the outer and inner surface temperatures are 22 C and 6 C, respectively? 3.32 Physicists have determined the theoretical value of the thermal conductivity of a carbon nanotube to be kcn,T 5000 W/m 䡠 K. (a) Assuming the actual thermal conductivity of the carbon nanotube is the same as its theoretical value, find the thermal contact resistance, Rt,c, that exists between the carbon nanotube and the top surfaces of the heated and sensing islands in Example 3.4 .

Tsur

Transistor case Ts,c, Pelec

Base plate, (k,ε ) Interface, Ac

W Enclosure

Air

T∞, h L

(a) If the air-filled aluminum-to-aluminum interface is characterized by an area of Ac 2 104 m2 and a roughness of 10 m, what is the maximum allowable power dissipation if the surface temperature of the case, Ts,c, is not to exceed 85 C? (b) The convection coefficient may be increased by subjecting the plate surface to a forced flow of air. Explore the effect of increasing the coefficient over the range 4 h 200 W/m2 䡠 K.

Porous Media 3.34 Ring-porous woods, such as oak, are characterized by grains. The dark grains consist of very low-density material that forms early in the springtime. The surrounding lighter-colored wood is composed of highdensity material that forms slowly throughout most of the growing season. Wood grain (low-density)

(b) Using the value of the thermal contact resistance calculated in part (a), plot the fraction of the total resistance between the heated and sensing islands that is due to the thermal contact resistances for island separation distances of 5 m s 20 m. 3.33 Consider a power transistor encapsulated in an aluminum case that is attached at its base to a square aluminum plate of thermal conductivity k 240 W/m 䡠 K, thickness L 6 mm, and width W 20 mm. The case is joined to the plate by screws that maintain a contact pressure of 1 bar, and the back surface of the plate transfers heat by natural convection and radiation to ambient air and large surroundings at T앝 Tsur 25 C. The surface has an emissivity of 0.9, and the convection coefficient is h 4 W/m2 䡠 K. The case is completely enclosed such that heat transfer may be assumed to occur exclusively through the base plate.

High-density material

Assuming the low-density material is highly porous and the oak is dry, determine the fraction of the oak crosssection that appears as being grained. Hint: Assume the thermal conductivity parallel to the grains is the same as the radial conductivity of Table A.3. 3.35 A batt of glass fiber insulation is of density 28 kg/m3. Determine the maximum and minimum possible values of the effective thermal conductivity of the insulation at T 300 K, and compare with the value reported in Table A.3.

200

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

3.36 Air usually constitutes up to half of the volume of commercial ice creams and takes the form of small spherical bubbles interspersed within a matrix of frozen matter. The thermal conductivity of ice cream that contains no air is kna 1.1 W/m 䡠 K at T –20 C. Determine the thermal conductivity of commercial ice cream characterized by 0.20, also at T –20 C. 3.37 Determine the density, specific heat, and thermal conductivity of a lightweight aggregate concrete that is composed of 65% stone mix concrete and 35% air by volume. Evaluate properties at T 300 K. 3.38 A one-dimensional plane wall of thickness L is constructed of a solid material with a linear, nonuniform porosity distribution described by (x) max(x/L). Plot the steady-state temperature distribution, T(x), for ks 10 W/m 䡠 K, kf 0.1 W/m 䡠 K, L 1 m, max 0.25, T(x 0) 30 C and qx 100 W/m2 using the expression for the minimum effective thermal conductivity of a porous medium, the expression for the maximum effective thermal conductivity of a porous medium, Maxwell’s expression, and for the case where keff(x) ks.

Alternative Conduction Analysis 3.39 The diagram shows a conical section fabricated from pure aluminum. It is of circular cross section having diameter D ax1/2, where a 0.5 m1/2. The small end is located at x1 25 mm and the large end at x2 125 mm. The end temperatures are T1 600 K and T2 400 K, while the lateral surface is well insulated. T2 T1

x1 x

x2

(a) Derive an expression for the temperature distribution T(x) in symbolic form, assuming one-dimensional conditions. Sketch the temperature distribution.

0

x1 T1 x2

T2

The sides are well insulated, while the top surface of the cone at x1 is maintained at T1 and the bottom surface at x2 is maintained at T2. (a) Obtain an expression for the temperature distribution T(x). (b) What is the rate of heat transfer across the cone if it is constructed of pure aluminum with x1 0.075 m, T1 100 C, x2 0.225 m, and T2 20 C? 3.41 From Figure 2.5 it is evident that, over a wide temperature range, the temperature dependence of the thermal conductivity of many solids may be approximated by a linear expression of the form k ko aT, where ko is a positive constant and a is a coefficient that may be positive or negative. Obtain an expression for the heat flux across a plane wall whose inner and outer surfaces are maintained at T0 and T1, respectively. Sketch the forms of the temperature distribution corresponding to a 0, a 0, and a 0. 3.42 Consider a tube wall of inner and outer radii ri and ro, whose temperatures are maintained at Ti and To, respectively. The thermal conductivity of the cylinder is temperature dependent and may be represented by an expression of the form k ko(1 aT), where ko and a are constants. Obtain an expression for the heat transfer per unit length of the tube. What is the thermal resistance of the tube wall? 3.43 Measurements show that steady-state conduction through a plane wall without heat generation produced a convex temperature distribution such that the midpoint temperature was To higher than expected for a linear temperature distribution.

T1 T ( x)

TL/2 ∆To

(b) Calculate the heat rate qx. 3.40 A truncated solid cone is of circular cross section, and its diameter is related to the axial coordinate by an expression of the form D ax3/2, where a 1.0 m1/2.

T2

L

䊏

201

Problems

Assuming that the thermal conductivity has a linear dependence on temperature, k ko(1 ␣T), where ␣ is a constant, develop a relationship to evaluate ␣ in terms of To, T1, and T2. 3.44 A device used to measure the surface temperature of an object to within a spatial resolution of approximately 50 nm is shown in the schematic. It consists of an extremely sharp-tipped stylus and an extremely small cantilever that is scanned across the surface. The probe tip is of circular cross section and is fabricated of polycrystalline silicon dioxide. The ambient temperature is measured at the pivoted end of the cantilever as T⬁ 25 C, and the device is equipped with a sensor to measure the temperature at the upper end of the sharp tip, Tsen. The thermal resistance between the sensing probe and the pivoted end is Rt 5 106 K/W. (a) Determine the thermal resistance between the surface temperature and the sensing temperature. (b) If the sensing temperature is Tsen 28.5 C, determine the surface temperature. Hint: Although nanoscale heat transfer effects may be important, assume that the conduction occurring in the air adjacent to the probe tip can be described by Fourier’s law and the thermal conductivity found in Table A.4. Tsen

T∞ = 25°C

Cantilever

(T앝 25 C) that maintains a convection coefficient of h 25 W/m2 䡠 K and to large surroundings for which Tsur T앝 25 C. The surface emissivity of calcium silicate is approximately 0.8. Compute and plot the temperature distribution in the insulation as a function of the dimensionless radial coordinate, (r r1)/(r2 r1), where r1 0.06 m and r2 is a variable (0.06 r2 0.20 m). Compute and plot the heat loss as a function of the insulation thickness for 0 (r2 r1) 0.14 m. 3.46 Consider the water heater described in Problem 1.48. We now wish to determine the energy needed to compensate for heat losses incurred while the water is stored at the prescribed temperature of 55 C. The cylindrical storage tank (with flat ends) has a capacity of 100 gal, and foamed urethane is used to insulate the side and end walls from ambient air at an annual average temperature of 20 C. The resistance to heat transfer is dominated by conduction in the insulation and by free convection in the air, for which h ⬇ 2 W/m2 䡠 K. If electric resistance heating is used to compensate for the losses and the cost of electric power is $0.18/kWh, specify tank and insulation dimensions for which the annual cost associated with the heat losses is less than $50. 3.47 To maximize production and minimize pumping costs, crude oil is heated to reduce its viscosity during transportation from a production field.

Stylus

Tsen

Surface

d = 100 nm Air

Tsurf

L = 50 nm

Cylindrical Wall 3.45 A steam pipe of 0.12-m outside diameter is insulated with a layer of calcium silicate. (a) If the insulation is 20 mm thick and its inner and outer surfaces are maintained at Ts,1 800 K and Ts,2 490 K, respectively, what is the heat loss per unit length (q) of the pipe? (b) We wish to explore the effect of insulation thickness on the heat loss q and outer surface temperature Ts,2, with the inner surface temperature fixed at Ts,1 800 K. The outer surface is exposed to an airflow

(a) Consider a pipe-in-pipe configuration consisting of concentric steel tubes with an intervening insulating material. The inner tube is used to transport warm crude oil through cold ocean water. The inner steel pipe (ks 35 W/m 䡠 K) has an inside diameter of Di,1 150 mm and wall thickness ti 10 mm while the outer steel pipe has an inside diameter of Di,2 250 mm and wall thickness to ti. Determine the maximum allowable crude oil temperature to ensure the polyurethane foam insulation (kp 0.075 W/m 䡠 K) between the two pipes does not exceed its maximum service temperature of Tp,max 70 C. The ocean water is at T앝,o –5 C and provides an external convection heat transfer coefficient of ho 500 W/m2 䡠 K. The convection coefficient associated with the flowing crude oil is hi 450 W/m2 䡠 K. (b) It is proposed to enhance the performance of the pipe-in-pipe device by replacing a thin (ta 5 mm) section of polyurethane located at the outside of the inner pipe with an aerogel insulation material (ka 0.012 W/m 䡠 K). Determine the maximum allowable crude oil temperature to ensure maximum polyurethane temperatures are below Tp,max 70 C.

202

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

3.48 A thin electrical heater is wrapped around the outer surface of a long cylindrical tube whose inner surface is maintained at a temperature of 5 C. The tube wall has inner and outer radii of 25 and 75 mm, respectively, and a thermal conductivity of 10 W/m 䡠 K. The thermal contact resistance between the heater and the outer surface of the tube (per unit length of the tube) is Rt,c 0.01 m 䡠 K/W. The outer surface of the heater is exposed to a fluid with T앝 10 C and a convection coefficient of h 100 W/m2 䡠 K. Determine the heater power per unit length of tube required to maintain the heater at To 25 C. 3.49 In Problem 3.48, the electrical power required to maintain the heater at To 25 C depends on the thermal conductivity of the wall material k, the thermal contact resistance Rt,c and the convection coefficient h. Compute and plot the separate effect of changes in k (1 k 200 W/m 䡠 K), Rt,c (0 Rt,c 0.1 m 䡠 K/W), and h (10 h 1000 W/m2 䡠 K) on the total heater power requirement, as well as the rate of heat transfer to the inner surface of the tube and to the fluid. 3.50 A stainless steel (AISI 304) tube used to transport a chilled pharmaceutical has an inner diameter of 36 mm and a wall thickness of 2 mm. The pharmaceutical and ambient air are at temperatures of 6 C and 23 C, respectively, while the corresponding inner and outer convection coefficients are 400 W/m2 䡠 K and 6 W/m2 䡠 K, respectively. (a) What is the heat gain per unit tube length? (b) What is the heat gain per unit length if a 10-mmthick layer of calcium silicate insulation (kins 0.050 W/m 䡠 K) is applied to the tube? 3.51 Superheated steam at 575 C is routed from a boiler to the turbine of an electric power plant through steel tubes (k 35 W/m 䡠 K) of 300-mm inner diameter and 30-mm wall thickness. To reduce heat loss to the surroundings and to maintain a safe-to-touch outer surface temperature, a layer of calcium silicate insulation (k 0.10 W/m 䡠 K) is applied to the tubes, while degradation of the insulation is reduced by wrapping it in a thin sheet of aluminum having an emissivity of 0.20. The air and wall temperatures of the power plant are 27 C. (a) Assuming that the inner surface temperature of a steel tube corresponds to that of the steam and the convection coefficient outside the aluminum sheet is 6 W/m2 䡠 K, what is the minimum insulation thickness needed to ensure that the temperature of the aluminum does not exceed 50 C? What is the corresponding heat loss per meter of tube length?

(b) Explore the effect of the insulation thickness on the temperature of the aluminum and the heat loss per unit tube length. 3.52 A thin electrical heater is inserted between a long circular rod and a concentric tube with inner and outer radii of 20 and 40 mm. The rod (A) has a thermal conductivity of kA 0.15 W/m 䡠 K, while the tube (B) has a thermal conductivity of kB 1.5 W/m 䡠 K and its outer surface is subjected to convection with a fluid of temperature T앝 15 C and heat transfer coefficient 50 W/m2 䡠 K. The thermal contact resistance between the cylinder surfaces and the heater is negligible. (a) Determine the electrical power per unit length of the cylinders (W/m) that is required to maintain the outer surface of cylinder B at 5 C. (b) What is the temperature at the center of cylinder A? 3.53 A wire of diameter D 2 mm and uniform temperature T has an electrical resistance of 0.01 /m and a current flow of 20 A. (a) What is the rate at which heat is dissipated per unit length of wire? What is the heat dissipation per unit volume within the wire? (b) If the wire is not insulated and is in ambient air and large surroundings for which T앝 Tsur 20 C, what is the temperature T of the wire? The wire has an emissivity of 0.3, and the coefficient associated with heat transfer by natural convection may be approximated by an expression of the form, h C[(T T앝)/D]1/4, where C 1.25 W/m7/4 䡠 K5/4. (c) If the wire is coated with plastic insulation of 2-mm thickness and a thermal conductivity of 0.25 W/m 䡠 K, what are the inner and outer surface temperatures of the insulation? The insulation has an emissivity of 0.9, and the convection coefficient is given by the expression of part (b). Explore the effect of the insulation thickness on the surface temperatures. 3.54 A 2-mm-diameter electrical wire is insulated by a 2-mm-thick rubberized sheath (k 0.13 W/m 䡠 K), and the wire/sheath interface is characterized by a thermal 3 104 m2 䡠 K/W. The concontact resistance of Rt,c vection heat transfer coefficient at the outer surface of the sheath is 10 W/m2 䡠 K, and the temperature of the ambient air is 20 C. If the temperature of the insulation may not exceed 50 C, what is the maximum allowable electrical power that may be dissipated per unit length of the conductor? What is the critical radius of the insulation?

䊏

203

Problems

3.55 Electric current flows through a long rod generating thermal energy at a uniform volumetric rate of q˙ 2 106 W/m3. The rod is concentric with a hollow ceramic cylinder, creating an enclosure that is filled with air. To = 25°C Tr

heater for which interfacial contact resistances are negligible.

Resistance heater q"h, Th

r3 Ceramic, k = 1.75 W/m•K Di = 40 mm Do = 120 mm

r2 r1

Enclosure, air space •

Rod, q, Dr = 20 mm

Internal flow

T∞,i, hi

The thermal resistance per unit length due to radiation between the enclosure surfaces is Rrad 0.30 m 䡠 K/W, and the coefficient associated with free convection in the enclosure is h 20 W/m2 䡠 K. (a) Construct a thermal circuit that can be used to calculate the surface temperature of the rod, Tr . Label all temperatures, heat rates, and thermal resistances, and evaluate each thermal resistance. (b) Calculate the surface temperature of the rod for the prescribed conditions. 3.56 The evaporator section of a refrigeration unit consists of thin-walled, 10-mm-diameter tubes through which refrigerant passes at a temperature of 18 C. Air is cooled as it flows over the tubes, maintaining a surface convection coefficient of 100 W/m2 䡠 K, and is subsequently routed to the refrigerator compartment. (a) For the foregoing conditions and an air temperature of 3 C, what is the rate at which heat is extracted from the air per unit tube length? (b) If the refrigerator’s defrost unit malfunctions, frost will slowly accumulate on the outer tube surface. Assess the effect of frost formation on the cooling capacity of a tube for frost layer thicknesses in the range 0 ␦ 4 mm. Frost may be assumed to have a thermal conductivity of 0.4 W/m 䡠 K. (c) The refrigerator is disconnected after the defrost unit malfunctions and a 2-mm-thick layer of frost has formed. If the tubes are in ambient air for which T앝 20 C and natural convection maintains a convection coefficient of 2 W/m2 䡠 K, how long will it take for the frost to melt? The frost may be assumed to have a mass density of 700 kg/m3 and a latent heat of fusion of 334 kJ/kg. 3.57 A composite cylindrical wall is composed of two materials of thermal conductivity kA and kB, which are separated by a very thin, electric resistance

B A Ambient air

T∞,o, ho

Liquid pumped through the tube is at a temperature T앝,i and provides a convection coefficient hi at the inner surface of the composite. The outer surface is exposed to ambient air, which is at T앝,o and provides a convection coefficient of ho. Under steady-state conditions, a uniform heat flux of qh is dissipated by the heater. (a) Sketch the equivalent thermal circuit of the system and express all resistances in terms of relevant variables. (b) Obtain an expression that may be used to determine the heater temperature, Th. (c) Obtain an expression for the ratio of heat flows to the outer and inner fluids, qo /qi. How might the variables of the problem be adjusted to minimize this ratio? 3.58 An electrical current of 700 A flows through a stainless steel cable having a diameter of 5 mm and an electrical resistance of 6 104 /m (i.e., per meter of cable length). The cable is in an environment having a temperature of 30 C, and the total coefficient associated with convection and radiation between the cable and the environment is approximately 25 W/m2 䡠 K. (a) If the cable is bare, what is its surface temperature? (b) If a very thin coating of electrical insulation is applied to the cable, with a contact resistance of 0.02 m2 䡠 K/W, what are the insulation and cable surface temperatures? (c) There is some concern about the ability of the insulation to withstand elevated temperatures. What thickness of this insulation (k 0.5 W/m 䡠 K) will yield the lowest value of the maximum insulation temperature? What is the value of the maximum temperature when this thickness is used?

204

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

3.59 A 0.20-m-diameter, thin-walled steel pipe is used to transport saturated steam at a pressure of 20 bars in a room for which the air temperature is 25 C and the convection heat transfer coefficient at the outer surface of the pipe is 20 W/m2 䡠 K. (a) What is the heat loss per unit length from the bare pipe (no insulation)? Estimate the heat loss per unit length if a 50-mm-thick layer of insulation (magnesia, 85%) is added. The steel and magnesia may each be assumed to have an emissivity of 0.8, and the steam-side convection resistance may be neglected. (b) The costs associated with generating the steam and installing the insulation are known to be $4/109 J and $100/m of pipe length, respectively. If the steam line is to operate 7500 h/yr, how many years are needed to pay back the initial investment in insulation? 3.60 An uninsulated, thin-walled pipe of 100-mm diameter is used to transport water to equipment that operates outdoors and uses the water as a coolant. During particularly harsh winter conditions, the pipe wall achieves a temperature of –15 C and a cylindrical layer of ice forms on the inner surface of the wall. If the mean water temperature is 3 C and a convection coefficient of 2000 W/m2 䡠 K is maintained at the inner surface of the ice, which is at 0 C, what is the thickness of the ice layer? 3.61 Steam flowing through a long, thin-walled pipe maintains the pipe wall at a uniform temperature of 500 K. The pipe is covered with an insulation blanket comprised of two different materials, A and B. The interface between the two materials may be assumed to have an infinite contact resistance, and the entire outer surface is exposed to air for which T앝 300 K and h 25 W/m2 䡠 K. r1 = 50 mm A

Ts,2(A)

kA = 2 W/m K •

kB = 0.25 W/m•K

Ts,2(B) Ts,1 = 500 K

r2 = 100 mm

B

T∞, h

(a) Sketch the thermal circuit of the system. Label (using the preceding symbols) all pertinent nodes and resistances. (b) For the prescribed conditions, what is the total heat loss from the pipe? What are the outer surface temperatures Ts,2(A) and Ts,2(B)?

3.62 A bakelite coating is to be used with a 10-mm-diameter conducting rod, whose surface is maintained at 200 C by passage of an electrical current. The rod is in a fluid at 25 C, and the convection coefficient is 140 W/m2 䡠 K. What is the critical radius associated with the coating? What is the heat transfer rate per unit length for the bare rod and for the rod with a coating of bakelite that corresponds to the critical radius? How much bakelite should be added to reduce the heat transfer associated with the bare rod by 25%?

Spherical Wall 3.63 A storage tank consists of a cylindrical section that has a length and inner diameter of L 2 m and Di 1 m, respectively, and two hemispherical end sections. The tank is constructed from 20-mm-thick glass (Pyrex) and is exposed to ambient air for which the temperature is 300 K and the convection coefficient is 10 W/m2 䡠 K. The tank is used to store heated oil, which maintains the inner surface at a temperature of 400 K. Determine the electrical power that must be supplied to a heater submerged in the oil if the prescribed conditions are to be maintained. Radiation effects may be neglected, and the Pyrex may be assumed to have a thermal conductivity of 1.4 W/m 䡠 K. 3.64 Consider the liquid oxygen storage system and the laboratory environmental conditions of Problem 1.49. To reduce oxygen loss due to vaporization, an insulating layer should be applied to the outer surface of the container. Consider using a laminated aluminum foil/glass mat insulation, for which the thermal conductivity and surface emissivity are k 0.00016 W/m 䡠 K and 0.20, respectively. (a) If the container is covered with a 10-mm-thick layer of insulation, what is the percentage reduction in oxygen loss relative to the uncovered container? (b) Compute and plot the oxygen evaporation rate (kg/s) as a function of the insulation thickness t for 0 t 50 mm. 3.65 A spherical Pyrex glass shell has inside and outside diameters of D1 0.1 m and D2 0.2 m, respectively. The inner surface is at Ts,1 100 C while the outer surface is at Ts,2 45 C. (a) Determine the temperature at the midpoint of the shell thickness, T(rm 0.075 m). (b) For the same surface temperatures and dimensions as in part (a), show how the midpoint temperature would change if the shell material were aluminum. 3.66 In Example 3.6, an expression was derived for the critical insulation radius of an insulated, cylindrical tube.

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205

Problems

Derive the expression that would be appropriate for an insulated sphere. 3.67 A hollow aluminum sphere, with an electrical heater in the center, is used in tests to determine the thermal conductivity of insulating materials. The inner and outer radii of the sphere are 0.15 and 0.18 m, respectively, and testing is done under steady-state conditions with the inner surface of the aluminum maintained at 250 C. In a particular test, a spherical shell of insulation is cast on the outer surface of the sphere to a thickness of 0.12 m. The system is in a room for which the air temperature is 20 C and the convection coefficient at the outer surface of the insulation is 30 W/m2 䡠 K. If 80 W are dissipated by the heater under steady-state conditions, what is the thermal conductivity of the insulation? 3.68 A spherical tank for storing liquid oxygen on the space shuttle is to be made from stainless steel of 0.80-m outer diameter and 5-mm wall thickness. The boiling point and latent heat of vaporization of liquid oxygen are 90 K and 213 kJ/kg, respectively. The tank is to be installed in a large compartment whose temperature is to be maintained at 240 K. Design a thermal insulation system that will maintain oxygen losses due to boiling below 1 kg/day. 3.69 A spherical, cryosurgical probe may be imbedded in diseased tissue for the purpose of freezing, and thereby destroying, the tissue. Consider a probe of 3-mm diameter whose surface is maintained at 30 C when imbedded in tissue that is at 37 C. A spherical layer of frozen tissue forms around the probe, with a temperature of 0 C existing at the phase front (interface) between the frozen and normal tissue. If the thermal conductivity of frozen tissue is approximately 1.5 W/m 䡠 K and heat transfer at the phase front may be characterized by an effective convection coefficient of 50 W/m2 䡠 K, what is the thickness of the layer of frozen tissue (assuming negligible perfusion)? 3.70 A spherical vessel used as a reactor for producing pharmaceuticals has a 10-mm-thick stainless steel wall (k 17 W/m 䡠 K) and an inner diameter of l m. The exterior surface of the vessel is exposed to ambient air (T앝 25 C) for which a convection coefficient of 6 W/m2 䡠 K may be assumed. (a) During steady-state operation, an inner surface temperature of 50 C is maintained by energy generated within the reactor. What is the heat loss from the vessel? (b) If a 20-mm-thick layer of fiberglass insulation (k 0.040 W/m 䡠 K) is applied to the exterior of the vessel and the rate of thermal energy generation is unchanged, what is the inner surface temperature of the vessel?

3.71 The wall of a spherical tank of 1-m diameter contains an exothermic chemical reaction and is at 200 C when the ambient air temperature is 25 C. What thickness of urethane foam is required to reduce the exterior temperature to 40 C, assuming the convection coefficient is 20 W/m2 䡠 K for both situations? What is the percentage reduction in heat rate achieved by using the insulation? 3.72 A composite spherical shell of inner radius r1 0.25 m is constructed from lead of outer radius r2 0.30 m and AISI 302 stainless steel of outer radius r3 0.31 m. The cavity is filled with radioactive wastes that generate heat at a rate of q˙ 5 105 W/m3. It is proposed to submerge the container in oceanic waters that are at a temperature of T앝 10 C and provide a uniform convection coefficient of h 500 W/m2 䡠 K at the outer surface of the container. Are there any problems associated with this proposal? 3.73 The energy transferred from the anterior chamber of the eye through the cornea varies considerably depending on whether a contact lens is worn. Treat the eye as a spherical system and assume the system to be at steady state. The convection coefficient ho is unchanged with and without the contact lens in place. The cornea and the lens cover one-third of the spherical surface area.

r1

r2 r3

Anterior chamber

T∞,i, hi

Cornea

k1

T∞,o, ho k2 Contact lens

Values of the parameters representing this situation are as follows: r1 10.2 mm r3 16.5 mm T앝,i 37 C k1 0.35 W/m 䡠 K hi 12 W/m2 䡠 K

r2 12.7 mm T앝,o 21 C k2 0.80 W/m 䡠 K ho 6 W/m2 䡠 K

(a) Construct the thermal circuits, labeling all potentials and flows for the systems excluding the contact lens and including the contact lens. Write resistance elements in terms of appropriate parameters. (b) Determine the heat loss from the anterior chamber with and without the contact lens in place. (c) Discuss the implication of your results.

206

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One-Dimensional, Steady-State Conduction

3.74 The outer surface of a hollow sphere of radius r2 is subjected to a uniform heat flux q2. The inner surface at r1 is held at a constant temperature Ts,1. (a) Develop an expression for the temperature distribution T(r) in the sphere wall in terms of q2, Ts,1, r1, r2, and the thermal conductivity of the wall material k. (b) If the inner and outer tube radii are r1 50 mm and r2 100 mm, what heat flux q2 is required to maintain the outer surface at Ts,2 50 C, while the inner surface is at Ts,1 20 C? The thermal conductivity of the wall material is k 10 W/m 䡠 K. 3.75 A spherical shell of inner and outer radii ri and ro, respectively, is filled with a heat-generating material that provides for a uniform volumetric generation rate (W/m3) of q˙. The outer surface of the shell is exposed to a fluid having a temperature T앝 and a convection coefficient h. Obtain an expression for the steady-state temperature distribution T(r) in the shell, expressing your result in terms of ri, ro, q˙, h, T앝, and the thermal conductivity k of the shell material. 3.76 A spherical tank of 3-m diameter contains a liquifiedpetroleum gas at 60 C. Insulation with a thermal conductivity of 0.06 W/m 䡠 K and thickness 250 mm is applied to the tank to reduce the heat gain. (a) Determine the radial position in the insulation layer at which the temperature is 0 C when the ambient air temperature is 20 C and the convection coefficient on the outer surface is 6 W/m2 䡠 K. (b) If the insulation is pervious to moisture from the atmospheric air, what conclusions can you reach about the formation of ice in the insulation? What effect will ice formation have on heat gain to the LP gas? How could this situation be avoided? 3.77 A transistor, which may be approximated as a hemispherical heat source of radius ro 0.1 mm, is embedded in a large silicon substrate (k 125 W/m 䡠 K) and dissipates heat at a rate q. All boundaries of the silicon are maintained at an ambient temperature of T앝 27 C, except for the top surface, which is well insulated.

ro

Silicon substrate

q T∞

Obtain a general expression for the substrate temperature distribution and evaluate the surface temperature of the heat source for q 4 W.

3.78 One modality for destroying malignant tissue involves imbedding a small spherical heat source of radius ro within the tissue and maintaining local temperatures above a critical value Tc for an extended period. Tissue that is well removed from the source may be assumed to remain at normal body temperature (Tb 37 C). Obtain a general expression for the radial temperature distribution in the tissue under steady-state conditions for which heat is dissipated at a rate q. If ro 0.5 mm, what heat rate must be supplied to maintain a tissue temperature of T Tc 42 C in the domain 0.5 r 5 mm? The tissue thermal conductivity is approximately 0.5 W/m 䡠 K. Assume negligible perfusion.

Conduction with Thermal Energy Generation 3.79 The air inside a chamber at T앝,i 50 C is heated convectively with hi 20 W/m2 䡠 K by a 200-mm-thick wall having a thermal conductivity of 4 W/m 䡠 K and a uniform heat generation of 1000 W/m3. To prevent any heat generated within the wall from being lost to the outside of the chamber at T앝,o 25 C with ho 5 W/m2 䡠 K, a very thin electrical strip heater is placed on the outer wall to provide a uniform heat flux, qo. Wall, k, q•

Strip heater, q"o Outside chamber

Inside chamber

T∞, o, ho

T∞, i, hi x

L

(a) Sketch the temperature distribution in the wall on T x coordinates for the condition where no heat generated within the wall is lost to the outside of the chamber. (b) What are the temperatures at the wall boundaries, T(0) and T(L), for the conditions of part (a)? (c) Determine the value of qo that must be supplied by the strip heater so that all heat generated within the wall is transferred to the inside of the chamber. (d) If the heat generation in the wall were switched off while the heat flux to the strip heater remained constant, what would be the steady-state temperature, T(0), of the outer wall surface? 3.80 Consider cylindrical and spherical shells with inner and outer surfaces at r1 and r2 maintained at uniform temperatures Ts,1 and Ts,2, respectively. If there is uniform heat generation within the shells, obtain expressions for the steady-state, one-dimensional radial distributions of the temperature, heat flux, and heat rate. Contrast your results with those summarized in Appendix C.

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207

Problems

3.81 A plane wall of thickness 0.1 m and thermal conductivity 25 W/m 䡠 K having uniform volumetric heat generation of 0.3 MW/m3 is insulated on one side, while the other side is exposed to a fluid at 92 C. The convection heat transfer coefficient between the wall and the fluid is 500 W/m2 䡠 K. Determine the maximum temperature in the wall. 3.82 Large, cylindrical bales of hay used to feed livestock in the winter months are D 2 m in diameter and are stored end-to-end in long rows. Microbial energy generation occurs in the hay and can be excessive if the farmer bales the hay in a too-wet condition. Assuming the thermal conductivity of baled hay to be k 0.04 W/m 䡠 K, determine the maximum steady-state . hay temperature for dry hay (q 1W/m3), moist hay . . 3 (q 10 W/m ), and wet hay (q 100 W/m3). Ambient conditions are T앝 0 C and h 25 W/m2 䡠 K. 3.83 Consider the cylindrical bales of hay in Problem 3.82. It is proposed to utilize the microbial energy generation associated with wet hay to heat water. Consider a 30-mm diameter, thin-walled tube inserted lengthwise through the middle of a cylindrical bale. The tube carries water at T앝,i 20 C with hi 200 W/m2 䡠 K. (a) Determine the steady-state heat transfer to the water per unit length of tube. (b) Plot the radial temperature distribution in the hay, T(r). (c) Plot the heat transfer to the water per unit length of tube for bale diameters of 0.2 m D 2 m. 3.84 Consider one-dimensional conduction in a plane composite wall. The outer surfaces are exposed to a fluid at 25 C and a convection heat transfer coefficient of 1000 W/m2 䡠 K. The middle wall B experiences uniform . heat generation qB, while there is no generation in walls A and C. The temperatures at the interfaces are T1 261 C and T2 211 C. T1

T2

T∞, h

T∞, h A

B

(b) Plot the temperature distribution, showing its important features. (c) Consider conditions corresponding to a loss of coolant at the exposed surface of material A (h 0). Determine T1 and T2 and plot the temperature distribution throughout the system. 3.85 Consider a plane composite wall that is composed of three materials (materials A, B, and C are arranged left to right) of thermal conductivities kA 0.24 W/m 䡠 K, kB 0.13 W/m 䡠 K, and kC 0.50 W/m 䡠 K. The thicknesses of the three sections of the wall are LA 20 mm, L B 13 mm, and LC 20 mm. A contact resistance of Rt,c 102 m2 䡠 K/W exists at the interface between materials A and B, as well as at the interface between materials B and C. The left face of the composite wall is insulated, while the right face is exposed to convective conditions characterized by h 10 W/m2 䡠 K, T앝 20 C. For Case 1, thermal energy is generated within . material A at the rate qA 5000 W/m3. For Case 2, thermal energy is generated within material C at the . rate qC 5000 W/m3. (a) Determine the maximum temperature within the composite wall under steady-state conditions for Case 1. (b) Sketch the steady-state temperature distribution on T x coordinates for Case 1. (c) Sketch the steady-state temperature distribution for Case 2 on the same T x coordinates used for Case 1. 3.86 An air heater may be fabricated by coiling Nichrome wire and passing air in cross flow over the wire. Consider a heater fabricated from wire of diameter D 1 mm, electrical resistivity e 106 䡠 m, thermal conductivity k 25 W/m 䡠 K, and emissivity 0.20. The heater is designed to deliver air at a temperature of T앝 50 C under flow conditions that provide a convection coefficient of h 250 W/m2 䡠 K for the wire. The temperature of the housing that encloses the wire and through which the air flows is Tsur 50 C. Wire (D, L, ρe, k, ε , Tmax)

Housing, Tsur Air

C

q• B ∆E

LA kA = 25 W/m•K kC = 50 W/m•K

2LB

LC

LA = 30 mm LB = 30 mm LC = 20 mm

(a) Assuming negligible contact resistance at the inter. faces, determine the volumetric heat generation qB and the thermal conductivity kB.

T∞ , h

I

If the maximum allowable temperature of the wire is Tmax 1200 C, what is the maximum allowable electric current I? If the maximum available voltage is E 110 V, what is the corresponding length L of wire that may be used in the heater and the power rating of the heater? Hint: In your solution, assume

208

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

negligible temperature variations within the wire, but after obtaining the desired results, assess the validity of this assumption. 3.87 Consider the composite wall of Example 3.7. In the Comments section, temperature distributions in the wall were determined assuming negligible contact resistance between materials A and B. Compute and plot the temperature distributions if the thermal contact resistance is Rt, c 104 m2 䡠 K/W.

3.90 A nuclear fuel element of thickness 2L is covered with a steel cladding of thickness b. Heat generated within . the nuclear fuel at a rate q is removed by a fluid at T앝, which adjoins one surface and is characterized by a convection coefficient h. The other surface is well insulated, and the fuel and steel have thermal conductivities of kƒ and ks, respectively. Nuclear fuel Steel

3.88 Consider uniform thermal energy generation inside a one-dimensional plane wall of thickness L with one surface held at Ts,1 and the other surface insulated.

Insulation

b

(a) Find an expression for the conduction heat flux to the cold surface and the temperature of the hot surface Ts,2, . expressing your results in terms of k, q, L, and Ts,1.

Case 1

Case 2

To

•

q, k

–L

0

+L

x

•

–L

To

•

q, k

A

B

0

+L

x

(a) Sketch the temperature distribution for Case 1 on T ⫺ x coordinates. Describe the key features of this distribution. Identify the location of the maximum temperature in the wall and calculate this temperature. (b) Sketch the temperature distribution for Case 2 on the same T ⫺ x coordinates. Describe the key features of this distribution. (c) What is the temperature difference between the two walls at x 0 for Case 2? (d) What is the location of the maximum temperature in the composite wall of Case 2? Calculate this temperature.

L

b

(a) Obtain an equation for the temperature distribution T(x) in the nuclear fuel. Express your results in . terms of q, kƒ, L, b, ks, h, and T앝. (b) Sketch the temperature distribution T(x) for the entire system. 3.91 Consider the clad fuel element of Problem 3.90. (a) Using appropriate relations from Tables C.1 and C.2, obtain an expression for the temperature distribution T(x) in the fuel element. For kf 60 W/m 䡠 K, L 15 mm, b 3 mm, ks 15 W/m 䡠 K, h 10,000 W/m2 䡠 K, and T앝 200 C, what are the largest and smallest temperatures in the fuel element if heat is generated uniformly at a volumetric rate of q˙ 2 107 W/m3? What are the corresponding locations?

Thin dielectric strip, R"t

q, k

T∞, h L

x

(b) Compare the heat flux found in part (a) with the heat flux associated with a plane wall without energy generation whose surface temperatures are Ts,1 and Ts,2. 3.89 A plane wall of thickness 2L and thermal conductivity k . experiences a uniform volumetric generation rate q. As shown in the sketch for Case 1, the surface at x L is perfectly insulated, while the other surface is maintained at a uniform, constant temperature To. For Case 2, a very thin dielectric strip is inserted at the midpoint of the wall (x 0) in order to electrically isolate the two sections, A and B. The thermal resistance of the strip is R t 0.0005 m2 䡠 K/W. The parameters associated with the wall are k 50 W/m 䡠 K, L . 20 mm, q 5 106 W/m3, and To 50⬚C.

Steel

(b) If the insulation is removed and equivalent convection conditions are maintained at each surface, what is the corresponding form of the temperature distribution in the fuel element? For the conditions of part (a), what are the largest and smallest temperatures in the fuel? What are the corresponding locations? (c) For the conditions of parts (a) and (b), plot the temperature distributions in the fuel element.

3.92 In Problem 3.79 the strip heater acts to guard against heat losses from the wall to the outside, and the required heat flux qo depends on chamber operating . conditions such as q and T앝,i. As a first step in designing a controller for the guard heater, compute . . and plot qo and T(0) as a function of q for 200 q 3 2000 W/m and T앝,i 30, 50, and 70 C. 3.93 The exposed surface (x 0) of a plane wall of thermal conductivity k is subjected to microwave radiation that causes volumetric heating to vary as x . . q(x) qo 1 L

冢

冣

䊏

Problems

. where qo (W/m3) is a constant. The boundary at x L is perfectly insulated, while the exposed surface is maintained at a constant temperature To. Determine the tem. perature distribution T(x) in terms of x, L, k, qo, and To. 3.94 A quartz window of thickness L serves as a viewing port in a furnace used for annealing steel. The inner surface (x 0) of the window is irradiated with a uniform heat flux qo due to emission from hot gases in the furnace. A fraction, , of this radiation may be assumed to be absorbed at the inner surface, while the remaining radiation is partially absorbed as it passes through the quartz. The volumetric heat generation due to this absorption may be described by an expression of the form . q(x) (1 )qo␣e␣x where ␣ is the absorption coefficient of the quartz. Convection heat transfer occurs from the outer surface (x L) of the window to ambient air at T앝 and is characterized by the convection coefficient h. Convection and radiation emission from the inner surface may be neglected, along with radiation emission from the outer surface. Determine the temperature distribution in the quartz, expressing your result in terms of the foregoing parameters.

209 (a) It is proposed that, under steady-state conditions, . the system operates with a generation rate of q 8 3 7 10 W/m and cooling system characteristics of T앝 95 C and h 7000 W/m2 䡠 K. Is this proposal satisfactory? . (b) Explore the effect of variations in q and h by plotting temperature distributions T(r) for a range of parameter values. Suggest an envelope of acceptable operating conditions. 3.98 A nuclear reactor fuel element consists of a solid cylindrical pin of radius r1 and thermal conductivity kf. The fuel pin is in good contact with a cladding material of outer radius r2 and thermal conductivity kc. Consider steady-state conditions for which uniform heat genera. tion occurs within the fuel at a volumetric rate q and the outer surface of the cladding is exposed to a coolant that is characterized by a temperature T앝 and a convection coefficient h. (a) Obtain equations for the temperature distributions Tf (r) and Tc(r) in the fuel and cladding, respectively. Express your results exclusively in terms of the foregoing variables.

3.95 For the conditions described in Problem 1.44, determine the temperature distribution, T(r), in the container, . expressing your result in terms of qo, ro, T앝, h, and the thermal conductivity k of the radioactive wastes.

(b) Consider a uranium oxide fuel pin for which kƒ 2 W/m 䡠 K and r1 6 mm and cladding for which . kc 25 W/m 䡠 K and r2 9 mm. If q 2 108 3 2 W/m , h 2000 W/m 䡠 K, and T앝 300 K, what is the maximum temperature in the fuel element?

3.96 A cylindrical shell of inner and outer radii, ri and ro, respectively, is filled with a heat-generating material that provides a uniform volumetric generation rate . (W/m3) of q. The inner surface is insulated, while the outer surface of the shell is exposed to a fluid at T앝 and a convection coefficient h.

(c) Compute and plot the temperature distribution, T(r), for values of h 2000, 5000, and 10,000 W/m2 䡠 K. If the operator wishes to maintain the centerline temperature of the fuel element below 1000 K, can she do so by adjusting the coolant flow and hence the value of h?

(a) Obtain an expression for the steady-state temperature distribution T(r) in the shell, expressing your . result in terms of ri, ro, q, h, T앝, and the thermal conductivity k of the shell material.

3.99 Consider the configuration of Example 3.8, where uniform volumetric heating within a stainless steel tube is induced by an electric current and heat is transferred by convection to air flowing through the tube. The tube wall has inner and outer radii of r1 25 mm and r2 35 mm, a thermal conductivity of k 15 W/m 䡠 K, an electrical resistivity of e 0.7 106 䡠 m, and a maximum allowable operating temperature of 1400 K.

(b) Determine an expression for the heat rate, q(ro), at . the outer radius of the shell in terms of q and shell dimensions. 3.97 The cross section of a long cylindrical fuel element in a nuclear reactor is shown. Energy generation occurs uniformly in the thorium fuel rod, which is of diameter D 25 mm and is wrapped in a thin aluminum cladding. Coolant

T∞, h

Thorium fuel rod

D

Thin aluminum cladding

(a) Assuming the outer tube surface to be perfectly insulated and the airflow to be characterized by a temperature and convection coefficient of T앝,1 400 K and h1 100 W/m2 䡠 K, determine the maximum allowable electric current I. (b) Compute and plot the radial temperature distribution in the tube wall for the electric current of part (a) and three values of h1 (100, 500, and 1000 W/m2 䡠 K). For each value of h1, determine the rate of heat transfer to the air per unit length of tube.

210

Chapter 3

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One-Dimensional, Steady-State Conduction

(c) In practice, even the best of insulating materials would be unable to maintain adiabatic conditions at the outer tube surface. Consider use of a refractory insulating material of thermal conductivity k 1.0 W/m 䡠 K and neglect radiation exchange at its outer surface. For h1 100 W/m2 䡠 K and the maximum allowable current determined in part (a), compute and plot the temperature distribution in the composite wall for two values of the insulation thickness (␦ 25 and 50 mm). The outer surface of the insulation is exposed to room air for which T앝, 2 300 K and h2 25 W/m2 䡠 K. For each insulation thickness, determine the rate of heat transfer per unit tube length to the inner airflow and the ambient air. 3.100 A high-temperature, gas-cooled nuclear reactor consists of a composite cylindrical wall for which a thorium fuel element (k ⬇ 57 W/m 䡠 K) is encased in graphite (k ⬇ 3 W/m 䡠 K) and gaseous helium flows through an annular coolant channel. Consider conditions for which the helium temperature is T앝 600 K and the convection coefficient at the outer surface of the graphite is h 2000 W/m2 䡠 K.

r1 = 8 mm r2 = 11 mm r3 = 14 mm

Coolant channel with helium flow (T∞, h) Graphite Thorium, q•

T1 T2 T3

(a) If thermal energy is uniformly generated in the fuel . element at a rate q 108 W/m3, what are the temperatures T1 and T2 at the inner and outer surfaces, respectively, of the fuel element? (b) Compute and plot the temperature distribution in . the composite wall for selected values of q. What . is the maximum allowable value of q? 3.101 A long cylindrical rod of diameter 200 mm with thermal conductivity of 0.5 W/m 䡠 K experiences uniform volumetric heat generation of 24,000 W/m3. The rod is encapsulated by a circular sleeve having an outer diameter of 400 mm and a thermal conductivity of 4 W/m 䡠 K. The outer surface of the sleeve is exposed to cross flow of air at 27 C with a convection coefficient of 25 W/m2 䡠 K. (a) Find the temperature at the interface between the rod and sleeve and on the outer surface. (b) What is the temperature at the center of the rod?

3.102 A radioactive material of thermal conductivity k is cast as a solid sphere of radius ro and placed in a liquid bath for which the temperature T앝 and convection coefficient h are known. Heat is uniformly generated within . the solid at a volumetric rate of q. Obtain the steadystate radial temperature distribution in the solid, . expressing your result in terms of ro, q, k, h, and T앝. 3.103 Radioactive wastes are packed in a thin-walled spherical container. The wastes generate thermal energy nonuni. . formly according to the relation q qo[1 (r/ro)2] where . . q is the local rate of energy generation per unit volume, q is a constant, and ro is the radius of the container. Steadystate conditions are maintained by submerging the container in a liquid that is at T앝 and provides a uniform convection coefficient h. ro

Coolant T∞, h

q• = q• o [1 – (r/ro)2]

Determine the temperature distribution, T(r), in the con. tainer. Express your result in terms of qo, ro, T앝, h, and the thermal conductivity k of the radioactive wastes. 3.104 Radioactive wastes (krw 20 W/m 䡠 K) are stored in a spherical, stainless steel (kss 15 W/m 䡠 K) container of inner and outer radii equal to ri 0.5 m and ro 0.6 m. Heat is generated volumetrically within the wastes at a . uniform rate of q 105 W/m3, and the outer surface of the container is exposed to a water flow for which h 1000 W/m2 䡠 K and T앝 25 C.

Water T∞, h

ri

Radioactive wastes, krw, q• Stainless steel,

ro

Ts, o

kss

Ts, i

(a) Evaluate the steady-state outer surface temperature, Ts,o. (b) Evaluate the steady-state inner surface temperature, Ts,i. (c) Obtain an expression for the temperature distribution, T(r), in the radioactive wastes. Express your . result in terms of ri, Ts,i, krw, and q. Evaluate the temperature at r 0.

䊏

211

Problems

(d) A proposed extension of the foregoing design involves storing waste materials having the same thermal conductivity but twice the heat generation . (q 2 105 W/m3) in a stainless steel container of equivalent inner radius (ri 0.5 m). Safety considerations dictate that the maximum system temperature not exceed 475 C and that the container wall thickness be no less than t 0.04 m and preferably at or close to the original design (t 0.1 m). Assess the effect of varying the outside convection coefficient to a maximum achievable value of h 5000 W/m2 䡠 K (by increasing the water velocity) and the container wall thickness. Is the proposed extension feasible? If so, recommend suitable operating and design conditions for h and t, respectively. 3.105 Unique characteristics of biologically active materials such as fruits, vegetables, and other products require special care in handling. Following harvest and separation from producing plants, glucose is catabolized to produce carbon dioxide, water vapor, and heat, with attendant internal energy generation. Consider a carton of apples, each of 80-mm diameter, which is ventilated with air at 5 C and a velocity of 0.5 m/s. The corresponding value of the heat transfer coefficient is 7.5 W/m2 䡠 K. Within each apple thermal energy is uniformly generated at a total rate of 4000 J/kg 䡠 day. The density and thermal conductivity of the apple are 840 kg/m3 and 0.5 W/m 䡠 K, respectively. Apple, 80 mm diameter

3.106 Consider the plane wall, long cylinder, and sphere shown schematically, each with the same characteristic length a, thermal conductivity k, and uniform volu. metric energy generation rate q. Plane wall

Long cylinder

•

Sphere

•

q, k

•

q, k

q, k

r=a a

x

a

x

(a) On the same graph, plot the steady-state dimen. sionless temperature, [T(x or r) T(a)]/[(qa2)/2k], versus the dimensionless characteristic length, x/a or r/a, for each shape. (b) Which shape has the smallest temperature difference between the center and the surface? Explain this behavior by comparing the ratio of the volumeto-surface area. (c) Which shape would be preferred for use as a nuclear fuel element? Explain why.

Extended Surfaces 3.107 The radiation heat gage shown in the diagram is made from constantan metal foil, which is coated black and is in the form of a circular disk of radius R and thickness t. The gage is located in an evacuated enclosure. The incident radiation flux absorbed by the foil, qi, diffuses toward the outer circumference and into the larger copper ring, which acts as a heat sink at the constant temperature T(R). Two copper lead wires are attached to the center of the foil and to the ring to complete a thermocouple circuit that allows for measurement of the temperature difference between the foil center and the foil edge, T T(0) T(R).

Air

T∞ = 5°C

(a) Determine the temperatures.

apple

center

and

q"i

surface

(b) For the stacked arrangement of apples within the crate, the convection coefficient depends on the velocity as h C1V 0.425, where C1 10.1 W/m2 䡠 K 䡠 (m/s)0.425. Compute and plot the center and surface temperatures as a function of the air velocity for 0.1 V 1 m/s.

Evacuated enclosure

R Foil

T(0)

T(R)

Copper ring

Copper wires

212

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Obtain the differential equation that determines T(r), the temperature distribution in the foil, under steady-state conditions. Solve this equation to obtain an expression relating T to qi. You may neglect radiation exchange between the foil and its surroundings.

nanowire that may be grown for conditions characterized by h 105 W/m2 䡠 K and T앝 8000 K. Assume properties of the nanowire are the same as for bulk silicon carbide. Gas absorption

3.108 Copper tubing is joined to the absorber of a flat-plate solar collector as shown. Cover plate

Solid deposition Evacuated space

q"rad

Nanowire

h, T∞

Absorber plate

Liquid catalyst Water

t

Tw Insulation

L Initial time

The aluminum alloy (2024-T6) absorber plate is 6 mm thick and well insulated on its bottom. The top surface of the plate is separated from a transparent cover plate by an evacuated space. The tubes are spaced a distance L of 0.20 m from each other, and water is circulated through the tubes to remove the collected energy. The water may be assumed to be at a uniform temperature of Tw 60 C. Under steady-state operating conditions for which the net radiation heat flux to the surface is qrad 800 W/m2, what is the maximum temperature on the plate and the heat transfer rate per unit length of tube? Note that qrad represents the net effect of solar radiation absorption by the absorber plate and radiation exchange between the absorber and cover plates. You may assume the temperature of the absorber plate directly above a tube to be equal to that of the water. 3.109 One method that is used to grow nanowires (nanotubes with solid cores) is to initially deposit a small droplet of a liquid catalyst onto a flat surface. The surface and catalyst are heated and simultaneously exposed to a higher-temperature, low-pressure gas that contains a mixture of chemical species from which the nanowire is to be formed. The catalytic liquid slowly absorbs the species from the gas through its top surface and converts these to a solid material that is deposited onto the underlying liquid-solid interface, resulting in construction of the nanowire. The liquid catalyst remains suspended at the tip of the nanowire. Consider the growth of a 15-nm-diameter silicon carbide nanowire onto a silicon carbide surface. The surface is maintained at a temperature of Ts 2400 K, and the particular liquid catalyst that is used must be maintained in the range 2400 K Tc 3000 K to perform its function. Determine the maximum length of a

Intermediate time

Maximum length

3.110 Consider the manufacture of photovoltaic silicon, as described in Problem 1.42. The thin sheet of silicon is pulled from the pool of molten material very slowly and is subjected to an ambient temperature of T앝 527 C within the growth chamber. A convection coefficient of h 7.5 W/m2 䡠 K is associated with the exposed surfaces of the silicon sheet when it is inside the growth chamber. Calculate the maximum allowable velocity of the silicon sheet Vsi. The latent heat of fusion for silicon is hsf 1.8 106 J/kg. It can be assumed that the thermal energy released due to solidification is removed by conduction along the sheet. 3.111 Copper tubing is joined to a solar collector plate of thickness t, and the working fluid maintains the temperature of the plate above the tubes at To. There is a uniform net radiation heat flux qrad to the top surface of the plate, while the bottom surface is well insulated. The top surface is also exposed to a fluid at T앝 that provides for a uniform convection coefficient h. Air

T∞, h q"rad To

To Absorber plate

t Working fluid

Working fluid

x 2L

(a) Derive the differential equation that governs the temperature distribution T(x) in the plate.

䊏

213

Problems

(b) Obtain a solution to the differential equation for appropriate boundary conditions. 3.112 A thin flat plate of length L, thickness t, and width W L is thermally joined to two large heat sinks that are maintained at a temperature To. The bottom of the plate is well insulated, while the net heat flux to the top surface of the plate is known to have a uniform value of qo. L

distance between the two legs of the sting, L L1 L2, to ensure that the sting temperature does not influence the junction temperature and, in turn, invalidate the gas temperature measurement. Consider two different types of thermocouple junctions consisting of (i) copper and constantan wires and (ii) chromel and alumel wires. Evaluate the thermal conductivity of copper and constantan at T 300 K. Use kCh 19 W/m 䡠 K and kAl 29 W/m 䡠 K for the thermal conductivities of the chromel and alumel wires, respectively.

x Heat sink

q"o

Thermocouple junction

Heat sink

To

To t

(a) Derive the differential equation that determines the steady-state temperature distribution T(x) in the plate. (b) Solve the foregoing equation for the temperature distribution, and obtain an expression for the rate of heat transfer from the plate to the heat sinks. 3.113 Consider the flat plate of Problem 3.112, but with the heat sinks at different temperatures, T(0) To and T(L) TL, and with the bottom surface no longer insulated. Convection heat transfer is now allowed to occur between this surface and a fluid at T앝, with a convection coefficient h. (a) Derive the differential equation that determines the steady-state temperature distribution T(x) in the plate. (b) Solve the foregoing equation for the temperature distribution, and obtain an expression for the rate of heat transfer from the plate to the heat sinks. (c) For qo 20,000 W/m2, To 100 C, TL 35 C, T앝 25 C, k 25 W/m 䡠 K, h 50 W/m2 䡠 K, L 100 mm, t 5 mm, and a plate width of W 30 mm, plot the temperature distribution and determine the sink heat rates, qx(0) and qx(L). On the same graph, plot three additional temperature distributions corresponding to changes in the following parameters, with the remaining parameters unchanged: (i) qo 30,000 W/m2, (ii) h 200 W/m2 䡠 K, and (iii) the value of qo for which qx(0) 0 when h 200 W/m2 䡠 K. 3.114 The temperature of a flowing gas is to be measured with a thermocouple junction and wire stretched between two legs of a sting, a wind tunnel test fixture. The junction is formed by butt-welding two wires of different material, as shown in the schematic. For wires of diameter D 125 m and a convection coefficient of h 700 W/m2 䡠 K, determine the minimum separation

L1 Gas h, T∞

L

Sting

L2 D

3.115 A bonding operation utilizes a laser to provide a constant heat flux, qo, across the top surface of a thin adhesivebacked, plastic film to be affixed to a metal strip as shown in the sketch. The metal strip has a thickness d 1.25 mm, and its width is large relative to that of the film. The thermophysical properties of the strip are 7850 kg/m3, cp 435 J/kg 䡠 K, and k 60 W/m 䡠 K. The thermal resistance of the plastic film of width w1 40 mm is negligible. The upper and lower surfaces of the strip (including the plastic film) experience convection with air at 25 C and a convection coefficient of 10 W/m2 䡠 K. The strip and film are very long in the direction normal to the page. Assume the edges of the metal strip are at the air temperature (T앝). Laser source, q"o Plastic film

T∞, h Metal strip

d

w1 w2 x T∞, h

(a) Derive an expression for the temperature distribution in the portion of the steel strip with the plastic film (w1/2 x w1/2). (b) If the heat flux provided by the laser is 10,000 W/m2, determine the temperature of the plastic film at the center (x 0) and its edges (x w1/2). (c) Plot the temperature distribution for the entire strip and point out its special features.

214

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

3.116 A thin metallic wire of thermal conductivity k, diameter D, and length 2L is annealed by passing an electrical current through the wire to induce a uniform volu. metric heat generation q. The ambient air around the wire is at a temperature T앝, while the ends of the wire at x L are also maintained at T앝. Heat transfer from the wire to the air is characterized by the convection coefficient h. Obtain expressions for the following:

experiences uniform volumetric energy generation at a . rate of q 10 106 W/m3. Air at Ta 80 C provides a convection coefficient of ha 35 W/m2 䡠 K on one side of the membrane, while hydrogen at Th 80 C, hh 235 W/m2 䡠 K flows on the opposite side of the membrane. The flow channels are 2L 3 mm wide. The membrane is clamped between bipolar plates, each of which is at a temperature Tbp 80 C. Membrane

(a) The steady-state temperature distribution T(x) along the wire, (b) The maximum wire temperature.

t

(c) The average wire temperature.

2L

3.117 A motor draws electric power Pelec from a supply line and delivers mechanical power Pmech to a pump through a rotating copper shaft of thermal conductivity ks, length L, and diameter D. The motor is mounted on a square pad of width W, thickness t, and thermal conductivity kp. The surface of the housing exposed to ambient air at T앝 is of area Ah, and the corresponding convection coefficient is hh. Opposite ends of the shaft are at temperatures of Th and T앝, and heat transfer from the shaft to the ambient air is characterized by the convection coefficient hs. The base of the pad is at T앝. T∞, hh

T∞, hs

Pelec

x

Bipolar plate, Tbp

(a) Derive the differential equation that governs the temperature distribution T(x) in the membrane. (b) Obtain a solution to the differential equation, assuming the membrane is at the bipolar plate temperature at x 0 and x 2L.

T∞

(c) Plot the temperature distribution T(x) from x 0 to x L for carbon nanotube loadings of 0% and 10% by volume. Comment on the ability of the carbon nanotubes to keep the membrane below its softening temperature of 85 C.

Pump

D Th

L

t

Pad, kp

W

Ta , ha

Th , hh

Motor housing, Th, Ah

Electric motor

Air

Hydrogen

Shaft, ks, Pmech

T∞

(a) Expressing your result in terms of Pelec, Pmech, ks, L, D, W, t, kp, Ah, hh, and hs, obtain an expression for (Th T앝).

3.119 Consider a rod of diameter D, thermal conductivity k, and length 2L that is perfectly insulated over one portion of its length, L x 0, and experiences convection with a fluid (T앝, h) over the other portion, 0 x L. One end is maintained at T1, while the other is separated from a heat sink at T3 by an interfa. cial thermal contact resistance Rt,c

(b) What is the value of Th if Pelec 25 kW, Pmech 15 kW, ks 400 W/m 䡠 K, L 0.5 m, D 0.05 m, W 0.7 m, t 0.05 m, kp 0.5 W/m 䡠 K, Ah 2 m2, hh 10 W/m2 䡠 K, hs 300 W/m2 䡠 K, and T1 T앝 25 C? 3.118 Consider the fuel cell stack of Problem 1.58. The t 0.42-mm-thick membranes have a nominal thermal conductivity of k 0.79 W/m 䡠 K that can be increased to keff,x 15.1 W/m 䡠 K by loading 10%, by volume, carbon nanotubes into the catalyst layers. The membrane

Insulation

R"t,c = 4 × 10–4 m2•K/W T2

–L

Rod 0 D = 5 mm L = 50 mm k = 100 W/m•K

T3

x +L T∞ = 20°C h = 500 W/m2•K

215

Problems

䊏

(a) Sketch the temperature distribution on T x coordinates and identify its key features. Assume that T1 T3 T앝.

Duct wall

Ambient air

Water

T∞,o, ho

T∞,i, hi

(b) Derive an expression for the midpoint temperature T2 in terms of the thermal and geometric parameters of the system. (c) For T1 200 C, T3 100 C, and the conditions provided in the schematic, calculate T2 and plot the temperature distribution. Describe key features of the distribution and compare it to your sketch of part (a). 3.120 A carbon nanotube is suspended across a trench of width s 5 m that separates two islands, each at T앝 300 K. A focused laser beam irradiates the nanotube at a distance from the left island, delivering q 10 W of energy to the nanotube. The nanotube temperature is measured at the midpoint of the trench using a point probe. The measured nanotube temperature is T1 324.5 K for 1 1.5 m and T2 326.4 K for 2 3.5 m.

Temperature measurement Laser irradiation s/2

ξ

T∞ Tsur 300 K

s 5 µm

Rt,c,L

Carbon nanotube

Rt,c,R

Determine the two contact resistances, Rt,c,L and Rt,c,R at the left and right ends of the nanotube, respectively. The experiment is performed in a vacuum with Tsur 300 K. The nanotube thermal conductivity and diameter are kcn 3100 W/m 䡠 K and D 14 nm, respectively. 3.121 A probe of overall length L 200 mm and diameter D 12.5 mm is inserted through a duct wall such that a portion of its length, referred to as the immersion length Li, is in contact with the water stream whose temperature, T앝, i, is to be determined. The convection coefficients over the immersion and ambient-exposed lengths are hi 1100 W/m2 䡠 K and ho 10 W/m2 䡠 K, respectively. The probe has a thermal conductivity of 177 W/m 䡠 K and is in poor thermal contact with the duct wall.

Sensor, Ttip Leads

D

Lo

Li L

(a) Derive an expression for evaluating the measurement error, Terr Ttip T앝,i, which is the difference between the tip temperature, Ttip, and the water temperature, T앝,i. Hint: Define a coordinate system with the origin at the duct wall and treat the probe as two fins extending inward and outward from the duct, but having the same base temperature. Use Case A results from Table 3.4. (b) With the water and ambient air temperatures at 80 and 20 C, respectively, calculate the measurement error, Terr, as a function of immersion length for the conditions Li /L 0.225, 0.425, and 0.625. (c) Compute and plot the effects of probe thermal conductivity and water velocity (hi) on the measurement error. 3.122 A rod of diameter D 25 mm and thermal conductivity k 60 W/m 䡠 K protrudes normally from a furnace wall that is at Tw 200 C and is covered by insulation of thickness Lins 200 mm. The rod is welded to the furnace wall and is used as a hanger for supporting instrumentation cables. To avoid damaging the cables, the temperature of the rod at its exposed surface, To, must be maintained below a specified operating limit of Tmax 100 C. The ambient air temperature is T앝 25 C, and the convection coefficient is h 15 W/m2 䡠 K. Air

T∞, h

Tw

D

To

Hot furnace wall

Insulation

Lins

Lo

(a) Derive an expression for the exposed surface temperature To as a function of the prescribed thermal and

216

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

geometrical parameters. The rod has an exposed length Lo, and its tip is well insulated.

(a) Calculate the steady-state temperature To of the rod at the midpoint of the heated portion in the coil.

(b) Will a rod with Lo 200 mm meet the specified operating limit? If not, what design parameters would you change? Consider another material, increasing the thickness of the insulation, and increasing the rod length. Also, consider how you might attach the base of the rod to the furnace wall as a means to reduce To.

(b) Calculate the temperature of the rod Tb at the edge of the heated portion.

3.123 A metal rod of length 2L, diameter D, and thermal conductivity k is inserted into a perfectly insulating wall, exposing one-half of its length to an airstream that is of temperature T앝 and provides a convection coefficient h at the surface of the rod. An electromagnetic field induces volumetric energy generation at . a uniform rate q within the embedded portion of the rod.

3.125 From Problem 1.71, consider the wire leads connecting the transistor to the circuit board. The leads are of thermal conductivity k, thickness t, width w, and length L. One end of a lead is maintained at a temperature Tc corresponding to the transistor case, while the other end assumes the temperature Tb of the circuit board. During steady-state operation, current flow through the leads provides for uniform volumetric heating in the amount . q, while there is convection cooling to air that is at T앝 and maintains a convection coefficient h.

Air T∞ = 20°C h = 100 W/m2•K

Tb

To

Transistor case(Tc) Wire lead(k)

T∞, h

Rod, D, k

x

•

q L

Circuit board(Tb)

L L = 50 mm D = 5 mm k = 25 W/m•K • q = 1 × 106 W/m3

x

(a) Derive an expression for the steady-state temperature Tb at the base of the exposed half of the rod. The exposed region may be approximated as a very long fin. (b) Derive an expression for the steady-state temperature To at the end of the embedded half of the rod. (c) Using numerical values provided in the schematic, plot the temperature distribution in the rod and describe key features of the distribution. Does the rod behave as a very long fin? 3.124 A very long rod of 5-mm diameter and uniform thermal conductivity k 25 W/m 䡠 K is subjected to a heat treatment process. The center, 30-mm-long portion of the rod within the induction heating coil experiences uniform volumetric heat generation of 7.5 106 W/m3. Induction heating coil

To

t w

Gap

(a) Derive an equation from which the temperature distribution in a wire lead may be determined. List all pertinent assumptions. (b) Determine the temperature distribution in a wire lead, expressing your results in terms of the prescribed variables. 3.126 Turbine blades mounted to a rotating disc in a gas turbine engine are exposed to a gas stream that is at T앝 1200 C and maintains a convection coefficient of h 250 W/m2 䡠 K over the blade. Blade tip

L

Gas stream

T∞, h

Tb

x •

Region experiencing q

30 mm

Very long rod, 5-mm dia.

The unheated portions of the rod, which protrude from the heating coil on either side, experience convection with the ambient air at T앝 20 C and h 10 W/m2 䡠 K. Assume that there is no convection from the surface of the rod within the coil.

Tb Rotating disk Air coolant

䊏

217

Problems

The blades, which are fabricated from Inconel, k ⬇ 20 W/m 䡠 K, have a length of L 50 mm. The blade profile has a uniform cross-sectional area of Ac 6 104 m2 and a perimeter of P 110 mm. A proposed blade-cooling scheme, which involves routing air through the supporting disc, is able to maintain the base of each blade at a temperature of Tb 300 C. (a) If the maximum allowable blade temperature is 1050 C and the blade tip may be assumed to be adiabatic, is the proposed cooling scheme satisfactory? (b) For the proposed cooling scheme, what is the rate at which heat is transferred from each blade to the coolant? 3.127 In a test to determine the friction coefficient associated with a disk brake, one disk and its shaft are rotated at a constant angular velocity , while an equivalent disk/shaft assembly is stationary. Each disk has an outer radius of r2 180 mm, a shaft radius of r1 20 mm, a thickness of t 12 mm, and a thermal conductivity of k 15 W/m 䡠 K. A known force F is applied to the system, and the corresponding torque required to maintain rotation is measured. The disk contact pressure may be assumed to be uniform (i.e., independent of location on the interface), and the disks may be assumed to be well insulated from the surroundings. t r2

ω

T1

r1

F

T∞, h

Tb

Ts (x) 2t

t y

To (x)

x x

In this problem we seek to determine conditions for which the transverse (y-direction) temperature difference within the extended surface is negligible compared to the temperature difference between the surface and the environment, such that the one-dimensional analysis of Section 3.6.1 is valid. (a) Assume that the transverse temperature distribution is parabolic and of the form

冢冣

T(y) To(x) y t Ts(x) To(x)

2

where Ts(x) is the surface temperature and To(x) is the centerline temperature at any x-location. Using Fourier’s law, write an expression for the conduction heat flux at the surface, qy (t), in terms of Ts and To. (b) Write an expression for the convection heat flux at the surface for the x-location. Equating the two expressions for the heat flux by conduction and convection, identify the parameter that determines the ratio (To Ts)/(Ts T앝). (c) From the foregoing analysis, develop a criterion for establishing the validity of the onedimensional assumption used to model an extended surface.

τ Disk interface, friction coefficient, µ

(a) Obtain an expression that may be used to evaluate from known quantities. (b) For the region r1 r r2, determine the radial temperature distribution T(r) in the disk, where T(r1) T1 is presumed to be known. (c) Consider test conditions for which F 200 N, 40 rad/s, 8 N 䡠 m, and T1 80 C. Evaluate the friction coefficient and the maximum disk temperature. 3.128 Consider an extended surface of rectangular cross section with heat flow in the longitudinal direction.

Simple Fins 3.129 A long, circular aluminum rod is attached at one end to a heated wall and transfers heat by convection to a cold fluid. (a) If the diameter of the rod is tripled, by how much would the rate of heat removal change? (b) If a copper rod of the same diameter is used in place of the aluminum, by how much would the rate of heat removal change? 3.130 A brass rod 100 mm long and 5 mm in diameter extends horizontally from a casting at 200 C. The rod is in an air environment with T앝 20 C and h 30 W/m2 䡠 K. What is the temperature of the rod 25, 50, and 100 mm from the casting?

218

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

3.131 The extent to which the tip condition affects the thermal performance of a fin depends on the fin geometry and thermal conductivity, as well as the convection coefficient. Consider an alloyed aluminum (k 180 W/m 䡠 K) rectangular fin of length L 10 mm, thickness t 1 mm, and width w t. The base temperature of the fin is Tb l00 C, and the fin is exposed to a fluid of temperature T앝 25 C. (a) Assuming a uniform convection coefficient of h 100 W/m2 䡠 K over the entire fin surface, determine the fin heat transfer rate per unit width qf , efficiency f , effectiveness f , thermal resistance per unit width Rt, f , and the tip temperature T(L) for Cases A and B of Table 3.4. Contrast your results with those based on an infinite fin approximation. (b) Explore the effect of variations in the convection coefficient on the heat rate for 10 h 1000 W/m2 䡠 K. Also consider the effect of such variations for a stainless steel fin (k 15 W/m 䡠 K). 3.132 A pin fin of uniform, cross-sectional area is fabricated of an aluminum alloy (k 160 W/m 䡠 K). The fin diameter is D 4 mm, and the fin is exposed to convective conditions characterized by h 220 W/m2 䡠 K. It is reported that the fin efficiency is f 0.65. Determine the fin length L and the fin effectiveness f. Account for tip convection. 3.133 The extent to which the tip condition affects the thermal performance of a fin depends on the fin geometry and thermal conductivity, as well as the convection coefficient. Consider an alloyed aluminum (k 180 W/m 䡠 K) rectangular fin whose base temperature is Tb 100 C. The fin is exposed to a fluid of temperature T앝 25 C, and a uniform convection coefficient of h 100 W/m2 䡠 K may be assumed for the fin surface. (a) For a fin of length L 10 mm, thickness t 1 mm, and width w t, determine the fin heat transfer rate per unit width qf , efficiency f, effectiveness f, thermal resistance per unit width Rt,f, and tip temperature T(L) for Cases A and B of Table 3.4. Contrast your results with those based on an infinite fin approximation. (b) Explore the effect of variations in L on the heat rate for 3 L 50 mm. Also consider the effect of such variations for a stainless steel fin (k 15 W/m 䡠 K). 3.134 A straight fin fabricated from 2024 aluminum alloy (k 185 W/m 䡠 K) has a base thickness of t 3 mm and a length of L 15 mm. Its base temperature is Tb 100 C, and it is exposed to a fluid for which T앝 20 C and h 50 W/m2 䡠 K. For the foregoing conditions and a fin of unit width, compare the fin heat

rate, efficiency, and volume for rectangular, triangular, and parabolic profiles. 3.135 Triangular and parabolic straight fins are subjected to the same thermal conditions as the rectangular straight fin of Problem 3.134. (a) Determine the length of a triangular fin of unit width and base thickness t 3 mm that will provide the same fin heat rate as the straight rectangular fin. Determine the ratio of the mass of the triangular straight fin to that of the rectangular straight fin. (b) Repeat part (a) for a parabolic straight fin. 3.136 Two long copper rods of diameter D 10 mm are soldered together end to end, with solder having a melting point of 650 C. The rods are in air at 25 C with a convection coefficient of 10 W/m2 䡠 K. What is the minimum power input needed to effect the soldering? 3.137 Circular copper rods of diameter D 1 mm and length L 25 mm are used to enhance heat transfer from a surface that is maintained at Ts,1 100 C. One end of the rod is attached to this surface (at x 0), while the other end (x 25 mm) is joined to a second surface, which is maintained at Ts,2 0 C. Air flowing between the surfaces (and over the rods) is also at a temperature of T앝 0 C, and a convection coefficient of h 100 W/m2 䡠 K is maintained. (a) What is the rate of heat transfer by convection from a single copper rod to the air? (b) What is the total rate of heat transfer from a 1 m 1 m section of the surface at 100 C, if a bundle of the rods is installed on 4-mm centers? 3.138 During the initial stages of the growth of the nanowire of Problem 3.109, a slight perturbation of the liquid catalyst droplet can cause it to be suspended on the top of the nanowire in an off-center position. The resulting nonuniform deposition of solid at the solid-liquid interface can be manipulated to form engineered shapes such as a nanospring, that is characterized by a spring radius r, spring pitch s, overall chord length Lc (length running along the spring), and end-to-end length L, as shown in the sketch. Consider a silicon carbide nanospring of diameter D 15 nm, r 30 nm, s 25 nm, and Lc 425 nm. From experiments, it is known that the average spring pitch s– varies with average tem– – perature T by the relation ds–/dT 0.1 nm/K. Using this information, a student suggests that a nanoactuator can be constructed by connecting one end of the nanospring to a small heater and raising the temperature of that end of the nano spring above its initial value. Calculate the actuation distance L for conditions where h 106 W/m2 䡠 K, T앝 Ti 25 C, with a base

䊏

temperature of Tb 50 C. If the base temperature can be controlled to within 1 C, calculate the accuracy to which the actuation distance can be controlled. Hint: Assume the spring radius does not change when the spring is heated. The overall spring length may be approximated by the formula, L

Lc s 2 兹 r2 (s2)2 L

x

Tb

• D

219

Problems

s

T∞, h

3.139 Consider two long, slender rods of the same diameter but different materials. One end of each rod is attached to a base surface maintained at 100 C, while the surfaces of the rods are exposed to ambient air at 20 C. By traversing the length of each rod with a thermocouple, it was observed that the temperatures of the rods were equal at the positions xA 0.15 m and xB 0.075 m, where x is measured from the base surface. If the thermal conductivity of rod A is known to be kA 70 W/m 䡠 K, determine the value of kB for rod B. 3.140 A 40-mm-long, 2-mm-diameter pin fin is fabricated of an aluminum alloy (k 140 W/m 䡠 K). (a) Determine the fin heat transfer rate for Tb 50 C, T앝 25 C, h 1000 W/m2 䡠 K, and an adiabatic tip condition. (b) An engineer suggests that by holding the fin tip at a low temperature, the fin heat transfer rate can be increased. For T(x L) 0 C, determine the new fin heat transfer rate. Other conditions are as in part (a). (c) Plot the temperature distribution, T(x), over the range 0 x L for the adiabatic tip case and the prescribed tip temperature case. Also show the ambient temperature in your graph. Discuss relevant features of the temperature distribution. (d) Plot the fin heat transfer rate over the range 0 h 1000 W/m2 䡠 K for the adiabatic tip case and the prescribed tip temperature case. For the prescribed tip temperature case, what would the

calculated fin heat transfer rate be if Equation 3.78 were used to determine qf rather than Equation 3.76? 3.141 An experimental arrangement for measuring the thermal conductivity of solid materials involves the use of two long rods that are equivalent in every respect, except that one is fabricated from a standard material of known thermal conductivity kA while the other is fabricated from the material whose thermal conductivity kB is desired. Both rods are attached at one end to a heat source of fixed temperature Tb, are exposed to a fluid of temperature T앝, and are instrumented with thermocouples to measure the temperature at a fixed distance x1 from the heat source. If the standard material is aluminum, with kA 200 W/m 䡠 K, and measurements reveal values of TA 75 C and TB 60 C at x1 for Tb 100 C and T앝 25 C, what is the thermal conductivity kB of the test material?

Fin Systems and Arrays 3.142 Finned passages are frequently formed between parallel plates to enhance convection heat transfer in compact heat exchanger cores. An important application is in electronic equipment cooling, where one or more air-cooled stacks are placed between heat-dissipating electrical components. Consider a single stack of rectangular fins of length L and thickness t, with convection conditions corresponding to h and T앝. 200 mm 100 mm

14 mm

To

x

L

Air T∞, h

TL

(a) Obtain expressions for the fin heat transfer rates, qf,o and qf,L, in terms of the base temperatures, To and TL. (b) In a specific application, a stack that is 200 mm wide and 100 mm deep contains 50 fins, each of length L 12 mm. The entire stack is made from aluminum, which is everywhere 1.0 mm thick. If temperature limitations associated with electrical components joined to opposite plates dictate maximum allowable plate temperatures of To 400 K

220

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

and TL 350 K, what are the corresponding maximum power dissipations if h 150 W/m2 䡠 K and T앝 300 K?

(a) Consider limitations for which the array has N 11 fins, in which case values of the fin thickness t 0.182 mm and pitch S 1.982 mm are obtained from the requirements that W (N 1)S t and S t 1.8 mm. If the maximum allowable chip temperature is Tc 85 C, what is the corresponding value of the chip power qc? An adiabatic fin tip condition may be assumed, and airflow along the outer surfaces of the heat sink may be assumed to provide a convection coefficient equivalent to that associated with airflow through the channels.

3.143 The fin array of Problem 3.142 is commonly found in compact heat exchangers, whose function is to provide a large surface area per unit volume in transferring heat from one fluid to another. Consider conditions for which the second fluid maintains equivalent temperatures at the parallel plates, To TL, thereby establishing symmetry about the midplane of the fin array. The heat exchanger is 1 m long in the direction of the flow of air (first fluid) and 1 m wide in a direction normal to both the airflow and the fin surfaces. The length of the fin passages between adjoining parallel plates is L 8 mm, whereas the fin thermal conductivity and convection coefficient are k 200 W/m 䡠 K (aluminum) and h 150 W/m2 䡠 K, respectively. (a) If the fin thickness and pitch are t 1 mm and S 4 mm, respectively, what is the value of the thermal resistance Rt,o for a one-half section of the fin array? (b) Subject to the constraints that the fin thickness and pitch may not be less than 0.5 and 3 mm, respectively, assess the effect of changes in t and S. 3.144 An isothermal silicon chip of width W 20 mm on a side is soldered to an aluminum heat sink (k 180 W/m 䡠 K) of equivalent width. The heat sink has a base thickness of Lb 3 mm and an array of rectangular fins, each of length Lf 15 mm. Airflow at T앝 20 C is maintained through channels formed by the fins and a cover plate, and for a convection coefficient of h 100 W/m2 䡠 K, a minimum fin spacing of 1.8 mm is dictated by limitations on the flow pressure drop. The solder joint has a thermal resistance of Rt, c 2 106 m2 䡠 K/W. Chip, Tc, qc

(b) With (S t) and h fixed at 1.8 mm and 100 W/m2 䡠 K, respectively, explore the effect of increasing the fin thickness by reducing the number of fins. With N 11 and S t fixed at 1.8 mm, but relaxation of the constraint on the pressure drop, explore the effect of increasing the airflow, and hence the convection coefficient. 3.145 As seen in Problem 3.109, silicon carbide nanowires of diameter D 15 nm can be grown onto a solid silicon carbide surface by carefully depositing droplets of catalyst liquid onto a flat silicon carbide substrate. Silicon carbide nanowires grow upward from the deposited drops, and if the drops are deposited in a pattern, an array of nanowire fins can be grown, forming a silicon carbide nano-heat sink. Consider finned and unfinned electronics packages in which an extremely small, 10 m 10 m electronics device is sandwiched between two d 100-nm-thick silicon carbide sheets. In both cases, the coolant is a dielectric liquid at 20 C. A heat transfer coefficient of h 1 105 W/m2 䡠 K exists on the top and bottom of the unfinned package and on all surfaces of the exposed silicon carbide fins, which are each L 300 nm long. Each nano-heat sink includes a 200 200 array of nanofins. Determine the maximum allowable heat rate that can be generated by the electronic device so that its temperature is maintained at Tt 85 C for the unfinned and finned packages.

Solder, R"t ,c

W

T∞, h

Cover plate

D

T∞, h

Heat sink, k

Tt

d

L

Lb t

W = 10 µm

Lf

S

T∞, h T∞, h

Air

T∞, h

Unfinned

Nano-finned

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221

Problems

3.146 As more and more components are placed on a single integrated circuit (chip), the amount of heat that is dissipated continues to increase. However, this increase is limited by the maximum allowable chip operating temperature, which is approximately 75 C. To maximize heat dissipation, it is proposed that a 4 4 array of copper pin fins be metallurgically joined to the outer surface of a square chip that is 12.7 mm on a side.

Top view

W

Pin fins, Dp

D 200 mm Tsur,t

ht, T∞,t

Lb 10 mm

Lf 25 mm w 80 mm hb, T∞,b t 5 mm

Tsur,b

Sideview

3.148 In Problem 3.146, the prescribed value of ho 1000 W/m2 䡠 K is large and characteristic of liquid cooling. In T∞,o, ho practice it would be far more preferable to use air coolLp Chip, ing, for which a reasonable upper limit to the convecqc, Tc Chip tion coefficient would be ho 250 W/m2 䡠 K. Assess the Lb effect of changes in the pin fin geometry on the chip heat rate if the remaining conditions of Problem 3.146, Contact Air W = 12.7 mm including a maximum allowable chip temperature of resistance, T∞,i, hi 75 C, remain in effect. Parametric variations that may R"t, c /Ac be considered include the total number of pins N in the Board, kb square array, the pin diameter Dp, and the pin length Lp. However, the product N1/2Dp should not exceed 9 mm (a) Sketch the equivalent thermal circuit for the pin– to ensure adequate airflow passage through the array. chip–board assembly, assuming one-dimensional, Recommend a design that enhances chip cooling. steady-state conditions and negligible contact resistance between the pins and the chip. In vari- 3.149 Water is heated by submerging 50-mm-diameter, thinable form, label appropriate resistances, temperawalled copper tubes in a tank and passing hot combustures, and heat rates. tion gases (Tg 750 K) through the tubes. To enhance heat transfer to the water, four straight fins of uniform (b) For the conditions prescribed in Problem 3.27, cross section, which form a cross, are inserted in each what is the maximum rate at which heat can be tube. The fins are 5 mm thick and are also made of dissipated in the chip when the pins are in place? copper (k 400 W/m 䡠 K). That is, what is the value of qc for Tc 75 C? The pin diameter and length are Dp 1.5 mm and D = 50 mm Lp 15 mm. 3.147 A homeowner’s wood stove is equipped with a top burner for cooking. The D 200-mm-diameter burner is fabricated of cast iron (k 65 W/m 䡠 K). The bottom (combustion) side of the burner has 8 straight fins of uniform cross section, arranged as shown in the sketch. A very thin ceramic coating ( 0.95) is applied to all surfaces of the burner. The top of the burner is exposed to room conditions (Tsur,t T앝,t 20 C, ht 40 W/m2 䡠 K), while the bottom of the burner is exposed to combustion conditions (Tsur,b T앝.b 450 C, hb 50 W/m2 䡠 K). Compare the top surface temperature of the finned burner to that which would exist for a burner without fins. Hint: Use the same expression for radiation heat transfer to the bottom of the finned burner as for the burner with no fins.

Ts = 350 K

Water

Fins (t = 5 mm)

Gases

Tg = 750 K

hg = 30 W/m2•K Tube wall

If the tube surface temperature is Ts 350 K and the gas-side convection coefficient is hg 30 W/m2 䡠 K, what is the rate of heat transfer to the water per meter of pipe length? 3.150 As a means of enhancing heat transfer from highperformance logic chips, it is common to attach a

222

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

heat sink to the chip surface in order to increase the surface area available for convection heat transfer. Because of the ease with which it may be manufactured (by taking orthogonal sawcuts in a block of material), an attractive option is to use a heat sink consisting of an array of square fins of width w on a side. The spacing between adjoining fins would be determined by the width of the sawblade, with the sum of this spacing and the fin width designated as the fin pitch S. The method by which the heat sink is joined to the chip would determine the interfacial contact resistance, Rt,c. Wc

(S w) 0.25 mm, and/or increasing Lƒ (subject to manufacturing constraints that Lƒ 10 mm). Assess the effect of such changes. 3.151 Because of the large number of devices in today’s PC chips, finned heat sinks are often used to maintain the chip at an acceptable operating temperature. Two fin designs are to be evaluated, both of which have base (unfinned) area dimensions of 53 mm 57 mm. The fins are of square cross section and fabricated from an extruded aluminum alloy with a thermal conductivity of 175 W/m 䡠 K. Cooling air may be supplied at 25 C, and the maximum allowable chip temperature is 75 C. Other features of the design and operating conditions are tabulated.

Heat sink Top View

T∞, h

Fin Dimensions Cross Section Design w ⴛ w (mm) A B

w Square fins

33 11

Length L (mm) 30 7

Convection Number of Coefficient Fins in Array (W/m2 䡠 K) 69 14 17

125 375

57 mm

Lf

L = 30 mm

S Heat sink Interface,

Lb

53 mm

R"t,c Chip,

3 mm × 3 mm Tb = 75°C cross section

qc, Tc

Consider a square chip of width Wc 16 mm and conditions for which cooling is provided by a dielectric liquid with T앝 25 C and h 1500 W/m2 䡠 K. The heat sink is fabricated from copper (k 400 W/m 䡠 K), and its characteristic dimensions are w 0.25 mm, S 0.50 mm, Lƒ 6 mm, and Lb 3 mm. The prescribed values of w and S represent minima imposed by manufacturing constraints and the need to maintain adequate flow in the passages between fins. (a) If a metallurgical joint provides a contact resistance of Rt,c 5 106 m2 䡠 K/W and the maximum allowable chip temperature is 85 C, what is the maximum allowable chip power dissipation qc? Assume all of the heat to be transferred through the heat sink. (b) It may be possible to increase the heat dissipation by increasing w, subject to the constraint that

54 pins, 9 × 6 array (Design A)

Determine which fin arrangement is superior. In your analysis, calculate the heat rate, efficiency, and effectiveness of a single fin, as well as the total heat rate and overall efficiency of the array. Since real estate inside the computer enclosure is important, compare the total heat rate per unit volume for the two designs. 3.152 Consider design B of Problem 3.151. Over time, dust can collect in the fine grooves that separate the fins. Consider the buildup of a dust layer of thickness Ld, as shown in the sketch. Calculate and plot the total heat rate for design B for dust layers in the range 0 Ld 5 mm. The thermal conductivity of the dust can be taken as kd = 0.032 W/m 䡠 K. Include the effects of convection from the fin tip.

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223

Problems

while air at T앝,o 25 C flows through the annular region formed by the larger concentric tube.

L = 7 mm Ld

r1 Dust

r2 Air

r3 1 mm 1 mm cross section

Tb = 75°C

•

Spider with 12 ribs

Tube

r2

r1 t

Insulating sleeve

Water

T∞,i, hi

3.153 A long rod of 20-mm diameter and a thermal conductivity of 1.5 W/m 䡠 K has a uniform internal volumetric thermal energy generation of 106 W/m3. The rod is covered with an electrically insulating sleeve of 2-mm thickness and thermal conductivity of 0.5 W/m 䡠 K. A spider with 12 ribs and dimensions as shown in the sketch has a thermal conductivity of 175 W/m 䡠 K, and is used to support the rod and to maintain concentricity with an 80mm-diameter tube. Air at T앝 25 C passes over the spider surface, and the convection coefficient is 20 W/m2 䡠 K. The outer surface of the tube is well insulated. We wish to increase volumetric heating within the rod, while not allowing its centerline temperature to exceed 100 C. Determine the impact of the following changes, which may be effected independently or concurrently: (i) increasing the air speed and hence the convection coefficient; (ii) changing the number and/or thickness of the ribs; and (iii) using an electrically nonconducting sleeve material of larger thermal conductivity (e.g., amorphous carbon or quartz). Recommend a realis. tic configuration that yields a significant increase in q. Rod, q

T∞,o, ho

r3

Air

T∞ = 25°C

t

(a) Sketch the equivalent thermal circuit of the heater and relate each thermal resistance to appropriate system parameters. (b) If hi 5000 W/m2 䡠 K and ho 200 W/m2 䡠 K, what is the heat rate per unit length? (c) Assess the effect of increasing the number of fins N and/or the fin thickness t on the heat rate, subject to the constraint that Nt 50 mm. 3.155 Determine the percentage increase in heat transfer associated with attaching aluminum fins of rectangular profile to a plane wall. The fins are 50 mm long, 0.5 mm thick, and are equally spaced at a distance of 4 mm (250 fins/m). The convection coefficient associated with the bare wall is 40 W/m2 䡠 K, while that resulting from attachment of the fins is 30 W/m2 䡠 K. 3.156 Heat is uniformly generated at the rate of 2 105 W/m3 in a wall of thermal conductivity 25 W/m 䡠 K and thickness 60 mm. The wall is exposed to convection on both sides, with different heat transfer coefficients and temperatures as shown. There are straight rectangular fins on the right-hand side of the wall, with dimensions as shown and thermal conductivity of 250 W/m 䡠 K. What is the maximum temperature that will occur in the wall?

r1 = 12 mm r2 = 17 mm r3 = 40 mm t = 4 mm L = r3 – r2 = 23 mm

3.154 An air heater consists of a steel tube (k 20 W/m 䡠 K), with inner and outer radii of r1 13 mm and r2 16 mm, respectively, and eight integrally machined longitudinal fins, each of thickness t 3 mm. The fins extend to a concentric tube, which is of radius r3 40 mm and insulated on its outer surface. Water at a temperature T앝,i 90 C flows through the inner tube,

Lf = 20 mm

k = 25 W/m•K q• = 2 105 W/m3 h1 = 50 W/m2•K T∞,1 = 30°C

t = 2 mm

δ = 2 mm 2L = 60 mm

kf = 250 W/m•K

h2 = 12 W/m2•K T∞,2 = 15°C

224

Chapter 3

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One-Dimensional, Steady-State Conduction

3.157 Aluminum fins of triangular profile are attached to a plane wall whose surface temperature is 250 C. The fin base thickness is 2 mm, and its length is 6 mm. The system is in ambient air at a temperature of 20 C, and the surface convection coefficient is 40 W/m2 䡠 K. (a) What are the fin efficiency and effectiveness? (b) What is the heat dissipated per unit width by a single fin? 3.158 An annular aluminum fin of rectangular profile is attached to a circular tube having an outside diameter of 25 mm and a surface temperature of 250 C. The fin is 1 mm thick and 10 mm long, and the temperature and the convection coefficient associated with the adjoining fluid are 25 C and 25 W/m2 䡠 K, respectively. (a) What is the heat loss per fin? (b) If 200 such fins are spaced at 5-mm increments along the tube length, what is the heat loss per meter of tube length? 3.159 Annular aluminum fins of rectangular profile are attached to a circular tube having an outside diameter of 50 mm and an outer surface temperature of 200 C. The fins are 4 mm thick and 15 mm long. The system is in ambient air at a temperature of 20 C, and the surface convection coefficient is 40 W/m2 䡠 K. (a) What are the fin efficiency and effectiveness? (b) If there are 125 such fins per meter of tube length, what is the rate of heat transfer per unit length of tube?

qi 105 W/m2. Assuming negligible contact resistance between the wall and the casing, determine the wall inner temperature Ti, the interface temperature T1, and the fin base temperature Tb. Determine these temperatures if the interface contact resistance is Rt, c 104 m2 䡠 K/W. 3.161 Consider the air-cooled combustion cylinder of Problem 3.160, but instead of imposing a uniform heat flux at the inner surface, consider conditions for which the time-averaged temperature of the combustion gases is Tg 1100 K and the corresponding convection coefficient is hg 150 W/m2 䡠 K. All other conditions, including the cylinder/casing contact resistance, remain the same. Determine the heat rate per unit length of cylinder (W/m), as well as the cylinder inner temperature Ti, the interface temperatures T1,i and T1,o, and the fin base temperature Tb. Subject to the constraint that the fin gap is fixed at ␦ 2 mm, assess the effect of increasing the fin thickness at the expense of reducing the number of fins. 3.162 Heat transfer from a transistor may be enhanced by inserting it in an aluminum sleeve (k 200 W/m 䡠 K) having 12 integrally machined longitudinal fins on its outer surface. The transistor radius and height are r1 2.5 mm and H 4 mm, respectively, while the fins are of length L r3 r2 8 mm and uniform thickness t 0.8 mm. The thickness of the sleeve base is r2 r1 1 mm, and the contact resistance of the sleeve-transistor interface is Rt,c 0.6 103 m2 䡠 K/W. Air at T앝 20 C flows over the fin surface, providing an approximately uniform convection coeffficient of h 30 W/m2 䡠 K.

3.160 It is proposed to air-cool the cylinders of a combustion chamber by joining an aluminum casing with annular fins (k 240 W/m 䡠 K) to the cylinder wall (k 50 W/m 䡠 K). Cylinder wall

Ti

t

Aluminum casing

T∞, h

T1 Tb

Transistor

R"t,c, T1

t = 2 mm q"i

H

δ = 2 mm

Sleeve with longitudinal fins

ri = 60 mm

T∞, h

r1 = 66 mm r2 = 70 mm ro = 95 mm

The air is at 320 K and the corresponding convection coefficient is 100 W/m2 䡠 K. Although heating at the inner surface is periodic, it is reasonable to assume steady-state conditions with a time-averaged heat flux of

r1 r2

r3

(a) When the transistor case temperature is 80 C, what is the rate of heat transfer from the sleeve? (b) Identify all of the measures that could be taken to improve design and/or operating conditions, such that heat dissipation may be increased while still maintaining a case temperature of 80 C. In words, assess the relative merits of each measure. Choose

䊏

225

Problems

what you believe to be the three most promising measures, and numerically assess the effect of corresponding changes in design and/or operating conditions on thermal performance. 3.163 Consider the conditions of Problem 3.149 but now allow for a tube wall thickness of 5 mm (inner and outer diameters of 50 and 60 mm), a fin-to-tube thermal contact resistance of 104 m2 䡠 K/W, and the fact that the water temperature, Tw 350 K, is known, not the tube surface temperature. The water-side convection coefficient is hw 2000 W/m2 䡠 K. Determine the rate of heat transfer per unit tube length (W/m) to the water. What would be the separate effect of each of the following design changes on the heat rate: (i) elimination of the contact resistance; (ii) increasing the number of fins from four to eight; and (iii) changing the tube wall and fin material from copper to AISI 304 stainless steel (k 20 W/m 䡠 K)? 3.164 A scheme for concurrently heating separate water and air streams involves passing them through and over an array of tubes, respectively, while the tube wall is heated electrically. To enhance gas-side heat transfer, annular fins of rectangular profile are attached to the outer tube surface. Attachment is facilitated with a dielectric adhesive that electrically isolates the fins from the current-carrying tube wall. Gas flow

(a) Assuming uniform volumetric heat generation within the tube wall, obtain expressions for the heat rate per unit tube length (W/m) at the inner (ri) and outer (ro) surfaces of the wall. Express your results in terms of the tube inner and outer surface temperatures, Ts,i and Ts,o, and other pertinent parameters. (b) Obtain expressions that could be used to determine Ts,i and Ts,o in terms of parameters associated with the water- and air-side conditions. (c) Consider conditions for which the water and air are at T앝,i T앝,o 300 K, with corresponding convection coefficients of hi 2000 W/m2 䡠 K and ho 100 W/m2 䡠 K. Heat is uniformly dissipated in a stainless steel tube (kw 15 W/m 䡠 K), having inner and outer radii of ri 25 mm and ro 30 mm, and aluminum fins (t ␦ 2 mm, rt 55 mm) are attached to the outer surface, with Rt,c 104 m2 䡠 K/W. Determine the heat rates and temperatures at the inner and outer surfaces as a func. tion of the rate of volumetric heating q. The upper . limit to q will be determined by the constraints that Ts,i not exceed the boiling point of water (100 C) and Ts,o not exceed the decomposition temperature of the adhesive (250 C).

The Bioheat Equation 3.165 Consider the conditions of Example 3.12, except that the person is now exercising (in the air environment), which increases the metabolic heat generation rate by a factor of 8, to 5600 W/m3. At what rate would the person have to perspire (in liters/s) to maintain the same skin temperature as in that example? 3.166 Consider the conditions of Example 3.12 for an air environment, except now the air and surroundings temperatures are both 15 C. Humans respond to cold by shivering, which increases the metabolic heat generation rate. What would the metabolic heat generation rate (per unit volume) have to be to maintain a comfortable skin temperature of 33 C under these conditions?

Liquid flow

Air

T∞,o, ho t

Ts,o

δ

Ts,i

rt T∞,i, hi

ri ro

I Tube, q•, k w Adhesive, R"t,c

3.167 Consider heat transfer in a forearm, which can be approximated as a cylinder of muscle of radius 50 mm (neglecting the presence of bones), with an outer layer of skin and fat of thickness 3 mm. There is metabolic heat generation and perfusion within the muscle. The metabolic heat generation rate, perfusion rate, arterial temperature, and properties of blood, muscle, and skin/fat layer are identical to those in Example 3.12.

226

Chapter 3

䊏

One-Dimensional, Steady-State Conduction with the flowing gases is h h1 h2 80 W/m2 䡠 K while the electrical resistance of the load is Re,load 4 .

The environment and surroundings are the same as for the air environment in Example 3.12. ri = 50 mm Skin/fat

Cover plate Heat sink 1, k

Re,load Air T∞,1, h1

Muscle δ sf = 3 mm

Thermoelectric module Lb

(a) Write the bioheat transfer equation in radial coordinates. Write the boundary conditions that express symmetry at the centerline of the forearm and specified temperature at the outer surface of the muscle. Solve the differential equation and apply the boundary conditions to find an expression for the temperature distribution. Note that the derivatives of the modified Bessel functions are given in Section 3.6.4. (b) Equate the heat flux at the outer surface of the muscle to the heat flux through the skin/fat layer and into the environment to determine the temperature at the outer surface of the muscle. (c) Find the maximum forearm temperature.

Thermoelectric Power Generation 3.168 For one of the M 48 modules of Example 3.13, determine a variety of different efficiency values concerning the conversion of waste heat to electrical energy. (a) Determine the thermodynamic efficiency, therm ⬅ PM1/q1. (b) Determine the figure of merit ZT for one module, and the thermoelectric efficiency, TE using Equation 3.128.

2L Lf

t

Air T∞,2, h2

Solder, Rt,c Heat sink 2, k

Cover plate S W

(a) Sketch the equivalent thermal circuit and determine the electric power generated by the module for the situation where the hot and cold gases provide convective heating and cooling directly to the module (no heat sinks). (b) Two heat sinks (k 180 W/m 䡠 K; see sketch), each with a base thickness of Lb 4 mm and fin length Lf 20 mm, are soldered to the upper and lower sides of the module. The fin spacing is 3 mm, while the solder joints each have a thermal 2.5 106 m2 䡠 K/W. Each resistance of Rt,c heat sink has N 11 fins, so that t 2.182 mm and S 5.182 mm, as determined from the requirements that W (N 1)S t and S t 3 mm. Sketch the equivalent thermal circuit and determine the electric power generated by the module. Compare the electric power generated to your answer for part (a). Assume adiabatic fin tips and convection coefficients that are the same as in part (a).

(c) Determine the Carnot efficiency, Carnot 1 – T2/T1. (d) Determine both the thermoelectric efficiency and the Carnot efficiency for the case where h1 h2 l 앝. (e) The energy conversion efficiency of thermoelectric devices is commonly reported by evaluating Equation 3.128, but with T앝,1 and T앝,2 used instead of T1 and T2, respectively. Determine the value of TE based on the inappropriate use of T앝,1 and T앝,2, and compare with your answers for parts (b) and (d). 3.169 One of the thermoelectric modules of Example 3.13 is installed between a hot gas at T앝,1 450 C and a cold gas at T앝,2 20 C. The convection coefficient associated

3.170 Thermoelectric modules have been used to generate electric power by tapping the heat generated by wood stoves. Consider the installation of the thermoelectric module of Example 3.13 on a vertical surface of a wood stove that has a surface temperature of 5 Ts 375 C. A thermal contact resistance of Rt,c 106 m2 䡠 K/W exists at the interface between the stove and the thermoelectric module, while the room air and walls are at T앝 Tsur 25 C. The exposed surface of the thermoelectric module has an emissivity of 0.90 and is subjected to a convection coefficient of h 15 W/m2 䡠 K. Sketch the equivalent thermal circuit and determine the electric power

䊏

227

Problems

generated by the module. The load electrical resistance is Re,load 3 . 3.171 The electric power generator for an orbiting satellite is composed of a long, cylindrical uranium heat source that is housed within an enclosure of square cross section. The only way for heat that is generated by the uranium to leave the enclosure is through four rows of the thermoelectric modules of Example 3.13. The thermoelectric modules generate electric power and also radiate heat into deep space characterized by Tsur 4 K. Consider the situation for which there are 20 modules in each row for a total of M 4 20 80 modules. The modules are wired in series with an electrical load of Re,load 250 , and have an emissivity of 0.93. Determine the electric power generated for E˙ g 1, 10, and 100 kW. Also determine the surface temperatures of the modules for the three thermal energy generation rates.

Tsur W 2L

•

Heat source, Eg Insulation Thermoelectric module,

Re, load

3.172 Rows of the thermoelectric modules of Example 3.13 are attached to the flat absorber plate of Problem 3.108. The rows of modules are separated by Lsep 0.5 m and the backs of the modules are cooled by water at a temperature of Tw 40 C, with h 45 W/m2 䡠 K. Cover plate Evacuated space Absorber plate

q″rad

W

Water Tw , h

Insulation

Lsep

Thermoelectric module

Determine the electric power produced by one row of thermoelectric modules connected in series electrically with a load resistance of 60 . Calculate the heat

transfer rate to the flowing water. Assume rows of 20 immediately adjacent modules, with the lengths of both the module rows and water tubing to be Lrow 20W where W 54 mm is the module dimension taken from Example 3.13. Neglect thermal contact resistances and the temperature drop across the tube wall, and assume that the high thermal conductivity tube wall creates a uniform temperature around the tube perimeter. Because of the thermal resistance provided by the thermoelectric modules, it is no longer appropriate to assume that the temperature of the absorber plate directly above a tube is equal to that of the water.

Micro- and Nanoscale Conduction 3.173 Determine the conduction heat transfer through an air layer held between two 10 mm 10 mm parallel aluminum plates. The plates are at temperatures Ts,1 305 K and Ts,2 295 K, respectively, and the air is at atmospheric pressure. Determine the conduction heat rate for plate spacings of L 1 mm, L 1 m, and L 10 nm. Assume a thermal accommodation coefficient of ␣t 0.92. 3.174 Determine the parallel plate separation distance L, above which the thermal resistance associated with molecule-surface collisions Rt,ms is less than 1% of the resistance associated with molecule–molecule collisions, Rt,mm for (i) air between steel plates with ␣t 0.92 and (ii) helium between clean aluminum plates with ␣t 0.02. The gases are at atmospheric pressure, and the temperature is T 300 K. 3.175 Determine the conduction heat flux through various plane layers that are subjected to boundary temperatures of Ts,1 301 K and Ts,2 299 K at atmospheric pressure. Hint: Do not account for micro- or nanoscale effects within the solid, and assume the thermal accommodation coefficient for an aluminum–air interface is ␣t 0.92. (a) Case A: The plane layer is aluminum. Determine the heat flux qx for Ltot 600 m and Ltot 600 nm. (b) Case B: Conduction occurs through an air layer. Determine the heat flux qx for Ltot 600 m and Ltot 600 nm. (c) Case C: The composite wall is composed of air held between two aluminum sheets. Determine the heat flux qx for Ltot 600 m (with aluminum sheet thicknesses of ␦ 40 m) and Ltot 600 nm (with aluminum sheet thicknesses of ␦ 40 nm). (d) Case D: The composite wall is composed of 7 air layers interspersed between 8 aluminum sheets.

228

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Determine the heat flux qx for Ltot 600 m (with aluminum sheet and air layer thicknesses of ␦ 40 m) and Ltot 600 nm (with aluminum sheet and air layer thicknesses of ␦ 40 nm).

Ts,1

Ts,1 Aluminum

Air

Ts,2 x Case A

Ts,2 x

Ltot

Case B

Ltot

Ts,1

Ts,1 Air

δ

Aluminum

δ Ts,2

Ts,2 x Case C

Ltot

Air x

Aluminum Case D

Ltot

3.176 The Knudsen number, Kn mfp/L, is a dimensionless parameter used to describe potential micro- or nanoscale effects. Derive an expression for the ratio of the thermal resistance due to molecule–surface collisions to the thermal resistance associated with molecule–molecule collisions, Rt,ms/Rt,mm, in terms of the Knudsen number, the thermal accommodation coefficient ␣t , and the ratio of specific heats ␥, for an ideal gas. Plot the critical Knudsen number, Kncrit, that is associated with Rt,ms /Rt,mm 0.01 versus ␣t, for ␥ 1.4 and 1.67 (corresponding to air and helium, respectively). 3.177 A nanolaminated material is fabricated with an atomic layer deposition process, resulting in a series of

stacked, alternating layers of tungsten and aluminum oxide, each layer being ␦ 0.5 nm thick. Each tungsten–aluminum oxide interface is associated with a thermal resistance of Rt,i 3.85 109 m2 䡠 K/W. The theoretical values of the thermal conductivities of the thin aluminum oxide and tungsten layers are kA 1.65 W/m 䡠 K and kT 6.10 W/m 䡠 K, respectively. The properties are evaluated at T 300 K. (a) Determine the effective thermal conductivity of the nanolaminated material. Compare the value of the effective thermal conductivity to the bulk thermal conductivities of aluminum oxide and tungsten, given in Tables A.1 and A.2. (b) Determine the effective thermal conductivity of the nanolaminated material assuming that the thermal conductivities of the tungsten and aluminum oxide layers are equal to their bulk values. 3.178 Gold is commonly used in semiconductor packaging to form interconnections (also known as interconnects) that carry electrical signals between different devices in the package. In addition to being a good electrical conductor, gold interconnects are also effective at protecting the heat-generating devices to which they are attached by conducting thermal energy away from the devices to surrounding, cooler regions. Consider a thin film of gold that has a cross section of 60 nm 250 nm. (a) For an applied temperature difference of 20 C, determine the energy conducted along a 1-mlong, thin-film interconnect. Evaluate properties at 300 K. (b) Plot the lengthwise (in the 1-m direction) and spanwise (in the thinnest direction) thermal conductivities of the gold film as a function of the film thickness L for 30 L 140 nm.

C H A P T E R

Two-Dimensional, Steady-State Conduction

4

230

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

T

o this point, we have restricted our attention to conduction problems in which the temperature gradient is significant for only one coordinate direction. However, in many cases such problems are grossly oversimplified if a one-dimensional treatment is used, and it is necessary to account for multidimensional effects. In this chapter, we consider several techniques for treating two-dimensional systems under steady-state conditions. We begin our consideration of two-dimensional, steady-state conduction by briefly reviewing alternative approaches to determining temperatures and heat rates (Section 4.1). The approaches range from exact solutions, which may be obtained for idealized conditions, to approximate methods of varying complexity and accuracy. In Section 4.2 we consider some of the mathematical issues associated with obtaining an exact solution. In Section 4.3, we present compilations of existing exact solutions for a variety of simple geometries. Our objective in Sections 4.4 and 4.5 is to show how, with the aid of a computer, numerical ( finite-difference or finite-element) methods may be used to accurately predict temperatures and heat rates within the medium and at its boundaries.

4.1 Alternative Approaches Consider a long, prismatic solid in which there is two-dimensional heat conduction (Figure 4.1). With two surfaces insulated and the other surfaces maintained at different temperatures, T1 ⬎ T2, heat transfer by conduction occurs from surface 1 to 2. According to Fourier’s law, Equation 2.3 or 2.4, the local heat flux in the solid is a vector that is everywhere perpendicular to lines of constant temperature (isotherms). The directions of the heat flux vector are represented by the heat flow lines of Figure 4.1, and the vector itself is the resultant of heat flux components in the x- and y-directions. These components are determined by Equation 2.6. Since the heat flow lines are, by definition, in the direction of heat flow, no heat can be conducted across a heat flow line, and they are therefore sometimes referred to as adiabats. Conversely, adiabatic surfaces (or symmetry lines) are heat flow lines. Recall that, in any conduction analysis, there exist two major objectives. The first objective is to determine the temperature distribution in the medium, which, for the present problem, necessitates determining T(x, y). This objective is achieved by solving the appropriate form of the heat equation. For two-dimensional, steady-state conditions with no generation and constant thermal conductivity, this form is, from Equation 2.22, ⭸2T ⭸2T ⫹ ⫽0 ⭸x2 ⭸y2

(4.1)

y q"y

T1

q" = iq"x + jq"y

T2 < T1

q"x Isotherms

Heat flow lines

Isotherm x

FIGURE 4.1 Two-dimensional conduction.

4.2

䊏

The Method of Separation of Variables

231

If Equation 4.1 can be solved for T(x, y), it is then a simple matter to satisfy the second major objective, which is to determine the heat flux components q⬙x and q⬙y by applying the rate equations (2.6). Methods for solving Equation 4.1 include the use of analytical, graphical, and numerical (finite-difference, finite-element, or boundary-element) approaches. The analytical method involves effecting an exact mathematical solution to Equation 4.1. The problem is more difficult than those considered in Chapter 3, since it now involves a partial, rather than an ordinary, differential equation. Although several techniques are available for solving such equations, the solutions typically involve complicated mathematical series and functions and may be obtained for only a restricted set of simple geometries and boundary conditions [1–5]. Nevertheless, the solutions are of value, since the dependent variable T is determined as a continuous function of the independent variables (x, y). Hence the solution could be used to compute the temperature at any point of interest in the medium. To illustrate the nature and importance of analytical techniques, an exact solution to Equation 4.1 is obtained in Section 4.2 by the method of separation of variables. Conduction shape factors and dimensionless conduction heat rates (Section 4.3) are compilations of existing solutions for geometries that are commonly encountered in engineering practice. In contrast to the analytical methods, which provide exact results at any point, graphical and numerical methods can provide only approximate results at discrete points. Although superseded by computer solutions based on numerical procedures, the graphical, or flux-plotting, method may be used to obtain a quick estimate of the temperature distribution. Its use is restricted to two-dimensional problems involving adiabatic and isothermal boundaries. The method is based on the fact that isotherms must be perpendicular to heat flow lines, as noted in Figure 4.1. Unlike the analytical or graphical approaches, numerical methods (Sections 4.4 and 4.5) may be used to obtain accurate results for complex, two- or three-dimensional geometries involving a wide variety of boundary conditions.

4.2 The Method of Separation of Variables To appreciate how the method of separation of variables may be used to solve twodimensional conduction problems, we consider the system of Figure 4.2. Three sides of a thin rectangular plate or a long rectangular rod are maintained at a constant temperature T1, while the fourth side is maintained at a constant temperature T2 ⫽ T1. Assuming negligible heat transfer from the surfaces of the plate or the ends of the rod, temperature gradients normal to the x–y plane may be neglected (⭸2T/⭸z2 ⬇ 0) and conduction heat transfer is primarily in the x- and y-directions. We are interested in the temperature distribution T(x, y), but to simplify the solution we introduce the transformation ⬅

T ⫺ T1 T2 ⫺ T1

(4.2)

Substituting Equation 4.2 into Equation 4.1, the transformed differential equation is then ⭸2 ⭸2 ⫹ ⫽0 ⭸x2 ⭸y2

The graphical method is described, and its use is demonstrated, in Section 4S.1.

(4.3)

232

Chapter 4

y

䊏

Two-Dimensional, Steady-State Conduction

T2, θ = 1

W

T1, θ = 0

T1, θ = 0

T(x, y)

0 0

L T1, θ = 0

x

FIGURE 4.2 Two-dimensional conduction in a thin rectangular plate or a long rectangular rod.

Since the equation is second order in both x and y, two boundary conditions are needed for each of the coordinates. They are (0, y) ⫽ 0 (L, y) ⫽ 0

and and

(x, 0) ⫽ 0 (x, W) ⫽ 1

Note that, through the transformation of Equation 4.2, three of the four boundary conditions are now homogeneous and the value of is restricted to the range from 0 to 1. We now apply the separation of variables technique by assuming that the desired solution can be expressed as the product of two functions, one of which depends only on x while the other depends only on y. That is, we assume the existence of a solution of the form (x, y) ⫽ X(x) 䡠 Y(y)

(4.4)

Substituting into Equation 4.3 and dividing by XY, we obtain 2 2 ⫺ 1 d X2 ⫽ 1 d Y2 (4.5) X dx Y dy and it is evident that the differential equation is, in fact, separable. That is, the left-hand side of the equation depends only on x and the right-hand side depends only on y. Hence the equality can apply in general (for any x or y) only if both sides are equal to the same constant. Identifying this, as yet unknown, separation constant as 2, we then have

d 2X ⫹ 2X ⫽ 0 (4.6) dx 2 d 2Y ⫺ 2Y ⫽ 0 (4.7) dy 2 and the partial differential equation has been reduced to two ordinary differential equations. Note that the designation of 2 as a positive constant was not arbitrary. If a negative value were selected or a value of 2 ⫽ 0 was chosen, it is readily shown (Problem 4.1) that it would be impossible to obtain a solution that satisfies the prescribed boundary conditions. The general solutions to Equations 4.6 and 4.7 are, respectively, X ⫽ C1 cos x ⫹ C2 sin x Y ⫽ C3e⫺y ⫹ C4e⫹y in which case the general form of the two-dimensional solution is ⫽ (C1 cos x ⫹ C2 sin x)(C3e⫺y ⫹ C4ey)

(4.8)

4.2

䊏

233

The Method of Separation of Variables

Applying the condition that (0, y) ⫽ 0, it is evident that C1 ⫽ 0. In addition from the requirement that (x, 0) ⫽ 0, we obtain C2 sin x(C3 ⫹ C4) ⫽ 0 which may only be satisfied if C3 ⫽ ⫺C4. Although the requirement could also be satisfied by having C2 ⫽ 0, this would result in (x, y) ⫽ 0, which does not satisfy the boundary condition (x, W) ⫽ 1. If we now invoke the requirement that (L, y) ⫽ 0, we obtain C2C4 sin L(ey ⫺ e⫺y) ⫽ 0 The only way in which this condition may be satisfied (and still have a nonzero solution) is by requiring that assume discrete values for which sin L ⫽ 0. These values must then be of the form ⫽ n L

n ⫽ 1, 2, 3, . . .

(4.9)

where the integer n ⫽ 0 is precluded, since it implies (x, y) ⫽ 0. The desired solution may now be expressed as ⫽ C2C4 sin nx (eny/L ⫺ e⫺ny/L) L

(4.10)

Combining constants and acknowledging that the new constant may depend on n, we obtain ny (x, y) ⫽ Cn sin nx sinh L L where we have also used the fact that (eny/L ⫺ e⫺ny/L) ⫽ 2 sinh (ny/L). In this form we have really obtained an infinite number of solutions that satisfy the differential equation and boundary conditions. However, since the problem is linear, a more general solution may be obtained from a superposition of the form (x, y) ⫽

⬁

sinh 兺 C sin nx L n

n⫽1

ny L

(4.11)

To determine Cn we now apply the remaining boundary condition, which is of the form (x, W) ⫽ 1 ⫽

⬁

兺C

n⫽1

n

sin nx sinh nW L L

(4.12)

Although Equation 4.12 would seem to be an extremely complicated relation for evaluating Cn, a standard method is available. It involves writing an infinite series expansion in terms of orthogonal functions. An infinite set of functions g1(x), g2(x), … , gn(x), … is said to be orthogonal in the domain a ⱕ x ⱕ b if

冕 g (x)g (x) dx ⫽ 0 b

m

a

n

m⫽n

(4.13)

234

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

Many functions exhibit orthogonality, including the trigonometric functions sin (nx/L) and cos (nx/L) for 0 ⱕ x ⱕ L. Their utility in the present problem rests with the fact that any function f(x) may be expressed in terms of an infinite series of orthogonal functions ⬁

兺 A g (x)

f (x) ⫽

(4.14)

n n

n⫽1

The form of the coefficients An in this series may be determined by multiplying each side of the equation by gm(x) and integrating between the limits a and b.

冕 f(x)g (x) dx ⫽ 冕 g (x) 兺 A g (x) dx b

앝

b

m

m

a

n n

a

(4.15)

n⫽1

However, from Equation 4.13 it is evident that all but one of the terms on the right-hand side of Equation 4.15 must be zero, leaving us with

冕 f(x)g (x) dx ⫽ A 冕 g (x) dx b

b

m

2 m

m

a

a

Hence, solving for Am, and recognizing that this holds for any An by switching m to n:

冕 f (x)g (x) dx A ⫽ 冕 g (x) dx b

n

a

n

(4.16)

b

2 n

a

The properties of orthogonal functions may be used to solve Equation 4.12 for Cn by formulating an infinite series for the appropriate form of f(x). From Equation 4.14 it is evident that we should choose f(x) ⫽ 1 and the orthogonal function gn(x) ⫽ sin (nx/L). Substituting into Equation 4.16 we obtain

冕 sin nxL dx 2 (⫺1) ⫹ 1 ⫽ A ⫽ n nx 冕 sin L dx L

n⫹1

0 L

n

2

0

Hence from Equation 4.14, we have 1⫽

n⫹1 2 (⫺1) ⫹ 1 sin nx n L n⫽1 ⬁

兺

(4.17)

which is simply the expansion of unity in a Fourier series. Comparing Equations 4.12 and 4.17 we obtain Cn ⫽

2[(⫺1)n⫹1 ⫹ 1] n sinh (nW/L)

n ⫽ 1, 2, 3, . . .

(4.18)

Substituting Equation 4.18 into Equation 4.11, we then obtain for the final solution 2 (x, y) ⫽

(⫺1)n⫹1 ⫹ 1 sinh (ny/L) sin nx n L sinh (nW/L) n⫽1 ⬁

兺

(4.19)

4.3

䊏

The Conduction Shape Factor and the Dimensionless Conduction Heat Rate

235

y W

θ =1

0.75 0.5 0.25

θ =0

0 0

θ =0

θ = 0.1

θ =0

L

x

FIGURE 4.3 Isotherms and heat flow lines for two-dimensional conduction in a rectangular plate.

Equation 4.19 is a convergent series, from which the value of may be computed for any x and y. Representative results are shown in the form of isotherms for a schematic of the rectangular plate (Figure 4.3). The temperature T corresponding to a value of may be obtained from Equation 4.2, and components of the heat flux may be determined by using Equation 4.19 with Equation 2.6. The heat flux components determine the heat flow lines, which are shown in the figure. We note that the temperature distribution is symmetric about x ⫽ L/2, with ⭸T/⭸x ⫽ 0 at that location. Hence, from Equation 2.6, we know the symmetry plane at x ⫽ L/2 is adiabatic and therefore is a heat flow line. However, note that the discontinuities prescribed at the upper corners of the plate are physically untenable. In reality, large temperature gradients could be maintained in proximity to the corners, but discontinuities could not exist. Exact solutions have been obtained for a variety of other geometries and boundary conditions, including cylindrical and spherical systems. Such solutions are presented in specialized books on conduction heat transfer [1–5].

4.3 The Conduction Shape Factor and the Dimensionless Conduction Heat Rate In general, finding analytical solutions to the two- or three-dimensional heat equation is time-consuming and, in many cases, not possible. Therefore, a different approach is often taken. For example, in many instances, two- or three-dimensional conduction problems may be rapidly solved by utilizing existing solutions to the heat diffusion equation. These solutions are reported in terms of a shape factor S or a steady-state dimensionless conduction heat rate, q*ss. The shape factor is defined such that q ⫽ Sk⌬T1⫺2

(4.20)

where ⌬T1⫺2 is the temperature difference between boundaries, as shown in, for example, Figure 4.2. It also follows that a two-dimensional conduction resistance may be expressed as Rt,cond(2D) ⫽ 1 Sk

(4.21)

236

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

Shape factors have been obtained analytically for numerous two- and three-dimensional systems, and results are summarized in Table 4.1 for some common configurations. Results are also available for other configurations [6–9]. In cases 1 through 8 and case 11, twodimensional conduction is presumed to occur between the boundaries that are maintained at uniform temperatures, with ⌬T1⫺2 ⫽ T1 ⫺ T2. In case 9, three-dimensional conduction exists in the corner region, while in case 10 conduction occurs between an isothermal disk (T1) and a semi-infinite medium of uniform temperature (T2) at locations well removed from the disk. Shape factors may also be defined for one-dimensional geometries, and from the results of Table 3.3, it follows that for plane, cylindrical, and spherical walls, respectively, the shape factors are A/L, 2L/ln(r2/r1), and 4r1r2/(r2 ⫺ r1). Cases 12 through 15 are associated with conduction from objects held at an isothermal temperature (T1) that are embedded within an infinite medium of uniform temperature (T2)

Shape factors for two-dimensional geometries may also be estimated with the graphical method that is described in Section 4S.1.

TABLE 4.1 Conduction shape factors and dimensionless conduction heat rates for selected systems. (a) Shape factors [q ⴝ Sk(T1 ⴚ T2)] System

Schematic

Restrictions

Shape Factor

z ⬎ D/2

2D 1 ⫺ D/4z

LⰇD

2L cosh⫺1 (2z/D)

LⰇD z ⬎ 3D/2

2L ln (4z/D)

LⰇD

2L ln (4L/D)

T2

Case 1 Isothermal sphere buried in a semiinfinite medium

z T1

D T2

Case 2 Horizontal isothermal cylinder of length L buried in a semi-infinite medium

z L T1

Case 3 Vertical cylinder in a semi-infinite medium

D T2 L T1 D

Case 4 Conduction between two cylinders of length L in infinite medium

T1

D1

D2 T2

w

L Ⰷ D1, D2 LⰇw

2L

冢

cosh⫺1

4w2 ⫺ D21 ⫺ D22 2D1D2

冣

4.3

TABLE 4.1

䊏

The Conduction Shape Factor and the Dimensionless Conduction Heat Rate

237

Continued

System Case 5 Horizontal circular cylinder of length L midway between parallel planes of equal length and infinite width

Schematic

Restrictions

T2

∞

∞

z

z Ⰷ D/2 LⰇz

D

z T1

∞

Shape Factor

2L ln (8z/D)

∞

T2

Case 6 Circular cylinder of length L centered in a square solid of equal length

T2 D w

w⬎D LⰇw

2L ln (1.08 w/D)

T1

Case 7 Eccentric circular cylinder of length L in a cylinder of equal length

T1

d D

T2

z

Case 8 Conduction through the edge of adjoining walls

2L D⬎d LⰇD

冢D

cosh⫺1

2

⫹ d2 ⫺ 4z2 2Dd

冣

T2

L D

D ⬎ 5L

0.54D

L Ⰶ length and width of wall

0.15L

None

2D

W w ⬍ 1.4

2L 0.785 ln (W/w)

W w ⬎ 1.4

2L 0.930 ln (W/w) ⫺ 0.050

T1 L

Case 9 Conduction through corner of three walls with a temperature difference ⌬T1⫺2 across the walls Case 10 Disk of diameter D and temperature T1 on a semi-infinite medium of thermal conductivity k and temperature T2

L

L L

D

T1

k T2

Case 11 Square channel of length L

L

T1 T2 w W

L ⰇW

238

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

TABLE 4.1 Continued (b) Dimensionless conduction heat rates [q ⴝ q*ss kAs(T1 ⴚ T2)/Lc; Lc ⬅ (As/4)1/2] System

Schematic

Case 12 Isothermal sphere of diameter D and temperature T1 in an infinite medium of temperature T2

T1

q*ss

D2

1

D 2 2

2兹2 ⫽ 0.900

2wL

0.932

D T2

Case 13 Infinitely thin, isothermal disk of diameter D and temperature T1 in an infinite medium of temperature T2

T1 D T2

Case 14 Infinitely thin rectangle of length L, width w, and temperature T1 in an infinite medium of temperature T2

Case 15 Cuboid shape of height d with a square footprint of width D and temperature T1 in an infinite medium of temperature T2

Active Area, As

L w

T1 T2

D

2D2 ⫹ 4Dd T1

d

T2

d/D

q*ss

0.1 1.0 2.0 10

0.943 0.956 0.961 1.111

at locations removed from the object. For these infinite medium cases, useful results may be obtained by defining a characteristic length Lc ⬅ (As /4)1/2

(4.22)

where As is the surface area of the object. Conduction heat transfer rates from the object to the infinite medium may then be reported in terms of a dimensionless conduction heat rate [10] q* ss ⬅ qLc /kAs(T1 ⫺ T2)

(4.23)

From Table 4.1, it is evident that the values of q*ss, which have been obtained analytically and numerically, are similar for a wide range of geometrical configurations. As a consequence of this similarity, values of q*ss may be estimated for configurations that are similar to those for which q*ss is known. For example, dimensionless conduction heat rates from cuboid shapes (case 15) over the range 0.1 ⱕ d/D ⱕ 10 may be closely approximated by interpolating the values of q*ss reported in Table 4.1. Additional procedures that may be exploited to estimate values of q*ss for other geometries are explained in [10]. Note that results for q*ss in Table 4.1b may be converted to expressions for S listed in Table 4.1a. For example, the shape factor of case 10 may be derived from the dimensionless conduction heat rate of case 13 (recognizing that the infinite medium can be viewed as two adjacent semi-infinite media).

4.3

䊏

The Conduction Shape Factor and the Dimensionless Conduction Heat Rate

239

The shape factors and dimensionless conduction heat rates reported in Table 4.1 are associated with objects that are held at uniform temperatures. For uniform heat flux conditions, the object’s temperature is no longer uniform but varies spatially with the coolest temperatures located near the periphery of the heated object. Hence, the temperature difference that is used to define S or q*ss is replaced by a temperature difference involving the spatially averaged surface temperature of the object (T1 ⫺ T2) or by the difference between the maximum surface temperature of the heated object and the far field temperature of the surrounding medium, (T1,max ⫺ T2). For the uniformly heated geometry of case 10 (a disk of diameter D in contact with a semi-infinite medium of thermal conductivity k and temperature T2), the values of S are 32D/16 and D/2 for temperature differences based on the average and maximum disk temperatures, respectively.

EXAMPLE 4.1 A metallic electrical wire of diameter d ⫽ 5 mm is to be coated with insulation of thermal conductivity k ⫽ 0.35 W/m 䡠 K. It is expected that, for the typical installation, the coated wire will be exposed to conditions for which the total coefficient associated with convection and radiation is h ⫽ 15 W/m2 䡠 K. To minimize the temperature rise of the wire due to ohmic heating, the insulation thickness is specified so that the critical insulation radius is achieved (see Example 3.5). During the wire coating process, however, the insulation thickness sometimes varies around the periphery of the wire, resulting in eccentricity of the wire relative to the coating. Determine the change in the thermal resistance of the insulation due to an eccentricity that is 50% of the critical insulation thickness.

SOLUTION Known: Wire diameter, convective conditions, and insulation thermal conductivity. Find: Thermal resistance of the wire coating associated with peripheral variations in the coating thickness. Schematic: d = 5 mm

tcr /2 D

tcr z

Insulation, k (a) Concentric wire

Assumptions: 1. Steady-state conditions. 2. Two-dimensional conduction.

(b) Eccentric wire

T∞, h

240

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

3. Constant properties. 4. Both the exterior and interior surfaces of the coating are at uniform temperatures.

Analysis: From Example 3.5, the critical insulation radius is 䡠 K ⫽ 0.023 m ⫽ 23 mm rcr ⫽ k ⫽ 0.35 W/m h 15 W/m2 䡠 K Therefore, the critical insulation thickness is tcr ⫽ rcr ⫺ d/2 ⫽ 0.023 m ⫺ 0.005 m ⫽ 0.021 m ⫽ 21 mm 2 The thermal resistance of the coating associated with the concentric wire may be evaluated using Equation 3.33 and is R⬘t,cond ⫽

ln[rcr/(d/2)] ln[0.023 m/(0.005 m/2)] ⫽ ⫽ 1.0 m 䡠 K/W 2k 2(0.35 W/m 䡠 K)

For the eccentric wire, the thermal resistance of the insulation may be evaluated using case 7 of Table 4.1, where the eccentricity is z ⫽ 0.5 ⫻ tcr ⫽ 0.5 ⫻ 0.021 m ⫽ 0.010 m cosh⫺1 R⬘t,cond(2D) ⫽ 1 ⫽ Sk cosh⫺1 ⫽

d ⫺ 4z 冢D ⫹2Dd 冣 2

2

2

2k

冢

(2 ⫻ 0.023 m)2 ⫹ (0.005 m)2 ⫺ 4(0.010 m)2 2 ⫻ (2 ⫻ 0.023 m) ⫻ 0.005 m

冣

2 ⫻ 0.35 W/m 䡠 K

⫽ 0.91 m 䡠 K/W Therefore, the reduction in the thermal resistance of the insulation is 0.10 m 䡠 K/W, or 10%. 䉰

Comments: 1. Reduction in the local insulation thickness leads to a smaller local thermal resistance of the insulation. Conversely, locations associated with thicker coatings have increased local thermal resistances. These effects offset each other, but not exactly; the maximum resistance is associated with the concentric wire case. For this application, eccentricity of the wire relative to the coating provides enhanced thermal performance relative to the concentric wire case. 2. The interior surface of the coating will be at nearly uniform temperature if the thermal conductivity of the wire is high relative to that of the insulation. Such is the case for metallic wire. However, the exterior surface temperature of the coating will not be perfectly uniform due to the variation in the local insulation thickness.

4.4

䊏

241

Finite-Difference Equations

4.4 Finite-Difference Equations As discussed in Sections 4.1 and 4.2, analytical methods may be used, in certain cases, to effect exact mathematical solutions to steady, two-dimensional conduction problems. These solutions have been generated for an assortment of simple geometries and boundary conditions and are well documented in the literature [1–5]. However, more often than not, two-dimensional problems involve geometries and/or boundary conditions that preclude such solutions. In these cases, the best alternative is often one that uses a numerical technique such as the finite-difference, finite-element, or boundary-element method. Another strength of numerical methods is that they can be readily extended to three-dimensional problems. Because of its ease of application, the finite-difference method is well suited for an introductory treatment of numerical techniques.

4.4.1

The Nodal Network

In contrast to an analytical solution, which allows for temperature determination at any point of interest in a medium, a numerical solution enables determination of the temperature at only discrete points. The first step in any numerical analysis must therefore be to select these points. Referring to Figure 4.4, this may be done by subdividing the medium of interest into a number of small regions and assigning to each a reference point that is at its center. ∆x

∆y

m, n + 1

y, n m, n

m + 1, n

x, m

m – 1, n

m, n – 1 (a)

∂T ___ ∂x

Tm,n – Tm –1, n = ______________ m–1/2,n

∂T ___ ∂x

m–1

T(x)

m

∆x Tm+1,n – Tm,n = ______________

m+1/2,n

∆x m – 12_

m + 12_

∆x

∆x

x (b)

FIGURE 4.4 Two-dimensional conduction. (a) Nodal network. (b) Finite-difference approximation.

m+1

242

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

The reference point is frequently termed a nodal point (or simply a node), and the aggregate of points is termed a nodal network, grid, or mesh. The nodal points are designated by a numbering scheme that, for a two-dimensional system, may take the form shown in Figure 4.4a. The x and y locations are designated by the m and n indices, respectively. Each node represents a certain region, and its temperature is a measure of the average temperature of the region. For example, the temperature of the node (m, n) of Figure 4.4a may be viewed as the average temperature of the surrounding shaded area. The selection of nodal points is rarely arbitrary, depending often on matters such as geometric convenience and the desired accuracy. The numerical accuracy of the calculations depends strongly on the number of designated nodal points. If this number is large (a fine mesh), accurate solutions can be obtained.

4.4.2

Finite-Difference Form of the Heat Equation

Determination of the temperature distribution numerically dictates that an appropriate conservation equation be written for each of the nodal points of unknown temperature. The resulting set of equations may then be solved simultaneously for the temperature at each node. For any interior node of a two-dimensional system with no generation and uniform thermal conductivity, the exact form of the energy conservation requirement is given by the heat equation, Equation 4.1. However, if the system is characterized in terms of a nodal network, it is necessary to work with an approximate, or finite-difference, form of this equation. A finite-difference equation that is suitable for the interior nodes of a two-dimensional system may be inferred directly from Equation 4.1. Consider the second derivative, ⭸2T/⭸x2. From Figure 4.4b, the value of this derivative at the (m, n) nodal point may be approximated as ⭸2T ⭸x2

冏

⭸T/⭸x兩 m⫹1/2,n ⫺ ⭸T/⭸x兩 m⫺1/2,n ⌬x

艐

m,n

(4.24)

The temperature gradients may in turn be expressed as a function of the nodal temperatures. That is, ⭸T ⭸x ⭸T ⭸x

冏 冏

艐

Tm⫹1,n ⫺ Tm,n ⌬x

(4.25)

艐

Tm,n ⫺ Tm⫺1,n ⌬x

(4.26)

m⫹1/2,n

m⫺1/2,n

Substituting Equations 4.25 and 4.26 into 4.24, we obtain ⭸2T ⭸x2

冏

艐

m,n

Tm⫹1,n ⫹ Tm⫺1,n ⫺ 2Tm,n (⌬x)2

(4.27)

Proceeding in a similar fashion, it is readily shown that ⭸2T ⭸y2

冏

艐

m,n

艐

⭸T/⭸y 兩m,n⫹1/2 ⫺ ⭸T/⭸y 兩m,n⫺1/2 ⌬y Tm,n⫹1 ⫹ Tm,n⫺1 ⫺ 2Tm,n (⌬y)2

(4.28)

4.4

䊏

243

Finite-Difference Equations

Using a network for which ⌬x ⫽ ⌬y and substituting Equations 4.27 and 4.28 into Equation 4.1, we obtain Tm,n⫹1 ⫹ Tm,n⫺1 ⫹ Tm⫹1,n ⫹ Tm⫺1,n ⫺ 4Tm,n ⫽ 0

(4.29)

Hence for the (m, n) node, the heat equation, which is an exact differential equation, is reduced to an approximate algebraic equation. This approximate, finite-difference form of the heat equation may be applied to any interior node that is equidistant from its four neighboring nodes. It requires simply that the temperature of an interior node be equal to the average of the temperatures of the four neighboring nodes.

4.4.3

The Energy Balance Method

In many cases, it is desirable to develop the finite-difference equations by an alternative method called the energy balance method. As will become evident, this approach enables one to analyze many different phenomena such as problems involving multiple materials, embedded heat sources, or exposed surfaces that do not align with an axis of the coordinate system. In the energy balance method, the finite-difference equation for a node is obtained by applying conservation of energy to a control volume about the nodal region. Since the actual direction of heat flow (into or out of the node) is often unknown, it is convenient to formulate the energy balance by assuming that all the heat flow is into the node. Such a condition is, of course, impossible, but if the rate equations are expressed in a manner consistent with this assumption, the correct form of the finite-difference equation is obtained. For steady-state conditions with generation, the appropriate form of Equation 1.12c is then E˙ in ⫹ E˙ g ⫽ 0

(4.30)

Consider applying Equation 4.30 to a control volume about the interior node (m, n) of Figure 4.5. For two-dimensional conditions, energy exchange is influenced by conduction between (m, n) and its four adjoining nodes, as well as by generation. Hence Equation 4.30 reduces to 4

兺q

(i) l (m,n)

⫹ q˙(⌬x 䡠 ⌬y 䡠 1) ⫽ 0

i⫽1

∆x m, n + 1 ∆y ∆y m – 1, n

m, n

m + 1, n

m, n – 1 ∆x

FIGURE 4.5 Conduction to an interior node from its adjoining nodes.

244

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

where i refers to the neighboring nodes, q(i) l (m, n) is the conduction rate between nodes, and unit depth is assumed. To evaluate the conduction rate terms, we assume that conduction transfer occurs exclusively through lanes that are oriented in either the x- or y-direction. Simplified forms of Fourier’s law may therefore be used. For example, the rate at which energy is transferred by conduction from node (m – 1, n) to (m, n) may be expressed as q(m⫺1,n) l (m,n) ⫽ k(⌬y 䡠 1)

Tm⫺1,n ⫺ Tm,n ⌬x

(4.31)

The quantity (⌬y 䡠 1) is the heat transfer area, and the term (Tm–1,n ⫺ Tm,n)/⌬x is the finitedifference approximation to the temperature gradient at the boundary between the two nodes. The remaining conduction rates may be expressed as q(m⫹1,n) l (m,n) ⫽ k(⌬y 䡠 1)

Tm⫹1,n ⫺ Tm,n ⌬x

(4.32)

q(m,n⫹1) l (m,n) ⫽ k(⌬x 䡠 1)

Tm,n⫹1 ⫺ Tm,n ⌬y

(4.33)

q(m,n⫺1) l (m,n) ⫽ k(⌬x 䡠 1)

Tm,n⫺1 ⫺ Tm,n ⌬y

(4.34)

Note that, in evaluating each conduction rate, we have subtracted the temperature of the (m, n) node from the temperature of its adjoining node. This convention is necessitated by the assumption of heat flow into (m, n) and is consistent with the direction of the arrows shown in Figure 4.5. Substituting Equations 4.31 through 4.34 into the energy balance and remembering that ⌬x ⫽ ⌬y, it follows that the finite-difference equation for an interior node with generation is Tm,n⫹1 ⫹ Tm,n⫺1 ⫹ Tm⫹1,n ⫹ Tm⫺1,n ⫹

q˙ (⌬x)2 ⫺ 4Tm,n ⫽ 0 k

(4.35)

If there is no internally distributed source of energy (q˙ ⫽ 0), this expression reduces to Equation 4.29. It is important to note that a finite-difference equation is needed for each nodal point at which the temperature is unknown. However, it is not always possible to classify all such points as interior and hence to use Equation 4.29 or 4.35. For example, the temperature may be unknown at an insulated surface or at a surface that is exposed to convective conditions. For points on such surfaces, the finite-difference equation must be obtained by applying the energy balance method. To further illustrate this method, consider the node corresponding to the internal corner of Figure 4.6. This node represents the three-quarter shaded section and exchanges energy by convection with an adjoining fluid at T앝. Conduction to the nodal region (m, n) occurs along four different lanes from neighboring nodes in the solid. The conduction heat rates qcond may be expressed as q(m⫺1,n)l(m,n) ⫽ k(⌬y 䡠 1)

Tm⫺1,n ⫺ Tm,n ⌬x

(4.36)

4.4

䊏

245

Finite-Difference Equations

m, n + 1 qcond

∆y qcond

m – 1, n qcond

qconv

qcond m, n – 1

m + 1, n

T∞, h

FIGURE 4.6 Formulation of the finite-difference equation for an internal corner of a solid with surface convection.

∆x

q(m,n⫹1)l(m,n) ⫽ k(⌬x 䡠 1)

Tm,n⫹1 ⫺ Tm,n ⌬y

(4.37)

冢⌬y2 䡠 1冣

Tm⫹1,n ⫺ Tm,n ⌬x

(4.38)

Tm,n⫺1 ⫺ Tm,n q(m,n⫺1)l(m,n) ⫽ k ⌬x 䡠 1 2 ⌬y

(4.39)

q(m⫹1,n)l(m,n) ⫽ k

冢 冣

Note that the areas for conduction from nodal regions (m ⫺ 1, n) and (m, n ⫹ 1) are proportional to ⌬y and ⌬x, respectively, whereas conduction from (m ⫹ 1, n) and (m, n – 1) occurs along lanes of width ⌬y/2 and ⌬x/2, respectively. Conditions in the nodal region (m, n) are also influenced by convective exchange with the fluid, and this exchange may be viewed as occurring along half-lanes in the x- and ydirections. The total convection rate qconv may be expressed as

冢

冣

冢

冣

⌬y q(⬁)l(m,n) ⫽ h ⌬x 䡠 1 (T⬁ ⫺ Tm,n) ⫹ h 䡠 1 (T⬁ ⫺ Tm,n) 2 2

(4.40)

Implicit in this expression is the assumption that the exposed surfaces of the corner are at a uniform temperature corresponding to the nodal temperature Tm,n. This assumption is consistent with the concept that the entire nodal region is characterized by a single temperature, which represents an average of the actual temperature distribution in the region. In the absence of transient, three-dimensional, and generation effects, conservation of energy, Equation 4.30, requires that the sum of Equations 4.36 through 4.40 be zero. Summing these equations and rearranging, we therefore obtain

冢

冣

Tm⫺1,n ⫹ Tm,n⫹1 ⫹ 1 (Tm⫹1,n ⫹ Tm,n⫺1) ⫹ h⌬x T⬁ ⫺ 3 ⫹ h⌬x Tm,n ⫽ 0 2 k k

(4.41)

where again the mesh is such that ⌬x ⫽ ⌬y. Nodal energy balance equations pertinent to several common geometries for situations where there is no internal energy generation are presented in Table 4.2.

246

Chapter 4

TABLE 4.2

䊏

Two-Dimensional, Steady-State Conduction

Summary of nodal finite-difference equations Finite-Difference Equation for ⌬x ⴝ ⌬y

Configuration m, n + 1 ∆y m, n m + 1, n

m – 1, n

(4.29)

Tm,n⫹1 ⫹ Tm,n⫺1 ⫹ Tm⫹1,n ⫹ Tm⫺1,n ⫺ 4Tm,n ⫽ 0 Case 1.

m, n – 1

∆x ∆x

Interior node

m, n + 1

2(Tm⫺1,n ⫹ Tm,n⫹1) ⫹ (Tm⫹1,n ⫹ Tm,n⫺1) m – 1, n

m + 1, n

m, n

⫹2

T∞, h

∆y

m, n – 1

Case 2.

冢

冣

h ⌬x h ⌬x T ⫺2 3⫹ Tm,n ⫽ 0 k 앝 k

(4.41)

Node at an internal corner with convection

m, n + 1 ∆y m, n

(2Tm⫺1,n ⫹ Tm,n⫹1 ⫹ Tm,n⫺1) ⫹

T∞, h

m – 1, n

Case 3.

m, n – 1

冢

冣

2h ⌬x h ⌬x T앝 ⫺ 2 ⫹ 2 Tm,n ⫽ 0 k k

(4.42)a

Node at a plane surface with convection

∆x T∞ , h

m – 1, n

(Tm,n⫺1 ⫹ Tm⫺1,n) ⫹ 2

m, n ∆y m, n – 1

Case 4.

∆x

冢

冣

h ⌬x h ⌬x ⫹ 1 Tm,n ⫽ 0 T ⫺2 k ⬁ k

(4.43)

Node at an external corner with convection

m, n + 1 ∆y m, n

(2Tm⫺1,n ⫹ Tm,n⫹1 ⫹ Tm,n⫺1) ⫹

q"

m – 1, n

m, n – 1

Case 5.

2q⬙ ⌬x ⫺4Tm,n ⫽ 0 k

(4.44)b

Node at a plane surface with uniform heat flux

∆x a,b

To obtain the finite-difference equation for an adiabatic surface (or surface of symmetry), simply set h or q⬙ equal to zero.

EXAMPLE 4.2 Using the energy balance method, derive the finite-difference equation for the (m, n) nodal point located on a plane, insulated surface of a medium with uniform heat generation.

4.4

䊏

247

Finite-Difference Equations

SOLUTION Known: Network of nodal points adjoining an insulated surface. Find: Finite-difference equation for the surface nodal point. Schematic: m, n + 1 q4

Insulated surface

•

k, q

y, n

m – 1, n

m, n q3

q1

x, m

q2

∆y = ∆ x ∆y Unit depth (normal to paper)

m, n – 1

∆x ___ 2

Assumptions: 1. Steady-state conditions. 2. Two-dimensional conduction. 3. Constant properties. 4. Uniform internal heat generation. Analysis: Applying the energy conservation requirement, Equation 4.30, to the control surface about the region (⌬x/2 䡠 ⌬y 䡠 1) associated with the (m, n) node, it follows that, with volumetric heat generation at a rate q˙ ,

冢

冣

q1 ⫹ q2 ⫹ q3 ⫹ q4 ⫹ q˙ ⌬x 䡠 ⌬y 䡠 1 ⫽ 0 2 where Tm⫺1,n ⫺ Tm,n ⌬x T m,n⫺1 ⫺ Tm,n q2 ⫽ k ⌬x 䡠 1 2 ⌬y

q1 ⫽ k(⌬y 䡠 1)

冢

冣

q3 ⫽ 0

冢

冣

Tm,n⫹1 ⫺ Tm,n q4 ⫽ k ⌬x 䡠 1 2 ⌬y Substituting into the energy balance and dividing by k/2, it follows that 2Tm⫺1,n ⫹ Tm,n⫺1 ⫹ Tm,n⫹1 ⫺ 4Tm,n ⫹

q˙(⌬x 䡠 ⌬y) ⫽0 k

䉰

Comments: 1. The same result could be obtained by using the symmetry condition, Tm⫹1,n ⫽ Tm⫺1,n, with the finite-difference equation (Equation 4.35) for an interior nodal point.

248

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

If q˙ ⫽ 0, the desired result could also be obtained by setting h ⫽ 0 in Equation 4.42 (Table 4.2). 2. As an application of the foregoing finite-difference equation, consider the following two-dimensional system within which thermal energy is uniformly generated at an unknown rate q˙. The thermal conductivity of the solid is known, as are convection conditions at one of the surfaces. In addition, temperatures have been measured at locations corresponding to the nodal points of a finite-difference mesh. Ta

•

k, q

Tb

Tc

∆y

Tc ⫽ 230.9⬚C

Td ⫽ 220.1⬚C

Te ⫽ 222.4⬚C

T⬁ ⫽ 200.0⬚C

⌬x ⫽ 10 mm

∆x

Td

Tb ⫽ 227.6⬚C

h ⫽ 50 WⲐm2 䡠 K

y x

Ta ⫽ 235.9⬚C

k ⫽ 1 WⲐm 䡠 K ⌬y ⫽ 10 mm

Te T∞, h

The generation rate can be determined by applying the finite-difference equation to node c. q˙ (⌬x 䡠 ⌬y) ⫽0 k q˙ (0.01 m)2 (2 ⫻ 227.6 ⫹ 222.4 ⫹ 235.9 ⫺ 4 ⫻ 230.9)⬚C ⫹ ⫽0 1 W/m 䡠 K 2Tb ⫹ Te ⫹ Ta ⫺ 4Tc ⫹

q˙ ⫽ 1.01 ⫻ 105 W/m3 From the prescribed thermal conditions and knowledge of q˙, we can also determine whether the conservation of energy requirement is satisfied for node e. Applying an energy balance to a control volume about this node, it follows that q1 ⫹ q2 ⫹ q3 ⫹ q4 ⫹ q˙ (⌬x/2 䡠 ⌬y/2 䡠 1) ⫽ 0 k(⌬x/2 䡠 1)

Tc ⫺ Te T ⫺ Te ⫹ 0 ⫹ h(⌬x/2 䡠 1)(T⬁ ⫺ Te) ⫹ k(⌬y/2 䡠 1) d ⌬y ⌬x ⫹ q˙(⌬x/2 䡠 ⌬y/2 䡠 1) ⫽ 0 ∆x •

k, q

Tc q1 ∆y

q2 q4 Td

Te q3 T∞, h

4.4

䊏

249

Finite-Difference Equations

If the energy balance is satisfied, the left-hand side of this equation will be identically equal to zero. Substituting values, we obtain (230.9 ⫺ 222.4)⬚C 0.010 m ⫹ 0 ⫹ 50 W/m2 䡠 K(0.005 m2) (200 ⫺ 222.4)⬚C (220.1 ⫺ 222.4)⬚C ⫹ 1 W/m 䡠 K(0.005 m2) ⫹ 1.01 ⫻ 105 W/m3(0.005)2 m3 ⫽ 0(?) 0.010 m 4.250 W ⫹ 0 ⫺ 5.600 W ⫺ 1.150 W ⫹ 2.525 W ⫽ 0(?) 0.025 W 艐 0 1 W/m 䡠 K(0.005 m2)

The inability to precisely satisfy the energy balance is attributable to temperature measurement errors, the approximations employed in developing the finite-difference equations, and the use of a relatively coarse mesh.

It is useful to note that heat rates between adjoining nodes may also be formulated in terms of the corresponding thermal resistances. Referring, for example, to Figure 4.6, the rate of heat transfer by conduction from node (m ⫺ 1, n) to (m, n) may be expressed as q(m⫺1,n) l (m,n) ⫽

Tm⫺1,n ⫺ Tm,n Tm⫺1,n ⫺ Tm,n ⫽ Rt,cond ⌬x/k (⌬y 䡠 1)

yielding a result that is equivalent to that of Equation 4.36. Similarly, the rate of heat transfer by convection to (m, n) may be expressed as q(⬁) l (m,n) ⫽

T⬁ ⫺ Tm,n T⬁ ⫺ Tm,n ⫽ Rt,conv {h[(⌬x/2) 䡠 1 ⫹ (⌬y/2) 䡠 1]}⫺1

which is equivalent to Equation 4.40. As an example of the utility of resistance concepts, consider an interface that separates two dissimilar materials and is characterized by a thermal contact resistance R⬙t,c (Figure 4.7). The rate of heat transfer from node (m, n) to (m, n ⫺ 1) may be expressed as q(m,n) l (m,n⫺1) ⫽

Tm,n ⫺ Tm,n⫺1 Rtot

(4.45)

where, for a unit depth, Rtot ⫽

R⬙t,c ⌬y/2 ⌬y/2 ⫹ ⫹ kA(⌬x 䡠 1) ⌬x 䡠 1 kB(⌬x 䡠 1)

(4.46)

∆x

∆y

(m, n)

Material A

kA R"t,c

∆y

(m, n – 1)

Material B

kB

FIGURE 4.7 Conduction between adjoining, dissimilar materials with an interface contact resistance.

250

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

4.5 Solving the Finite-Difference Equations Once the nodal network has been established and an appropriate finite-difference equation has been written for each node, the temperature distribution may be determined. The problem reduces to one of solving a system of linear, algebraic equations. In this section, we formulate the system of linear, algebraic equations as a matrix equation and briefly discuss its solution by the matrix inversion method. We also present some considerations for verifying the accuracy of the solution.

4.5.1

Formulation as a Matrix Equation

Consider a system of N finite-difference equations corresponding to N unknown temperatures. Identifying the nodes by a single integer subscript, rather than by the double subscript (m, n), the procedure for performing a matrix inversion begins by expressing the equations as

...

...

...

...

...

...

a11T1 ⫹ a12T2 ⫹ a13T3 ⫹ … ⫹ a1NTN ⫽ C1 a21T1 ⫹ a22T2 ⫹ a23T3 ⫹ … ⫹ a2NTN ⫽ C2 aN1T1 ⫹ aN2T2 ⫹ aN3T3 ⫹ … ⫹ aNNTN ⫽ CN

(4.47)

where the quantities a11, a12, . . . , C1, . . . are known coefficients and constants involving quantities such as ⌬x, k, h, and T앝. Using matrix notation, these equations may be expressed as [A][T] ⫽ [C]

(4.48)

where

aNN

T⬅

TN

,

C⬅

C1 C2 ...

,

T1 T2 ...

aN1 aN2 …

a1N a2N ...

a12 … a22 … ...

...

A⬅

a11 a21

CN

The coefficient matrix [A] is square (N ⫻ N), and its elements are designated by a double subscript notation, for which the first and second subscripts refer to rows and columns, respectively. The matrices [T] and [C] have a single column and are known as column vectors. Typically, they are termed the solution and right-hand side vectors, respectively. If the matrix multiplication implied by the left-hand side of Equation 4.48 is performed, Equations 4.47 are obtained. Numerous mathematical methods are available for solving systems of linear, algebraic equations [11, 12], and many computational software programs have the built-in capability to solve Equation 4.48 for the solution vector [T]. For small matrices, the solution can be found using a programmable calculator or by hand. One method suitable for hand or computer calculation is the Gauss–Seidel method, which is presented in Appendix D.

4.5

䊏

4.5.2

251

Solving the Finite-Difference Equations

Verifying the Accuracy of the Solution

It is good practice to verify that a numerical solution has been correctly formulated by performing an energy balance on a control surface surrounding all nodal regions whose temperatures have been evaluated. The temperatures should be substituted into the energy balance equation, and if the balance is not satisfied to a high degree of precision, the finitedifference equations should be checked for errors. Even when the finite-difference equations have been properly formulated and solved, the results may still represent a coarse approximation to the actual temperature field. This behavior is a consequence of the finite spacings (⌬x, ⌬y) between nodes and of finite-difference approximations, such as k(⌬y 䡠 1)(Tm⫺1,n ⫺ Tm,n)/⌬x, to Fourier’s law of conduction, ⫺k(dy 䡠 1)⭸T/⭸x. The finite-difference approximations become more accurate as the nodal network is refined (⌬x and ⌬y are reduced). Hence, if accurate results are desired, grid studies should be performed, whereby results obtained for a fine grid are compared with those obtained for a coarse grid. One could, for example, reduce ⌬x and ⌬y by a factor of 2, thereby increasing the number of nodes and finite-difference equations by a factor of 4. If the agreement is unsatisfactory, further grid refinements could be made until the computed temperatures no longer depend significantly on the choice of ⌬x and ⌬y. Such grid-independent results would provide an accurate solution to the physical problem. Another option for validating a numerical solution involves comparing results with those obtained from an exact solution. For example, a finite-difference solution of the physical problem described in Figure 4.2 could be compared with the exact solution given by Equation 4.19. However, this option is limited by the fact that we seldom seek numerical solutions to problems for which there exist exact solutions. Nevertheless, if we seek a numerical solution to a complex problem for which there is no exact solution, it is often useful to test our finitedifference procedures by applying them to a simpler version of the problem.

EXAMPLE 4.3 A major objective in advancing gas turbine engine technologies is to increase the temperature limit associated with operation of the gas turbine blades. This limit determines the permissible turbine gas inlet temperature, which, in turn, strongly influences overall system performance. In addition to fabricating turbine blades from special, high-temperature, high-strength superalloys, it is common to use internal cooling by machining flow channels within the blades and routing air through the channels. We wish to assess the effect of such a scheme by approximating the blade as a rectangular solid in which rectangular channels are machined. The blade, which has a thermal conductivity of k ⫽ 25 W/m 䡠 K, is 6 mm thick, and each channel has a 2 mm ⫻ 6 mm rectangular cross section, with a 4-mm spacing between adjoining channels. Combustion gases

T∞,o, ho

Air channel T∞,i, hi

2 mm 6 mm 4 mm

Combustion gases

6 mm

Turbine blade, k

T∞,o, ho

252

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

Under operating conditions for which ho ⫽ 1000 W/m2 䡠 K, T앝,o ⫽ 1700 K, hi ⫽ 200 W/m2 䡠 K, and T앝,i ⫽ 400 K, determine the temperature field in the turbine blade and the rate of heat transfer per unit length to the channel. At what location is the temperature a maximum?

SOLUTION Known: Dimensions and operating conditions for a gas turbine blade with embedded channels. Find: Temperature field in the blade, including a location of maximum temperature. Rate of heat transfer per unit length to the channel. Schematic: T∞,o, ho 1

2

3

7

8

9

Symmetry adiabat 13

14

15

19

20

21

4

5

6

10

11

12

16

17

18

x ∆y = 1 mm

∆x = 1 mm

Symmetry adiabat

T∞,i, hi

Symmetry adiabat

y

Assumptions: 1. Steady-state, two-dimensional conduction. 2. Constant properties. Analysis: Adopting a grid space of ⌬x ⫽ ⌬y ⫽ 1 mm and identifying the three lines of symmetry, the foregoing nodal network is constructed. The corresponding finite-difference equations may be obtained by applying the energy balance method to nodes 1, 6, 18, 19, and 21 and by using the results of Table 4.2 for the remaining nodes. Heat transfer to node 1 occurs by conduction from nodes 2 and 7, as well as by convection from the outer fluid. Since there is no heat transfer from the region beyond the symmetry adiabat, application of an energy balance to the one-quarter section associated with node 1 yields a finite-difference equation of the form Node 1:

冢

T2 ⫹ T7 ⫺ 2 ⫹

冣

ho⌬x h ⌬x T1 ⫽ ⫺ o T⬁,o k k

A similar result may be obtained for nodal region 6, which is characterized by equivalent surface conditions (2 conduction, 1 convection, 1 adiabatic). Nodes 2 to 5 correspond to case 3 of Table 4.2, and choosing node 3 as an example, it follows that Node 3:

T2 ⫹ T4 ⫹ 2T9 ⫺2

冢h k⌬x ⫹ 2冣T ⫽ ⫺ 2hk⌬x T o

o

3

⬁,o

4.5

䊏

253

Solving the Finite-Difference Equations

Nodes 7, 12, 13, and 20 correspond to case 5 of Table 4.2, with q⬙ ⫽ 0, and choosing node 12 as an example, it follows that Node 12:

T6 ⫹ 2T11 ⫹ T18 ⫺ 4T12 ⫽ 0

Nodes 8 to 11 and 14 are interior nodes (case 1), in which case the finite-difference equation for node 8 is Node 8:

T2 ⫹ T7 ⫹ T9 ⫹ T14 ⫺ 4T8 ⫽ 0

Node 15 is an internal corner (case 2) for which Node 15:

冢

2T9 ⫹ 2T14 ⫹ T16 ⫹ T21 ⫺ 2 3 ⫹

冣

hi ⌬x h ⌬x T15 ⫽ ⫺ 2 i T⬁,i k k

while nodes 16 and 17 are situated on a plane surface with convection (case 3): Node 16:

冢h k⌬x ⫹ 2冣T

2T10 ⫹ T15 ⫹ T17 ⫺ 2

i

16

⫽⫺

2hi ⌬x T⬁,i k

In each case, heat transfer to nodal regions 18 and 21 is characterized by conduction from two adjoining nodes and convection from the internal flow, with no heat transfer occurring from an adjoining adiabat. Performing an energy balance for nodal region 18, it follows that Node 18:

冢

T12 ⫹ T17 ⫺ 2 ⫹

冣

hi ⌬x h ⌬x T18 ⫽ ⫺ i T⬁, i k k

The last special case corresponds to nodal region 19, which has two adiabatic surfaces and experiences heat transfer by conduction across the other two surfaces. Node 19:

T13 ⫹ T20 ⫺ 2T19 ⫽ 0

The equations for nodes 1 through 21 may be solved simultaneously using IHT, another commercial code, or a handheld calculator. The following results are obtained: T1

T2

T3

T4

T5

T6

1526.0 K

1525.3 K

1523.6 K

1521.9 K

1520.8 K

1520.5 K

T7

T8

T9

T10

T11

T12

1519.7 K

1518.8 K

1516.5 K

1514.5 K

1513.3 K

1512.9 K

T13

T14

T15

T16

T17

T18

1515.1 K

1513.7 K

1509.2 K

1506.4 K

1505.0 K

1504.5 K

T19

T20

T21

1513.4 K

1511.7 K

1506.0 K

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

The temperature field may also be represented in the form of isotherms, and four such lines of constant temperature are shown schematically. Also shown are heat flux lines that have been carefully drawn so that they are everywhere perpendicular to the isotherms and coincident with the symmetry adiabat. The surfaces that are exposed to the combustion gases and air are not isothermal, and therefore the heat flow lines are not perpendicular to these boundaries.

1521.7

1517.4

Symmetry adiabat

1513.1 1508.9

As expected, the maximum temperature exists at the location farthest removed from the coolant, which corresponds to node 1. Temperatures along the surface of the turbine blade exposed to the combustion gases are of particular interest. The finite-difference predictions are plotted below (with straight lines connecting the nodal temperatures). 1528

1526 T (K)

254

1524

1522

1520

0

1

2

3

4

5

x (mm)

The rate of heat transfer per unit length of channel may be calculated in two ways. Based on heat transfer from the blade to the air, it is q⬘ ⫽ 4hi[(⌬y/2)(T21 ⫺ T앝,i) ⫹ (⌬y/2 ⫹ ⌬x/2)(T15 ⫺ T앝,i) ⫹ (⌬x)(T16 ⫺ T⬁,i) ⫹ ⌬x(T17 ⫺ T⬁,i) ⫹ (⌬x/2)(T18 ⫺ T⬁,i)] Alternatively, based on heat transfer from the combustion gases to the blade, it is q⬘ ⫽ 4ho[(⌬x/2)(T⬁,o ⫺ T1) ⫹ (⌬x)(T⬁,o ⫺ T2) ⫹ (⌬x)(T⬁,o ⫺ T3) ⫹ (⌬x)(T⬁,o ⫺ T4) ⫹ (⌬x)(T⬁,o ⫺ T5) ⫹ (⌬x/2)(T⬁,o ⫺ T6)]

4.5

䊏

255

Solving the Finite-Difference Equations

where the factor of 4 originates from the symmetry conditions. In both cases, we obtain q⬘ ⫽ 3540.6 W/m

䉰

Comments: 1. In matrix notation, following Equation 4.48, the equations for nodes 1 through 21 are of the form [A][T] ⫽ [C], where ⎡ −a ⎢ 1 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 1 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 [A] = ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢⎣ 0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

−b 1 0 1 −b 1 0 1 −b

1

0

0 0 1

0 0 0

0 0 0

2 0 0

0 2 0

0 0 2

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0 0 −4 2 1 −4

0 0 0 1

0 0 0 0

2 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 1

0 0 0

0 0 0

1 0 0

0 1 0

0 0 1

0 0 0

0 0 0

0 0 0

0 2

0 0

0 0

0 0

1 0

0 1

0 0

0 0 1

0 0 0

0 0 0

1 0 0

0 0 0 1

0 0 0 0

1 −b 1 0 1 −a 0 0 0 0 0 0

0 0 0

1 0 0

0 1 0

0 0 1

0 0 0

0 0 0

1 −4 1 0 0 1 −4 1 0 0 1 −4

0 0

0 0

0 0

0 0

1 0

0 1

0 0

0 0

0 0

2 −4 0 0 0 −4

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

1 0 0

0 2 0

0 0 2

0 0 0

0 0 0

1 −4 1 0 0 2 −c 1 0 0 1 −d

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

2 0

0 1

0 0

0 0

0 0

1 0

−d 1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

1 0

0 2

0 0

0 0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1 0 0 0 0 −e 0 −2 1 0 1 −4 0

0

1

0⎤ 0⎥ ⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0 ⎥⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 1⎥ ⎥ 0 ⎥ 0⎥ ⎥ 0 ⎥ 0⎥ ⎥ 1⎥ −e ⎥⎦

⎡ −f ⎤ ⎢ −2f ⎥ ⎥ ⎢ ⎢−2f ⎥ ⎥ ⎢ ⎢ −2f ⎥ ⎢ −2f ⎥ ⎥ ⎢ ⎢ −f ⎥ ⎢0 ⎥ ⎥ ⎢ ⎢0 ⎥ ⎢0 ⎥ ⎥ ⎢ ⎢0 ⎥ [C] = ⎢0 ⎥ ⎥ ⎢ ⎢0 ⎥ ⎥ ⎢ ⎢0 ⎥ ⎢0 ⎥ ⎢ ⎥ ⎢ −2g ⎥ ⎢ −2g ⎥ ⎢ ⎥ ⎢ −2g ⎥ ⎢ −g ⎥ ⎢ ⎥ ⎢0 ⎥ ⎢0 ⎥ ⎢ ⎥ ⎢⎣ −g ⎥⎦

With ho⌬x/k ⫽ 0.04 and hi⌬x/k ⫽ 0.008, the following coefficients in the equations can be calculated: a ⫽ 2.04, b ⫽ 4.08, c ⫽ 6.016, d ⫽ 4.016, e ⫽ 2.008, f ⫽ 68, and g ⫽ 3.2. By framing the equations as a matrix equation, standard tools for solving matrix equations may be used. 2. To ensure that no errors have been made in formulating and solving the finite-difference equations, the calculated temperatures should be used to verify that conservation of energy is satisfied for a control surface surrounding all nodal regions. This check has already been performed, since it was shown that the heat transfer rate from the combustion gases to the blade is equal to that from the blade to the air. 3. The accuracy of the finite-difference solution may be improved by refining the grid. If, for example, we halve the grid spacing (⌬x ⫽ ⌬y ⫽ 0.5 mm), thereby increasing the number of unknown nodal temperatures to 65, we obtain the following results for selected temperatures and the heat rate: T1 ⫽ 1525.9 K, T6 ⫽ 1520.5 K, T18 ⫽ 1504.5 K, T19 ⫽ 1513.5 K, q⬘ ⫽ 3539.9 W/m

T15 ⫽ 1509.2 K, T21 ⫽ 1505.7 K,

Agreement between the two sets of results is excellent. Of course, use of the finer mesh increases setup and computation time, and in many cases the results obtained from a coarse grid are satisfactory. Selection of the appropriate grid is a judgment that the engineer must make.

256

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

4. In the gas turbine industry, there is great interest in adopting measures that reduce blade temperatures. Such measures could include use of a different alloy of larger thermal conductivity and/or increasing coolant flow through the channel, thereby increasing hi. Using the finite-difference solution with ⌬x ⫽ ⌬y ⫽ 1 mm, the following results are obtained for parametric variations of k and hi: k (W/m 䡠 K)

hi (W/m2 䡠 K)

T1 (K)

qⴕ (W/m)

25 50 25 50

200 200 1000 1000

1526.0 1523.4 1154.5 1138.9

3540.6 3563.3 11,095.5 11,320.7

Why do increases in k and hi reduce temperature in the blade? Why is the effect of the change in hi more significant than that of k? 5. Note that, because the exterior surface of the blade is at an extremely high temperature, radiation losses to its surroundings may be significant. In the finite-difference analysis, such effects could be considered by linearizing the radiation rate equation (see Equations 1.8 and 1.9) and treating radiation in the same manner as convection. However, because the radiation coefficient hr depends on the surface temperature, an iterative finite-difference solution would be necessary to ensure that the resulting surface temperatures correspond to the temperatures at which hr is evaluated at each nodal point. 6. See Example 4.3 in IHT. This problem can also be solved using Tools, FiniteDifference Equations in the Advanced section of IHT. 7. A second software package accompanying this text, Finite-Element Heat Transfer (FEHT), may also be used to solve one- and two-dimensional forms of the heat equation. This example is provided as a solved model in FEHT and may be accessed through Examples on the Toolbar.

4.6 Summary The primary objective of this chapter was to develop an appreciation for the nature of a twodimensional conduction problem and the methods that are available for its solution. When confronted with a two-dimensional problem, one should first determine whether an exact solution is known. This may be done by examining some of the excellent references in which exact solutions to the heat equation are obtained [1–5]. One may also want to determine whether the shape factor or dimensionless conduction heat rate is known for the system of interest [6–10]. However, often, conditions are such that the use of a shape factor, dimensionless conduction heat rate, or an exact solution is not possible, and it is necessary to use a finite-difference or finite-element solution. You should therefore appreciate the inherent nature of the discretization process and know how to formulate and solve the finite-difference

䊏

257

Problems

equations for the discrete points of a nodal network. You may test your understanding of related concepts by addressing the following questions. • What is an isotherm? What is a heat flow line? How are the two lines related geometrically? • What is an adiabat? How is it related to a line of symmetry? How is it intersected by an isotherm? • What parameters characterize the effect of geometry on the relationship between the heat rate and the overall temperature difference for steady conduction in a two-dimensional system? How are these parameters related to the conduction resistance? • What is represented by the temperature of a nodal point, and how does the accuracy of a nodal temperature depend on prescription of the nodal network?

References 1. Schneider, P. J., Conduction Heat Transfer, AddisonWesley, Reading, MA, 1955. 2. Carslaw, H. S., and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, London, 1959. 3. Özisik, M. N., Heat Conduction, Wiley Interscience, New York, 1980. 4. Kakac, S., and Y. Yener, Heat Conduction, Hemisphere Publishing, New York, 1985. 5. Poulikakos, D., Conduction Heat Transfer, PrenticeHall, Englewood Cliffs, NJ, 1994. 6. Sunderland, J. E., and K. R. Johnson, Trans. ASHRAE, 10, 237–241, 1964. 7. Kutateladze, S. S., Fundamentals of Heat Transfer, Academic Press, New York, 1963.

8. General Electric Co. (Corporate Research and Development), Heat Transfer Data Book, Section 502, General Electric Company, Schenectady, NY, 1973. 9. Hahne, E., and U. Grigull, Int. J. Heat Mass Transfer, 18, 751–767, 1975. 10. Yovanovich, M. M., in W. M. Rohsenow, J. P. Hartnett, and Y. I. Cho, Eds., Handbook of Heat Transfer, McGraw-Hill, New York, 1998, pp. 3.1–3.73. 11. Gerald, C. F., and P. O. Wheatley, Applied Numerical Analysis, Pearson Education, Upper Saddle River, NJ, 1998. 12. Hoffman, J. D., Numerical Methods for Engineers and Scientists, McGraw-Hill, New York, 1992.

Problems Exact Solutions 4.1 In the method of separation of variables (Section 4.2) for two-dimensional, steady-state conduction, the separation constant 2 in Equations 4.6 and 4.7 must be a positive constant. Show that a negative or zero value of 2 will result in solutions that cannot satisfy the prescribed boundary conditions. 4.2 A two-dimensional rectangular plate is subjected to prescribed boundary conditions. Using the results of the exact solution for the heat equation presented in Section 4.2, calculate the temperature at the midpoint (1, 0.5) by considering the first five nonzero terms of the infinite series that must be evaluated. Assess the error resulting from using only the first three terms of the infinite series. Plot the temperature distributions T(x, 0.5) and T(1.0, y).

y (m) T2 = 150°C 1

T1 = 50°C 0 0

T1 = 50°C

2

T1 = 50°C x (m)

4.3 Consider the two-dimensional rectangular plate of Problem 4.2 having a thermal conductivity of 50 W/m 䡠 K. Beginning with the exact solution for the temperature distribution, derive an expression for the heat transfer rate per unit thickness from the plate along the lower surface (0 ⱕ x ⱕ 2, y ⫽ 0). Evaluate the heat rate considering the first five nonzero terms of the infinite series.

258

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

4.4 A two-dimensional rectangular plate is subjected to the boundary conditions shown. Derive an expression for the steady-state temperature distribution T(x, y). y T = Ax

b T=0

T=0

4.8 Consider Problem 4.5 for the case where the plate is of square cross section, W ⫽ L.

x

a

T=0

4.5 A two-dimensional rectangular plate is subjected to prescribed temperature boundary conditions on three sides and a uniform heat flux into the plate at the top surface. Using the general approach of Section 4.2, derive an expression for the temperature distribution in the plate. q"s

y

T1

0

L

0

(a) Derive an expression for the shape factor, Smax, associated with the maximum top surface temperature, such that q ⫽ Smax k (T2,max ⫺ T1) where T2,max is the maximum temperature along y ⫽ W. (b) Derive an expression for the shape factor, Savg, associated with the average top surface tempera– – ture, q ⫽ Savg k(T 2 ⫺ T1) where T2 is the average temperature along y ⫽ W. (c) Evaluate the shape factors that can be used to determine the maximum and average temperatures along y ⫽ W. Evaluate the maximum and average temperatures for T1 ⫽ 0°C, L ⫽ W ⫽ 10 mm, k ⫽ 20 W/m 䡠 K, and q⬙s ⫽ 1000 W/m2.

W

T1

An experiment for the configuration shown yields a heat transfer rate per unit length of q⬘conv ⫽ 110 W/m for surface temperatures of T1 ⫽ 53°C and T2 ⫽ 15°C, respectively. For inner and outer cylinders of diameters d ⫽ 20 mm and D ⫽ 60 mm, and an eccentricity factor of z ⫽ 10 mm, determine the value of keff. The actual thermal conductivity of the fluid is k ⫽ 0.255 W/m 䡠 K.

x

T1

Shape Factors and Dimensionless Conduction Heat Rates 4.6 Using the thermal resistance relations developed in Chapter 3, determine shape factor expressions for the following geometries: (a) Plane wall, cylindrical shell, and spherical shell. (b) Isothermal sphere of diameter D buried in an infinite medium. 4.7 Free convection heat transfer is sometimes quantified by writing Equation 4.20 as qconv ⫽ Skeff ⌬T1⫺2, where keff is an effective thermal conductivity. The ratio keff /k is greater than unity because of fluid motion driven by buoyancy forces, as represented by the dashed streamlines.

4.9 Radioactive wastes are temporarily stored in a spherical container, the center of which is buried a distance of 10 m below the earth’s surface. The outside diameter of the container is 2 m, and 500 W of heat are released as a result of radioactive decay. If the soil surface temperature is 20°C, what is the outside surface temperature of the container under steady-state conditions? On a sketch of the soil–container system drawn to scale, show representative isotherms and heat flow lines in the soil. 4.10 Based on the dimensionless conduction heat rates for cases 12–15 in Table 4.1b, find shape factors for the following objects having temperature T1, located at the surface of a semi-infinite medium having temperature T2. The surface of the semi-infinite medium is adiabatic. (a) A buried hemisphere, flush with the surface. (b) A disk on the surface. Compare your result to Table 4.1a, case 10. (c) A square on the surface. (d) A buried cube, flush with the surface.

T1

D

D

z D

g

T1

T2

d

(a)

T2

T1

T2 (b) and (c)

D T1

T2 (d)

4.11 Determine the heat transfer rate between two particles of diameter D ⫽ 100 m and temperatures T1 ⫽ 300.1 K

259

Problems

䊏

and T2 ⫽ 299.9 K, respectively. The particles are in contact and are surrounded by air. Air

D

4.15 A small water droplet of diameter D ⫽ 100 m and temperature Tmp ⫽ 0°C falls on a nonwetting metal surface that is at temperature Ts ⫽ –15°C. Determine how long it will take for the droplet to freeze completely. The latent heat of fusion is hsf ⫽ 334 kJ/kg. Air

Water droplet D, Tmp

T1

T2

4.12 A two-dimensional object is subjected to isothermal conditions at its left and right surfaces, as shown in the schematic. Both diagonal surfaces are adiabatic and the depth of the object is L ⫽ 100 mm.

y θ = π/2

x

T1 T2

a

Nonwetting metal, Ts

4.16 A tube of diameter 50 mm having a surface temperature of 85°C is embedded in the center plane of a concrete slab 0.1 m thick with upper and lower surfaces at 20°C. Using the appropriate tabulated relation for this configuration, find the shape factor. Determine the heat transfer rate per unit length of the tube. 4.17 Pressurized steam at 450 K flows through a long, thinwalled pipe of 0.5-m diameter. The pipe is enclosed in a concrete casing that is of square cross section and 1.5 m on a side. The axis of the pipe is centered in the casing, and the outer surfaces of the casing are maintained at 300 K. What is the heat loss per unit length of pipe?

b

(a) Determine the two-dimensional shape factor for the object for a ⫽ 10 mm, b ⫽ 12 mm. (b) Determine the two-dimensional shape factor for the object for a ⫽ 10 mm, b ⫽ 15 mm. (c) Use the alternative conduction analysis of Section 3.2 to estimate the shape factor for parts (a) and (b). Compare the values of the approximate shape factors of the alternative conduction analysis to the two-dimensional shape factors of parts (a) and (b). (d) For T1 ⫽ 100°C and T2 ⫽ 60°C, determine the heat transfer rate per unit depth for k ⫽ 15 W/m 䡠 K for parts (a) and (b). 4.13 An electrical heater 100 mm long and 5 mm in diameter is inserted into a hole drilled normal to the surface of a large block of material having a thermal conductivity of 5 W/m 䡠 K. Estimate the temperature reached by the heater when dissipating 50 W with the surface of the block at a temperature of 25°C. 4.14 Two parallel pipelines spaced 0.5 m apart are buried in soil having a thermal conductivity of 0.5 W/m 䡠 K. The pipes have outer diameters of 100 and 75 mm with surface temperatures of 175°C and 5°C, respectively. Estimate the heat transfer rate per unit length between the two pipelines.

4.18 The temperature distribution in laser-irradiated materials is determined by the power, size, and shape of the laser beam, along with the properties of the material being irradiated. The beam shape is typically Gaussian, and the local beam irradiation flux (often referred to as the laser fluence) is q⬙(x, y) ⫽ q⬙(x ⫽ y ⫽ 0)exp(⫺xⲐrb)2 exp(⫺yⲐrb)2 The x- and y-coordinates determine the location of interest on the surface of the irradiated material. Consider the case where the center of the beam is located at x ⫽ y ⫽ r ⫽ 0. The beam is characterized by a radius rb, defined as the radial location where the local fluence is q⬙(rb) ⫽ q⬙(r ⫽ 0)/e 艐 0.368q⬙(r ⫽ 0). A shape factor for Gaussian heating is S ⫽ 21/2rb, where S is defined in terms of T1,max ⫺ T2 [Nissin, Y. I., A. Lietoila, R. G. Gold, and J. F. Gibbons, J. Appl. Phys., 51, 274, 1980]. Calculate the maximum steadystate surface temperature associated with irradiation of a material of thermal conductivity k ⫽ 27 W/m 䡠 K and absorptivity ␣ ⫽ 0.45 by a Gaussian beam with rb ⫽ 0.1 mm and power P ⫽ 1 W. Compare your result with the maximum temperature that would occur if the irradiation was from a circular beam of the same diameter and power, but characterized by a uniform fluence (a flat beam). Also calculate the average temperature of the irradiated surface for the uniform fluence case. The temperature far from the irradiated spot is T2 ⫽ 25°C.

260

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

4.19 Hot water at 85°C flows through a thin-walled copper tube of 30-mm diameter. The tube is enclosed by an eccentric cylindrical shell that is maintained at 35°C and has a diameter of 120 mm. The eccentricity, defined as the separation between the centers of the tube and shell, is 20 mm. The space between the tube and shell is filled with an insulating material having a thermal conductivity of 0.05 W/m 䡠 K. Calculate the heat loss per unit length of the tube, and compare the result with the heat loss for a concentric arrangement. 4.20 A furnace of cubical shape, with external dimensions of 0.35 m, is constructed from a refractory brick (fireclay). If the wall thickness is 50 mm, the inner surface temperature is 600°C, and the outer surface temperature is 75°C, calculate the heat loss from the furnace. 4.21 Laser beams are used to thermally process materials in a wide range of applications. Often, the beam is scanned along the surface of the material in a desired pattern. Consider the laser heating process of Problem 4.18, except now the laser beam scans the material at a scanning velocity of U. A dimensionless maximum surface temperature can be well correlated by an expression of the form [Nissin, Y. I., A. Lietoila, R. G. Gold, and J. F. Gibbons, J. Appl. Phys., 51, 274, 1980] T1,max,U⫽0 ⫺ T2 ⫽ 1 ⫹ 0.301Pe ⫺ 0.0108Pe2 T1,max,U⫽0 ⫺ T2 for the range 0 ⬍ Pe ⬍ 10, where Pe is a dimensionless velocity known as the Peclet number. For this problem, Pe ⫽ Urb /兹2␣ where ␣ is the thermal diffusivity of the material. The maximum material temperature does not occur directly below the laser beam, but at a lag distance ␦ behind the center of the moving beam. The dimensionless lag distance can be correlated to Pe by [Sheng, I. C., and Y. Chen, J. Thermal Stresses, 14, 129, 1991] ␦U 1.55 ␣ ⫽ 0.944Pe (a) For the laser beam size and shape and material of Problem 4.18, determine the laser power required to achieve T1,max ⫽ 200°C for U ⫽ 2 m/s. The density and specific heat of the material are ⫽ 2000 kg/m3 and c ⫽ 800 J/kg 䡠 K, respectively.

is evacuated, eliminating conduction and convection across the gap. Small cylindrical pillars, each L ⫽ 0.2 mm long and D ⫽ 0.15 mm in diameter, are inserted between the glass sheets to ensure that the glass does not break due to stresses imposed by the pressure difference across each glass sheet. A con⬙ ⫽ 1.5 ⫻ 10⫺6 m2 䡠 K/W exists tact resistance of Rt,c between the pillar and the sheet. For nominal glass temperatures of T1 ⫽ 20°C and T2 ⫽ ⫺10°C, determine the conduction heat transfer through an individual stainless steel pillar. 4.23 A pipeline, used for the transport of crude oil, is buried in the earth such that its centerline is a distance of 1.5 m below the surface. The pipe has an outer diameter of 0.5 m and is insulated with a layer of cellular glass 100 mm thick. What is the heat loss per unit length of pipe when heated oil at 120°C flows through the pipe and the surface of the earth is at a temperature of 0°C? 4.24 A long power transmission cable is buried at a depth (ground-to-cable-centerline distance) of 2 m. The cable is encased in a thin-walled pipe of 0.1-m diameter, and, to render the cable superconducting (with essentially zero power dissipation), the space between the cable and pipe is filled with liquid nitrogen at 77 K. If the pipe is covered with a superinsulator (ki ⫽ 0.005 W/m 䡠 K) of 0.05-m thickness and the surface of the earth (kg ⫽ 1.2 W/m 䡠 K) is at 300 K, what is the cooling load (W/m) that must be maintained by a cryogenic refrigerator per unit pipe length? 4.25 A small device is used to measure the surface temperature of an object. A thermocouple bead of diameter D ⫽ 120 m is positioned a distance z ⫽ 100 m from the surface of interest. The two thermocouple wires, each of diameter d ⫽ 25 m and length L ⫽ 300 m, are held by a large manipulator that is at a temperature of Tm ⫽ 23°C. Manipulator, Tm d L Air

(b) Determine the lag distance ␦ associated with U ⫽ 2 m/s. (c) Plot the required laser power to achieve Tmax,1 ⫽ 200⬚C for 0 ⱕ U ⱕ 2 m/s.

Shape Factors with Thermal Circuits 4.22 A double-glazed window consists of two sheets of glass separated by an L ⫽ 0.2-mm-thick gap. The gap

z

D, Ttc

Thermocouple bead

Ts

If the thermocouple registers a temperature of Ttc ⫽ 29°C, what is the surface temperature? The thermal

䊏

261

Problems

conductivities of the chromel and alumel thermocouple wires are kCh ⫽ 19 W/m 䡠 K and kAl ⫽ 29 W/m 䡠 K, respectively. You may neglect radiation and convection effects. 4.26 A cubical glass melting furnace has exterior dimensions of width W ⫽ 5 m on a side and is constructed from refractory brick of thickness L ⫽ 0.35 m and thermal conductivity k ⫽ 1.4 W/m 䡠 K. The sides and top of the furnace are exposed to ambient air at 25°C, with free convection characterized by an average coefficient of h ⫽ 5 W/m2 䡠 K. The bottom of the furnace rests on a framed platform for which much of the surface is exposed to the ambient air, and a convection coefficient of h ⫽ 5 W/m2 䡠 K may be assumed as a first approximation. Under operating conditions for which combustion gases maintain the inner surfaces of the furnace at 1100°C, what is the heat loss from the furnace? 4.27 A hot fluid passes through circular channels of a cast iron platen (A) of thickness LA ⫽ 30 mm which is in poor contact with the cover plates (B) of thickness LB ⫽ 7.5 mm. The channels are of diameter D ⫽ 15 mm with a centerline spacing of Lo ⫽ 60 mm. The thermal conductivities of the materials are kA ⫽ 20 W/m 䡠 K and kB ⫽ 75 W/m 䡠 K, while the contact resistance between the two materials is R⬙t,c ⫽ 2.0 ⫻ 10⫺4 m2 䡠 K/W. The hot fluid is at Ti ⫽ 150°C, and the convection coefficient is 1000 W/m2 䡠 K. The cover plate is exposed to ambient air at T앝 ⫽ 25°C with a convection coefficient of 200 W/m2 䡠 K. The shape factor between one channel and the platen top and bottom surfaces is 4.25. Air

T∞, h

Ts

Cover plate, B

LB

4.28 An aluminum heat sink (k ⫽ 240 W/m 䡠 K), used to cool an array of electronic chips, consists of a square channel of inner width w ⫽ 25 mm, through which liquid flow may be assumed to maintain a uniform surface temperature of T1 ⫽ 20⬚C. The outer width and length of the channel are W ⫽ 40 mm and L ⫽ 160 mm, respectively. Chip, Tc

Rt,c Heat sink

Coolant

w

L

W

If N ⫽ 120 chips attached to the outer surfaces of the heat sink maintain an approximately uniform surface temperature of T2 ⫽ 50⬚C and all of the heat dissipated by the chips is assumed to be transferred to the coolant, what is the heat dissipation per chip? If the contact resistance between each chip and the heat sink is Rt,c ⫽ 0.2 K/W, what is the chip temperature? 4.29 Hot water is transported from a cogeneration power station to commercial and industrial users through steel pipes of diameter D ⫽ l50 mm, with each pipe centered in concrete (k ⫽ 1.4 W/m ⭈ K) of square cross section (w ⫽ 300 mm). The outer surfaces of the concrete are exposed to ambient air for which T앝 ⫽ 0⬚C and h ⫽ 25 W/m2 䡠 K.

R"t,c

Concrete, k

D

Contact resistance

T2

T1

To Fluid

Air

Ti, hi

T∞, h

LA Platen, A

Lo

Cover plate, B

LB

T1

R"t,c Water

Air

w D

Ts

L

•

Ti, m

T∞, h

(b) Determine the outer surface temperature of the cover plate, Ts.

(a) If the inlet temperature of water flowing through the pipe is Ti ⫽ 90⬚C, what is the heat loss per unit length of pipe in proximity to the inlet? The temperature of the pipe T1 may be assumed to be that of the inlet water.

(c) Comment on the effects that changing the centerline spacing will have on q⬘i and Ts. How would insulating the lower surface affect q⬘i and Ts?

(b) If the difference between the inlet and outlet temperatures of water flowing through a 100-m-long pipe is not to exceed 5⬚C, estimate the minimum

(a) Determine the heat rate from a single channel per unit length of the platen normal to the page, q⬘i.

262

Chapter 4

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Two-Dimensional, Steady-State Conduction

. allowable flow rate m . A value of c ⫽ 4207 J/kg 䡠 K may be used for the specific heat of the water.

material exceeds the fluid temperature, attachment of a fin depresses the junction temperature Tj below the original temperature of the base, and heat flow from the base material to the fin is two-dimensional.

4.30 A long constantan wire of 1-mm diameter is butt welded to the surface of a large copper block, forming a thermocouple junction. The wire behaves as a fin, permitting heat to flow from the surface, thereby depressing the sensing junction temperature Tj below that of the block To.

T∞, h

Tj

D

Air

Aluminum or stainless steel base

Thermocouple wire, D

T∞, h

Tb

Aluminum pin fin

Tj Copper block, To

(a) If the wire is in air at 25°C with a convection coefficient of 10 W/m2 䡠 K, estimate the measurement error (Tj ⫺ To) for the thermocouple when the block is at 125°C. (b) For convection coefficients of 5, 10, and 25 W/m2 䡠 K, plot the measurement error as a function of the thermal conductivity of the block material over the range 15 to 400 W/m 䡠 K. Under what circumstances is it advantageous to use smaller diameter wire? 4.31 A hole of diameter D ⫽ 0.25 m is drilled through the center of a solid block of square cross section with w ⫽ 1 m on a side. The hole is drilled along the length, l ⫽ 2 m, of the block, which has a thermal conductivity of k ⫽ 150 W/m 䡠 K. The four outer surfaces are exposed to ambient air, with T앝,2 ⫽ 25°C and h2 ⫽ 4 W/m2 䡠 K, while hot oil flowing through the hole is characterized by T앝,1 ⫽ 300°C and h1 ⫽ 50 W/m2 䡠 K. Determine the corresponding heat rate and surface temperatures.

D = 0.25 m

h1, T∞,1

Consider conditions for which a long aluminum pin fin of diameter D ⫽ 5 mm is attached to a base material whose temperature far from the junction is maintained at Tb ⫽ 100°C. Fin convection conditions correspond to h ⫽ 50 W/m2 䡠 K and T앝 ⫽ 25°C. (a) What are the fin heat rate and junction temperature when the base material is (i) aluminum (k ⫽ 240 W/m 䡠 K) and (ii) stainless steel (k ⫽ 15 W/m 䡠 K)? (b) Repeat the foregoing calculations if a thermal contact resistance of R⬙t, j ⫽ 3 ⫻ 10⫺5 m2 䡠 K/W is associated with the method of joining the pin fin to the base material. (c) Considering the thermal contact resistance, plot the heat rate as a function of the convection coefficient over the range 10 ⱕ h ⱕ 100 W/m2 䡠 K for each of the two materials. 4.33 An igloo is built in the shape of a hemisphere, with an inner radius of 1.8 m and walls of compacted snow that are 0.5 m thick. On the inside of the igloo, the surface heat transfer coefficient is 6 W/m2 䡠 K; on the outside, under normal wind conditions, it is 15 W/m2 䡠 K. The thermal conductivity of compacted snow is 0.15 W/m 䡠 K. The temperature of the ice cap on which the igloo sits is ⫺20°C and has the same thermal conductivity as the compacted snow.

h2, T∞,2 Arctic wind, T∞

w=1m

4.32 In Chapter 3 we assumed that, whenever fins are attached to a base material, the base temperature is unchanged. What in fact happens is that, if the temperature of the base

Igloo

Tair

Ice cap, Tic

䊏

263

Problems

(a) Assuming that the occupants’ body heat provides a continuous source of 320 W within the igloo, calculate the inside air temperature when the outside air temperature is T앝 ⫽ ⫺40°C. Be sure to consider heat losses through the floor of the igloo. (b) Using the thermal circuit of part (a), perform a parameter sensitivity analysis to determine which variables have a significant effect on the inside air temperature. For instance, for very high wind conditions, the outside convection coefficient could double or even triple. Does it make sense to construct the igloo with walls half or twice as thick? 4.34 Consider the thin integrated circuit (chip) of Problem 3.150. Instead of attaching the heat sink to the chip surface, an engineer suggests that sufficient cooling might be achieved by mounting the top of the chip onto a large copper (k ⫽ 400 W/m 䡠 K) surface that is located nearby. The metallurgical joint between the chip and the substrate provides a contact resistance of R⬙t,c ⫽ 5 ⫻ 10⫺6 m2 䡠 K/W, and the maximum allowable chip temperature is 85°C. If the large substrate temperature is T2 ⫽ 25°C at locations far from the chip, what is the maximum allowable chip power dissipation qc? 4.35 An electronic device, in the form of a disk 20 mm in diameter, dissipates 100 W when mounted flush on a large aluminum alloy (2024) block whose temperature is maintained at 27°C. The mounting arrangement is such that a contact resistance of R⬙t,c ⫽ 5 ⫻ 10⫺5 m2 䡠 K/W exists at the interface between the device and the block. Air

T∞, h Electronic device, Td, P

Pin fins (30), D = 1.5 mm L = 15 mm

Copper, 5-mm thickness Device

Epoxy,

R"t ,c

Epoxy, Aluminum block, Tb

R"t ,c

(a) Calculate the temperature the device will reach, assuming that all the power generated by the device must be transferred by conduction to the block. (b) To operate the device at a higher power level, a circuit designer proposes to attach a finned heat sink to the top of the device. The pin fins and base material are fabricated from copper (k ⫽ 400 W/m 䡠 K) and are exposed to an airstream at 27°C for which the convection coefficient is 1000 W/m2 䡠 K. For the device temperature computed in part (a), what is the permissible operating power?

4.36 The elemental unit of an air heater consists of a long circular rod of diameter D, which is encapsulated by a finned sleeve and in which thermal energy is generated by ohmic heating. The N fins of thickness t and length L are integrally fabricated with the square sleeve of width w. Under steady-state operating conditions, the rate of thermal energy generation corresponds to the rate of heat transfer to airflow over the sleeve. Fins, N Sleeve, ks Airflow

T∞, h

t D Ts w

Heater • (q, kh)

L

(a) Under conditions for which a uniform surface temperature Ts is maintained around the circumference of the heater and the temperature T앝 and convection coefficient h of the airflow are known, obtain an expression for the rate of heat transfer per unit length to the air. Evaluate the heat rate for Ts ⫽ 300⬚C, D ⫽ 20 mm, an aluminum sleeve (ks ⫽ 240 W/m 䡠 K), w ⫽ 40 mm, N ⫽ 16, t ⫽ 4 mm, L ⫽ 20 mm, T앝 ⫽ 50⬚C, and h ⫽ 500 W/m2 䡠 K. (b) For the foregoing heat rate and a copper heater of thermal conductivity kh ⫽ 400 W/m 䡠 K, what is the required volumetric heat generation within the heater and its corresponding centerline temperature? (c) With all other quantities unchanged, explore the effect of variations in the fin parameters (N, L, t) on the heat rate, subject to the constraint that the fin thickness and the spacing between fins cannot be less than 2 mm. 4.37 For a small heat source attached to a large substrate, the spreading resistance associated with multidimensional conduction in the substrate may be approximated by the expression [Yovanovich, M. M., and V. W. Antonetti, in Adv. Thermal Modeling Elec. Comp. and Systems, Vol. 1, A. Bar-Cohen and A. D. Kraus, Eds., Hemisphere, NY, 79–128, 1988] Rt(sp) ⫽

1 ⫺ 1.410 Ar ⫹ 0.344 A3r ⫹ 0.043 A5r ⫹ 0.034 A7r 4ksub A1/2 s, h

where Ar ⫽ As,h /As,sub is the ratio of the heat source area to the substrate area. Consider application of the expression to an in-line array of square chips of width Lh ⫽ 5 mm on a side and pitch Sh ⫽ 10 mm. The interface

264

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

between the chips and a large substrate of thermal conductivity ksub ⫽ 80 W/m 䡠 K is characterized by a thermal contact resistance of R⬙t,c ⫽ 0.5 ⫻ 10⫺4 m2 䡠 K/W. Top view Substrate, ksub

the sketch, the boundary condition changes from specified heat flux q⬙s (into the domain) to convection, at the location of the node (m, n). Write the steadystate, two-dimensional finite difference equation at this node.

Chip, Th Side view

Air

q"s

T∞, h Sh

h, T∞

Lh

Lh

m, n Sh

Substrate

∆y

R"t,c

∆x

If a convection heat transfer coefficient of h ⫽ 100 W/m2 䡠 K is associated with airflow (T앝 ⫽ 15⬚C) over the chips and substrate, what is the maximum allowable chip power dissipation if the chip temperature is not to exceed Th ⫽ 85⬚C?

4.42 Determine expressions for q(m⫺1,n) → (m,n), q(m⫹1,n) → (m,n), q(m,n⫹1) → (m,n) and q(m,n⫺1) → (m,n) for conduction associated with a control volume that spans two different materials. There is no contact resistance at the interface between the materials. The control volumes are L units long into the page. Write the finite difference equation under steadystate conditions for node (m, n).

Finite-Difference Equations: Derivations ∆x

4.38 Consider nodal configuration 2 of Table 4.2. Derive the finite-difference equations under steady-state conditions for the following situations. (a) The horizontal boundary of the internal corner is perfectly insulated and the vertical boundary is subjected to the convection process (T앝, h). (b) Both boundaries of the internal corner are perfectly insulated. How does this result compare with Equation 4.41?

∆y

•

(m 1, n)

•

(m, n 1)

•

(m, n)

•

(m, n 1)

Material A kA

•

(m 1, n) Material B kB

4.39 Consider nodal configuration 3 of Table 4.2. Derive the finite-difference equations under steady-state conditions for the following situations. (a) The boundary is insulated. Explain how Equation 4.42 can be modified to agree with your result. (b) The boundary is subjected to a constant heat flux. 4.40 Consider nodal configuration 4 of Table 4.2. Derive the finite-difference equations under steady-state conditions for the following situations. (a) The upper boundary of the external corner is perfectly insulated and the side boundary is subjected to the convection process (T앝, h). (b) Both boundaries of the external corner are perfectly insulated. How does this result compare with Equation 4.43? 4.41 One of the strengths of numerical methods is their ability to handle complex boundary conditions. In

4.43 Consider heat transfer in a one-dimensional (radial) cylindrical coordinate system under steady-state conditions with volumetric heat generation. (a) Derive the finite-difference equation for any interior node m. (b) Derive the finite-difference equation for the node n located at the external boundary subjected to the convection process (T앝, h). 4.44 In a two-dimensional cylindrical configuration, the radial (⌬r) and angular (⌬) spacings of the nodes are uniform. The boundary at r ⫽ ri is of uniform temperature Ti. The boundaries in the radial direction are adiabatic (insulated) and exposed to surface convection (T앝 , h), as illustrated. Derive the finite-difference equations for (i) node 2, (ii) node 3, and (iii) node 1.

䊏

5

4

T∞, h

265

Problems

6

2

1

(b) Node (m, n) at the tip of a cutting tool with the upper surface exposed to a constant heat flux q⬙o, and the diagonal surface exposed to a convection cooling process with the fluid at T앝 and a heat transfer coefficient h. Assume ⌬x ⫽ ⌬y.

3

∆r

q"o

∆φ

∆φ Uniform temperature surface, Ti

m + 1, n

m, n 45°

ri

∆y

∆x m + 1, n – 1

4.45 Upper and lower surfaces of a bus bar are convectively cooled by air at T앝, with hu ⫽ hl. The sides are cooled by maintaining contact with heat sinks at To, through a thermal contact resistance of R⬙t,c. The bar is of thermal conductivity k, and its width is twice its thickness L. T∞, hu 1

2

To

•

2 4

3

∆y

∆ x = ∆y

∆y 8

9

10

11

12

13

14

15

To

Consider steady-state conditions for which heat is uni. formly generated at a volumetric rate q due to passage of an electric current. Using the energy balance method, derive finite-difference equations for nodes 1 and 13.

kB

Derive the finite-difference equation, assuming no internal generation. 4.48 Consider the two-dimensional grid (⌬x ⫽ ⌬y) representing steady-state conditions with no internal volumetric generation for a system with thermal conductivity k. One of the boundaries is maintained at a constant temperature Ts while the others are adiabatic.

4.46 Derive the nodal finite-difference equations for the following configurations. (a) Node (m, n) on a diagonal boundary subjected to convection with a fluid at T앝 and a heat transfer coefficient h. Assume ⌬x ⫽ ⌬y.

∆y

y

12

11

10

9

8

13

4

5

6

7

14

3

15

2

∆x

m + 1, n + 1 ∆y

T∞, h m, n

m + 1, n m, n – 1 ∆x

kA

L

R"t,c

T∞, hl

Material A

Material B

4

7

m – 1, n – 1

0

1

5

6

R"t,c

4.47 Consider the nodal point 0 located on the boundary between materials of thermal conductivity kA and kB.

q, k

3

∆x

T∞, h

x

16

Insulation

Isothermal boundary, Ts

1 Insulation

Derive an expression for the heat rate per unit length normal to the page crossing the isothermal boundary (Ts). 4.49 Consider a one-dimensional fin of uniform crosssectional area, insulated at its tip, x ⫽ L. (See Table 3.4,

266

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

case B). The temperature at the base of the fin Tb and of the adjoining fluid T앝, as well as the heat transfer coefficient h and the thermal conductivity k, are known. (a) Derive the finite-difference equation for any interior node m. (b) Derive the finite-difference equation for a node n located at the insulated tip.

Finite-Difference Equations: Analysis 4.50 Consider the network for a two-dimensional system without internal volumetric generation having nodal temperatures shown below. If the grid spacing is 125 mm and the thermal conductivity of the material is 50 W/m 䡠 K, calculate the heat rate per unit length normal to the page from the isothermal surface (Ts).

1

2

3

4

5

6 7

Node

Ti (°C)

1 2 3 4 5 6 7

120.55 120.64 121.29 123.89 134.57 150.49 147.14

Ts = 100°C

4.51 An ancient myth describes how a wooden ship was destroyed by soldiers who reflected sunlight from their polished bronze shields onto its hull, setting the ship ablaze. To test the validity of the myth, a group of college students are given mirrors and they reflect sunlight onto a 100 mm ⫻ 100 mm area of a t ⫽ 10-mm-thick plywood mockup characterized by k ⫽ 0.8 W/m 䡠 K. The bottom of the mockup is in water at Tw ⫽ 20°C, while the air temperature is T앝 ⫽ 25°C. The surroundings are at Tsur ⫽ 23°C. The wood has an emissivity of ⫽ 0.90; both the front and back surfaces of the plywood are characterized by h ⫽ 5 W/m2 䡠 K. The absorbed irradiation from the N students’ mirrors is GS,N ⫽ 70,000 W/m2 on the front surface of the mockup. Tsur 23°C

T∞ 25°C h 5 W/m2·K

L2 800 mm

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

Irradiation location A

H = 300 mm

Irradiation location B L1 500 mm

Tw 20°C

(a) A debate ensues concerning where the beam should be focused, location A or location B. Using a finite

difference method with ⌬x ⫽ ⌬y ⫽ 100 mm and treating the wood as a two-dimensional extended surface (Figure 3.17a), enlighten the students as to whether location A or location B will be more effective in igniting the wood by determining the maximum local steady-state temperature. (b) Some students wonder whether the same technique can be used to melt a stainless steel hull. Repeat part (a) considering a stainless steel mockup of the same dimensions with k ⫽ 15 W/m 䡠 K and ⫽ 0.2. The value of the absorbed irradiation is the same as in part (a). 4.52 Consider the square channel shown in the sketch operating under steady-state conditions. The inner surface of the channel is at a uniform temperature of 600 K, while the outer surface is exposed to convection with a fluid at 300 K and a convection coefficient of 50 W/m2 䡠 K. From a symmetrical element of the channel, a twodimensional grid has been constructed and the nodes labeled. The temperatures for nodes 1, 3, 6, 8, and 9 are identified. T∞ = 300 K h = 50 W/m2• K

1

2

3

5

6

7

4

∆ x = ∆y = 0.01 m 8

T = 600 K

9

y x

k = 1 W/m•K T1 = 430 K T3 = 394 K

T8 = T9 = 600 K T6 = 492 K

(a) Beginning with properly defined control volumes, derive the finite-difference equations for nodes 2, 4, and 7 and determine the temperatures T2, T4, and T7 (K). (b) Calculate the heat loss per unit length from the channel. 4.53 A long conducting rod of rectangular cross section (20 mm ⫻ 30 mm) and thermal conductivity k ⫽ 20 W/m 䡠 K experiences uniform heat generation at a . rate q ⫽ 5 ⫻ 107 W/m3, while its surfaces are maintained at 300 K. (a) Using a finite-difference method with a grid spacing of 5 mm, determine the temperature distribution in the rod. (b) With the boundary conditions unchanged, what heat generation rate will cause the midpoint temperature to reach 600 K?

䊏

267

Problems

4.54 A flue passing hot exhaust gases has a square cross section, 300 mm to a side. The walls are constructed of refractory brick 150 mm thick with a thermal conductivity of 0.85 W/m 䡠 K. Calculate the heat loss from the flue per unit length when the interior and exterior surfaces are maintained at 350 and 25°C, respectively. Use a grid spacing of 75 mm. 4.55 Steady-state temperatures (K) at three nodal points of a long rectangular rod are as shown. The rod experiences a uniform volumetric generation rate of 5 ⫻ 107 W/m3 and has a thermal conductivity of 20 W/m 䡠 K. Two of its sides are maintained at a constant temperature of 300 K, while the others are insulated. 5 mm 1

2

398.0 5 mm

348.5

3

374.6

Uniform temperature, 300 K

(a) Determine the temperatures at nodes 1, 2, and 3. (b) Calculate the heat transfer rate per unit length (W/m) from the rod using the nodal temperatures. Compare this result with the heat rate calculated from knowledge of the volumetric generation rate and the rod dimensions. 4.56 Functionally graded materials are intentionally fabricated to establish a spatial distribution of properties in the final product. Consider an L ⫻ L two-dimensional object with L ⫽ 20 mm. The thermal conductivity distribution of the functionally graded material is k(x) ⫽ 20 W/m 䡠 K ⫹ (7070 W/m5/2 䡠 K) x3/2. Two sets of boundary conditions, denoted as cases 1 and 2, are applied.

Case 1 — — — 2 — — —

Surface

Boundary Condition

1 2 3 4 1 2 3 4

T ⫽ 100°C T ⫽ 50°C Adiabatic Adiabatic Adiabatic Adiabatic T ⫽ 50°C T ⫽ 100°C

(a) Determine the spatially averaged value of the thermal conductivity k. Use this value to estimate the heat rate per unit length for cases 1 and 2. (b) Using a grid spacing of 2 mm, determine the heat rate per unit depth for case 1. Compare your result to the estimated value calculated in part (a). (c) Using a grid spacing of 2 mm, determine the heat rate per unit depth for case 2. Compare your result to the estimated value calculated in part (a). 4.57 Steady-state temperatures at selected nodal points of the symmetrical section of a flow channel are known to be T2 ⫽ 95.47⬚C, T3 ⫽ 117.3⬚C, T5 ⫽ 79.79⬚C, T6 ⫽ 77.29⬚C, T8 ⫽ 87.28⬚C, and T10 ⫽ 77.65⬚C. The wall experiences uniform volumetric heat generation of . q ⫽ 106 W/m3 and has a thermal conductivity of k ⫽ 10 W/m 䡠 K. The inner and outer surfaces of the channel experience convection with fluid temperatures of T앝,i ⫽ 50⬚C and T앝,o ⫽ 25⬚C and convection coefficients of hi ⫽500 W/m2 䡠 K and ho ⫽ 250 W/m2 䡠 K. y 1

T∞,i, hi

2

Surface B

Insulation 4

3

5

•

k, q 6

Symmetry plane

∆x = ∆y = 25 mm 7

8

9

10

x

Surface A

T∞,o, ho

Surface 3

Surface 2 Surface 1 y

k(x) x Surface 4

(a) Determine the temperatures at nodes 1, 4, 7, and 9. (b) Calculate the heat rate per unit length (W/m) from the outer surface A to the adjacent fluid. (c) Calculate the heat rate per unit length from the inner fluid to surface B. (d) Verify that your results are consistent with an overall energy balance on the channel section. 4.58 Consider an aluminum heat sink (k ⫽ 240 W/m 䡠 K), such as that shown schematically in Problem 4.28. The

268

Chapter 4

Two-Dimensional, Steady-State Conduction

䊏

inner and outer widths of the square channel are w ⫽ 20 mm and W ⫽ 40 mm, respectively, and an outer surface temperature of Ts ⫽ 50⬚C is maintained by the array of electronic chips. In this case, it is not the inner surface temperature that is known, but conditions (T앝, h) associated with coolant flow through the channel, and we wish to determine the rate of heat transfer to the coolant per unit length of channel. For this purpose, consider a symmetrical section of the channel and a two-dimensional grid with ⌬x ⫽ ⌬y ⫽ 5 mm. (a) For T앝 ⫽ 20⬚C and h ⫽ 5000 W/m2 䡠 K, determine the unknown temperatures, T1, . . ., T7, and the rate of heat transfer per unit length of channel, q⬘. (b) Assess the effect of variations in h on the unknown temperatures and the heat rate. Heat sink, k T4

Ts

T∞ , h

T1

T5

T2

T6

T3

T7

Ts

Calculate the heat transfer per unit depth into the page, q⬘, using ⌬x ⫽ ⌬y ⫽ ⌬r ⫽ 10 mm and ⌬ ⫽ /8. The base of the rectangular subdomain is held at Th ⫽ 20°C, while the vertical surface of the cylindrical subdomain and the surface at outer radius ro are at Tc ⫽ 0°C. The remaining surfaces are adiabatic, and the thermal conductivity is k ⫽ 10 W/m 䡠 K. 4.60 Consider the two-dimensional tube of a noncircular cross section formed by rectangular and semicylindrical subdomains patched at the common dashed control surfaces in a manner similar to that described in Problem 4.59. Note that, along the dashed control surfaces, temperatures in the two subdomains are identical and local conduction heat fluxes to the semicylindrical subdomain are identical to local conduction heat fluxes from the rectangular subdomain. The bottom of the domain is held at Ts ⫽ 100°C by condensing steam, while the flowing fluid is characterized by the temperature and convection coefficient shown in the sketch. The remaining surfaces are insulated, and the thermal conductivity is k ⫽ 15 W/m 䡠 K.

k = 15 W/m⋅K

Coolant, T∞, h

4.59 Conduction within relatively complex geometries can sometimes be evaluated using the finite-difference methods of this text that are applied to subdomains and patched together. Consider the two-dimensional domain formed by rectangular and cylindrical subdomains patched at the common, dashed control surface. Note that, along the dashed control surface, temperatures in the two subdomains are identical and local conduction heat fluxes to the cylindrical subdomain are identical to local conduction heat fluxes from the rectangular subdomain.

Tc = 0°C

Adiabatic surfaces

T∞,i = 20°C hi = 240 W/m2·K

r Di = 40 mm

y

t = 10 mm

x L = Do = 80 mm

Find the heat transfer rate per unit length of tube, q⬘, using ⌬x ⫽ ⌬y ⫽ ⌬r ⫽ 10 mm and ⌬ ⫽ /8. Hint: Take advantage of the symmetry of the problem by considering only half of the entire domain. 4.61 The steady-state temperatures (°C) associated with selected nodal points of a two-dimensional system having a thermal conductivity of 1.5 W/m 䡠 K are shown on the accompanying grid. Insulated boundary

ro = 50 mm

T2

129.4 0.1 m

y

ri = 30 mm

H = 30 mm W = 20 mm x Th = 20°C

Ts = 100°C

0.1 m 172.9

137.0 103.5

T1

132.8

Isothermal boundary

T0 = 200°C

45.8

T3

67.0

T∞ = 30°C h = 50 W/m2•K

䊏

269

Problems

(a) Determine the temperatures at nodes 1, 2, and 3. (b) Calculate the heat transfer rate per unit thickness normal to the page from the system to the fluid. 4.62 A steady-state, finite-difference analysis has been performed on a cylindrical fin with a diameter of 12 mm and a thermal conductivity of 15 W/m 䡠 K. The convection process is characterized by a fluid temperature of 25°C and a heat transfer coefficient of 25 W/m2 䡠 K. T∞, h

T0

T1

T2

T3

D

T0 = 100.0°C T1 = 93.4°C T2 = 89.5°C

∆x

x

(a) The temperatures for the first three nodes, separated by a spatial increment of x ⫽ 10 mm, are given in the sketch. Determine the fin heat rate. (b) Determine the temperature at node 3, T3. 4.63 Consider the two-dimensional domain shown. All surfaces are insulated except for the isothermal surfaces at x ⫽ 0 and L.

T2

2H/3

H

T1 0.8 L

y

(a) Determine the temperatures at nodes 1, 2, 3, and 4. Estimate the midpoint temperature. (b) Reducing the mesh size by a factor of 2, determine the corresponding nodal temperatures. Compare your results with those from the coarser grid. (c) From the results for the finer grid, plot the 75, 150, and 250°C isotherms. 4.65 Consider a long bar of square cross section (0.8 m to the side) and of thermal conductivity 2 W/m 䡠 K. Three of these sides are maintained at a uniform temperature of 300°C. The fourth side is exposed to a fluid at 100°C for which the convection heat transfer coefficient is 10 W/m2 䡠 K. (a) Using an appropriate numerical technique with a grid spacing of 0.2 m, determine the midpoint temperature and heat transfer rate between the bar and the fluid per unit length of the bar. (b) Reducing the grid spacing by a factor of 2, determine the midpoint temperature and heat transfer rate. Plot the corresponding temperature distribution across the surface exposed to the fluid. Also, plot the 200 and 250°C isotherms. 4.66 Consider a two-dimensional, straight triangular fin of length L ⫽ 50 mm and base thickness t ⫽ 20 mm. The thermal conductivity of the fin is k ⫽ 25 W/m 䡠 K. The base temperature is Tb ⫽ 50°C, and the fin is exposed to convection conditions characterized by h ⫽ 50 W/m2 䡠 K, T앝 ⫽ 20°C. Using a finite difference mesh with ⌬x ⫽ 10 mm and ⌬y ⫽ 2 mm, and taking advantage of symmetry, determine the fin efficiency, f. Compare your value of the fin efficiency with that reported in Figure 3.19.

x L = 50 mm

L = 5H/3 6 5 4 3 2 1

(a) Use a one-dimensional analysis to estimate the shape factor S. (b) Estimate the shape factor using a finite difference analysis with ⌬x ⫽ ⌬y ⫽ 0.05L. Compare your answer with that of part (a), and explain the difference between the two solutions.

t = 20 mm

15 14 13 12

18 17 16

100°C

T∞ = 20°C h = 50 W/m2·K

x

2

Air duct

T2 = 30°C

T1 = 80°C

200°C 3

4 1.5L

x 300°C

21

4.67 A common arrangement for heating a large surface area is to move warm air through rectangular ducts below the surface. The ducts are square and located midway between the top and bottom surfaces that are exposed to room air and insulated, respectively.

y

50°C

20 19

y

4.64 Consider two-dimensional, steady-state conduction in a square cross section with prescribed surface temperatures.

1

11 10 9 8 7

L

L L

Concrete

270

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

For the condition when the floor and duct temperatures are 30 and 80°C, respectively, and the thermal conductivity of concrete is 1.4 W/m 䡠 K, calculate the heat rate from each duct, per unit length of duct. Use a grid spacing with ⌬x ⫽ 2 ⌬y, where ⌬y ⫽ 0.125L and L ⫽150 mm. 4.68 Consider the gas turbine cooling scheme of Example 4.3. In Problem 3.23, advantages associated with applying a thermal barrier coating (TBC) to the exterior surface of a turbine blade are described. If a 0.5-mm-thick zirconia coating (k ⫽ 1.3 W/m 䡠 K, R⬙t,c ⫽ 10⫺4 m2 䡠 K/W) is applied to the outer surface of the air-cooled blade, determine the temperature field in the blade for the operating conditions of Example 4.3. 4.69 A long, solid cylinder of diameter D ⫽ 25 mm is formed of an insulating core that is covered with a very thin (t ⫽ 50 m), highly polished metal sheathing of thermal conductivity k ⫽ 25 W/m 䡠 K. Electric current flows through the stainless steel from one end of the cylinder to the other, inducing uniform volumetric heating within the . sheathing of q ⫽ 5 ⫻ 106 W/m3. As will become evident in Chapter 6, values of the convection coefficient between the surface and air for this situation are spatially nonuniform, and for the airstream conditions of the experiment, the convection heat transfer coefficient varies with the angle as h() ⫽ 26 ⫹ 0.637 ⫺ 8.922 for 0 ⱕ ⬍ /2 and h() ⫽ 5 for /2 ⱕ ⱕ .

device to nonintrusively determine the surface temperature distribution. Predict the temperature distribution of the painted surface, accounting for radiation heat transfer with large surroundings at Tsur ⫽ 25°C. 4.71 Consider using the experimental methodology of Problem 4.70 to determine the convection coefficient distribution about an airfoil of complex shape.

Tsur = 25°C

3 Air T∞ = 25°C

Location Metal sheathing q• = 5 106 W/m3 1 k = 25 W/m • K

(b) Accounting for -direction conduction in the stainless steel, determine temperatures in the stainless steel at increments of ⌬ ⫽ /20 for 0 ⱕ ⱕ . Compare the temperature distribution with that of part (a). Hint: The temperature distribution is symmetrical about the horizontal centerline of the cylinder. 4.70 Consider Problem 4.69. An engineer desires to measure the surface temperature of the thin sheathing by painting it black ( ⫽ 0.98) and using an infrared measurement

9 10 11

Insulation

1 29

28 27

26 25 24 23 22

12

13

14

15

16 21 20 19 18 17

Accounting for conduction in the metal sheathing and radiation losses to the large surroundings, determine the convection heat transfer coefficients at the locations shown. The surface locations at which the temperatures are measured are spaced 2 mm apart. The thickness of the metal sheathing is t ⫽ 20 m, the volumetric gener. ation rate is q ⫽ 20 ⫻ 106 W/m3, the sheathing’s thermal conductivity is k ⫽ 25 W/m 䡠 K, and the emissivity of the painted surface is ⫽ 0.98. Compare your results to cases where (i) both conduction along the sheathing and radiation are neglected, and (ii) when only radiation is neglected.

θ

(a) Neglecting conduction in the -direction within the stainless steel, plot the temperature distribution T() for 0 ⱕ ⱕ for T앝 ⫽ 25°C.

7 8

Metal sheathing

D = 25 mm

Insulation

2

30

t = 50 µm Air T∞ = 25°C

5 6

4

Temperature Temperature Temperature (°C) Location (°C) Location (°C) 27.77

11

34.29

21

31.13

2

27.67

12

36.78

22

30.64

3

27.71

13

39.29

23

30.60

4

27.83

14

41.51

24

30.77

5

28.06

15

42.68

25

31.16

6

28.47

16

42.84

26

31.52

7

28.98

17

41.29

27

31.85

8

29.67

18

37.89

28

31.51

9

30.66

19

34.51

29

29.91

10

32.18

20

32.36

30

28.42

4.72 A thin metallic foil of thickness 0.25 mm with a pattern of extremely small holes serves as an acceleration grid to control the electrical potential of an ion beam. Such a grid is used in a chemical vapor deposition (CVD) process for the fabrication of semiconductors. The top surface of the grid is exposed to a uniform heat flux

䊏

271

Problems

caused by absorption of the ion beam, q⬙s ⫽ 600 W/m2. The edges of the foil are thermally coupled to watercooled sinks maintained at 300 K. The upper and lower surfaces of the foil experience radiation exchange with the vacuum enclosure walls maintained at 300 K. The effective thermal conductivity of the foil material is 40 W/m 䡠 K, and its emissivity is 0.45.

Vacuum enclosure, Tsur Ion beam, q"s

Grid hole pattern Grid

x

L = 115 mm Water-cooled electrode sink, Tsink

Assuming one-dimensional conduction and using a finite-difference method representing the grid by 10 nodes in the x-direction, estimate the temperature distribution for the grid. Hint: For each node requiring an energy balance, use the linearized form of the radiation rate equation, Equation 1.8, with the radiation coefficient hr, Equation 1.9, evaluated for each node. 4.73 A long bar of rectangular cross section, 0.4 m ⫻ 0.6 m on a side and having a thermal conductivity of 1.5 W/m 䡠 K, is subjected to the boundary conditions shown.

w

w/4 w/2 w/2 T2

(a) Using a finite-difference method with a mesh size of ⌬x ⫽ ⌬y ⫽ 40 mm, calculate the unknown nodal temperatures and the heat transfer rate per width of groove spacing (w) and per unit length normal to the page. (b) With a mesh size of ⌬x ⫽ ⌬y ⫽ 10 mm, repeat the foregoing calculations, determining the temperature field and the heat rate. Also, consider conditions for which the bottom surface is not at a uniform temperature T2 but is exposed to a fluid at T앝 ⫽ 20°C. With ⌬x ⫽ ⌬y ⫽ 10 mm, determine the temperature field and heat rate for values of h ⫽ 5, 200, and 1000 W/m2 䡠 K, as well as for h → 앝. 4.75 Refer to the two-dimensional rectangular plate of Problem 4.2. Using an appropriate numerical method with ⌬x ⫽ ⌬y ⫽ 0.25 m, determine the temperature at the midpoint (1, 0.5). 4.76 The shape factor for conduction through the edge of adjoining walls for which D ⬎ L/5, where D and L are the wall depth and thickness, respectively, is shown in Table 4.1. The two-dimensional symmetrical element of the edge, which is represented by inset (a), is bounded by the diagonal symmetry adiabat and a section of the wall thickness over which the temperature distribution is assumed to be linear between T1 and T2.

Uniform temperature, T = 200°C

T∞, h Insulated

T1

w

y T2

T2 T2 T2 T2

Linear temperature distribution

Symmetry adiabat

T2 Uniform temperature, T = 200°C

Two of the sides are maintained at a uniform temperature of 200°C. One of the sides is adiabatic, and the remaining side is subjected to a convection process with T앝 ⫽ 30°C and h ⫽ 50 W/m2 䡠 K. Using an appropriate numerical technique with a grid spacing of 0.1 m, determine the temperature distribution in the bar and the heat transfer rate between the bar and the fluid per unit length of the bar. 4.74 The top surface of a plate, including its grooves, is maintained at a uniform temperature of T1 ⫽ 200°C. The lower surface is at T2 ⫽ 20°C, the thermal conductivity is 15 W/m 䡠 K, and the groove spacing is 0.16 m.

∆y ∆x

T1

x

(a)

T1 T2

y T2 a

b

L

L a

b

x

n•L (b)

(a) Using the nodal network of inset (a) with L ⫽ 40 mm, determine the temperature distribution in the element for T1 ⫽ 100°C and T2 ⫽ 0°C. Evaluate the heat rate

272

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

per unit depth (D ⫽ 1 m) if k ⫽ 1 W/m 䡠 K. Determine the corresponding shape factor for the edge, and compare your result with that from Table 4.1. (b) Choosing a value of n ⫽ 1 or n ⫽ 1.5, establish a nodal network for the trapezoid of inset (b) and determine the corresponding temperature field. Assess the validity of assuming linear temperature distributions across sections a–a and b–b. 4.77 The diagonal of a long triangular bar is well insulated, while sides of equivalent length are maintained at uniform temperatures Ta and Tb.

Ta = 100°C Insulation

Tb = 0°C

(a) Establish a nodal network consisting of five nodes along each of the sides. For one of the nodes on the diagonal surface, define a suitable control volume and derive the corresponding finite-difference equation. Using this form for the diagonal nodes and appropriate equations for the interior nodes, find the temperature distribution for the bar. On a scale drawing of the shape, show the 25, 50, and 75°C isotherms. (b) An alternate and simpler procedure to obtain the finite-difference equations for the diagonal nodes follows from recognizing that the insulated diagonal surface is a symmetry plane. Consider a square 5 ⫻ 5 nodal network, and represent its diagonal as a symmetry line. Recognize which nodes on either side of the diagonal have identical temperatures. Show that you can treat the diagonal nodes as “interior” nodes and write the finite-difference equations by inspection. 4.78 A straight fin of uniform cross section is fabricated from a material of thermal conductivity 50 W/m 䡠 K, thickness w ⫽ 6 mm, and length L ⫽ 48 mm, and it is very long in the direction normal to the page. The convection heat transfer coefficient is 500 W/m2 䡠 K with an ambient air temperature of T앝 ⫽ 30°C. The base of the fin is maintained at Tb ⫽ 100°C, while the fin tip is well insulated. T∞, h w

Tb T∞, h L

Insulated

(a) Using a finite-difference method with a space increment of 4 mm, estimate the temperature distribution within the fin. Is the assumption of onedimensional heat transfer reasonable for this fin? (b) Estimate the fin heat transfer rate per unit length normal to the page. Compare your result with the one-dimensional, analytical solution, Equation 3.81. (c) Using the finite-difference mesh of part (a), compute and plot the fin temperature distribution for values of h ⫽ 10, 100, 500, and 1000 W/m2 䡠 K. Determine and plot the fin heat transfer rate as a function of h. 4.79 A rod of 10-mm diameter and 250-mm length has one end maintained at 100°C. The surface of the rod experiences free convection with the ambient air at 25°C and a convection coefficient that depends on the difference between the temperature of the surface and the ambient air. Specifically, the coefficient is prescribed by a correlation of the form, hfc ⫽ 2.89[0.6 ⫹ 0.624 (T ⫺ T앝)1/6]2, where the units are hfc (W/m2 䡠 K) and T (K). The surface of the rod has an emissivity ⫽ 0.2 and experiences radiation exchange with the surroundings at Tsur ⫽ 25°C. The fin tip also experiences free convection and radiation exchange. Tsur = 25°C Quiescent air,

T∞ = 25°C Stainless steel rod

Tb = 100°C

k = 14 W/m•K, ε = 0.2

D= 10 mm

L = 250 mm x

Assuming one-dimensional conduction and using a finite-difference method representing the fin by five nodes, estimate the temperature distribution for the fin. Determine also the fin heat rate and the relative contributions of free convection and radiation exchange. Hint: For each node requiring an energy balance, use the linearized form of the radiation rate equation, Equation 1.8, with the radiation coefficient hr, Equation 1.9, evaluated for each node. Similarly, for the convection rate equation associated with each node, the free convection coefficient hfc must be evaluated for each node. 4.80 A simplified representation for cooling in very large-scale integration (VLSI) of microelectronics is shown in the sketch. A silicon chip is mounted in a dielectric substrate, and one surface of the system is convectively cooled, while the remaining surfaces are well insulated from the surroundings. The problem is rendered two-dimensional

䊏

273

Problems

by assuming the system to be very long in the direction perpendicular to the paper. Under steady-state operation, electric power dissipation in the chip provides for uni. form volumetric heating at a rate of q . However, the heating rate is limited by restrictions on the maximum temperature that the chip is allowed to achieve.

(b) The grid spacing used in the foregoing finite-difference solution is coarse, resulting in poor precision for the temperature distribution and heat removal rate. Investigate the effect of grid spacing by considering spatial increments of 50 and 25 m. (c) Consistent with the requirement that a ⫹ b ⫽ 400 m, can the heat sink dimensions be altered in a manner that reduces the overall thermal resistance?

Coolant Chip kc = 50 W/m•K q• = 107 W/m3

T∞ = 20°C h = 500 W/m2•K H/4 L/3 Substrate, ks = 5 W/m•K

H= 12 mm

4.82 A plate (k ⫽ 10 W/m 䡠 K) is stiffened by a series of longitudinal ribs having a rectangular cross section with length L ⫽ 8 mm and width w ⫽ 4 mm. The base of the plate is maintained at a uniform temperature Tb ⫽ 45°C, while the rib surfaces are exposed to air at a temperature of T앝 ⫽ 25°C and a convection coefficient of h ⫽ 600 W/m2 䡠 K.

L = 27 mm

y

For the conditions shown on the sketch, will the maximum temperature in the chip exceed 85°C, the maximum allowable operating temperature set by industry standards? A grid spacing of 3 mm is suggested. 4.81 A heat sink for cooling computer chips is fabricated from copper (ks ⫽ 400 W/m 䡠 K), with machined microchannels passing a cooling fluid for which T ⫽ 25°C and h ⫽ 30,000 W/m2 䡠 K. The lower side of the sink experiences no heat removal, and a preliminary heat sink design calls for dimensions of a ⫽ b ⫽ ws ⫽ wf ⫽ 200 m. A symmetrical element of the heat path from the chip to the fluid is shown in the inset. y Tc Chips, Tc

a

ws

wf

Sink, ks Microchannel

b

T∞, h Insulation

x ws ___ 2

wf ___ 2

(a) Using the symmetrical element with a square nodal network of ⌬x ⫽ ⌬y ⫽ 100 m, determine the corresponding temperature field and the heat rate q⬘ to the coolant per unit channel length (W/m) for a maximum allowable chip temperature Tc, max ⫽ 75°C. Estimate the corresponding thermal resistance between the chip surface and the fluid, R⬘t,c⫺ƒ (m 䡠 K/W). What is the maximum allowable heat dissipation for a chip that measures 10 mm ⫻ 10 mm on a side?

T∞, h

Rib Plate

w

Tb

x L

T∞, h

(a) Using a finite-difference method with ⌬x ⫽ ⌬y ⫽ 2 mm and a total of 5 ⫻ 3 nodal points and regions, estimate the temperature distribution and the heat rate from the base. Compare these results with those obtained by assuming that heat transfer in the rib is one-dimensional, thereby approximating the behavior of a fin. (b) The grid spacing used in the foregoing finitedifference solution is coarse, resulting in poor precision for estimates of temperatures and the heat rate. Investigate the effect of grid refinement by reducing the nodal spacing to ⌬x ⫽ ⌬y ⫽ 1 mm (a 9 ⫻ 3 grid) considering symmetry of the center line. (c) Investigate the nature of two-dimensional conduction in the rib and determine a criterion for which the one-dimensional approximation is reasonable. Do so by extending your finite-difference analysis to determine the heat rate from the base as a function of the length of the rib for the range 1.5 ⱕ L/w ⱕ 10, keeping the length L constant. Compare your results with those determined by approximating the rib as a fin. 4.83 The bottom half of an I-beam providing support for a furnace roof extends into the heating zone. The web is well insulated, while the flange surfaces experience

274

Chapter 4

Two-Dimensional, Steady-State Conduction

䊏

(a) Using a grid spacing of 30 mm and the Gauss-Seidel iteration method, determine the nodal temperatures and the heat rate per unit length normal to the page into the bar from the air.

convection with hot gases at T앝 ⫽ 400°C and a convection coefficient of h ⫽ 150 W/m2 䡠 K. Consider the symmetrical element of the flange region (inset a), assuming that the temperature distribution across the web is uniform at Tw ⫽ 100°C. The beam thermal conductivity is 10 W/m 䡠 K, and its dimensions are wƒ ⫽ 80 mm, ww ⫽ 30 mm, and L ⫽ 30 mm.

(b) Determine the effect of grid spacing on the temperature field and heat rate. Specifically, consider a grid spacing of 15 mm. For this grid, explore the effect of changes in h on the temperature field and the isotherms.

Oven roof I-beam

Insulation Flange Gases T∞, h

Assume uniform

Web

y

Uniform ?

wo

w ___w

4.85 A long trapezoidal bar is subjected to uniform temperatures on two surfaces, while the remaining surfaces are well insulated. If the thermal conductivity of the material is 20 W/m 䡠 K, estimate the heat transfer rate per unit length of the bar using a finite-difference method. Use the Gauss–Seidel method of solution with a space increment of 10 mm.

T∞, h

2

Insulation

Tw T2 = 25°C

L

(b)

50 mm

x

wf ___ 2

20 mm

T∞, h

(a)

T1 = 100°C

(a) Calculate the heat transfer rate per unit length to the beam using a 5 ⫻ 4 nodal network. (b) Is it reasonable to assume that the temperature distribution across the web–flange interface is uniform? Consider the L-shaped domain of inset (b) and use a fine grid to obtain the temperature distribution across the web–flange interface. Make the distance wo ⱖ ww /2. 4.84 A long bar of rectangular cross section is 60 mm ⫻ 90 mm on a side and has a thermal conductivity of 1 W/m 䡠 K. One surface is exposed to a convection process with air at 100°C and a convection coefficient of 100 W/m2 䡠 K, while the remaining surfaces are maintained at 50°C.

4.86 Small-diameter electrical heating elements dissipating 50 W/m (length normal to the sketch) are used to heat a ceramic plate of thermal conductivity 2 W/m 䡠 K. The upper surface of the plate is exposed to ambient air at 30°C with a convection coefficient of 100 W/m2 䡠 K, while the lower surface is well insulated. Air

T∞, h Ceramic plate

Ts Ts = 50°C

Ts

Heating element

y

6 mm

x 2 mm

T∞, h

30 mm

24 mm

24 mm

(a) Using the Gauss–Seidel method with a grid spacing of ⌬x ⫽ 6 mm and ⌬y ⫽ 2 mm, obtain the temperature distribution within the plate. (b) Using the calculated nodal temperatures, sketch four isotherms to illustrate the temperature distribution in the plate. (c) Calculate the heat loss by convection from the plate to the fluid. Compare this value with the element dissipation rate.

䊏

275

Problems

(d) What advantage, if any, is there in not making ⌬x ⫽ ⌬y for this situation? (e) With ⌬x ⫽ ⌬y ⫽ 2 mm, calculate the temperature field within the plate and the rate of heat transfer from the plate. Under no circumstances may the temperature at any location in the plate exceed 400°C. Would this limit be exceeded if the airflow were terminated and heat transfer to the air were by natural convection with h ⫽ 10 W/m2 䡠 K?

Special Applications: Finite Element Analysis 4.87 A straight fin of uniform cross section is fabricated from a material of thermal conductivity k ⫽ 5 W/m 䡠 K, thickness w ⫽ 20 mm, and length L ⫽ 200 mm. The fin is very long in the direction normal to the page. The base of the fin is maintained at Tb ⫽ 200°C, and the tip condition allows for convection (case A of Table 3.4), with h ⫽ 500 W/m2 䡠 K and T앝 ⫽ 25°C. T∞ = 100°C h = 500 W/m2•K Tb = 200°C

k = 5 W/m•K

T∞, h

q'f

w = 20 mm x

L = 200 mm

T∞, h

(a) Assuming one-dimensional heat transfer in the fin, calculate the fin heat rate, q⬘f (W/m), and the tip temperature TL. Calculate the Biot number for the fin to determine whether the one-dimensional assumption is valid. (b) Using the finite-element method of FEHT, perform a two-dimensional analysis on the fin to determine the fin heat rate and tip temperature. Compare your results with those from the one-dimensional, analytical solution of part (a). Use the View/Temperature Contours option to display isotherms, and discuss key features of the corresponding temperature field and heat flow pattern. Hint: In drawing the outline of the fin, take advantage of symmetry. Use a fine mesh near the base and a coarser mesh near the tip. Why? (c) Validate your FEHT model by comparing predictions with the analytical solution for a fin with thermal conductivities of k ⫽ 50 W/m 䡠 K and 500 W/m 䡠 K. Is the one-dimensional heat transfer assumption valid for these conditions?

4.88 Consider the long rectangular bar of Problem 4.84 with the prescribed boundary conditions. (a) Using the finite-element method of FEHT, determine the temperature distribution. Use the View/ Temperature Contours command to represent the isotherms. Identify significant features of the distribution. (b) Using the View/Heat Flows command, calculate the heat rate per unit length (W/m) from the bar to the airstream. (c) Explore the effect on the heat rate of increasing the convection coefficient by factors of two and three. Explain why the change in the heat rate is not proportional to the change in the convection coefficient. 4.89 Consider the long rectangular rod of Problem 4.53, which experiences uniform heat generation while its surfaces are maintained at a fixed temperature. (a) Using the finite-element method of FEHT, determine the temperature distribution. Use the View/ Temperature Contours command to represent the isotherms. Identify significant features of the distribution. (b) With the boundary conditions unchanged, what heat generation rate will cause the midpoint temperature to reach 600 K? 4.90 Consider the symmetrical section of the flow channel of . Problem 4.57, with the prescribed values of q , k, T앝,i, T앝,o, hi, and ho. Use the finite-element method of FEHT to obtain the following results. (a) Determine the temperature distribution in the symmetrical section, and use the View/Temperature Contours command to represent the isotherms. Identify significant features of the temperature distribution, including the hottest and coolest regions and the region with the steepest gradients. Describe the heat flow field. (b) Using the View/Heat Flows command, calculate the heat rate per unit length (W/m) from the outer surface A to the adjacent fluid. (c) Calculate the heat rate per unit length from the inner fluid to surface B. (d) Verify that your results are consistent with an overall energy balance on the channel section. 4.91 The hot-film heat flux gage shown schematically may be used to determine the convection coefficient of an adjoining fluid stream by measuring the electric power dissipation per unit area, P⬙e (W/m2), and the average surface temperature, Ts,f , of the film. The power dissipated in the film is transferred directly to the fluid by convection, as well as by conduction into the substrate.

276

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

If substrate conduction is negligible, the gage measurements can be used to determine the convection coefficient without application of a correction factor. Your assignment is to perform a two-dimensional, steadystate conduction analysis to estimate the fraction of the power dissipation that is conducted into a 2-mm-thick quartz substrate of width W ⫽ 40 mm and thermal conductivity k ⫽ 1.4 W/m 䡠 K. The thin, hot-film gage has a width of w ⫽ 4 mm and operates at a uniform power dissipation of 5000 W/m2. Consider cases for which the fluid temperature is 25°C and the convection coefficient is 500, 1000, and 2000 W/m2 䡠 K. Ts, f

Hot-thin film, P"e = 5000 W/m2

Fluid

T∞, h

Quartz substrate k = 1.4 W/m•K

w = 4 mm

2 mm

(b) Determine the effect of grid spacing on the temperature field and heat loss per unit length to the air. Specifically, consider a grid spacing of 25 mm and plot appropriately spaced isotherms on a schematic of the system. Explore the effect of changes in the convection coefficients on the temperature field and heat loss. 4.93 Electronic devices dissipating electrical power can be cooled by conduction to a heat sink. The lower surface of the sink is cooled, and the spacing of the devices ws, the width of the device wd, and the thickness L and thermal conductivity k of the heat sink material each affect the thermal resistance between the device and the cooled surface. The function of the heat sink is to spread the heat dissipated in the device throughout the sink material. ws = 48 mm

P"e

Device, Td = 85°C

wd = 18 mm

T∞ , h

L = 24 mm

W = 40 mm w/2

W/2

Sink material,

Use the finite-element method of FEHT to analyze a symmetrical half-section of the gage and the quartz substrate. Assume that the lower and end surfaces of the substrate are perfectly insulated, while the upper surface experiences convection with the fluid. (a) Determine the temperature distribution and the conduction heat rate into the region below the hot film for the three values of h. Calculate the fractions of electric power dissipation represented by these rates. Hint: Use the View/Heat Flow command to find the heat rate across the boundary elements. (b) Use the View/Temperature Contours command to view the isotherms and heat flow patterns. Describe the heat flow paths, and comment on features of the gage design that influence the paths. What limitations on applicability of the gage have been revealed by your analysis? 4.92 Consider the system of Problem 4.54. The interior surface is exposed to hot gases at 350°C with a convection coefficient of 100 W/m2 䡠 K, while the exterior surface experiences convection with air at 25°C and a convection coefficient of 5 W/m2 䡠 K. (a) Using a grid spacing of 75 mm, calculate the temperature field within the system and determine the heat loss per unit length by convection from the outer surface of the flue to the air. Compare this result with the heat gained by convection from the hot gases to the air.

k = 300 W/m•K

Cooled surface, Ts = 25°C

(a) Beginning with the shaded symmetrical element, use a coarse (5 ⫻ 5) nodal network to estimate the thermal resistance per unit depth between the device and lower surface of the sink, R⬘t,d⫺s (m 䡠 K/W). How does this value compare with thermal resistances based on the assumption of one-dimensional conduction in rectangular domains of (i) width wd and length L and (ii) width ws and length L? (b) Using nodal networks with grid spacings three and five times smaller than that in part (a), determine the effect of grid size on the precision of the thermal resistance calculation. (c) Using the finer nodal network developed for part (b), determine the effect of device width on the thermal resistance. Specifically, keeping ws and L fixed, find the thermal resistance for values of wd /ws ⫽ 0.175, 0.275, 0.375, and 0.475. 4.94 Consider one-dimensional conduction in a plane

composite wall. The exposed surfaces of materials A and B are maintained at T1 ⫽ 600 K and T2 ⫽ 300 K, respectively. Material A, of thickness La ⫽ 20 mm, has a temperature-dependent thermal conductivity of ka ⫽ ko [1 ⫹ ␣(T ⫺ To)], where ko ⫽ 4.4 W/m 䡠 K, ␣ ⫽ 0.008 K⫺1, To ⫽ 300 K, and T is in kelvins. Material B is of thickness Lb ⫽ 5 mm and has a thermal conductivity of kb ⫽ 1 W/m 䡠 K.

䊏

277

Problems

that of IHT, or the finite-element method of FEHT to obtain the following results.

kb

ka = ka(T)

T1 = 600 K

T2 = 300 K

Fluid

T∞,o = 25°C ho = 250 W/m2•K Temperature uniformity of 5°C required

A

B L

x

La

Heating channel T∞,i = 200°C hi = 500 W/m2•K

L

La + Lb L

(a) Calculate the heat flux through the composite wall by assuming material A to have a uniform thermal conductivity evaluated at the average temperature of the section. (b) Using a space increment of 1 mm, obtain the finitedifference equations for the internal nodes and calculate the heat flux considering the temperaturedependent thermal conductivity for Material A. If the IHT software is employed, call-up functions from Tools/Finite-Difference Equations may be used to obtain the nodal equations. Compare your result with that obtained in part (a). (c) As an alternative to the finite-difference method of part (b), use the finite-element method of FEHT to calculate the heat flux, and compare the result with that from part (a). Hint: In the Specify/Material Properties box, properties may be entered as a function of temperature (T), the space coordinates (x, y), or time (t). See the Help section for more details. 4.95 A platen of thermal conductivity k ⫽ 15 W/m 䡠 K is heated by flow of a hot fluid through channels of width L ⫽ 20 mm, with T앝,i ⫽ 200⬚C and hi ⫽ 500 W/m2 䡠 K. The upper surface of the platen is used to heat a process fluid at T앝,o ⫽ 25⬚C with a convection coefficient of ho ⫽ 250 W/m2 䡠 K. The lower surface of the platen is insulated. To heat the process fluid uniformly, the temperature of the platen’s upper surface must be uniform to within 5⬚C. Use a finite-difference method, such as

Platen, k = 15 W/m•K

L /2 Insulation

W

(a) Determine the maximum allowable spacing W between the channel centerlines that will satisfy the specified temperature uniformity requirement. (b) What is the corresponding heat rate per unit length from a flow channel? 4.96 Consider the cooling arrangement for the very large-scale integration (VLSI) chip of Problem 4.93. Use the finiteelement method of FEHT to obtain the following results. (a) Determine the temperature distribution in the chipsubstrate system. Will the maximum temperature exceed 85°C? (b) Using the FEHT model developed for part (a), determine the volumetric heating rate that yields a maximum temperature of 85°C. (c) What effect would reducing the substrate thickness have on the maximum operating temperature? For a . volumetric generation rate of q ⫽ 107 W/m3, reduce the thickness of the substrate from 12 to 6 mm, keeping all other dimensions unchanged. What is the maximum system temperature for these conditions? What fraction of the chip power generation is removed by convection directly from the chip surface?

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4S.1 The Graphical Method The graphical method may be employed for two-dimensional problems involving adiabatic and isothermal boundaries. Although the approach has been superseded by computer solutions based on numerical procedures, it may be used to obtain a first estimate of the temperature distribution and to develop a physical appreciation for the nature of the temperature field and heat flow.

4S.1.1 Methodology of Constructing a Flux Plot The rationale for the graphical method comes from the fact that lines of constant temperature must be perpendicular to lines that indicate the direction of heat flow (see Figure 4.1). The objective of the graphical method is to systematically construct such a network of isotherms and heat flow lines. This network, commonly termed a flux plot, is used to infer the temperature distribution and the rate of heat flow through the system. Consider a square, two-dimensional channel whose inner and outer surfaces are maintained at T1 and T2, respectively. A cross section of the channel is shown in Figure 4S.1a. A procedure for constructing the flux plot, a portion of which is shown in Figure 4S.1b, is as follows. 1. The first step is to identify all relevant lines of symmetry. Such lines are determined by thermal, as well as geometrical, conditions. For the square channel of Figure 4S.1a, such lines include the designated vertical, horizontal, and diagonal lines. For this system it is therefore possible to consider only one-eighth of the configuration, as shown in Figure 4S.1b. 2. Lines of symmetry are adiabatic in the sense that there can be no heat transfer in a direction perpendicular to the lines. They are therefore heat flow lines and should be treated as such. Since there is no heat flow in a direction perpendicular to a heat flow line, it can be termed an adiabat. 3. After all known lines of constant temperature are identified, an attempt should be made to sketch lines of constant temperature within the system. Note that isotherms should always be perpendicular to adiabats. 4. Heat flow lines should then be drawn with an eye toward creating a network of curvilinear squares. This is done by having the heat flow lines and isotherms intersect at right angles and by requiring that all sides of each square be of approximately the same length. It is often impossible to satisfy this second requirement exactly, and it is more realistic to strive for equivalence between the sums of the opposite sides of each square, as shown in Figure 4S.1c. Assigning the x-coordinate to the direction of heat

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4S.1

䊏

The Graphical Method

y

b a

T1 T2 Adiabats

∆x

x

T2 qi c

Symmetry lines

∆y ∆Tj

d

(c)

(a)

T1 qi qi

∆Tj Isotherms (b)

FIGURE 4S.1 Two-dimensional conduction in a square channel of length l. (a) Symmetry planes. (b) Flux plot. (c) Typical curvilinear square.

flow and the y-coordinate to the direction normal to this flow, the requirement may be expressed as x ⬅ ab cd 艐 y ⬅ ac bd 2 2

(4S.1)

It is difficult to create a satisfactory network of curvilinear squares in the first attempt, and several iterations must often be made. This trial-and-error process involves adjusting the isotherms and adiabats until satisfactory curvilinear squares are obtained for most of the network.1 Once the flux plot has been obtained, it may be used to infer the temperature distribution in the medium. From a simple analysis, the heat transfer rate may then be obtained.

4S.1.2

Determination of the Heat Transfer Rate

The rate at which energy is conducted through a lane, which is the region between adjoining adiabats, is designated as qi. If the flux plot is properly constructed, the value of qi will be approximately the same for all lanes and the total heat transfer rate may be expressed as q艐

M

兺q

i

Mqi

(4S.2)

i1

where M is the number of lanes associated with the plot. From the curvilinear square of Figure 4S.1c and the application of Fourier’s law, qi may be expressed as qi 艐 kAi

1

Tj x

艐 k(y 䡠 l)

Tj x

(4S.3)

In certain regions, such as corners, it may be impossible to approach the curvilinear square requirements. However, such difficulties generally have a small effect on the overall accuracy of the results obtained from the flux plot.

4S.1

䊏

W-3

The Graphical Method

where Tj is the temperature difference between successive isotherms, Ai is the conduction heat transfer area for the lane, and l is the length of the channel normal to the page. However, since the temperature increment is approximately the same for all adjoining isotherms, the overall temperature difference between boundaries, T12, may be expressed as T12

N

兺 T 艐 N T j

j

(4S.4)

j1

where N is the total number of temperature increments. Combining Equations 4S.2 through 4S.4 and recognizing that x ⬇ y for curvilinear squares, we obtain q 艐 Ml k T12 N

(4S.5)

The manner in which a flux plot may be used to obtain the heat transfer rate for a twodimensional system is evident from Equation 4S.5. The ratio of the number of heat flow lanes to the number of temperature increments (the value of M/N) may be obtained from the plot. Recall that specification of N is based on step 3 of the foregoing procedure, and the value, which is an integer, may be made large or small depending on the desired accuracy. The value of M is then a consequence of following step 4. Note that M is not necessarily an integer, since a fractional lane may be needed to arrive at a satisfactory network of curvilinear squares. For the network of Figure 4S.1b, N 6 and M 5. Of course, as the network, or mesh, of curvilinear squares is made finer, N and M increase and the estimate of M/N becomes more accurate.

4S.1.3

The Conduction Shape Factor

Equation 4S.5 may be used to define the shape factor, S, of a two-dimensional system. That is, the heat transfer rate may be expressed as q SkT12

(4S.6)

S ⬅ Ml N

(4S.7)

where, for a flux plot,

From Equation 4S.6, it also follows that a two-dimensional conduction resistance may be expressed as Rt,cond(2D) 1 Sk

(4S.8)

Shape factors have been obtained for numerous two-dimensional systems, and results are summarized in Table 4.1 for some common configurations. In cases 1 through 9 and case 11, conduction is presumed to occur between boundaries that are maintained at uniform temperatures, with T12 ⬅ T1 T2. In case 10 conduction is between an isothermal surface (T1) and a semi-infinite medium of uniform temperature (T2) at locations well removed from the surface. Shape factors may also be defined for one-dimensional geometries, and from the

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4S.1

䊏

The Graphical Method

results of Table 3.3, it follows that for plane, cylindrical, and spherical walls, respectively, the shape factors are A/L, 2L/ln (r2/r1), and 4r1r2/(r2 r1). Results are available for many other configurations [1–4].

EXAMPLE 4S.1 A hole of diameter D 0.25 m is drilled through the center of a solid block of square cross section with w 1 m on a side. The hole is drilled along the length, l 2 m, of the block, which has a thermal conductivity of k 150 W/m 䡠 K. A warm fluid passing through the hole maintains an inner surface temperature of T1 75°C, while the outer surface of the block is kept at T2 25°C. 1. Using the flux plot method, determine the shape factor for the system. 2. What is the rate of heat transfer through the block?

SOLUTION Known: Dimensions and thermal conductivity of a block with a circular hole drilled along its length. Find: 1. Shape factor. 2. Heat transfer rate for prescribed surface temperatures. Schematic: k = 150 W/m•K

T2 = 25°C T1 = 75°C D1 = 0.25 m

w =1m

Symmetrical section

w =1m

Assumptions: 1. Steady-state conditions. 2. Two-dimensional conduction. 3. Constant properties. 4. Ends of block are well insulated. Analysis: 1. The flux plot may be simplified by identifying lines of symmetry and reducing the system to the one-eighth section shown in the schematic. Using a fairly coarse grid involving N 6 temperature increments, the flux plot was generated. The resulting network of curvilinear squares is as follows.

4S.2

䊏

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The Gauss-Seidel Method: Example of Usage

T1

Line of symmetry and adiabat N=1 2 3 4 5

6

M=1 T2 2

3 Line of symmetry and adiabat

With the number of heat flow lanes for the section corresponding to M 3, it follows from Equation 4S.7 that the shape factor for the entire block is S 8 Ml 8 3 2 m 8 m N 6

䉰

where the factor of 8 results from the number of symmetrical sections. The accuracy of this result may be determined by referring to Table 4.1, where, for the prescribed system, case 6, it follows that S

2L 2 2 m 8.59 m ln (1.08 w/D) ln (1.08 1 m/0.25 m)

Hence the result of the flux plot underpredicts the shape factor by approximately 7%. Note that, although the requirement l w is not satisfied for this problem, the shape factor result from Table 4.1 remains valid if there is negligible axial conduction in the block. This condition is satisfied if the ends are insulated. 2. Using S 8.59 m with Equation 4S.6, the heat rate is q Sk (T1 T2) q 8.59 m 150 W/m 䡠 K (75 25)C 64.4 kW

䉰

Comments: The accuracy of the flux plot may be improved by using a finer grid (increasing the value of N). How would the symmetry and heat flow lines change if the vertical sides were insulated? If one vertical and one horizontal side were insulated? If both vertical and one horizontal side were insulated?

4S.2 The Gauss-Seidel Method: Example of Usage The Gauss-Seidel method, described in Appendix D, is utilized in the following example.

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4S.1

䊏

The Graphical Method

EXAMPLE 4S.2 A large industrial furnace is supported on a long column of fireclay brick, which is 1 m 1 m on a side. During steady-state operation, installation is such that three surfaces of the column are maintained at 500 K while the remaining surface is exposed to an airstream for which T앝 300 K and h 10 W/m2 䡠 K. Using a grid of x y 0.25 m, determine the two-dimensional temperature distribution in the column and the heat rate to the airstream per unit length of column.

SOLUTION Known: Dimensions and surface conditions of a support column. Find: Temperature distribution and heat rate per unit length. Schematic: ∆x = 0.25 m

Ts = 500 K

∆y = 0.25 m 1

2

1

3

4

3

5

6

5

7

8

7

Fireclay brick

Ts = 500 K

Ts = 500 K

Air

T∞ = 300 K h = 10 W/m2•K

Assumptions: 1. Steady-state conditions. 2. Two-dimensional conduction. 3. Constant properties. 4. No internal heat generation. Properties: Table A.3, fireclay brick (T ⬇ 478 K): k 1 W/m 䡠 K. Analysis: The prescribed grid consists of 12 nodal points at which the temperature is unknown. However, the number of unknowns is reduced to 8 through symmetry, in which case the temperature of nodal points to the left of the symmetry line must equal the temperature of those to the right.

4S.2

䊏

The Gauss-Seidel Method: Example of Usage

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Nodes 1, 3, and 5 are interior points for which the finite-difference equations may be inferred from Equation 4.29. Hence Node 1:

T2 T3 1000 4T1 0

Node 3:

T1 T4 T5 500 4T3 0

Node 5:

T3 T6 T7 500 4T5 0

Equations for points 2, 4, and 6 may be obtained in a like manner or, since they lie on a symmetry adiabat, by using Equation 4.42 with h 0. Hence Node 2:

2T1 T4 500 4T2 0

Node 4:

T2 2T3 T6 4T4 0

Node 6:

T4 2T5 T8 4T6 0

From Equation 4.42 and the fact that h x/k 2.5, it also follows that Node 7:

2T5 T8 2000 9T7 0

Node 8:

2T6 2T7 1500 9T8 0

Having the required finite-difference equations, the temperature distribution will be determined by using the Gauss–Seidel iteration method. Referring to the arrangement of finite-difference equations, it is evident that the order is already characterized by diagonal dominance. This behavior is typical of finite-difference solutions to conduction problems. We therefore begin with step 2 and express the equations in explicit form T 1(k) 0.25T 2(k1) 0.25T (k1) 250 3 T 2(k) 0.50T 1(k) 0.25T 4(k1) 125 (k) (k1) T (k) 0.25T 5(k1) 125 3 0.25T 1 0.25T 4 (k1) T 4(k) 0.25T 2(k) 0.50T (k) 3 0.25T 6

T 5(k) 0.25T 3(k) 0.25T (k1) 0.25T 7(k1) 125 6 (k1) T 6(k) 0.25T 4(k) 0.50T (k) 5 0.25T 8

T 7(k) 0.2222T 5(k) 0.1111T (k1) 222.22 8 T 8(k) 0.2222T 6(k) 0.2222T (k) 7 166.67 Having the finite-difference equations in the required form, the iteration procedure may be implemented by using a table that has one column for the iteration (step) number and a column for each of the nodes labeled as Ti. The calculations proceed as follows: 1. For each node, the initial temperature estimate is entered on the row for k 0. Values are selected rationally to reduce the number of required iterations. 2. Using the N finite-difference equations and values of Ti from the first and second rows, the new values of Ti are calculated for the first iteration (k 1). These new values are entered on the second row. 3. This procedure is repeated to calculate T i(k) from the previous values of T i(k1) and the current values of T i(k), until the temperature difference between iterations meets the prescribed criterion, 0.2 K, at every nodal point.

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4S.1

䊏

The Graphical Method

k

T1

T2

T3

T4

T5

T6

T7

T8

0 1 2 3 4 5 6 7 8

480 477.5 480.8 484.6 487.0 488.1 488.7 489.0 489.1

470 471.3 475.7 480.6 482.9 484.0 484.5 484.8 485.0

440 451.9 462.5 467.6 469.7 470.8 471.4 471.7 471.9

430 441.3 453.1 457.4 459.6 460.7 461.3 461.6 461.8

400 428.0 432.6 434.3 435.5 436.1 436.5 436.7 436.8

390 411.8 413.9 415.9 417.2 417.9 418.3 418.5 418.6

370 356.2 355.8 356.2 356.6 356.7 356.9 356.9 356.9

350 337.3 337.7 338.3 338.6 338.8 338.9 339.0 339.0

The results given in row 8 are in excellent agreement with those that would be obtained by an exact solution of the matrix equation, although better agreement could be obtained by reducing the value of . However, given the approximate nature of the finite-difference equations, the results still represent approximations to the actual temperatures. The accuracy of the approximation may be improved by using a finer grid (increasing the number of nodes). The heat rate from the column to the airstream may be computed from the expression

冢Lq冣 2h 冤冢x2 冣 (T T ) x (T T ) 冢x2 冣 (T T )冥 앝

s

7

앝

8

앝

where the factor of 2 outside the brackets originates from the symmetry condition. Hence

冢Lq冣 2 10 W/m 䡠 K[0.125 m (200 K) 2

0.25 m (56.9 K) 0.125 m (39.0 K)] 882 W/m

䉰

Comments: 1. To ensure that no errors have been made in formulating the finite-difference equations or in effecting their solution, a check should be made to verify that the results satisfy conservation of energy for the nodal network. For steady-state conditions, the requirement dictates that the rate of energy inflow be balanced by the rate of outflow for a control surface surrounding all nodal regions whose temperatures have been evaluated. Ts q1(1) q1(2)

Ts

q3

q2

1

2

3

4

5

6

7

8

q5

q7(1)

q7(2) T∞, h

q8

䊏

W-9

References

For the one-half symmetrical section shown schematically above, it follows that conduction into the nodal regions must be balanced by convection from the regions. Hence (2) (1) (2) q(1) 1 q1 q2 q3 q5 q7 q7 q8

The cumulative conduction rate is then

冤

qcond (T T1) (T T1) x (Ts T2) k x s y s L y x 2 y y

(Ts T3) (T T5) y (Ts T7) y s x x 2 x

冥

192.1 W/m and the convection rate is

冤

冥

qconv h x(T7 T앝) x (T8 T앝) 191.0 W/m L 2 Agreement between the conduction and convection rates is excellent, confirming that mistakes have not been made in formulating and solving the finite-difference equations. Note that convection transfer from the entire bottom surface (882 W/m) is obtained by adding transfer from the edge node at 500 K (250 W/m) to that from the interior nodes (191.0 W/m) and multiplying by 2 from symmetry. 2. Although the computed temperatures satisfy the finite-difference equations, they do not provide us with the exact temperature field. Remember that the equations are approximations whose accuracy may be improved by reducing the grid size (increasing the number of nodal points). 3. See Example 4S.2 in the Advanced section of IHT. 4. A second software package accompanying this text, Finite-Element Heat Transfer (FEHT), may also be used to solve one- and two-dimensional forms of the heat equation. This example is provided as a solved model in FEHT and may be accessed through Examples on the Toolbar.

References 1. Sunderland, J. E., and K. R. Johnson, Trans. ASHRAE, 10, 237–241, 1964. 2. Kutateladze, S. S., Fundamentals of Heat Transfer, Academic Press, New York, 1963.

3. General Electric Co. (Corporate Research and Development), Heat Transfer Data Book, Section 502, General Electric Company, Schenectady, NY, 1973. 4. Hahne, E., and U. Grigull, Int. J. Heat Mass Transfer, 18, 751–767, 1975.

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4S.1

The Graphical Method

䊏

Problems (b) Using the flux plot method, estimate the shape factor and the heat transfer rate through the strut per unit length.

Flux Plotting 4S.1 A long furnace, constructed from refractory brick with a thermal conductivity of 1.2 W/m • K, has the cross section shown with inner and outer surface temperatures of 600 and 60°C, respectively. Determine the shape factor and the heat transfer rate per unit length using the flux plot method.

1m2m

(c) Sketch the 25, 50, and 75°C isotherms. (d) Consider the same geometry, but now with the 0.1-m-wide surfaces insulated, the 45° surface maintained at T1 100°C, and the 0.2-m-wide surfaces maintained at T2 0°C. Using the flux plot method, estimate the corresponding shape factor and the heat rate per unit length. Sketch the 25, 50, and 75°C isotherms. 4S.4 A hot liquid flows along a V-groove in a solid whose top and side surfaces are well insulated and whose bottom surface is in contact with a coolant.

1.5 m

W/4

2.5 m

T1 W/4

4S.2 A hot pipe is embedded eccentrically as shown in a material of thermal conductivity 0.5 W/m 䡠 K. Using the flux plot method, determine the shape factor and the heat transfer per unit length when the pipe and outer surface temperatures are 150 and 35°C, respectively.

20 mm 40 mm

4S.3 A supporting strut fabricated from a material with a thermal conductivity of 75 W/m 䡠 K has the cross section shown. The end faces are at different temperatures T1 100°C and T2 0°C, while the remaining sides are insulated.

W/2 T2 W

Accordingly, the V-groove surface is at a temperature T1, which exceeds that of the bottom surface, T2. Construct an appropriate flux plot and determine the shape factor of the system. 4S.5 A very long conduit of inner circular cross section and a thermal conductivity of 1 W/m 䡠 K passes a hot fluid, which maintains the inner surface at T1 50°C. The outer surfaces of square cross section are insulated or maintained at a uniform temperature of T2 20°C, depending on the application. Find the shape factor and the heat rate for each case. T2

T2 T2

T2

0.2 m

T1

40 mm

T1

0.1 m 0.2 m

P T2 45°

T1 0.1 m

(a) Estimate the temperature at the location P.

120 mm

4S.6 A long support column of trapezoidal cross section is well insulated on its sides, and temperatures of 100 and 0°C are maintained at its top and bottom surfaces, respectively. The column is fabricated from AISI 1010 steel,

䊏

W-11

Problems

and its widths at the top and bottom surfaces are 0.3 and 0.6 m, respectively. 0.3 m

0.3 m

4S.8 The two-dimensional, square shapes, 1 m to a side, are maintained at uniform temperatures, T1 100°C and T2 0°C, on portions of their boundaries and are well insulated elsewhere.

H

0.3 m

T1

T1

0.6 m

(a) Using the flux plot method, determine the heat transfer rate per unit length of the column. (b) If the trapezoidal column is replaced by a bar of rectangular cross section 0.3 m wide and the same material, what height H must the bar be to provide an equivalent thermal resistance? 4S.7 Hollow prismatic bars fabricated from plain carbon steel are 1 m long with top and bottom surfaces, as well as both ends, well insulated. For each bar, find the shape factor and the heat rate per unit length of the bar when T1 500 K and T2 300 K. 100 mm

100 mm

T2

T2

T2

35 mm 100 mm

T1

100 mm 35 mm

T2

35 mm

T1

35 mm

T2

T1

T2

T2 (a)

(b)

Use the flux plot method to estimate the heat rate per unit length normal to the page if the thermal conductivity is 50 W/m 䡠 K.

C H A P T E R

Transient Conduction

5

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280

Chapter 5

䊏

Transient Conduction

I

n our treatment of conduction we have gradually considered more complicated conditions. We began with the simple case of one-dimensional, steady-state conduction with no internal generation, and we subsequently considered more realistic situations involving multidimensional and generation effects. However, we have not yet considered situations for which conditions change with time. We now recognize that many heat transfer problems are time dependent. Such unsteady, or transient, problems typically arise when the boundary conditions of a system are changed. For example, if the surface temperature of a system is altered, the temperature at each point in the system will also begin to change. The changes will continue to occur until a steadystate temperature distribution is reached. Consider a hot metal billet that is removed from a furnace and exposed to a cool airstream. Energy is transferred by convection and radiation from its surface to the surroundings. Energy transfer by conduction also occurs from the interior of the metal to the surface, and the temperature at each point in the billet decreases until a steady-state condition is reached. The final properties of the metal will depend significantly on the time-temperature history that results from heat transfer. Controlling the heat transfer is one key to fabricating new materials with enhanced properties. Our objective in this chapter is to develop procedures for determining the time dependence of the temperature distribution within a solid during a transient process, as well as for determining heat transfer between the solid and its surroundings. The nature of the procedure depends on assumptions that may be made for the process. If, for example, temperature gradients within the solid may be neglected, a comparatively simple approach, termed the lumped capacitance method, may be used to determine the variation of temperature with time. The method is developed in Sections 5.1 through 5.3. Under conditions for which temperature gradients are not negligible, but heat transfer within the solid is one-dimensional, exact solutions to the heat equation may be used to compute the dependence of temperature on both location and time. Such solutions are considered for finite solids (plane walls, long cylinders and spheres) in Sections 5.4 through 5.6 and for semi-infinite solids in Section 5.7. Section 5.8 presents the transient thermal response of a variety of objects subject to a step change in either surface temperature or surface heat flux. In Section 5.9, the response of a semi-infinite solid to periodic heating conditions at its surface is explored. For more complex conditions, finite-difference or finite-element methods must be used to predict the time dependence of temperatures within the solid, as well as heat rates at its boundaries (Section 5.10).

5.1 The Lumped Capacitance Method A simple, yet common, transient conduction problem is one for which a solid experiences a sudden change in its thermal environment. Consider a hot metal forging that is initially at a uniform temperature Ti and is quenched by immersing it in a liquid of lower temperature T앝 Ti (Figure 5.1). If the quenching is said to begin at time t 0, the temperature of the solid will decrease for time t 0, until it eventually reaches T앝. This reduction is due to convection heat transfer at the solid–liquid interface. The essence of the lumped capacitance method is the assumption that the temperature of the solid is spatially uniform at any instant during the transient process. This assumption implies that temperature gradients within the solid are negligible.

5.1

䊏

281

The Lumped Capacitance Method

Ti

t 1

T∞, h x

L

FIGURE 5.3 Effect of Biot number on steady-state temperature distribution in a plane wall with surface convection.

284

Chapter 5

䊏

Transient Conduction

The quantity (hL/k) appearing in Equation 5.9 is a dimensionless parameter. It is termed the Biot number, and it plays a fundamental role in conduction problems that involve surface convection effects. According to Equation 5.9 and as illustrated in Figure 5.3, the Biot number provides a measure of the temperature drop in the solid relative to the temperature difference between the solid’s surface and the fluid. From Equation 5.9, it is also evident that the Biot number may be interpreted as a ratio of thermal resistances. In particular, if Bi 1, the resistance to conduction within the solid is much less than the resistance to convection across the fluid boundary layer. Hence, the assumption of a uniform temperature distribution within the solid is reasonable if the Biot number is small. Although we have discussed the Biot number in the context of steady-state conditions, we are reconsidering this parameter because of its significance to transient conduction problems. Consider the plane wall of Figure 5.4, which is initially at a uniform temperature Ti and experiences convection cooling when it is immersed in a fluid of T앝 Ti. The problem may be treated as one-dimensional in x, and we are interested in the temperature variation with position and time, T(x, t). This variation is a strong function of the Biot number, and three conditions are shown in Figure 5.4. Again, for Bi 1 the temperature gradients in the solid are small and the assumption of a uniform temperature distribution, T(x, t) ⬇ T(t) is reasonable. Virtually all the temperature difference is between the solid and the fluid, and the solid temperature remains nearly uniform as it decreases to T앝. For moderate to large values of the Biot number, however, the temperature gradients within the solid are significant. Hence T T(x, t). Note that for Bi 1, the temperature difference across the solid is much larger than that between the surface and the fluid. We conclude this section by emphasizing the importance of the lumped capacitance method. Its inherent simplicity renders it the preferred method for solving transient heating and cooling problems. Hence, when confronted with such a problem, the very first thing that one should do is calculate the Biot number. If the following condition is satisfied Bi

hLc 0.1 k

(5.10)

the error associated with using the lumped capacitance method is small. For convenience, it is customary to define the characteristic length of Equation 5.10 as the ratio of the solid’s

T(x, 0) = Ti

T(x, 0) = Ti

T∞, h t

T∞, h

T∞ –L

L x

T∞ –L

Bi > 1 T = T(x, t)

FIGURE 5.4 Transient temperature distributions for different Biot numbers in a plane wall symmetrically cooled by convection.

5.2

䊏

Validity of the Lumped Capacitance Method

285

volume to surface area Lc ⬅ V/As. Such a definition facilitates calculation of Lc for solids of complicated shape and reduces to the half-thickness L for a plane wall of thickness 2L (Figure 5.4), to ro /2 for a long cylinder, and to ro /3 for a sphere. However, if one wishes to implement the criterion in a conservative fashion, Lc should be associated with the length scale corresponding to the maximum spatial temperature difference. Accordingly, for a symmetrically heated (or cooled) plane wall of thickness 2L, Lc would remain equal to the half-thickness L. However, for a long cylinder or sphere, Lc would equal the actual radius ro, rather than ro /2 or ro /3. Finally, we note that, with Lc ⬅ V/As, the exponent of Equation 5.6 may be expressed as hL k t hL hAs t c ␣t2 ht c c 2 Vc cLc k k Lc Lc or hAs t Bi 䡠 Fo Vc

(5.11)

where Fo ⬅ ␣t2 Lc

(5.12)

is termed the Fourier number. It is a dimensionless time, which, with the Biot number, characterizes transient conduction problems. Substituting Equation 5.11 into 5.6, we obtain T T앝 exp(Bi 䡠 Fo) i Ti T앝

(5.13)

EXAMPLE 5.1 A thermocouple junction, which may be approximated as a sphere, is to be used for temperature measurement in a gas stream. The convection coefficient between the junction surface and the gas is h 400 W/m2 䡠 K, and the junction thermophysical properties are k 20 W/m 䡠 K, c 400 J/kg 䡠 K, and 8500 kg/m3. Determine the junction diameter needed for the thermocouple to have a time constant of 1 s. If the junction is at 25 C and is placed in a gas stream that is at 200 C, how long will it take for the junction to reach 199 C?

SOLUTION Known: Thermophysical properties of thermocouple junction used to measure temperature of a gas stream. Find: 1. Junction diameter needed for a time constant of 1 s. 2. Time required to reach 199 C in gas stream at 200 C.

286

Chapter 5

䊏

Transient Conduction

Schematic: Leads

T∞ = 200°C h = 400 W/m2•K

Gas stream

Thermocouple junction Ti = 25°C

k = 20 W/m•K c = 400 J/kg•K ρ = 8500 kg/m3

D

Assumptions: 1. Temperature of junction is uniform at any instant. 2. Radiation exchange with the surroundings is negligible. 3. Losses by conduction through the leads are negligible. 4. Constant properties. Analysis: 1. Because the junction diameter is unknown, it is not possible to begin the solution by determining whether the criterion for using the lumped capacitance method, Equation 5.10, is satisfied. However, a reasonable approach is to use the method to find the diameter and to then determine whether the criterion is satisfied. From Equation 5.7 and the fact that As D2 and V D3/6 for a sphere, it follows that t

1 D c 6 hD2 3

Rearranging and substituting numerical values, 2 6h 䡠 K 1 s 7.06 104 m D c t 6 400 W/m 3 8500 kg/m 400 J/kg 䡠 K

䉰

With Lc ro /3 it then follows from Equation 5.10 that Bi

h(ro /3) 400 W/m2 䡠 K 3.53 104 m 2.35 103 k 3 20 W/m 䡠 K

Accordingly, Equation 5.10 is satisfied (for Lc ro, as well as for Lc ro /3) and the lumped capacitance method may be used to an excellent approximation. 2. From Equation 5.5 the time required for the junction to reach T 199 C is (D3/6)c Ti T앝 Dc Ti T앝 ln ln T T앝 6h T T앝 h(D2) 3 4 8500 kg/m 7.06 10 m 400 J/kg 䡠 K 25 200 t ln 199 200 6 400 W/m2 䡠 K t 5.2 s 艐 5t t

䉰

Comments: Heat transfer due to radiation exchange between the junction and the surroundings and conduction through the leads would affect the time response of the junction and would, in fact, yield an equilibrium temperature that differs from T앝.

5.3

䊏

287

General Lumped Capacitance Analysis

5.3 General Lumped Capacitance Analysis Although transient conduction in a solid is commonly initiated by convection heat transfer to or from an adjoining fluid, other processes may induce transient thermal conditions within the solid. For example, a solid may be separated from large surroundings by a gas or vacuum. If the temperatures of the solid and surroundings differ, radiation exchange could cause the internal thermal energy, and hence the temperature, of the solid to change. Temperature changes could also be induced by applying a heat flux at a portion, or all, of the surface or by initiating thermal energy generation within the solid. Surface heating could, for example, be applied by attaching a film or sheet electrical heater to the surface, while thermal energy could be generated by passing an electrical current through the solid. Figure 5.5 depicts the general situation for which thermal conditions within a solid may be influenced simultaneously by convection, radiation, an applied surface heat flux, and internal energy generation. It is presumed that, initially (t 0), the temperature of the solid Ti differs from that of the fluid T앝, and the surroundings Tsur , and that both surface and volu. metric heating (qs and q) are initiated. The imposed heat flux qs and the convection–radiation heat transfer occur at mutually exclusive portions of the surface, As(h) and As(c,r), respectively, and convection–radiation transfer is presumed to be from the surface. Moreover, although convection and radiation have been prescribed for the same surface, the surfaces may, in fact, differ (As,c As,r). Applying conservation of energy at any instant t, it follows from Equation 1.12c that qs As,h E˙ g (qconv qrad )As(c,r) Vc dT dt

(5.14)

or, from Equations 1.3a and 1.7, qs As,h E˙ g [h(T T앝) (T 4 T 4sur)]As(c,r) Vc dT dt

(5.15)

Equation 5.15 is a nonlinear, first-order, nonhomogeneous, ordinary differential equation that cannot be integrated to obtain an exact solution.1 However, exact solutions may be obtained for simplified versions of the equation.

Surroundings

Tsur ρ, c, V, T (0) = Ti

q"rad

q"s

•

•

Eg, Est

T∞, h q"conv

As, h

1

As(c, r)

FIGURE 5.5 Control surface for general lumped capacitance analysis.

An approximate, finite-difference solution may be obtained by discretizing the time derivative (Section 5.10) and marching the solution out in time.

288

Chapter 5

5.3.1

䊏

Transient Conduction

Radiation Only

If there is no imposed heat flux or generation and convection is either nonexistent (a vacuum) or negligible relative to radiation, Equation 5.15 reduces to 4 Vc dT As,r (T 4 T sur ) dt

(5.16)

Separating variables and integrating from the initial condition to any time t, it follows that As,r Vc

冕 dt 冕 T t

T

0

Ti

dT T4

(5.17)

4 sur

Evaluating both integrals and rearranging, the time required to reach the temperature T becomes t

冏

冦

冏 冏

T T T Ti Vc ln sur ln sur 3 T T Tsur Ti 4As,r Tsur sur

冤 冢TT 冣 tan 冢TT 冣冥冧

2 tan1

1

sur

i

冏 (5.18)

sur

This expression cannot be used to evaluate T explicitly in terms of t, Ti, and Tsur, nor does it readily reduce to the limiting result for Tsur 0 (radiation to deep space). However, returning to Equation 5.17, its solution for Tsur 0 yields t

5.3.2

冢

Vc 1 1 3As,r T 3 T 3i

冣

(5.19)

Negligible Radiation

An exact solution to Equation 5.15 may also be obtained if radiation may be neglected and all quantities (other than T, of course) are independent of time. Introducing a temperature difference ⬅ T T앝, where d/dt dT/dt, Equation 5.15 reduces to a linear, first-order, nonhomogeneous differential equation of the form d

a b 0 dt

(5.20)

where a ⬅ (hAs,c /Vc) and b ⬅ [(qs As,h E˙ g)/Vc]. Although Equation 5.20 may be solved by summing its homogeneous and particular solutions, an alternative approach is to eliminate the nonhomogeneity by introducing the transformation ⬅ ba

(5.21)

5.3

䊏

289

General Lumped Capacitance Analysis

Recognizing that d/dt d/dt, Equation 5.21 may be substituted into (5.20) to yield d

a 0 dt

(5.22)

Separating variables and integrating from 0 to t (i to ), it follows that exp(at) i

(5.23)

T T앝 (b/a) exp(at) Ti T앝 (b/a)

(5.24)

T T exp(at) b/a [1 exp(at)] Ti T Ti T

(5.25)

or substituting for and ,

Hence

As it must, Equation 5.25 reduces to Equation 5.6 when b 0 and yields T Ti at t 0. As t l 앝, Equation 5.25 reduces to (T T앝) (b/a), which could also be obtained by performing an energy balance on the control surface of Figure 5.5 for steady-state conditions.

5.3.3

Convection Only with Variable Convection Coefficient

In some cases, such as those involving free convection or boiling, the convection coefficient h varies with the temperature difference between the object and the fluid. In these situations, the convection coefficient can often be approximated with an expression of the form h C(T T앝)n

(5.26)

where n is a constant and the parameter C has units of W/m2 䡠 K(1 n). If radiation, surface heating, and volumetric generation are negligible, Equation 5.15 may be written as C(T T앝)nAs,c(T T앝) CAs,c(T T앝)1 n Vc dT dt

(5.27)

Substituting and d/dt dT/dt into the preceding expression, separating variables and integrating yields

冤

冥

nCAs,cni t 1 i Vc

1/n

(5.28)

It can be shown that Equation 5.28 reduces to Equation 5.6 if the heat transfer coefficient is independent of temperature, n 0.

5.3.4

Additional Considerations

In some cases the ambient or surroundings temperature may vary with time. For example, if the container of Figure 5.1 is insulated and of finite volume, the liquid temperature will

290

Chapter 5

䊏

Transient Conduction

increase as the metal forging is cooled. An analytical solution for the time-varying solid (and liquid) temperature is presented in Example 11.8. As evident in Examples 5.2 through 5.4, the heat equation can be solved numerically for a wide variety of situations involving variable properties or time-varying boundary conditions, internal energy generation rates, or surface heating or cooling.

EXAMPLE 5.2 Consider the thermocouple and convection conditions of Example 5.1, but now allow for radiation exchange with the walls of a duct that encloses the gas stream. If the duct walls are at 400 C and the emissivity of the thermocouple bead is 0.9, calculate the steady-state temperature of the junction. Also, determine the time for the junction temperature to increase from an initial condition of 25 C to a temperature that is within 1 C of its steady-state value.

SOLUTION Known: Thermophysical properties and diameter of the thermocouple junction used to measure temperature of a gas stream passing through a duct with hot walls. Find: 1. Steady-state temperature of the junction. 2. Time required for the thermocouple to reach a temperature that is within 1 C of its steady-state value. Schematic: Hot duct wall, Tsur = 400°C Gas stream

Junction, T(t) Ti = 25°C, D = 0.7 mm ρ = 8500 kg/m3 c = 400 J/kg•K ε = 0.9

T∞ = 200°C

h = 400 W/m2•K

Assumptions: Same as Example 5.1, but radiation transfer is no longer treated as negligible and is approximated as exchange between a small surface and large surroundings. Analysis: 1. For steady-state conditions, the energy balance on the thermocouple junction has the form E˙ in E˙ out 0 Recognizing that net radiation to the junction must be balanced by convection from the junction to the gas, the energy balance may be expressed as 4 T 4) h(T T앝)]As 0 [(T sur

5.3

䊏

291

General Lumped Capacitance Analysis

Substituting numerical values, we obtain T 218.7 C

䉰

2. The temperature-time history, T(t), for the junction, initially at T(0) Ti 25 C, follows from the energy balance for transient conditions, E˙ in E˙ out E˙ st From Equation 5.15, the energy balance may be expressed as 4 )]As Vc dT [h(T T앝) (T 4 T sur dt

The solution to this first-order differential equation can be obtained by numerical integration, giving the result, T(4.9 s) 217.7 C. Hence, the time required to reach a temperature that is within 1 C of the steady-state value is t 4.9 s.

䉰

Comments: 1. The effect of radiation exchange with the hot duct walls is to increase the junction temperature, such that the thermocouple indicates an erroneous gas stream temperature that exceeds the actual temperature by 18.7 C. The time required to reach a temperature that is within 1 C of the steady-state value is slightly less than the result of Example 5.l, which only considers convection heat transfer. Why is this so? 2. The response of the thermocouple and the indicated gas stream temperature depend on the velocity of the gas stream, which in turn affects the magnitude of the convection coefficient. Temperature–time histories for the thermocouple junction are shown in the following graph for values of h 200, 400, and 800 W/m2 䡠 K. Junction temperature, T (°C)

260 220 180

800 400 200 h (W/m2•K)

140 100 60 20

0

2

6 4 Elapsed time, t (s)

8

10

The effect of increasing the convection coefficient is to cause the junction to indicate a temperature closer to that of the gas stream. Further, the effect is to reduce the time required for the junction to reach the near-steady-state condition. What physical explanation can you give for these results? 3. The IHT software includes an integral function, Der(T, t), that can be used to represent the temperature–time derivative and to integrate first-order differential equations.

292

Chapter 5

䊏

Transient Conduction

EXAMPLE 5.3 A 3-mm-thick panel of aluminum alloy (k 177 W/m 䡠 K, c 875 J/kg 䡠 K, and 2770 kg/m3) is finished on both sides with an epoxy coating that must be cured at or above Tc 150 C for at least 5 min. The production line for the curing operation involves two steps: (1) heating in a large oven with air at T앝,o 175 C and a convection coefficient of ho 40 W/m2 䡠 K, and (2) cooling in a large chamber with air at T앝,c 25 C and a convection coefficient of hc 10 W/m2 䡠 K. The heating portion of the process is conducted over a time interval te, which exceeds the time tc required to reach 150 C by 5 min (te tc 300 s). The coating has an emissivity of 0.8, and the temperatures of the oven and chamber walls are 175 and 25 C, respectively. If the panel is placed in the oven at an initial temperature of 25 C and removed from the chamber at a safe-to-touch temperature of 37 C, what is the total elapsed time for the two-step curing operation?

SOLUTION Known: Operating conditions for a two-step heating/cooling process in which a coated aluminum panel is maintained at or above a temperature of 150 C for at least 5 min. Find: Total time tt required for the two-step process. Schematic: Tsur,o = 175°C

Tsur,c = 25°C

2L = 3 mm

As

ho, T∞,o = 175°C

Epoxy, ε = 0.8

Aluminum, T(0) = Ti,o = 25°C Step 1: Heating (0 ≤ t ≤ tc)

hc, T∞,c = 25°C

T(tt) = 37°C Step 2: Cooling (tc< t ≤ tt)

Assumptions: 1. Panel temperature is uniform at any instant. 2. Thermal resistance of epoxy is negligible. 3. Constant properties. Analysis: To assess the validity of the lumped capacitance approximation, we begin by calculating Biot numbers for the heating and cooling processes. Bih

ho L (40 W/m2 䡠 K)(0.0015 m) 3.4 104 k 177 W/m 䡠 K

Bic

hc L (10 W/m2 䡠 K)(0.0015 m) 8.5 105 k 177 W/m 䡠 K

5.3

䊏

293

General Lumped Capacitance Analysis

Hence the lumped capacitance approximation is excellent. To determine whether radiation exchange between the panel and its surroundings should be considered, the radiation heat transfer coefficient is determined from Equation 1.9. A representative value of hr for the heating process is associated with the cure condition, in which case hr,o (Tc Tsur,o)(T 2c T 2sur,o) 0.8 5.67 108 W/m2 䡠 K4(423 448)K(4232 4482)K2 15 W/m2 䡠 K Using Tc 150 C with Tsur,c 25 C for the cooling process, we also obtain hr,c 8.8 W/m2 䡠 K. Since the values of hr,o and hr,c are comparable to those of ho and hc, respectively, radiation effects must be considered. With V 2LAs and As,c As,r 2As, Equation 5.15 may be expressed as 4 [h(T T앝 ) (T 4 T sur )] cL dT dt

Selecting a suitable time increment, t, the equation may be integrated numerically to obtain the panel temperature at t t, 2t, 3t, and so on. Selecting t 10 s, calculations for the heating process are extended to te tc 300 s, which is 5 min beyond the time required for the panel to reach Tc 150 C. At te the cooling process is initiated and continued until the panel temperature reaches 37 C at t tt. The integration was performed using IHT, and results of the calculations are plotted as follows: 200 175 150

∆t(T >150°C)

T (°C)

125 Cooling Heating

100 75 50 25

0

tc

300

te

600 t (s)

900 tt

1200

The total time for the two-step process is tt 989 s

䉰

with intermediate times of tc 124 s and te 424 s.

Comments: 1. The duration of the two-step process may be reduced by increasing the convection coefficients and/or by reducing the period of extended heating. The second option is made possible by the fact that, during a portion of the cooling period, the panel

294

Chapter 5

䊏

Transient Conduction

temperature remains above 150 C. Hence, to satisfy the cure requirement, it is not necessary to extend heating for as much as 5 min from t tc. If the convection coefficients are increased to ho hc 100 W/m2 䡠 K and an extended heating period of 300 s is maintained, the numerical integration yields tc 58 s and tt 445 s. The corresponding time interval over which the panel temperature exceeds 150 C is t(T 150 C) 306 s (58 s t 364 s). If the extended heating period is reduced to 294 s, the numerical integration yields tc 58 s, tt 439 s, and t(T 150 C) 300 s. Hence the total process time is reduced, while the curing requirement is still satisfied. 2. Generally, the accuracy of a numerical integration improves with decreasing t, but at the expense of increased computation time. In this case, however, results obtained for t 1 s are virtually identical to those obtained for t 10 s, indicating that the larger time interval is sufficient to accurately depict the temperature history. 3. The complete solution for this example is provided as a ready-to-solve model in the Advanced section of IHT, using Models, Lumped Capacitance. The model can be used to check the results of Comment 1 or to independently explore modifications of the cure process. 4. If the Biot numbers were not small, it would be inappropriate to apply the lumped capacitance method. For moderate or large Biot numbers, temperatures near the solid’s centerline would continue to increase for some time after the conclusion of heating, as thermal energy near the solid’s surface propagates inward. The temperatures near the centerline would subsequently reach a maximum and would then decrease to the steady-state value. Correlations for the maximum temperature experienced at the panel’s centerline, along with the time at which these maximum temperatures are reached, have been correlated for a broad range of Bih and Bic values [1].

EXAMPLE 5.4 Air to be supplied to a hospital operating room is first purified by forcing it through a singlestage compressor. As it travels through the compressor, the air temperature initially increases due to compression, then decreases as it is returned to atmospheric pressure. Harmful pathogen particles in the air will also be heated and subsequently cooled, and they will be destroyed if their maximum temperature exceeds a lethal temperature Td. Consider spherical pathogen particles (D 10 m, 900 kg/m3, c 1100 J/kg 䡠 K, and k 0.2 W/m 䡠 K) that are dispersed in unpurified air. During the process, the air temperature may be described by an expression of the form T앝(t) 125 C 100 C 䡠 cos(2t/tp), where tp is the process time associated with flow through the compressor. If tp 0.004 s, and the initial and lethal pathogen temperatures are Ti 25 C and Td 220 C, respectively, will the pathogens be destroyed? The value of the convection heat transfer coefficient associated with the pathogen particles is h 4600 W/m2 䡠 K.

SOLUTION Known: Air temperature versus time, convection heat transfer coefficient, pathogen geometry, size, and properties.

5.3

䊏

295

General Lumped Capacitance Analysis

Find: Whether the pathogens are destroyed for tp 0.004 s. Schematic: Airstream

T∞(t) 125°C 100°C •cos(2πt/t π p) h 4600 W/m2 •K

Pathogen k 0.2 W/m •K c 1100 J/kg •K ρ 900 kg/m3

D 10 µm

Td 220°C

Assumptions: 1. Constant properties. 2. Negligible radiation. Analysis: The Biot number associated with a spherical pathogen particle is Bi

h(D/6) 4600 W/m2 䡠 K (10 106 m/6) 0.038 k 0.2 W/m 䡠 K

Hence, the lumped capacitance approximation is valid and we may apply Equation 5.2. dT hAs [T T (t)] 6h [T 125 C 100 C 䡠 cos(2t/t )] 앝 p dt Vc cD

(1)

The solution to this first-order differential equation may be obtained analytically, or by numerical integration.

Pathogen and air temperature, T (°C)

Numerical Integration A numerical solution of Equation 1 may be obtained by specifying the initial particle temperature, Ti, and using IHT or an equivalent numerical solver to integrate the equation. The plot of the numerical solution follows. 250 200 150 Pathogen 100 Air 50 0

0

0.001

0.002 Elapsed time, t (s)

0.003

0.004

296

Chapter 5

䊏

Transient Conduction

Inspection of the predicted pathogen temperatures yields Tmax 212 C 220 C 䉰

Hence, the pathogen is not destroyed.

Analytical Solution Equation 1 is a linear nonhomogeneous differential equation, therefore the solution can be found as the sum of a homogeneous and a particular solution, T Th Tp. The homogeneous part, Th, corresponds to the homogeneous differential equation, dTh /dt (6h/cD)Th, which has the familiar solution, Th c0 exp(6ht/cD). The particular solution, Tp, can then be found using the method of undetermined coefficients; for a nonhomogeneous term that includes a cosine function and a constant term, the particular solution is assumed to be of the form Tp c1 cos(2t/tp) c2 sin(2t/tp) c3. Substituting this expression into Equation 1 yields values for the coefficients, resulting in

冤 冢 冣

冢 冣冥

2cD 2t Tp 125 C 100 C A cos 2t tp 6htp sin tp

(2)

where A

(6h/cD)2 (6h/cD)2 (2/tp)2

The initial condition, T(0) Ti, is then applied to the complete solution, T Th Tp, to yield c0 100 C(A 1). Thus, the particle temperature is

冢

冦

冣 冤 冢 冣

冢 冣冥冧 (3)

2cD

sin 2t T(t) 125 C 100 C (A 1) exp 6ht A cos 2t t tp cD 6htp p

To find the maximum pathogen temperature, we could differentiate Equation 3 and set the result equal to zero. This yields a lengthy, implicit equation for the critical time tcrit at which the maximum temperature is reached. The maximum temperature may then be found by substituting t tcrit into Equation 3. Alternatively, Equation 3 can be plotted or T(t) may be tabulated to find Tmax 212 C 220 C Hence, the pathogen is not destroyed.

䉰

Comments: 1. The analytical and numerical solutions agree, as they must. 2. As evident in the previous plot, the air and pathogen particles initially have the same temperature, Ti 25 C. The pathogen thermal response lags that of the air since a temperature difference must exist between the air and the particle in order for the pathogen to be heated or cooled. As required by Equation 1 and as evident in the plot, the maximum particle temperature is reached when there is no temperature difference between the air and the pathogen.

5.3

䊏

297

General Lumped Capacitance Analysis

Pathogen and air temperature, T (°C)

3. The maximum pathogen temperature may be increased by extending the duration of the process. For a process time of tp 0.008 s, the air and pathogen particle temperatures are as follows. 250 200 150 Pathogen 100 Air 50 0

0

0.002

0.004

0.006

0.008

Elapsed time, t (s)

The maximum particle temperature is now Tmax 221 C Td 220 C, and the pathogen would be killed. However, because the duration of the cycle is twice as long as originally specified, approximately half of the air could be supplied to the operating room compared to the tp 0.004 s case. A trade-off exists between the amount of air that can be delivered to the operating room and its purity. 4. The maximum possible radiation heat transfer coefficient may be calculated based on the extreme temperatures of the problem and by assuming a particle emissivity of unity. Hence, hr,max (Tmax Tmin )(T 2max T 2min ) 5.67 108 W/m2 䡠 K4 (498 298)K (4982 2982)K2 15.2 W/m2 䡠 K Since hr,max h, radiation heat transfer is negligible. 5. The Der(T, t) function of the IHT software was used to generate the numerical solution for this problem. See Comment 3 of Example 5.2. If one is familiar with a numerical solver such as IHT, it is often much faster to obtain a numerical solution than an analytical solution, as is the case in this example. Moreover, if one seeks maximum or minimum values of the dependent variable or variables, such as the pathogen temperature in this example, it is often faster to determine the maxima or minima by inspection, rather than with an analytical solution. However, analytical solutions often explicitly show parameter dependencies and can provide insights that numerical solutions might obscure. 6. A time increment of t 0.00001 s was used to generate the numerical solutions. Generally, the accuracy of a numerical integration improves with decreasing t, but at the expense of increased computation time. For this example, results for t 0.000005 s are virtually identical to those obtained for the larger time increment, indicating that either increment is sufficient to accurately depict the temperature history and to determine the maximum particle temperature. 7. Assumption of instantaneous pathogen death at the lethal temperature is an approximation. Pathogen destruction also depends on the duration of exposure to the high temperatures [2].

298

Chapter 5

䊏

Transient Conduction

5.4 Spatial Effects Situations frequently arise for which the Biot number is not small, and we must cope with the fact that temperature gradients within the medium are no longer negligible. Use of the lumped capacitance method would yield incorrect results, so alternative approaches, presented in the remainder of this chapter, must be utilized. In their most general form, transient conduction problems are described by the heat equation, Equation 2.19, for rectangular coordinates or Equations 2.26 and 2.29, respectively, for cylindrical and spherical coordinates. The solutions to these partial differential equations provide the variation of temperature with both time and the spatial coordinates. However, in many problems, such as the plane wall of Figure 5.4, only one spatial coordinate is needed to describe the internal temperature distribution. With no internal generation and the assumption of constant thermal conductivity, Equation 2.19 then reduces to ⭸2T 1 ⭸T ⭸x2 ␣ ⭸t

(5.29)

To solve Equation 5.29 for the temperature distribution T(x, t), it is necessary to specify an initial condition and two boundary conditions. For the typical transient conduction problem of Figure 5.4, the initial condition is T(x, 0) Ti and the boundary conditions are ⭸T ⭸x and k

⭸T ⭸x

冏

xL

冏

0

(5.30)

(5.31)

x0

h[T(L, t) T앝]

(5.32)

Equation 5.30 presumes a uniform temperature distribution at time t 0; Equation 5.31 reflects the symmetry requirement for the midplane of the wall; and Equation 5.32 describes the surface condition experienced for time t 0. From Equations 5.29 through 5.32, it is evident that, in addition to depending on x and t, temperatures in the wall also depend on a number of physical parameters. In particular T T(x, t, Ti, T앝, L, k, ␣, h)

(5.33)

The foregoing problem may be solved analytically or numerically. These methods will be considered in subsequent sections, but first it is important to note the advantages that may be obtained by nondimensionalizing the governing equations. This may be done by arranging the relevant variables into suitable groups. Consider the dependent variable T. If the temperature difference ⬅ T T앝 is divided by the maximum possible temperature difference i ⬅ Ti T앝, a dimensionless form of the dependent variable may be defined as T T앝 * ⬅ i Ti T앝

(5.34)

5.5

䊏

299

The Plane Wall with Convection

Accordingly, * must lie in the range 0 * 1. A dimensionless spatial coordinate may be defined as x* ⬅ x L

(5.35)

where L is the half-thickness of the plane wall, and a dimensionless time may be defined as t* ⬅ ␣t2 ⬅ Fo L

(5.36)

where t* is equivalent to the dimensionless Fourier number, Equation 5.12. Substituting the definitions of Equations 5.34 through 5.36 into Equations 5.29 through 5.32, the heat equation becomes ⭸2* ⭸* (5.37) ⭸x*2 ⭸Fo and the initial and boundary conditions become

and ⭸* ⭸x*

冏

*(x*, 0) 1

(5.38)

⭸* ⭸x*

0

(5.39)

Bi *(1, t*)

(5.40)

x*1

冏

x*0

where the Biot number is Bi ⬅ hL/k. In dimensionless form the functional dependence may now be expressed as * f(x*, Fo, Bi)

(5.41)

Recall that a similar functional dependence, without the x* variation, was obtained for the lumped capacitance method, as shown in Equation 5.13. Comparing Equations 5.33 and 5.41, the considerable advantage associated with casting the problem in dimensionless form becomes apparent. Equation 5.41 implies that for a prescribed geometry, the transient temperature distribution is a universal function of x*, Fo, and Bi. That is, the dimensionless solution has a prescribed form that does not depend on the particular value of Ti, T앝, L, k, ␣, or h. Since this generalization greatly simplifies the presentation and utilization of transient solutions, the dimensionless variables are used extensively in subsequent sections.

5.5 The Plane Wall with Convection Exact, analytical solutions to transient conduction problems have been obtained for many simplified geometries and boundary conditions and are well documented [3–6]. Several mathematical techniques, including the method of separation of variables (Section 4.2), may be used for this purpose, and typically the solution for the dimensionless temperature distribution, Equation 5.41, is in the form of an infinite series. However, except for very small values of the Fourier number, this series may be approximated by a single term, considerably simplifying its evaluation.

300

Chapter 5

5.5.1

䊏

Transient Conduction

Exact Solution

Consider the plane wall of thickness 2L (Figure 5.6a). If the thickness is small relative to the width and height of the wall, it is reasonable to assume that conduction occurs exclusively in the x-direction. If the wall is initially at a uniform temperature, T(x, 0) Ti, and is suddenly immersed in a fluid of T앝 Ti, the resulting temperatures may be obtained by solving Equation 5.37 subject to the conditions of Equations 5.38 through 5.40. Since the convection conditions for the surfaces at x* 1 are the same, the temperature distribution at any instant must be symmetrical about the midplane (x* 0). An exact solution to this problem is of the form [4]

*

兺C

n

exp (2n Fo) cos (n x*)

(5.42a)

n1

where Fo ␣t/L2, the coefficient Cn is Cn

4 sin n 2n sin (2n)

(5.42b)

and the discrete values of n (eigenvalues) are positive roots of the transcendental equation n tan n Bi

(5.42c)

The first four roots of this equation are given in Appendix B.3. The exact solution given by Equation 5.42a is valid for any time, 0 Fo 앝.

5.5.2

Approximate Solution

It can be shown (Problem 5.43) that for values of Fo 0.2, the infinite series solution, Equation 5.42a, can be approximated by the first term of the series, n 1. Invoking this approximation, the dimensionless form of the temperature distribution becomes * C1 exp (21 Fo) cos (1x*)

(5.43a)

* * o cos (1x*)

(5.43b)

or where *o ⬅ (To T앝)/(Ti T앝) represents the midplane (x* 0) temperature 2 * o C1 exp (1 Fo)

T(x, 0) = Ti

T∞, h

r r* = __ ro

(5.44)

T(r, 0) = Ti

T∞, h

T∞, h ro

L

L x* = _x L (a)

(b)

FIGURE 5.6 One-dimensional systems with an initial uniform temperature subjected to sudden convection conditions: (a) Plane wall. (b) Infinite cylinder or sphere.

Graphical representations of the one-term approximations are presented in Section 5S.1.

5.5

䊏

301

The Plane Wall with Convection

An important implication of Equation 5.43b is that the time dependence of the temperature at any location within the wall is the same as that of the midplane temperature. The coefficients C1 and 1 are evaluated from Equations 5.42b and 5.42c, respectively, and are given in Table 5.1 for a range of Biot numbers.

TABLE 5.1 Coefficients used in the one-term approximation to the series solutions for transient one-dimensional conduction Plane Wall

Infinite Cylinder

Sphere

Bia

1 (rad)

C1

1 (rad)

C1

1 (rad)

C1

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.0998 0.1410 0.1723 0.1987 0.2218 0.2425 0.2615 0.2791 0.2956 0.3111

1.0017 1.0033 1.0049 1.0066 1.0082 1.0098 1.0114 1.0130 1.0145 1.0161

0.1412 0.1995 0.2440 0.2814 0.3143 0.3438 0.3709 0.3960 0.4195 0.4417

1.0025 1.0050 1.0075 1.0099 1.0124 1.0148 1.0173 1.0197 1.0222 1.0246

0.1730 0.2445 0.2991 0.3450 0.3854 0.4217 0.4551 0.4860 0.5150 0.5423

1.0030 1.0060 1.0090 1.0120 1.0149 1.0179 1.0209 1.0239 1.0268 1.0298

0.15 0.20 0.25 0.30 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.3779 0.4328 0.4801 0.5218 0.5932 0.6533 0.7051 0.7506 0.7910 0.8274 0.8603

1.0237 1.0311 1.0382 1.0450 1.0580 1.0701 1.0814 1.0919 1.1016 1.1107 1.1191

0.5376 0.6170 0.6856 0.7465 0.8516 0.9408 1.0184 1.0873 1.1490 1.2048 1.2558

1.0365 1.0483 1.0598 1.0712 1.0932 1.1143 1.1345 1.1539 1.1724 1.1902 1.2071

0.6609 0.7593 0.8447 0.9208 1.0528 1.1656 1.2644 1.3525 1.4320 1.5044 1.5708

1.0445 1.0592 1.0737 1.0880 1.1164 1.1441 1.1713 1.1978 1.2236 1.2488 1.2732

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

1.0769 1.1925 1.2646 1.3138 1.3496 1.3766 1.3978 1.4149 1.4289

1.1785 1.2102 1.2287 1.2402 1.2479 1.2532 1.2570 1.2598 1.2620

1.5994 1.7887 1.9081 1.9898 2.0490 2.0937 2.1286 2.1566 2.1795

1.3384 1.4191 1.4698 1.5029 1.5253 1.5411 1.5526 1.5611 1.5677

2.0288 2.2889 2.4556 2.5704 2.6537 2.7165 1.7654 2.8044 2.8363

1.4793 1.6227 1.7202 1.7870 1.8338 1.8673 1.8920 1.9106 1.9249

20.0 30.0 40.0 50.0 100.0 앝

1.4961 1.5202 1.5325 1.5400 1.5552 1.5708

1.2699 1.2717 1.2723 1.2727 1.2731 1.2733

2.2881 2.3261 2.3455 2.3572 2.3809 2.4050

1.5919 1.5973 1.5993 1.6002 1.6015 1.6018

2.9857 3.0372 3.0632 3.0788 3.1102 3.1415

1.9781 1.9898 1.9942 1.9962 1.9990 2.0000

Bi hL/k for the plane wall and hro /k for the infinite cylinder and sphere. See Figure 5.6.

a

302

Chapter 5

5.5.3

䊏

Transient Conduction

Total Energy Transfer

In many situations it is useful to know the total energy that has left (or entered) the wall up to any time t in the transient process. The conservation of energy requirement, Equation 1.12b, may be applied for the time interval bounded by the initial condition (t 0) and any time t 0 Ein Eout Est

(5.45)

Equating the energy transferred from the wall Q to Eout and setting Ein 0 and Est E(t) E(0), it follows that Q [E(t) E(0)] or

冕

Q c[T(x, t) Ti ]dV

(5.46a)

(5.46b)

where the integration is performed over the volume of the wall. It is convenient to nondimensionalize this result by introducing the quantity Qo cV(Ti T앝)

(5.47)

which may be interpreted as the initial internal energy of the wall relative to the fluid temperature. It is also the maximum amount of energy transfer that could occur if the process were continued to time t 앝. Hence, assuming constant properties, the ratio of the total energy transferred from the wall over the time interval t to the maximum possible transfer is Q Qo

t) T ] dV 1 冕(1 *)dV 冕 [T(x, T T V V i

i

(5.48)

앝

Employing the approximate form of the temperature distribution for the plane wall, Equation 5.43b, the integration prescribed by Equation 5.48 can be performed to obtain Q sin 1 1 * 1 o Qo

(5.49)

where *o can be determined from Equation 5.44, using Table 5.1 for values of the coefficients C1 and 1.

5.5.4

Additional Considerations

Because the mathematical problem is precisely the same, the foregoing results may also be applied to a plane wall of thickness L that is insulated on one side (x* 0) and experiences convective transport on the other side (x* 1). This equivalence is a consequence of the fact that, regardless of whether a symmetrical or an adiabatic requirement is prescribed at x* 0, the boundary condition is of the form ⭸*/⭸x* 0. Also note that the foregoing results may be used to determine the transient response of a plane wall to a sudden change in surface temperature. The process is equivalent to having

5.6

䊏

303

Radial Systems with Convection

an infinite convection coefficient, in which case the Biot number is infinite (Bi 앝) and the fluid temperature T앝 is replaced by the prescribed surface temperature Ts.

5.6 Radial Systems with Convection For an infinite cylinder or sphere of radius ro (Figure 5.6b), which is at an initial uniform temperature and experiences a change in convective conditions, results similar to those of Section 5.5 may be developed. That is, an exact series solution may be obtained for the time dependence of the radial temperature distribution, and a one-term approximation may be used for most conditions. The infinite cylinder is an idealization that permits the assumption of one-dimensional conduction in the radial direction. It is a reasonable approximation for cylinders having L/ro 10.

5.6.1

Exact Solutions

For a uniform initial temperature and convective boundary conditions, the exact solutions [4], applicable at any time (Fo 0), are as follows. Infinite Cylinder

In dimensionless form, the temperature is *

兺C

n

exp (2nFo)J0(nr*)

(5.50a)

n1

where Fo ␣t/r2o, J ( ) Cn 2 2 1 n 2 n J 0 (n) J 1 (n)

(5.50b)

and the discrete values of n are positive roots of the transcendental equation n

J1(n) Bi J0(n)

(5.50c)

where Bi hro /k. The quantities J1 and J0 are Bessel functions of the first kind, and their values are tabulated in Appendix B.4. Roots of the transcendental equation (5.50c) are tabulated by Schneider [4]. Sphere

Similarly, for the sphere *

兺C

n

n1

exp (2nFo) 1 sin (nr*) nr*

(5.51a)

where Fo ␣t/ro2, Cn

4[sin (n) n cos (n)] 2n sin (2n)

(5.51b)

304

Chapter 5

䊏

Transient Conduction

and the discrete values of n are positive roots of the transcendental equation 1 n cot n Bi

(5.51c)

where Bi hro /k. Roots of the transcendental equation are tabulated by Schneider [4].

5.6.2

Approximate Solutions

For the infinite cylinder and sphere, the foregoing series solutions can again be approximated by a single term, n 1, for Fo 0.2. Hence, as for the case of the plane wall, the time dependence of the temperature at any location within the radial system is the same as that of the centerline or centerpoint. Infinite Cylinder

The one-term approximation to Equation 5.50a is * C1 exp (21Fo)J0(1r*)

(5.52a)

* * o J0(1r*)

(5.52b)

or

where *o represents the centerline temperature and is of the form 2 * o C1 exp (1Fo)

(5.52c)

Values of the coefficients C1 and 1 have been determined and are listed in Table 5.1 for a range of Biot numbers. Sphere

From Equation 5.51a, the one-term approximation is * C1 exp (21Fo) 1 sin (1r*) 1r*

(5.53a)

1 sin ( r*) * * o 1 1r*

(5.53b)

or

where *o represents the center temperature and is of the form 2 * o C1 exp (1Fo)

(5.53c)

Values of the coefficients C1 and 1 have been determined and are listed in Table 5.1 for a range of Biot numbers.

5.6.3

Total Energy Transfer

As in Section 5.5.3, an energy balance may be performed to determine the total energy transfer from the infinite cylinder or sphere over the time interval t t. Substituting from

Graphical representations of the one-term approximations are presented in Section 5S.1.

5.6

䊏

Radial Systems with Convection

305

the approximate solutions, Equations 5.52b and 5.53b, and introducing Qo from Equation 5.47, the results are as follows. Infinite Cylinder

Q 2* 1 o J1(1) 1 Qo

(5.54)

Q 3* 1 3o [sin (1) 1 cos (1)] Qo 1

(5.55)

Sphere

Values of the center temperature *o are determined from Equation 5.52c or 5.53c, using the coefficients of Table 5.1 for the appropriate system.

5.6.4

Additional Considerations

As for the plane wall, the foregoing results may be used to predict the transient response of long cylinders and spheres subjected to a sudden change in surface temperature. Namely, an infinite Biot number would be prescribed, and the fluid temperature T앝 would be replaced by the constant surface temperature Ts.

EXAMPLE 5.5 Consider a steel pipeline (AISI 1010) that is 1 m in diameter and has a wall thickness of 40 mm. The pipe is heavily insulated on the outside, and, before the initiation of flow, the walls of the pipe are at a uniform temperature of 20 C. With the initiation of flow, hot oil at 60 C is pumped through the pipe, creating a convective condition corresponding to h 500 W/m2 䡠 K at the inner surface of the pipe. 1. What are the appropriate Biot and Fourier numbers 8 min after the initiation of flow? 2. At t 8 min, what is the temperature of the exterior pipe surface covered by the insulation? 3. What is the heat flux q(W/m2) to the pipe from the oil at t 8 min? 4. How much energy per meter of pipe length has been transferred from the oil to the pipe at t 8 min?

SOLUTION Known: Wall subjected to sudden change in convective surface condition. Find: 1. Biot and Fourier numbers after 8 min. 2. Temperature of exterior pipe surface after 8 min. 3. Heat flux to the wall at 8 min. 4. Energy transferred to pipe per unit length after 8 min.

306

Chapter 5

䊏

Transient Conduction

Schematic: T(x, 0) = Ti = –20°C

T(L, t)

T(0, t)

T∞ = 60°C h = 500 W/m2•K

Insulation Steel, AISI 1010

Oil

L = 40 mm x

Assumptions: 1. Pipe wall can be approximated as plane wall, since thickness is much less than diameter. 2. Constant properties. 3. Outer surface of pipe is adiabatic. Properties: Table A.1, steel type AISI 1010 [T (20 60) C/2 ⬇ 300 K]: 7832 kg/m3, c 434 J/kg 䡠 K, k 63.9 W/m 䡠 K, ␣ 18.8 106 m2/s. Analysis: 1. At t 8 min, the Biot and Fourier numbers are computed from Equations 5.10 and 5.12, respectively, with Lc L. Hence 2 Bi hL 500 W/m 䡠 K 0.04 m 0.313 k 63.9 W/m 䡠 K

Fo ␣t2 L

18.8 106 m2s 8 min 60 s/min 5.64 (0.04 m)2

䉰 䉰

2. With Bi 0.313, use of the lumped capacitance method is inappropriate. However, since Fo 0.2 and transient conditions in the insulated pipe wall of thickness L correspond to those in a plane wall of thickness 2L experiencing the same surface condition, the desired results may be obtained from the one-term approximation for a plane wall. The midplane temperature can be determined from Equation 5.44 * o

To T앝 C1 exp (21Fo) Ti T앝

where, with Bi 0.313, C1 1.047 and 1 0.531 rad from Table 5.1. With Fo 5.64, 2 * o 1.047 exp [(0.531 rad) 5.64] 0.214

Hence after 8 min, the temperature of the exterior pipe surface, which corresponds to the midplane temperature of a plane wall, is T(0, 8 min) T앝 * o (Ti T앝) 60 C 0.214(20 60) C 42.9 C

䉰

5.6

䊏

Radial Systems with Convection

307

3. Heat transfer to the inner surface at x L is by convection, and at any time t the heat flux may be obtained from Newton’s law of cooling. Hence at t 480 s, qx(L, 480 s) ⬅ qL h[T(L, 480 s) T앝] Using the one-term approximation for the surface temperature, Equation 5.43b with x* 1 has the form * *o cos (1) T(L, t) T앝 (Ti T앝)*o cos (1) T(L, 8 min) 60 C (20 60) C 0.214 cos(0.531 rad) T(L, 8 min) 45.2 C The heat flux at t 8 min is then qL 500 W/m2 䡠 K (45.2 60) C 7400 W/m2

䉰

4. The energy transfer to the pipe wall over the 8-min interval may be obtained from Equations 5.47 and 5.49. With Q sin(1) 1 *o 1 Qo Q sin(0.531 rad) 1

0.214 0.80 Qo 0.531 rad it follows that Q 0.80 cV(Ti T) or with a volume per unit pipe length of V DL, Q 0.80 cDL(Ti T앝) Q 0.80 7832 kg/m3 434 J/kg 䡠 K

1 m 0.04 m (20 60) C Q 2.73 107 J/m

䉰

Comments: 1. The minus sign associated with q and Q simply implies that the direction of heat transfer is from the oil to the pipe (into the pipe wall). 2. The solution for this example is provided as a ready-to-solve model in the Advanced section of IHT, which uses the Models, Transient Conduction, Plane Wall option. Since the IHT model uses a multiple-term approximation to the series solution, the results are more accurate than those obtained from the foregoing one-term approximation. IHT Models for Transient Conduction are also provided for the radial systems treated in Section 5.6.

308

Chapter 5

䊏

Transient Conduction

EXAMPLE 5.6 A new process for treatment of a special material is to be evaluated. The material, a sphere of radius ro 5 mm, is initially in equilibrium at 400 C in a furnace. It is suddenly removed from the furnace and subjected to a two-step cooling process. Step 1 Cooling in air at 20 C for a period of time ta until the center temperature reaches a critical value, Ta(0, ta ) 335 C. For this situation, the convection heat transfer coefficient is ha 10 W/m2 䡠 K. After the sphere has reached this critical temperature, the second step is initiated. Step 2 Cooling in a well-stirred water bath at 20 C, with a convection heat transfer coefficient of hw 6000 W/m2 䡠 K. The thermophysical properties of the material are 3000 kg/m3, k 20 W/m 䡠 K, c 1000 J/kg 䡠 K, and ␣ 6.66 106 m2/s. 1. Calculate the time ta required for step 1 of the cooling process to be completed. 2. Calculate the time tw required during step 2 of the process for the center of the sphere to cool from 335 C (the condition at the completion of step 1) to 50 C.

SOLUTION Known: Temperature requirements for cooling a sphere. Find: 1. Time ta required to accomplish desired cooling in air. 2. Time tw required to complete cooling in water bath. Schematic: T∞ = 20°C ha = 10 W/m2•K

T∞ = 20°C hw = 6000 W/m2•K

Air

Ti = 400°C Ta(0, ta) = 335°C

Water

Sphere, ro = 5 mm ρ = 3000 kg/m3 c = 1 kJ/kg•K α = 6.66 × 10–6 m2/s k = 20 W/m•K

Step 1

Assumptions: 1. One-dimensional conduction in r. 2. Constant properties.

Ti = 335°C Tw(0, tw) = 50°C Step 2

5.6

䊏

309

Radial Systems with Convection

Analysis: 1. To determine whether the lumped capacitance method can be used, the Biot number is calculated. From Equation 5.10, with Lc ro /3, Bi

ha ro 10 W/m2 䡠 K 0.005 m 8.33 104 3k 3 20 W/m 䡠 K

Accordingly, the lumped capacitance method may be used, and the temperature is nearly uniform throughout the sphere. From Equation 5.5 it follows that ta

Vc i roc Ti T앝 ln ln ha As a 3ha Ta T앝

where V (4/3)r 3o and As 4r 2o. Hence ta

3000 kg/m3 0.005 m 1000 J/kg 䡠 K 400 20 ln 94 s 335 20 3 10 W/m2 䡠 K

䉰

2. To determine whether the lumped capacitance method may also be used for the second step of the cooling process, the Biot number is again calculated. In this case Bi

h w ro 6000 W/m2 䡠 K 0.005 m 0.50 3k 3 20 W/m 䡠 K

and the lumped capacitance method is not appropriate. However, to an excellent approximation, the temperature of the sphere is uniform at t ta and the one-term approximation may be used for the calculations. The time tw at which the center temperature reaches 50 C, that is, T(0, tw) 50 C, can be obtained by rearranging Equation 5.53c Fo 12 ln 1

冤 C 冥 1 ln 冤 C1 T(0,T t) T T 冥 *o

w

2 1

1

1

i

where tw Fo r o2 /␣. With the Biot number now defined as Bi

h w ro 6000 W/m2 䡠 K 0.005 m 1.50 k 20 W/m 䡠 K

Table 5.1 yields C1 1.376 and 1 1.800 rad. It follows that Fo

冤

冥

(50 20) C 1 ln 1

0.82 2 1.376 (335 20) C (1.800 rad)

and r2 (0.005 m)2 3.1s tw Fo ␣o 0.82 6.66 106 m2/s Note that, with Fo 0.82, use of the one-term approximation is justified.

䉰

310

Chapter 5

䊏

Transient Conduction

Comments: 1. If the temperature distribution in the sphere at the conclusion of step 1 were not uniform, the one-term approximation could not be used for the calculations of step 2. 2. The surface temperature of the sphere at the conclusion of step 2 may be obtained from Equation 5.53b. With o* 0.095 and r* 1, *(ro)

T(ro) T앝 0.095 sin (1.800 rad) 0.0514 Ti T앝 1.800 rad

and T(ro) 20 C 0.0514(335 20) C 36 C The infinite series, Equation 5.51a, and its one-term approximation, Equation 5.53b, may be used to compute the temperature at any location in the sphere and at any time t ta. For (t ta ) 0.2(0.005 m)2/6.66 106 m2/s 0.75 s, a sufficient number of terms must be retained to ensure convergence of the series. For (t ta ) 0.75 s, satisfactory convergence is provided by the one-term approximation. Computing and plotting the temperature histories for r 0 and r ro, we obtain the following results for 0 (t ta ) 5 s: 400

300

T (°C)

r* = 1 200

r* = 0

100 50 0

0

1

2

3

4

5

t – ta (s)

3. The IHT Models, Transient Conduction, Sphere option could be used to analyze the cooling processes experienced by the sphere in air and water, steps 1 and 2. The IHT Models, Lumped Capacitance option may only be used to analyze the air-cooling process, step 1.

5.7 The Semi-Infinite Solid An important simple geometry for which analytical solutions may be obtained is the semiinfinite solid. Since, in principle, such a solid extends to infinity in all but one direction, it is characterized by a single identifiable surface (Figure 5.7). If a sudden change of conditions is imposed at this surface, transient, one-dimensional conduction will occur within the

5.7

䊏

311

The Semi-Infinite Solid

Case (1)

Case (2)

Case (3)

T(x, 0) = Ti T(0, t) = Ts

T(x, 0) = Ti –k ∂ T/∂ x⎥x = 0 = q"o

T(x, 0) = Ti –k ∂ T/∂ x⎥x = 0 = h[T∞ – T(0, t)]

Ts

T∞, h q"o

x

x

x

T(x, t) T∞

Ts t

t

Ti

Ti x

t

Ti x

x

FIGURE 5.7 Transient temperature distributions in a semi-infinite solid for three surface conditions: constant surface temperature, constant surface heat flux, and surface convection.

solid. The semi-infinite solid provides a useful idealization for many practical problems. It may be used to determine transient heat transfer near the surface of the earth or to approximate the transient response of a finite solid, such as a thick slab. For this second situation the approximation would be reasonable for the early portion of the transient, during which temperatures in the slab interior (well removed from the surface) are essentially uninfluenced by the change in surface conditions. These early portions of the transient might correspond to very small Fourier numbers, and the approximate solutions of Sections 5.5 and 5.6 would not be valid. Although the exact solutions of the preceding sections could be used to determine the temperature distributions, many terms might be required to evaluate the infinite series expressions. The following semi-infinite solid solutions often eliminate the need to evaluate the cumbersome infinite series exact solutions at small Fo. It will be shown that a plane wall of thickness 2L can be accurately approximated as a semi-infinite solid for Fo ␣t/L2 0.2. The heat equation for transient conduction in a semi-infinite solid is given by Equation 5.29. The initial condition is prescribed by Equation 5.30, and the interior boundary condition is of the form T(x l 앝, t) Ti

(5.56)

Closed-form solutions have been obtained for three important surface conditions, instantaneously applied at t 0 [3, 4]. These conditions are shown in Figure 5.7. They include application of a constant surface temperature Ts Ti, application of a constant surface heat flux qo, and exposure of the surface to a fluid characterized by T앝 Ti and the convection coefficient h. The solution for case 1 may be obtained by recognizing the existence of a similarity variable , through which the heat equation may be transformed from a partial differential equation, involving two independent variables (x and t), to an ordinary differential equation expressed in terms of the single similarity variable. To confirm that such a

312

Chapter 5

䊏

Transient Conduction

requirement is satisfied by ⬅ x/(4␣t)1/2, we first transform the pertinent differential operators, such that ⭸T dT ⭸ 1 dT ⭸x d ⭸x (4␣t)1/2 d

冤 冥

2 ⭸2T ⭸T ⭸ 1 d T2 d 2 d ⭸x ⭸x 4␣t d ⭸x ⭸T dT ⭸ x dT 1/2 d ⭸t d ⭸t 2t(4␣t)

Substituting into Equation 5.29, the heat equation becomes d 2T 2 dT d d2

(5.57)

With x 0 corresponding to 0, the surface condition may be expressed as T( 0) Ts

(5.58)

and with x l 앝, as well as t 0, corresponding to l 앝, both the initial condition and the interior boundary condition correspond to the single requirement that T( l 앝) Ti

(5.59)

Since the transformed heat equation and the initial/boundary conditions are independent of x and t, ⬅ x/(4␣t)1/2 is, indeed, a similarity variable. Its existence implies that, irrespective of the values of x and t, the temperature may be represented as a unique function of . The specific form of the temperature dependence, T(), may be obtained by separating variables in Equation 5.57, such that d(dT/d) 2 d (dT/d) Integrating, it follows that ln(dT/d) 2 C1 or dT C exp (2) 1 d Integrating a second time, we obtain

冕 exp(u ) du C

T C1

2

2

0

where u is a dummy variable. Applying the boundary condition at 0, Equation 5.58, it follows that C2 Ts and

冕 exp(u ) du T

T C1

2

s

0

5.7

䊏

313

The Semi-Infinite Solid

From the second boundary condition, Equation 5.59, we obtain Ti C1

冕 exp(u ) du T 앝

2

s

0

or, evaluating the definite integral, C1

2(Ti Ts) 1/2

Hence the temperature distribution may be expressed as T Ts (2/1/2) Ti Ts

冕 exp (u ) du ⬅ erf

2

(5.60)

0

where the Gaussian error function, erf , is a standard mathematical function that is tabulated in Appendix B. Note that erf() asymptotically approaches unity as becomes infinite. Thus, at any nonzero time, temperatures everywhere are predicted to have changed from Ti (become closer to Ts). The infinite speed at which boundary-condition information propagates into the semi-infinite solid is physically unrealistic, but this limitation of Fourier’s law is not important except at extremely small time scales, as discussed in Section 2.3. The surface heat flux may be obtained by applying Fourier’s law at x 0, in which case qs k

⭸T ⭸x

冏

k(Ti Ts)

x0

d(erf ) ⭸ d ⭸x

冏

0

qs k(Ts Ti)(2/1/2)exp(2)(4␣t)1/2 兩0 qs

k(Ts Ti) (␣t)1/2

(5.61)

Analytical solutions may also be obtained for the case 2 and case 3 surface conditions, and results for all three cases are summarized as follows. Case 1

Constant Surface Temperature: T(0, t) Ts

冢

T(x, t) Ts x erf Ti Ts 2兹␣t qs(t) Case 2

冣

(5.60)

k(Ts Ti)

(5.61)

兹␣t

Constant Surface Heat Flux: qs qo T(x, t) Ti

冢 冣

冢

2 q x 2qo(␣t/)1/2 x exp x o erfc k 4␣t k 2兹␣t

冣

(5.62)

Chapter 5

䊏

Transient Conduction

Surface Convection: k

Case 3

冢

T(x, t) Ti x erfc T앝 Ti 2兹␣t

⭸T ⭸x

冏

x0

h[T앝 T(0, t)]

冣

冤 冢

2 exp hx h ␣t k k2

h兹␣t x

冣冥冤erfc 冢2兹␣t k 冣冥

(5.63)

The complementary error function, erfc w, is defined as erfc w ⬅ 1 erf w. Temperature histories for the three cases are shown in Figure 5.7, and distinguishing features should be noted. With a step change in the surface temperature, case 1, temperatures within the medium monotonically approach Ts with increasing t, while the magnitude of the surface temperature gradient, and hence the surface heat flux, decreases as t1/2. A thermal penetration depth ␦p can be defined as the depth to which significant temperature effects propagate within a medium. For example, defining ␦p as the x-location at which (T – Ts)/ (Ti – Ts) 0.90, Equation 5.60 results in ␦p 2.3兹␣t.2 Hence, the penetration depth increases as t1/2 and is larger for materials with higher thermal diffusivity. For a fixed surface heat flux (case 2), Equation 5.62 reveals that T(0, t) Ts(t) increases monotonically as t1/2. For surface convection (case 3), the surface temperature and temperatures within the medium approach the fluid temperature T앝 with increasing time. As Ts approaches T앝, there is, of course, a reduction in the surface heat flux, qs(t) h[T Ts(t)]. Specific temperature histories computed from Equation 5.63 are plotted in Figure 5.8. The result corresponding to h 앝 is equivalent to that associated with a sudden change in surface temperature, case 1. That is, for h 앝, the surface instantaneously achieves the imposed fluid temperature (Ts T앝), and with the second term on the right-hand side of Equation 5.63 reducing to zero, the result is equivalent to Equation 5.60. An interesting permutation of case 1 occurs when two semi-infinite solids, initially at uniform temperatures TA,i and TB,i, are placed in contact at their free surfaces (Figure 5.9). 1.0 0.5

T – Ti ______ T∞ – Ti

314

T∞ T(x, t) h

∞ 3 0.4 0.5

1

x 2

0.1 0.3 0.2

0.05

0.1

h √α t = 0.05 _____ k

0.01 0

0.5

1.0

x _____ 2 √ αt

1.5

FIGURE 5.8 Temperature histories in a semi-infinite solid with surface convection [4]. (Adapted with permission.)

To apply the semi-infinite approximation to a plane wall of thickness 2L, it is necessary that ␦p L. Substituting ␦p L into the expression for the thermal penetration depth yields Fo 0.19 ⬇ 0.2. Hence, a plane wall of thickness 2L can be accurately approximated as a semi-infinite solid for Fo ␣t/L2 0.2. This restriction will also be demonstrated in Section 5.8. 2

5.7

䊏

315

The Semi-Infinite Solid

T B

TA, i

kB, ρB, cB q"s, B t Ts

t

q"s, A A

TB, i

kA, ρA, cA

FIGURE 5.9 Interfacial contact between two semiinfinite solids at different initial temperatures.

x

If the contact resistance is negligible, the requirement of thermal equilibrium dictates that, at the instant of contact (t 0), both surfaces must assume the same temperature Ts, for which TB,i Ts TA,i. Since Ts does not change with increasing time, it follows that the transient thermal response and the surface heat flux of each of the solids are determined by Equations 5.60 and 5.61, respectively. The equilibrium surface temperature of Figure 5.9 may be determined from a surface energy balance, which requires that qs,A qs,B

(5.64)

Substituting from Equation 5.61 for qs,A and qs,B and recognizing that the x-coordinate of Figure 5.9 requires a sign change for qs,A, it follows that kA(Ts TA,i) 1/2

(␣At)

kB(Ts TB,i) (␣Bt)1/2

(5.65)

or, solving for Ts, Ts

1/2 (kc)1/2 A TA,i (kc)B TB,i 1/2 (kc)1/2 A (kc)B

(5.66)

Hence the quantity m ⬅ (kc)1/2 is a weighting factor that determines whether Ts will more closely approach TA,i (mA mB) or TB,i (mB mA).

EXAMPLE 5.7 On a hot and sunny day, the concrete deck surrounding a swimming pool is at a temperature of Td 55 C. A swimmer walks across the dry deck to the pool. The soles of the swimmer’s dry feet are characterized by an Lsf 3-mm-thick skin/fat layer of thermal conductivity ksf 0.3 W/m 䡠 K. Consider two types of concrete decking; (i) a dense stone mix and (ii) a lightweight aggregate characterized by density, specific heat, and thermal conductivity of lw 1495 kg/m3, cp,lw 880 J/kg 䡠 K, and klw 0.28 W/m 䡠 K, respectively. The density and specific heat of the skin/fat layer may be approximated to be those of liquid water, and the skin/fat layer is at an initial temperature of Tsf,i 37 C. What is the temperature of the bottom of the swimmer’s feet after an elapsed time of t 1 s?

316

Chapter 5

Transient Conduction

䊏

SOLUTION Known: Concrete temperature, initial foot temperature, and thickness of skin/fat layer on the sole of the foot. Skin/fat and lightweight aggregate concrete properties. Find: The temperature of the bottom of the swimmer’s feet after 1 s. Schematic: Tsf,i = 37°C Lsf ⫽ 3 mm x

Foot Skin/fat Ts

Concrete deck Td,i = 55°C

Assumptions: 1. One-dimensional conduction in the x-direction. 2. Constant and uniform properties. 3. Negligible contact resistance. Properties: Table A.3 stone mix concrete (T 300 K): sm 2300 kg/m3, ksm 1.4 W/m 䡠 K, csm 880 J/kg 䡠 K. Table A.6 water (T 310 K): sf 993 kg/m3, csf 4178 J/kg 䡠 K. Analysis: If the skin/fat layer and the deck are both semi-infinite media, from Equation 5.66 the surface temperature Ts is constant when the swimmer’s foot is in contact with the deck. For the lightweight aggregate concrete decking, the thermal penetration depth at t 1 s is K 1s 冪k ct 2.3冪14950.28kg/mW/m 䡠880 J/kg 䡠 K

␦p,lw 2.3兹␣lwt 2.3

lw

3

lw lw

1.06 103 m 1.06 mm

Since the thermal penetration depth is relatively small, it is reasonable to assume that the lightweight aggregate deck behaves as a semi-infinite medium. Similarly, the thermal penetration depth in the stone mix concrete is ␦p,sm 1.91 mm, and the thermal penetration depth associated with the skin/fat layer of the foot is ␦p,sf 0.62 mm. Hence, it is reasonable to assume that the stone mix concrete deck responds as a semi-infinite medium, and, since ␦p,sf Lsf, it is also correct to assume that the skin/fat layer behaves as a semi-infinite medium. Therefore, Equation 5.66 may be used to determine the surface temperature of the swimmer’s foot for exposure to the two types of concrete decking. For the lightweight aggregate, Ts,lw

1/2 (kc)1/2 lw Td,i (kc)sf Tsf,i 1/2 (kc)1/2 lw (kc)sf

W/m 䡠 K 1495 kg/m 880 J/kg 䡠 K) 55 C 冤(0.28

(0.3 W/m 䡠 K 993 kg/m 4178 J/kg 䡠 K) 37 C冥 43.3 C (0.28 W/m 䡠 K 1495 kg/m 880 J/kg 䡠 K) 冤 (0.3 W/m 䡠 K 993 kg/m 4178 J/kg 䡠 K) 冥 3

1/2

3

1/2

3

3

1/2

1/2

䉰

5.8

䊏

317

Objects with Constant Surface Temperatures or Surface Heat Fluxes

Repeating the calculation for the stone mix concrete gives Ts,sm 47.8 C.

䉰

Comments: 1. The lightweight aggregate concrete feels cooler to the swimmer, relative to the stone mix concrete. Specifically, the temperature rise from the initial skin/fat temperature that is associated with the stone mix concrete is Tsm Tsm – Tsf,i 47.8 C – 37 C 10.8 C, whereas the temperature rise associated with the lightweight aggregate is Tlw Tlw – Tsf,i 43.3 C – 37 C 6.3 C. 2. The thermal penetration depths associated with an exposure time of t 1 s are small. Stones and air pockets within the concrete may be of the same size as the thermal penetration depth, making the uniform property assumption somewhat questionable. The predicted foot temperatures should be viewed as representative values.

5.8 Objects with Constant Surface Temperatures or Surface Heat Fluxes In Sections 5.5 and 5.6, the transient thermal response of plane walls, cylinders, and spheres to an applied convection boundary condition was considered in detail. It was pointed out that the solutions in those sections may be used for cases involving a step change in surface temperature by allowing the Biot number to be infinite. In Section 5.7, the response of a semi-infinite solid to a step change in surface temperature, or to an applied constant heat flux, was determined. This section will conclude our discussion of transient heat transfer in one-dimensional objects experiencing constant surface temperature or constant surface heat flux boundary conditions. A variety of approximate solutions are presented.

5.8.1

Constant Temperature Boundary Conditions

In the following discussion, the transient thermal response of objects to a step change in surface temperature is considered. Insight into the thermal response of objects to an applied constant temperature boundary condition may be obtained by casting the heat flux in Equation 5.61 into the nondimensional form

Semi-Infinite Solid

q* ⬅

qs Lc k(Ts Ti)

(5.67)

where Lc is a characteristic length and q* is the dimensionless conduction heat rate that was introduced in Section 4.3. Substituting Equation 5.67 into Equation 5.61 yields q*

1 兹Fo

(5.68)

Chapter 5

䊏

Transient Conduction

where the Fourier number is defined as Fo ⬅ ␣t/L2c. Note that the value of qs is independent of the choice of the characteristic length, as it must be for a semi-infinite solid. Equation 5.68 is plotted in Figure 5.10a, and since q* Fo1/2, the slope of the line is 1/2 on the log-log plot. Interior Heat Transfer: Plane Wall, Cylinder, and Sphere Results for heat transfer to the interior of a plane wall, cylinder, and sphere are also shown in Figure 5.10a. These results are generated by using Fourier’s law in conjunction with Equations 5.42, 5.50, and 5.51 for Bi l 앝. As in Sections 5.5 and 5.6, the characteristic length is Lc L or ro for a plane wall of thickness 2L or a cylinder (or sphere) of radius ro, respectively. For each geometry, q* initially follows the semi-infinite solid solution but at some point decreases rapidly as the objects approach their equilibrium temperature and qs (t l 앝) l 0. The value of q* is expected to decrease more rapidly for geometries that possess large surface area to volume ratios, and this trend is evident in Figure 5.10a.

100

Exterior objects, Lc = (As/4)1/2 Semi-infinite solid

q*

10

1

Interior, Lc = L or ro sphere 0.1

infinite cylinder plane wall

0.01

0.0001

0.001

0.01

0.1

1

10

Fo = ␣t/L2c (a) 100

Exterior objects, Lc = (As/4)1/2 Semi-infinite solid 10

q*

318

1

Interior, Lc = L or ro sphere

0.1

infinite cylinder plane wall

0.01 0.0001

0.001

0.01

0.1

Fo = ␣t/L2c (b)

1

10

FIGURE 5.10 Transient dimensionless conduction heat rates for a variety of geometries. (a) Constant surface temperature. Results for the geometries of Table 4.1 lie within the shaded region and are from Yovanovich [7]. (b) Constant surface heat flux.

5.8

䊏

Objects with Constant Surface Temperatures or Surface Heat Fluxes

319

Additional results are shown in Figure 5.10a for objects that are embedded in an exterior (surrounding) medium of infinite extent. The infinite medium is initially at temperature Ti, and the surface temperature of the object is suddenly changed to Ts. For the exterior cases, Lc is the characteristic length used in Section 4.3, namely Lc (As /4)1/2. For the sphere in a surrounding infinite medium, the exact solution for q*(Fo) is [7] Exterior Heat Transfer: Various Geometries

1 (5.69)

1 兹Fo As seen in the figure, for all of the exterior cases q* closely mimics that of the sphere when the appropriate length scale is used in its definition, regardless of the object’s shape. Moreover, in a manner consistent with the interior cases, q* initially follows the semi-infinite solid solution. In contrast to the interior cases, q* eventually reaches the nonzero, steady-state value of q*ss that is listed in Table 4.1. Note that qs in Equation 5.67 is the average surface heat flux for geometries that have nonuniform surface heat flux. As seen in Figure 5.10a, all of the thermal responses collapse to that of the semiinfinite solid for early times, that is, for Fo less than approximately 103. This remarkable consistency reflects the fact that temperature variations are confined to thin layers adjacent to the surface of any object at early times, regardless of whether internal or external heat transfer is of interest. At early times, therefore, Equations 5.60 and 5.61 may be used to predict the temperatures and heat transfer rates within the thin regions adjacent to the boundaries of any object. For example, predicted local heat fluxes and local dimensionless temperatures using the semi-infinite solid solutions are within approximately 5% of the predictions obtained from the exact solutions for the interior and exterior heat transfer cases involving spheres when Fo 103. q*

5.8.2

Constant Heat Flux Boundary Conditions

When a constant surface heat flux is applied to an object, the resulting surface temperature history is often of interest. In this case, the heat flux in the numerator of Equation 5.67 is now constant, and the temperature difference in the denominator, Ts Ti, increases with time. Semi-Infinite Solid In the case of a semi-infinite solid, the surface temperature history can be found by evaluating Equation 5.62 at x 0, which may be rearranged and combined with Equation 5.67 to yield

q* 1 2

冪Fo

(5.70)

As for the constant temperature case, q* Fo1/2, but with a different coefficient. Equation 5.70 is presented in Figure 5.10b. A second set of results is shown in Figure 5.10b for the interior cases of the plane wall, cylinder, and sphere. As for the constant surface temperature results of Figure 5.10a, q* initially follows the semiinfinite solid solution and subsequently decreases more rapidly, with the decrease occurring first for the sphere, then the cylinder, and finally the plane wall. Compared to the constant surface temperature case, the rate at which q* decreases is not as dramatic, since steadystate conditions are never reached; the surface temperature must continue to increase with

Interior Heat Transfer: Plane Wall, Cylinder, and Sphere

320

Chapter 5

䊏

Transient Conduction

time. At late times (large Fo), the surface temperature increases linearly with time, yielding q* Fo1, with a slope of 1 on the log-log plot. Results for heat transfer between a sphere and an exterior infinite medium are also presented in Figure 5.10b. The exact solution for the embedded sphere is

Exterior Heat Transfer: Various Geometries

q* [1 exp(Fo) erfc(Fo1/2)]1

(5.71)

As in the constant surface temperature case of Figure 5.10a, this solution approaches steady-state conditions, with qss 1. For objects of other shapes that are embedded within an infinite medium, q* would follow the semi-infinite solid solution at small Fo. At larger Fo, q* must asymptotically approach the value of qss given in Table 4.1 where Ts in Equation 5.67 is the average surface temperature for geometries that have nonuniform surface temperatures.

5.8.3

Approximate Solutions

Simple expressions have been developed for q*(Fo) [8]. These expressions may be used to approximate all the results included in Figure 5.10 over the entire range of Fo. These expressions are listed in Table 5.2, along with the corresponding exact solutions. Table 5.2a is for the constant surface temperature case, while Table 5.2b is for the constant surface heat flux situation. For each of the geometries listed in the left-hand column, the tables provide the length scale to be used in the definition of both Fo and q*, the exact solution for q*(Fo), the approximation solutions for early times (Fo 0.2) and late times (Fo 0.2), and the maximum percentage error associated with use of the approximations (which occurs at Fo ⬇ 0.2 for all results except the external sphere with constant heat flux).

EXAMPLE 5.8 Derive an expression for the ratio of the total energy transferred from the isothermal surfaces of a plane wall to the interior of the plane wall, Q/Qo, that is valid for Fo 0.2. Express your results in terms of the Fourier number Fo.

SOLUTION Known: Plane wall with constant surface temperatures. Find: Expression for Q/Qo as a function of Fo ␣t/L2. Schematic: Ti

Ts L

L x

5.8 䊏

TABLE 5.2a Summary of transient heat transfer results for constant surface temperature casesa [8]

Geometry Semi-infinite

Length Scale, Lc

Exact Solutions

L (arbitrary)

1 兹Fo

Interior Cases

앝

Plane wall of thickness 2L

L

Infinite cylinder

ro

2

兺 exp( 앝

2

Various shapes (Table 4.1, cases 12–15)

None

1 兹Fo

2 exp(21 Fo)

1 /2

1.7

2 n

Fo)

J0(n) 0

1 0.50 0.65 Fo 兹Fo

2 exp(21 Fo)

1 2.4050

0.8

Fo)

n n

1 1 兹Fo

2 exp(21 Fo)

1

6.3

兺 exp(

2 n

n1

Exterior Cases Sphere

Use exact solution.

Maximum Error (%)

n (n 12)

兺 exp( 앝

ro

Fo ⱖ 0.2

Fo)

n1

Sphere

Fo ⬍ 0.2

2 n

n1

2

Approximate Solutions

ro

1

1 兹Fo

(As /4)1/2

None

Use exact solution. 1

q* ss, 兹 Fo

q* ss from Table 4.1

None 7.1

q* ⬅ q⬙s Lc /k(Ts Ti) and Fo ⬅ ␣t/L2c , where Lc is the length scale given in the table, Ts is the object surface temperature, and Ti is (a) the initial object temperature for the interior cases and (b) the temperature of the infinite medium for the exterior cases. a

Objects with Constant Surface Temperatures or Surface Heat Fluxes

q*(Fo)

321

322 Chapter 5

TABLE 5.2b Summary of transient heat transfer results for constant surface heat flux casesa [8] q*(Fo)

Semi-infinite Interior Cases Plane wall of thickness 2L

1 2

L (arbitrary)

冪Fo 1

2 n

n1

2 n

兺 冤 冥 冤3Fo 15 2 兺 exp( Fo)冥 앝 exp( Fo) n 1 2Fo 2 4 2n n1

1

앝

1

2

Infinite cylinder

ro

Sphere

ro

n1

Exterior Cases Sphere Various shapes (Table 4.1, cases 12–15)

2 n

2 n

Maximum Error (%)

Fo ⱖ 0.2

Use exact solution.

冤Fo 13 2 兺 exp( Fo)冥 앝

L

Fo ⬍ 0.2

n n J1(n) 0 tan(n) n

None

Fo

冤Fo 13冥

冪Fo 8 1 2冪Fo 4

冤2Fo 14冥 冤3Fo 15冥

1 2

冪

1 2

1

5.3 1

2.1

1

ro

[1 exp(Fo)erfc(Fo 1/2)]1

1 2

冪Fo 4

0.77

1 兹Fo

(As /4)1/2

None

1 2

冪Fo 4

0.77

q*ss 兹Fo

4.5

3.2

Unknown

q* ⬅ qs Lc /k(Ts Ti) and Fo ⬅ ␣t/L2c, where Lc is the length scale given in the table, Ts is the object surface temperature, and Ti is (a) the initial object temperature for the interior cases and (b) the temperature of the infinite medium for the exterior cases.

a

Transient Conduction

Geometry

Exact Solutions

䊏

Approximate Solutions Length Scale, Lc

5.8

䊏

Objects with Constant Surface Temperatures or Surface Heat Fluxes

323

Assumptions: 1. One-dimensional conduction. 2. Constant properties. 3. Validity of the approximate solution of Table 5.2a. Analysis: From Table 5.2a for a plane wall of thickness 2L and Fo 0.2, q*

qs L 1 where Fo ␣t2 k(Ts Ti ) 兹Fo L

Combining the preceding equations yields qs

k(Ts Ti) 兹␣t

Recognizing that Q is the accumulated heat that has entered the wall up to time t, we can write

冕 t

qs dt Q t0 ␣ Qo Lc(Ts Ti) L兹␣

冕t t

1/2

t0

dt 2 兹Fo 兹

䉰

Comments: 1. The exact solution for Q/Qo at small Fourier number involves many terms that would need to be evaluated in the infinite series expression. Use of the approximate solution simplifies the evaluation of Q/Qo considerably. 2. At Fo 0.2, Q/Qo ⬇ 0.5. Approximately half of the total possible change in thermal energy of the plane wall occurs during Fo 0.2. 3. Although the Fourier number may be viewed as a dimensionless time, it has an important physical interpretation for problems involving heat transfer by conduction through a solid concurrent with thermal energy storage in the solid. Specifically, as suggested by the solution, the Fourier number provides a measure of the amount of energy stored in the solid at any time.

EXAMPLE 5.9 A proposed cancer treatment utilizes small, composite nanoshells whose size and composition are carefully specified so that the particles efficiently absorb laser irradiation at particular wavelengths [9]. Prior to treatment, antibodies are attached to the nanoscale particles. The doped particles are then injected into the patient’s bloodstream and are distributed throughout the body. The antibodies are attracted to malignant sites, and therefore carry and adhere the nanoshells only to cancerous tissue. After the particles have come to rest within the tumor, a laser beam penetrates through the tissue between the skin and the cancer, is absorbed by the nanoshells, and, in turn, heats and destroys the cancerous tissues.

324

Chapter 5

䊏

Transient Conduction

Consider an approximately spherical tumor of diameter Dt 3 mm that is uniformly infiltrated with nanoshells that are highly absorptive of incident radiation from a laser located outside the patient’s body. Mirror Laser Nanoshell impregnated tumor

1. Estimate the heat transfer rate from the tumor to the surrounding healthy tissue for a steady-state treatment temperature of Tt,ss 55 C at the surface of the tumor. The thermal conductivity of healthy tissue is approximately k 0.5 W/m 䡠 K, and the body temperature is Tb 37 C. 2. Find the laser power necessary to sustain the tumor surface temperature at Tt,ss 55 C if the tumor is located d 20 mm beneath the surface of the skin, and the laser heat flux decays exponentially, ql (x) ql,o(1 ) ex, between the surface of the body and the tumor. In the preceding expression, ql,o is the laser heat flux outside the body, 0.05 is the reflectivity of the skin surface, and 0.02 mm1 is the extinction coefficient of the tissue between the tumor and the surface of the skin. The laser beam has a diameter of Dl 5 mm. 3. Neglecting heat transfer to the surrounding tissue, estimate the time at which the tumor temperature is within 3 C of Tt,ss 55 C for the laser power found in part 2. Assume the tissue’s density and specific heat are that of water. 4. Neglecting the thermal mass of the tumor but accounting for heat transfer to the surrounding tissue, estimate the time needed for the surface temperature of the tumor to reach Tt 52 C.

SOLUTION Known: Size of a small sphere; thermal conductivity, reflectivity, and extinction coefficient of tissue; depth of sphere below the surface of the skin. Find: 1. Heat transferred from the tumor to maintain its surface temperature at Tt,ss 55 C. 2. Laser power needed to sustain the tumor surface temperature at Tt,ss 55 C. 3. Time for the tumor to reach Tt 52 C when heat transfer to the surrounding tissue is neglected. 4. Time for the tumor to reach Tt 52 C when heat transfer to the surrounding tissue is considered and the thermal mass of the tumor is neglected.

5.8

䊏

Objects with Constant Surface Temperatures or Surface Heat Fluxes

325

Schematic: Laser beam, q"l,o

Dl = 5 mm

Skin, = 0.05 x Tumor

d = 20 mm

Healthy tissue Tb = 37°C k = 0.5 W/m•K κ = 0.02 mm1 Dt = 3 mm

Assumptions: 1. One-dimensional conduction in the radial direction. 2. Constant properties. 3. Healthy tissue can be treated as an infinite medium. 4. The treated tumor absorbs all irradiation incident from the laser. 5. Lumped capacitance behavior for the tumor. 6. Neglect potential nanoscale heat transfer effects. 7. Neglect the effect of perfusion. 3 Properties: Table A.6, water (320 K, assumed): v1 f 989.1 kg/m , cp 4180 J/kg 䡠 K.

Analysis: 1. The steady-state heat loss from the spherical tumor may be determined by evaluating the dimensionless heat rate from the expression for case 12 of Table 4.1: q 2kDt(Tt,ss Tb) 2 0.5 W/m 䡠 K 3 103 m (55 37) C 0.170 W

䉰

2. The laser irradiation will be absorbed over the projected area of the tumor, D2t/4. To determine the laser power corresponding to q 0.170 W, we first write an energy balance for the sphere. For a control surface about the sphere, the energy absorbed from the laser irradiation is offset by heat conduction to the healthy tissue, q 0.170 W ⬇ ql(x d)Dt2/4, where, ql(x d) ql,o (1 )e⫺d and the laser power is Pl ql,oD2l /4. Hence, Pl qD2l ed/[(1 )D2t ] 1 0.170 W (5 103 m)2 e(0.02 mm 20 mm)/[(1 0.05) (3 103 m)2] 0.74 W 䉰 3. The general lumped capacitance energy balance, Equation 5.14, may be written ql (x d)D2t /4 q Vcp dT dt

Chapter 5

䊏

Transient Conduction

Separating variables and integrating between appropriate limits, q Vc

冕 dt 冕dT t

Tt

t0

Tb

yields Vcp 989.1 kg/m3 (/6) (3 103 m)3 4180 J/kg 䡠 K t q (Tt Tb) 0.170 W

(52 C 37 C) or t 5.16 s

䉰

4. Using Equation 5.71, q/2kDt(Tt Tb) q* [1 exp(Fo)erfc(Fo1/2)]1 which may be solved by trial-and-error to yield Fo 10.3 4␣t/D2t. Then, with ␣ k/cp 0.50 W/m 䡠 K/(989.1 kg/m3 4180 J/kg 䡠 K) 1.21 107 m2/s, we find t FoD2t /4␣ 10.3 (3 103 m)2 /(4 1.21 107 m2/s) 192 s

䉰

Comments: 1. The analysis does not account for blood perfusion. The flow of blood would lead to advection of warmed fluid away from the tumor (and relatively cool blood to the vicinity of the tumor), increasing the power needed to reach the desired treatment temperature. 2. The laser power needed to treat various-sized tumors, calculated as in parts 1 and 2 of the problem solution, is shown below. Note that as the tumor becomes smaller, a higher-powered laser is needed, which may seem counterintuitive. The power required to heat the tumor, which is the same as the heat loss calculated in part 1, increases in direct proportion to the diameter, as might be expected. However, since the laser power flux remains constant, a smaller tumor cannot absorb as much energy (the energy absorbed has a D2t dependence). Less of the overall laser power is utilized to heat the tumor, and the required laser power increases for smaller tumors. 2.5

2 Laser power, Pl (W)

326

1.5

1

0.5

1

2 3 Tumor diameter, Dt (mm)

4

5.9

䊏

327

Periodic Heating

3. To determine the actual time needed for the tumor temperature to approach steadystate conditions, a numerical solution of the heat diffusion equation applied to the surrounding tissue, coupled with a solution for the temperature history within the tumor, would be required. However, we see that significantly more time is needed for the surrounding tissue to reach steady-state conditions than to increase the temperature of the isolated spherical tumor. This is due to the fact that higher temperatures propagate into a large volume when heating of the surrounding tissue is considered, while in contrast the thermal mass of the tumor is limited by the tumor’s size. Hence, the actual time to heat both the tumor and the surrounding tissue will be slightly greater than 192 s. 4. Since temperatures are likely to increase at a considerable distance from the tumor, the assumption that the surroundings are of infinite size would need to be checked by inspecting results of the proposed numerical solution described in Comment 3.

5.9 Periodic Heating In the preceding discussion of transient heat transfer, we have considered objects that experience constant surface temperature or constant surface heat flux boundary conditions. In many practical applications the boundary conditions are not constant, and analytical solutions have been obtained for situations where the conditions vary with time. One situation involving nonconstant boundary conditions is periodic heating, which describes various applications, such as thermal processing of materials using pulsed lasers, and occurs naturally in situations such as those involving the collection of solar energy. Consider, for example, the semi-infinite solid of Figure 5.11a. For a surface temperature history described by T(0, t) Ti T sin t, the solution of Equation 5.29 subject to the interior boundary condition given by Equation 5.56 is T(x, t) Ti exp[x兹/2␣] sin[t x兹/2␣] T

(5.72)

This solution applies after sufficient time has passed to yield a quasi-steady state for which all temperatures fluctuate periodically about a time-invariant mean value. At locations in the solid, the fluctuations have a time lag relative to the surface temperature.

T(0, t) = Ti ∆Tsin(t)

Ti

∆T

␦p

x

(a)

qs(0, t) = ∆qs ∆qssin(t)

y

w x

␦p

(b)

FIGURE 5.11 Schematic of (a) a periodically heated, onedimensional semi-infinite solid and (b) a periodically heated strip attached to a semi-infinite solid.

328

Chapter 5

䊏

Transient Conduction

In addition, the amplitude of the fluctuations within the material decays exponentially with distance from the surface. Consistent with the earlier definition of the thermal penetration depth, ␦p can be defined as the x-location at which the amplitude of the temperature fluctuation is reduced by approximately 90% relative to that of the surface. This ␣/. The heat flux at the surface may be determined by applying results in ␦p 4 兹苶 Fourier’s law at x 0, yielding qs(t) kT兹/␣ sin(t /4)

(5.73)

Equation 5.73 reveals that the surface heat flux is periodic, with a time-averaged value of zero. Periodic heating can also occur in two- or three-dimensional arrangements, as shown in Figure 5.11b. Recall that for this geometry, a steady state can be attained with constant heating of the strip placed upon a semi-infinite solid (Table 4.1, case 13). In a similar manner, a quasi-steady state may be achieved when sinusoidal heating (qs qs qs sin t) is applied to the strip. Again, a quasi-steady state is achieved for which all temperatures fluctuate about a time-invariant mean value. The solution of the two-dimensional, transient heat diffusion equation for the twodimensional configuration shown in Figure 5.11b has been obtained, and the relationship between the amplitude of the applied sinusoidal heating and the amplitude of the temperature response of the heated strip can be approximated as [10] T 艐

冤

冥

冤

冥

qs qs 1 ln(/2) ln(w2/4␣) C1 1 ln(/2) C2 Lk 2 Lk 2

(5.74)

where the constant C1 depends on the thermal contact resistance at the interface between the heated strip and the underlying material. Note that the amplitude of the temperature fluctuation, T, corresponds to the spatially averaged temperature of the rectangular strip of length L and width w. The heat flux from the strip to the semi-infinite medium is assumed to be spatially uniform. The approximation is valid for L w. For the system of Figure 5.11b, the thermal penetration depth is smaller than that of Figure 5.11a because of the lateral spreading of thermal energy and is ␦p 艐兹␣/.

EXAMPLE 5.10 A nanostructured dielectric material has been fabricated, and the following method is used to measure its thermal conductivity. A long metal strip 3000 angstroms thick, w 100 m wide, and L 3.5 mm long is deposited by a photolithography technique on the top surface of a d 300-m-thick sample of the new material. The strip is heated periodically by an electric current supplied through two connector pads. The heating rate is qs(t) qs qs sin(t), where qs is 3.5 mW. The instantaneous, spatially averaged temperature of the metal strip is found experimentally by measuring the time variation of its electrical resistance, R(t) E(t)/I(t), and by knowing how the electrical resistance of the metal varies with temperature. The measured temperature of the metal strip is periodic; it has an amplitude of T 1.37 K at a relatively low heating frequency of 2 rad/s and 0.71 K at a frequency of 200 rad/s. Determine the thermal conductivity of the nanostructured dielectric material. The density and specific heats of the conventional version of the material are 3100 kg/m3 and 820 J/kg 䡠 K, respectively.

5.9

䊏

329

Periodic Heating

SOLUTION Known: Dimensions of a thin metal strip, the frequency and amplitude of the electric power dissipated within the strip, the amplitude of the induced oscillating strip temperature, and the thickness of the underlying nanostructured material. Find: The thermal conductivity of the nanostructured material. Schematic: Heated metal strip

Connector pad

L I⫹

E⫹

E⫺

x

I⫺

y d

Sample z

Assumptions: 1. Two-dimensional transient conduction in the x- and z-directions. 2. Constant properties. 3. Negligible radiation and convection losses from the metal strip and top surface of the sample. 4. The nanostructured material sample is a semi-infinite solid. 5. Uniform heat flux at the interface between the heated strip and the nanostructured material. Analysis: Substitution of T 1.37 K at 2 rad/s and T 0.71 K at 200 rad/s into Equation 5.74 results in two equations that may be solved simultaneously to yield C2 5.35

k 1.11 W/m 䡠 K

䉰

The thermal diffusivity is ␣ 4.37 107 m2/s, while the thermal penetration depths ␣/, resulting in ␦p 260 m and ␦p 26 m at 2 rad/s are estimated by ␦p ⬇ 兹苶 and 200 rad/s, respectively.

Comments: 1. The foregoing experimental technique, which is widely used to measure the thermal conductivity of microscale devices and nanostructured materials, is referred to as the 3 method [10]. 2. Because this technique is based on measurement of a temperature that fluctuates about a mean value that is approximately the same as the temperature of the surroundings, the measured value of k is relatively insensitive to radiation heat transfer losses from the top of the metal strip. Likewise, the technique is insensitive to thermal contact resistances that may exist at the interface between the sensing strip and the underlying material since these effects cancel when measurements are made at two different excitation frequencies [10].

330

Chapter 5

䊏

Transient Conduction

3. The specific heat and density are not strongly dependent on the nanostructure of most solids, and properties of conventional material may be used. 4. The thermal penetration depth is less than the sample thickness. Therefore, treating the sample as a semi-infinite solid is a valid approach. Thinner samples could be used if higher heating frequencies were employed.

5.10 Finite-Difference Methods Analytical solutions to transient problems are restricted to simple geometries and boundary conditions, such as the one-dimensional cases considered in the preceding sections. For some simple two- and three-dimensional geometries, analytical solutions are still possible. However, in many cases the geometry and/or boundary conditions preclude the use of analytical techniques, and recourse must be made to finite-difference (or finite-element) methods. Such methods, introduced in Section 4.4 for steady-state conditions, are readily extended to transient problems. In this section we consider explicit and implicit forms of finite-difference solutions to transient conduction problems.

5.10.1

Discretization of the Heat Equation: The Explicit Method

Once again consider the two-dimensional system of Figure 4.4. Under transient conditions with constant properties and no internal generation, the appropriate form of the heat equation, Equation 2.21, is 1 ⭸T ⭸2T ⭸2T ␣ ⭸t ⭸x2 ⭸y2

(5.75)

To obtain the finite-difference form of this equation, we may use the central-difference approximations to the spatial derivatives prescribed by Equations 4.27 and 4.28. Once again the m and n subscripts may be used to designate the x- and y-locations of discrete nodal points. However, in addition to being discretized in space, the problem must be discretized in time. The integer p is introduced for this purpose, where t pt

(5.76)

and the finite-difference approximation to the time derivative in Equation 5.75 is expressed as ⭸T ⭸t

冏

艐 m, n

p 1 p T m, n T m, n t

(5.77)

The superscript p is used to denote the time dependence of T, and the time derivative is expressed in terms of the difference in temperatures associated with the new (p 1) and previous ( p) times. Hence calculations must be performed at successive times separated by the interval t, and just as a finite-difference solution restricts temperature determination to discrete points in space, it also restricts it to discrete points in time.

Analytical solutions for some simple two- and three-dimensional geometries are found in Section 5S.2.

5.10

䊏

331

Finite-Difference Methods

If Equation 5.77 is substituted into Equation 5.75, the nature of the finite-difference solution will depend on the specific time at which temperatures are evaluated in the finite-difference approximations to the spatial derivatives. In the explicit method of solution, these temperatures are evaluated at the previous ( p) time. Hence Equation 5.77 is considered to be a forward-difference approximation to the time derivative. Evaluating terms on the right-hand side of Equations 4.27 and 4.28 at p and substituting into Equation 5.75, the explicit form of the finite-difference equation for the interior node (m, n) is p 1 p p p p p p p 1 T m, n T m, n T m 1, n T m1, n 2T m, n T m, n 1 T m, n1 2T m,n ␣ t (x)2 (y)2

(5.78)

Solving for the nodal temperature at the new (p 1) time and assuming that x y, it follows that p 1 p p p p p T m, n Fo(T m 1, n T m1, n T m, n 1 T m, n1) (1 4Fo)T m, n

(5.79)

where Fo is a finite-difference form of the Fourier number Fo ␣ t2 (x)

(5.80)

This approach can easily be extended to one- or three-dimensional systems. If the system is one-dimensional in x, the explicit form of the finite-difference equation for an interior node m reduces to p p

T m1 ) (1 2Fo)T mp T mp 1 Fo(T m 1

(5.81)

Equations 5.79 and 5.81 are explicit because unknown nodal temperatures for the new time are determined exclusively by known nodal temperatures at the previous time. Hence calculation of the unknown temperatures is straightforward. Since the temperature of each interior node is known at t 0 ( p 0) from prescribed initial conditions, the calculations begin at t t ( p 1), where Equation 5.79 or 5.81 is applied to each interior node to determine its temperature. With temperatures known for t t, the appropriate finite-difference equation is then applied at each node to determine its temperature at t 2 t ( p 2). In this way, the transient temperature distribution is obtained by marching out in time, using intervals of t. The accuracy of the finite-difference solution may be improved by decreasing the values of x and t. Of course, the number of interior nodal points that must be considered increases with decreasing x, and the number of time intervals required to carry the solution to a prescribed final time increases with decreasing t. Hence the computation time increases with decreasing x and t. The choice of x is typically based on a compromise between accuracy and computational requirements. Once this selection has been made, however, the value of t may not be chosen independently. It is, instead, determined by stability requirements. An undesirable feature of the explicit method is that it is not unconditionally stable. In a transient problem, the solution for the nodal temperatures should continuously approach final (steady-state) values with increasing time. However, with the explicit method, this solution may be characterized by numerically induced oscillations, which are physically impossible. The oscillations may become unstable, causing the solution to diverge from the actual steady-state conditions. To prevent such erroneous results, the prescribed value of t must be maintained below a certain limit, which depends on x and other parameters of the system. This dependence is termed a stability criterion, which may be obtained mathematically or demonstrated from a thermodynamic argument (see Problem 5.108). For the problems of interest in this text, the criterion is determined by requiring that the coefficient associated with the node of interest at the previous time is greater than or equal to zero.

332

Chapter 5

䊏

Transient Conduction

p In general, this is done by collecting all terms involving T m,n to obtain the form of the coefficient. This result is then used to obtain a limiting relation involving Fo, from which the maximum allowable value of t may be determined. For example, with Equations 5.79 and 5.81 already expressed in the desired form, it follows that the stability criterion for a onedimensional interior node is (1 2Fo) 0, or

Fo 1 2

(5.82)

and for a two-dimensional node, it is (1 4Fo) 0, or Fo 1 4

(5.83)

For prescribed values of x and ␣, these criteria may be used to determine upper limits to the value of t. Equations 5.79 and 5.81 may also be derived by applying the energy balance method of Section 4.4.3 to a control volume about the interior node. Accounting for changes in thermal energy storage, a general form of the energy balance equation may be expressed as E˙ in E˙ g E˙ st

(5.84)

In the interest of adopting a consistent methodology, it is again assumed that all heat flow is into the node. To illustrate application of Equation 5.84, consider the surface node of the onedimensional system shown in Figure 5.12. To more accurately determine thermal conditions near the surface, this node has been assigned a thickness that is one-half that of the interior nodes. Assuming convection transfer from an adjoining fluid and no generation, it follows from Equation 5.84 that hA(T앝 T 0p ) kA (T 1p T 0p ) cA x x 2

T 0p 1 T 0p t

or, solving for the surface temperature at t t, (T 1p T 0p ) T 0p T 0p 1 2h t (T앝 T 0p ) 2␣ t 2 c x x x A T∞, h T0

T1

T2

T3

•

qconv

E st

qcond

∆x ___ 2

∆x

FIGURE 5.12 Surface node with convection and one-dimensional transient conduction.

5.10

䊏

333

Finite-Difference Methods

Recognizing that (2ht/cx) 2(hx/k)(␣t/x2) 2 Bi Fo and grouping terms involving T 0p, it follows that T 0p 1 2Fo(T 1p Bi T앝) (1 2Fo 2Bi Fo)T 0p

(5.85)

The finite-difference form of the Biot number is Bi h x k

(5.86)

Recalling the procedure for determining the stability criterion, we require that the coefficient for T 0p be greater than or equal to zero. Hence 1 2Fo 2Bi Fo 0 or Fo(1 Bi) 1 2

(5.87)

Since the complete finite-difference solution requires the use of Equation 5.81 for the interior nodes, as well as Equation 5.85 for the surface node, Equation 5.87 must be contrasted with Equation 5.82 to determine which requirement is more stringent. Since Bi 0, it is apparent that the limiting value of Fo for Equation 5.87 is less than that for Equation 5.82. To ensure stability for all nodes, Equation 5.87 should therefore be used to select the maximum allowable value of Fo, and hence t, to be used in the calculations. Forms of the explicit finite-difference equation for several common geometries are presented in Table 5.3a. Each equation may be derived by applying the energy balance method to a control volume about the corresponding node. To develop confidence in your ability to apply this method, you should attempt to verify at least one of these equations.

EXAMPLE 5.11 A fuel element of a nuclear reactor is in the shape of a plane wall of thickness 2L 20 mm and is convectively cooled at both surfaces, with h 1100 W/m2 䡠 K and T앝 250 C. At normal operating power, heat is generated uniformly within the element at a volumetric rate of q· 1 107 W/m3. A departure from the steady-state conditions associated with normal operation will occur if there is a change in the generation rate. Consider a sudden change to q· 2 2 107 W/m3, and use the explicit finite-difference method to determine the fuel element temperature distribution after 1.5 s. The fuel element thermal properties are k 30 W/m 䡠 K and ␣ 5 106 m2/s.

SOLUTION Known: Conditions associated with heat generation in a rectangular fuel element with surface cooling. Find: Temperature distribution 1.5 s after a change in operating power.

334

Transient, two-dimensional finite-difference equations (x y)

Chapter 5

TABLE 5.3

(a) Explicit Method Configuration

Finite-Difference Equation

Stability Criterion

(b) Implicit Method

䊏

∆y

m, n m – 1, n

m + 1, n

∆x

p 1 p p Fo(Tm 1,n

Tm1,n Tm,n p p

Tm,n 1 Tm,n1) p

(1 4Fo)Tm,n

m, n – 1

1. Interior node

m, n + 1

p 1 p p 3Fo(Tm 1,n

2Tm1,n Tm,n p p

2Tm,n 1 Tm,n1 2Bi T앝)

Fo

1 4

(5.83)

p 1 p 1 p 1 (1 4Fo)Tm,n Fo(Tm 1,n

Tm1,n p 1 p 1 p

Tm,n 1

Tm,n1 ) Tm,n

(5.95)

(5.79)

∆x

m – 1, n

2

m, n m + 1, n

∆y

T∞, h m, n – 1

m, n + 1 ∆y

T∞, h m, n

m – 1, n

p

(1 4Fo 43 Bi Fo)T m,n

(5.89)

p 1 3Fo 䡠 (1 4Fo(1 3Bi))Tm,n p 1 p 1 p 1 p 1 (Tm 1,n 2T m1,n 2Tm,n 1

Tm,n1 ) 4 p Tm,n 3 Bi Fo T앝 (5.98)

(5.91)

p 1 (1 2Fo(2 Bi))Tm,n p 1 p 1 p 1 Fo(2Tm1,n Tm,n 1

Tm,n1 ) p

2Bi Fo T앝 Tm,n

1

Fo(3 Bi)

3 4

Fo(2 Bi)

1 2

(5.88)

2

2. Node at interior corner with convection p 1 p p Fo(2Tm1,n

Tm,n 1 Tm,n p

Tm,n1 2Bi T앝) p

(1 4Fo2Bi Fo)Tm,n

(5.90)

(5.99)

m, n – 1

3. Node at plane surface with convectiona

∆x

T∞, h

m – 1, n

m, n ∆y

p 1 p p Tm,n 2Fo(Tm1,n

Tm,n1

2Bi T앝) p

(1 4Fo 4Bi Fo)Tm,n (5.92)

m, n – 1 ∆x

a

4. Node at exterior corner with convection

Fo(1 Bi)

1 4

(5.93)

p 1 (1 4Fo(1 Bi))Tm,n p 1 p 1 2Fo(Tm1,n

Tm,n1 ) p Tm,n 4Bi Fo T앝

To obtain the finite-difference equation and/or stability criterion for an adiabatic surface (or surface of symmetry), simply set Bi equal to zero.

(5.100)

Transient Conduction

m, n + 1

5.10

䊏

335

Finite-Difference Methods

Schematic: Fuel element q•1 = 1 × 107 W/m3 q•2 = 2 × 107 W/m3 α = 5 × 10–6 m2/s k = 30 W/m•K

m–1

m

T∞ = 250°C h = 1100 W/m2•K Coolant

Symmetry adiabat

1

m+1

2

3

4 5

0 •

qcond

5

4 •

•

E g, E st

Eg, qcond

qcond

•

Est

L = 10 mm

qconv

x ∆ x = _L_ 5

L ∆ x = __ ___ 2 10

Assumptions: 1. One-dimensional conduction in x. 2. Uniform generation. 3. Constant properties. Analysis: A numerical solution will be obtained using a space increment of x 2 mm. Since there is symmetry about the midplane, the nodal network yields six unknown nodal temperatures. Using the energy balance method, Equation 5.84, an explicit finite-difference equation may be derived for any interior node m. kA

p T m1 T mp T p T mp T p 1 Tmp

q˙ A x A x c m

kA m 1 x x t

Solving for T p 1 and rearranging, m

冤

p p T mp 1 Fo T m1

T m 1

冥

q˙ (x)2

(1 2Fo)T mp k

(1)

This equation may be used for node 0, with T pm1 T pm 1, as well as for nodes 1, 2, 3, and 4. Applying energy conservation to a control volume about node 5, hA(T앝 T 5p ) k A

T 4p T 5p T p 1 T 5p

q˙ A x A x c 5 x 2 2 t

or

冤

T 5p 1 2Fo T 4p Bi T앝

冥

q˙ (x)2

(1 2Fo 2Bi Fo)T 5p 2k

(2)

Since the most restrictive stability criterion is associated with Equation 2, we select Fo from the requirement that Fo(1 Bi) 1 2

336

Chapter 5

䊏

Transient Conduction

Hence, with Bi h x k

1100 W/m2 䡠 K (0.002 m) 0.0733 30 W/m 䡠 K

it follows that Fo 0.466 or t

Fo(x)2 0.466(2 103 m)2 0.373 s ␣ 5 106 m2/s

To be well within the stability limit, we select t 0.3 s, which corresponds to Fo

5 106 m2/s(0.3 s) 0.375 (2 103 m)2

Substituting numerical values, including q˙ q˙ 2 2 107 W/m3, the nodal equations become T 0p 1 0.375(2T 1p 2.67) 0.250T 0p T 1p 1 0.375(T 0p T 2p 2.67) 0.250T 1p T 2p 1 0.375(T 1p T 3p 2.67) 0.250T 2p T 3p 1 0.375(T 2p T 4p 2.67) 0.250T 3p T 4p 1 0.375(T 3p T 5p 2.67) 0.250T 4p T 5p 1 0.750(T 4p 19.67) 0.195T 5p To begin the marching solution, the initial temperature distribution must be known. This distribution is given by Equation 3.47, with q˙ q˙ 1. Obtaining Ts T5 from Equation 3.51, T5 T앝

7 3 q˙ L m 340.91 C 250 C 10 W/m 0.01 h 1100 W/m2 䡠 K

it follows that

冢

2

冣

T(x) 16.67 1 x 2 340.91 C L Computed temperatures for the nodal points of interest are shown in the first row of the accompanying table. Using the finite-difference equations, the nodal temperatures may be sequentially calculated with a time increment of 0.3 s until the desired final time is reached. The results are illustrated in rows 2 through 6 of the table and may be contrasted with the new steady-state condition (row 7), which was obtained by using Equations 3.47 and 3.51 with q˙ q˙2:

5.10

䊏

337

Finite-Difference Methods

Tabulated Nodal Temperatures p

t(s)

T0

T1

T2

T3

T4

T5

0 1 2 3 4 5 앝

0 0.3 0.6 0.9 1.2 1.5 앝

357.58 358.08 358.58 359.08 359.58 360.08 465.15

356.91 357.41 357.91 358.41 358.91 359.41 463.82

354.91 355.41 355.91 356.41 356.91 357.41 459.82

351.58 352.08 352.58 353.08 353.58 354.07 453.15

346.91 347.41 347.91 348.41 348.89 349.37 443.82

340.91 341.41 341.88 342.35 342.82 343.27 431.82

Comments: 1. It is evident that, at 1.5 s, the wall is in the early stages of the transient process and that many additional calculations would have to be made to reach steady-state conditions with the finite-difference solution. The computation time could be reduced slightly by using the maximum allowable time increment (t 0.373 s), but with some loss of accuracy. In the interest of maximizing accuracy, the time interval should be reduced until the computed results become independent of further reductions in t. Extending the finite-difference solution, the time required to achieve the new steady-state condition may be determined, with temperature histories computed for the midplane (0) and surface (5) nodes having the following forms: 480 465.1

T0

T (°C)

440 431.8

T5 400

360

320

0

100

200 t (s)

300

400

With steady-state temperatures of T0 465.15 C and T5 431.82 C, it is evident that the new equilibrium condition is reached within 250 s of the step change in operating power. 2. This problem can be solved using Tools, Finite-Difference Equations, One-Dimensional, Transient in the Advanced section of IHT. The problem may also be solved using FiniteElement Heat Transfer (FEHT).

5.10.2

Discretization of the Heat Equation: The Implicit Method

In the explicit finite-difference scheme, the temperature of any node at t t may be calculated from knowledge of temperatures at the same and neighboring nodes for the preceding time t. Hence determination of a nodal temperature at some time is independent of

338

Chapter 5

䊏

Transient Conduction

temperatures at other nodes for the same time. Although the method offers computational convenience, it suffers from limitations on the selection of t. For a given space increment, the time interval must be compatible with stability requirements. Frequently, this dictates the use of extremely small values of t, and a very large number of time intervals may be necessary to obtain a solution. A reduction in the amount of computation time may often be realized by employing an implicit, rather than explicit, finite-difference scheme. The implicit form of a finite-difference equation may be derived by using Equation 5.77 to approximate the time derivative, while evaluating all other temperatures at the new (p 1) time, instead of the previous (p) time. Equation 5.77 is then considered to provide a backward-difference approximation to the time derivative. In contrast to Equation 5.78, the implicit form of the finite-difference equation for the interior node of a two-dimensional system is then p 1 p p 1 p 1 p 1 1 T m, n T m, n T m 1, n T m1, n 2T m, n ␣ t (x)2

p 1 p 1 p 1 T m, n 1 T m, n1 2T m, n

(y)2

(5.94)

Rearranging and assuming x y, it follows that p 1 p 1 p 1 p 1 p 1 p (1 4Fo)T m, n Fo(T m 1, n T m1, n T m, n 1 T m, n1) T m, n

(5.95)

From Equation 5.95 it is evident that the new temperature of the (m, n) node depends on the new temperatures of its adjoining nodes, which are, in general, unknown. Hence, to determine the unknown nodal temperatures at t t, the corresponding nodal equations must be solved simultaneously. Such a solution may be effected by using Gauss–Seidel iteration or matrix inversion, as discussed in Section 4.5 and Appendix D. The marching solution would then involve simultaneously solving the nodal equations at each time t t, 2t, . . . , until the desired final time was reached. Relative to the explicit method, the implicit formulation has the important advantage of being unconditionally stable. That is, the solution remains stable for all space and time intervals, in which case there are no restrictions on x and t. Since larger values of t may therefore be used with an implicit method, computation times may often be reduced, with little loss of accuracy. Nevertheless, to maximize accuracy, t should be sufficiently small to ensure that the results are independent of further reductions in its value. The implicit form of a finite-difference equation may also be derived from the energy balance method. For the surface node of Figure 5.12, it is readily shown that (1 2Fo 2Fo Bi)T 0p 1 2Fo T 1p 1 2Fo Bi T앝 T 0p

(5.96)

For any interior node of Figure 5.12, it may also be shown that p 1 p 1

T m 1 ) T mp (1 2Fo)T mp 1 Fo (T m1

(5.97)

Forms of the implicit finite-difference equation for other common geometries are presented in Table 5.3b. Each equation may be derived by applying the energy balance method.

5.10

䊏

339

Finite-Difference Methods

EXAMPLE 5.12 A thick slab of copper initially at a uniform temperature of 20 C is suddenly exposed to radiation at one surface such that the net heat flux is maintained at a constant value of 3 105 W/m2. Using the explicit and implicit finite-difference techniques with a space increment of x 75 mm, determine the temperature at the irradiated surface and at an interior point that is 150 mm from the surface after 2 min have elapsed. Compare the results with those obtained from an appropriate analytical solution.

SOLUTION Known: Thick slab of copper, initially at a uniform temperature, is subjected to a constant net heat flux at one surface. Find: 1. Using the explicit finite-difference method, determine temperatures at the surface and 150 mm from the surface after an elapsed time of 2 min. 2. Repeat the calculations using the implicit finite-difference method. 3. Determine the same temperatures analytically. Schematic: q"o = 3 × 105 W/m2

0

q"o

m–1

1

q"cond

q"cond

∆x ___ 2

x

m

m+1 q"cond

∆x = 75 mm

Assumptions: 1. One-dimensional conduction in x. 2. For the analytical solution, the thick slab may be approximated as a semi-infinite medium with constant surface heat flux. For the finite-difference solutions, implementation of the boundary condition T(x l 앝) Ti will be discussed below in this example. 3. Constant properties. Properties: Table A.1, copper (300 K): k 401 W/m 䡠 K, ␣ 117 106 m2/s. Analysis: 1. An explicit form of the finite-difference equation for the surface node may be obtained by applying an energy balance to a control volume about the node. qo A kA

T 1p T 0p T p 1 T 0p A x c 0 x 2 t

340

Chapter 5

䊏

Transient Conduction

or T 0p 1 2Fo

冢qkx T 冣 (1 2Fo)T 0

p 1

p 0

The finite-difference equation for any interior node is given by Equation 5.81. Both the surface and interior nodes are governed by the stability criterion Fo 1 2 Noting that the finite-difference equations are simplified by choosing the maxi1 mum allowable value of Fo, we select Fo 2. Hence (x)2 (0.075 m)2 24 s t Fo ␣ 1 2 117 106 m2/s With qo x 3 105 W/m2 (0.075 m) 56.1 C k 401 W/m 䡠 K the finite-difference equations become T 0p 1 56.1 C T 1p

and

T mp 1

p p T m 1

T m1 2

for the surface and interior nodes, respectively. Performing the calculations, the results are tabulated as follows:

Explicit Finite-Difference Solution for Fo 2 1

p

t(s)

T0

T1

T2

T3

T4

0 1 2 3 4 5

0 24 48 72 96 120

20 76.1 76.1 104.2 104.2 125.2

20 20 48.1 48.1 69.1 69.1

20 20 20 34.0 34.0 48.1

20 20 20 20 27.0 27.0

20 20 20 20 20 23.5

After 2 min, the surface temperature and the desired interior temperature are T0 125.2 C and T2 48.1 C. It can be seen from the explicit finite-difference solution that, with each successive time step, one more nodal temperature changes from its initial condition. For this reason, it is not necessary to formally implement the second boundary condition T(x l 앝) T. Also note that calculation of identical temperatures at successive times for the same node is an idiosyncrasy of using the maximum allowable value of Fo with the explicit finitedifference technique. The actual physical condition is, of course, one in which the temperature changes continuously with time. The idiosyncrasy is diminished and the accuracy of the calculations is improved by reducing the value of Fo.

5.10

䊏

341

Finite-Difference Methods

To determine the extent to which the accuracy may be improved by reducing Fo, 1 let us redo the calculations for Fo 4 (t 12 s). The finite-difference equations are then of the form T 0p 1 1 (56.1 C T 1p) 1T 0p 2 2 p p T mp 1 1(T m 1

T m1 ) 1T mp 4 2 and the results of the calculations are tabulated as follows:

Explicit Finite-Difference Solution for Fo 4 1

p

t(s)

T0

T1

T2

0 1 2 3 4 5 6 7 8 9 10

0 12 24 36 48 60 72 84 96 108 120

20 48.1 62.1 72.6 81.4 89.0 95.9 102.3 108.1 113.6 118.8

20 20 27.0 34.0 40.6 46.7 52.5 57.9 63.1 67.9 72.6

20 20 20 21.8 24.4 27.5 30.7 34.1 37.6 41.0 44.4

T3

T4

20 20 20 20 2

SIXTH EDITION

Introduction to Heat Transfer THEODORE L. BERGMAN Department of Mechanical Engineering University of Connecticut

ADRIENNE S. LAVINE Mechanical and Aerospace Engineering Department University of California, Los Angeles

FRANK P. INCROPERA College of Engineering University of Notre Dame

DAVID P. DEWITT School of Mechanical Engineering Purdue University

JOHN WILEY & SONS, INC.

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This book was typeset in 10.5/12 Times Roman by MPS Limited, a Macmillan Company and printed and bound by R. R. Donnelley (Jefferson City). The cover was printed by R. R. Donnelley (Jefferson City). Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations. Our company is built on a foundation of principles that include responsibility to the communities we serve and where we live and work. In 2008, we launched a Corporate Citizenship Initiative, a global effort to address the environmental, social, economic, and ethical challenges we face in our business. Among the issues we are addressing are carbon impact, paper specifications and procurement, ethical conduct within our business and among our vendors, and community and charitable support. For more information, please visit our website: www.wiley.com/go/citizenship. The paper in this book was manufactured by a mill whose forest management programs include sustained yield harvesting of its timberlands. Sustained yield-harvesting principles ensure that the number of trees cut each year does not exceed the amount of new growth. This book is printed on acid-free paper. Copyright © 2011, 2007, 2002 by John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008. Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. If you have chosen to adopt this textbook for use in your course, please accept this book as your complimentary desk copy. Outside of the United States, please contact your local representative.

ISBN 13 978-0470-50196-2 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Preface

In the Preface to the previous edition, we posed questions regarding trends in engineering education and practice, and whether the discipline of heat transfer would remain relevant. After weighing various arguments, we concluded that the future of engineering was bright and that heat transfer would remain a vital and enabling discipline across a range of emerging technologies including but not limited to information technology, biotechnology, pharmacology, and alternative energy generation. Since we drew these conclusions, many changes have occurred in both engineering education and engineering practice. Driving factors have been a contracting global economy, coupled with technological and environmental challenges associated with energy production and energy conversion. The impact of a weak global economy on higher education has been sobering. Colleges and universities around the world are being forced to set priorities and answer tough questions as to which educational programs are crucial, and which are not. Was our previous assessment of the future of engineering, including the relevance of heat transfer, too optimistic? Faced with economic realities, many colleges and universities have set clear priorities. In recognition of its value and relevance to society, investment in engineering education has, in many cases, increased. Pedagogically, there is renewed emphasis on the fundamental principles that are the foundation for lifelong learning. The important and sometimes dominant role of heat transfer in many applications, particularly in conventional as well as in alternative energy generation and concomitant environmental effects, has reaffirmed its relevance. We believe our previous conclusions were correct: The future of engineering is bright, and heat transfer is a topic that is crucial to address a broad array of technological and environmental challenges. In preparing this edition, we have sought to incorporate recent heat transfer research at a level that is appropriate for an undergraduate student. We have strived to include new examples and problems that motivate students with interesting applications, but whose solutions are based firmly on fundamental principles. We have remained true to the pedagogical approach of previous editions by retaining a rigorous and systematic methodology for problem solving. We have attempted to continue the tradition of providing a text that will serve as a valuable, everyday resource for students and practicing engineers throughout their careers.

iv

Preface

Approach and Organization Previous editions of the text have adhered to four learning objectives: 1. The student should internalize the meaning of the terminology and physical principles associated with heat transfer. 2. The student should be able to delineate pertinent transport phenomena for any process or system involving heat transfer. 3. The student should be able to use requisite inputs for computing heat transfer rates and/or material temperatures. 4. The student should be able to develop representative models of real processes and systems and draw conclusions concerning process/system design or performance from the attendant analysis. Moreover, as in previous editions, specific learning objectives for each chapter are clarified, as are means by which achievement of the objectives may be assessed. The summary of each chapter highlights key terminology and concepts developed in the chapter and poses questions designed to test and enhance student comprehension. It is recommended that problems involving complex models and/or exploratory, whatif, and parameter sensitivity considerations be addressed using a computational equationsolving package. To this end, the Interactive Heat Transfer (IHT) package available in previous editions has been updated. Specifically, a simplified user interface now delineates between the basic and advanced features of the software. It has been our experience that most students and instructors will use primarily the basic features of IHT. By clearly identifying which features are advanced, we believe students will be motivated to use IHT on a daily basis. A second software package, Finite Element Heat Transfer (FEHT), developed by F-Chart Software (Madison, Wisconsin), provides enhanced capabilities for solving two-dimensional conduction heat transfer problems. To encourage use of IHT, a Quickstart User’s Guide has been installed in the software. Students and instructors can become familiar with the basic features of IHT in approximately one hour. It has been our experience that once students have read the Quickstart guide, they will use IHT heavily, even in courses other than heat transfer. Students report that IHT significantly reduces the time spent on the mechanics of lengthy problem solutions, reduces errors, and allows more attention to be paid to substantive aspects of the solution. Graphical output can be generated for homework solutions, reports, and papers. As in previous editions, some homework problems require a computer-based solution. Other problems include both a hand calculation and an extension that is computer based. The latter approach is time-tested and promotes the habit of checking a computer-generated solution with a hand calculation. Once validated in this manner, the computer solution can be utilized to conduct parametric calculations. Problems involving both hand- and computer-generated solutions are identified by enclosing the exploratory part in a red rectangle, as, for example, (b) , (c) , or (d) . This feature also allows instructors who wish to limit their assignments of computer-based problems to benefit from the richness of these problems without assigning their computer-based parts. Solutions to problems for which the number is highlighted (for example, 1.19 ) are entirely computer based.

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v

What’s New in the Sixth Edition In the previous edition, Chapter 1 Introduction was modified to emphasize the relevance of heat transfer in various contemporary applications. Responding to today’s challenges involving energy production and its environmental impact, an expanded discussion of the efficiency of energy conversion and the production of greenhouse gases has been added. Chapter 1 has also been modified to embellish the complementary nature of heat transfer and thermodynamics. The existing treatment of the first law of thermodynamics is augmented with a new section on the relationship between heat transfer and the second law of thermodynamics as well as the efficiency of heat engines. Indeed, the influence of heat transfer on the efficiency of energy conversion is a recurring theme throughout this edition. The coverage of micro- and nanoscale effects in Chapter 2 Introduction to Conduction has been updated, reflecting recent advances. For example, the description of the thermophysical properties of composite materials is enhanced, with a new discussion of nanofluids. Chapter 3 One-Dimensional, Steady-State Conduction has undergone extensive revision and includes new material on conduction in porous media, thermoelectric power generation, and micro- as well as nanoscale systems. Inclusion of these new topics follows recent fundamental discoveries and is presented through the use of the thermal resistance network concept. Hence the power and utility of the resistance network approach is further emphasized in this edition. Chapter 4 Two-Dimensional, Steady-State Conduction has been reduced in length. Today, systems of linear, algebraic equations are readily solved using standard computer software or even handheld calculators. Hence the focus of the shortened chapter is on the application of heat transfer principles to derive the systems of algebraic equations to be solved and on the discussion and interpretation of results. The discussion of Gauss–Seidel iteration has been moved to an appendix for instructors wishing to cover that material. Chapter 5 Transient Conduction was substantially modified in the previous edition and has been augmented in this edition with a streamlined presentation of the lumpedcapacitance method. Chapter 6 Introduction to Convection includes clarification of how temperature-dependent properties should be evaluated when calculating the convection heat transfer coefficient. The fundamental aspects of compressible flow are introduced to provide the reader with guidelines regarding the limits of applicability of the treatment of convection in the text. Chapter 7 External Flow has been updated and reduced in length. Specifically, presentation of the similarity solution for flow over a flat plate has been simplified. New results for flow over noncircular cylinders have been added, replacing the correlations of previous editions. The discussion of flow across banks of tubes has been shortened, eliminating redundancy without sacrificing content. Chapter 8 Internal Flow entry length correlations have been updated, and the discussion of micro- and nanoscale convection has been modified and linked to the content of Chapter 3. Changes to Chapter 9 Free Convection include a new correlation for free convection from flat plates, replacing a correlation from previous editions. The discussion of boundary layer effects has been modified. Aspects of condensation included in Chapter 10 Boiling and Condensation have been updated to incorporate recent advances in, for example, external condensation on finned tubes. The effects of surface tension and the presence of noncondensable gases in modifying

Chapter-by-Chapter Content Changes

vi

Preface

condensation phenomena and heat transfer rates are elucidated. The coverage of forced convection condensation and related enhancement techniques has been expanded, again reflecting advances reported in the recent literature. The content of Chapter 11 Heat Exchangers is experiencing a resurgence in interest due to the critical role such devices play in conventional and alternative energy generation technologies. A new section illustrates the applicability of heat exchanger analysis to heat sink design and materials processing. Much of the coverage of compact heat exchangers included in the previous edition was limited to a specific heat exchanger. Although general coverage of compact heat exchangers has been retained, the discussion that is limited to the specific heat exchanger has been relegated to supplemental material, where it is available to instructors who wish to cover this topic in greater depth. The concepts of emissive power, irradiation, radiosity, and net radiative flux are now introduced early in Chapter 12 Radiation: Processes and Properties, allowing early assignment of end-of-chapter problems dealing with surface energy balances and properties, as well as radiation detection. The coverage of environmental radiation has undergone substantial revision, with the inclusion of separate discussions of solar radiation, the atmospheric radiation balance, and terrestrial solar irradiation. Concern for the potential impact of anthropogenic activity on the temperature of the earth is addressed and related to the concepts of the chapter. Much of the modification to Chapter 13 Radiation Exchange Between Surfaces emphasizes the difference between geometrical surfaces and radiative surfaces, a key concept that is often difficult for students to appreciate. Increased coverage of radiation exchange between multiple blackbody surfaces, included in older editions of the text, has been returned to Chapter 13. In doing so, radiation exchange between differentially small surfaces is briefly introduced and used to illustrate the limitations of the analysis techniques included in Chapter 13. Problem Sets Approximately 225 new end-of-chapter problems have been developed for this edition. An effort has been made to include new problems that (a) are amenable to short solutions or (b) involve finite-difference solutions. A significant number of solutions to existing end-of-chapter problems have been modified due to the inclusion of the new convection correlations in this edition.

Classroom Coverage The content of the text has evolved over many years in response to a variety of factors. Some factors are obvious, such as the development of powerful, yet inexpensive calculators and software. There is also the need to be sensitive to the diversity of users of the text, both in terms of (a) the broad background and research interests of instructors and (b) the wide range of missions associated with the departments and institutions at which the text is used. Regardless of these and other factors, it is important that the four previously identified learning objectives be achieved. Mindful of the broad diversity of users, the authors’ intent is not to assemble a text whose content is to be covered, in entirety, during a single semester- or quarter-long course. Rather, the text includes both (a) fundamental material that we believe must be covered and (b) optional material that instructors can use to address specific interests or that can be

Preface

vii

covered in a second, intermediate heat transfer course. To assist instructors in preparing a syllabus for a first course in heat transfer, we have several recommendations. Chapter 1 Introduction sets the stage for any course in heat transfer. It explains the linkage between heat transfer and thermodynamics, and it reveals the relevance and richness of the subject. It should be covered in its entirety. Much of the content of Chapter 2 Introduction to Conduction is critical in a first course, especially Section 2.1 The Conduction Rate Equation, Section 2.3 The Heat Diffusion Equation, and Section 2.4 Boundary and Initial Conditions. It is recommended that Chapter 2 be covered in its entirety. Chapter 3 One-Dimensional, Steady-State Conduction includes a substantial amount of optional material from which instructors can pick-and-choose or defer to a subsequent, intermediate heat transfer course. The optional material includes Section 3.1.5 Porous Media, Section 3.7 The Bioheat Equation, Section 3.8 Thermoelectric Power Generation, and Section 3.9 Micro- and Nanoscale Conduction. Because the content of these sections is not interlinked, instructors may elect to cover any or all of the optional material. The content of Chapter 4 Two-Dimensional, Steady-State Conduction is important because both (a) fundamental concepts and (b) powerful and practical solution techniques are presented. We recommend that all of Chapter 4 be covered in any introductory heat transfer course. The optional material in Chapter 5 Transient Conduction is Section 5.9 Periodic Heating. Also, some instructors do not feel compelled to cover Section 5.10 Finite-Difference Methods in an introductory course, especially if time is short. The content of Chapter 6 Introduction to Convection is often difficult for students to absorb. However, Chapter 6 introduces fundamental concepts and lays the foundation for the subsequent convection chapters. It is recommended that all of Chapter 6 be covered in an introductory course. Chapter 7 External Flow introduces several important concepts and presents convection correlations that students will utilize throughout the remainder of the text and in subsequent professional practice. Sections 7.1 through 7.5 should be included in any first course in heat transfer. However, the content of Section 7.6 Flow Across Banks of Tubes, Section 7.7 Impinging Jets, and Section 7.8 Packed Beds is optional. Since the content of these sections is not interlinked, instructors may select from any of the optional topics. Likewise, Chapter 8 Internal Flow includes matter that is used throughout the remainder of the text and by practicing engineers. However, Section 8.7 Heat Transfer Enhancement, and Section 8.8 Flow in Small Channels may be viewed as optional. Buoyancy-induced flow and heat transfer is covered in Chapter 9 Free Convection. Because free convection thermal resistances are typically large, they are often the dominant resistance in many thermal systems and govern overall heat transfer rates. Therefore, most of Chapter 9 should be covered in a first course in heat transfer. Optional material includes Section 9.7 Free Convection Within Parallel Plate Channels and Section 9.9 Combined Free and Forced Convection. In contrast to resistances associated with free convection, thermal resistances corresponding to liquid-vapor phase change are typically small, and they can sometimes be neglected. Nonetheless, the content of Chapter 10 Boiling and Condensation that should be covered in a first heat transfer course includes Sections 10.1 through 10.4, Sections 10.6 through 10.8, and Section 10.11. Section 10.5 Forced Convection Boiling may be material appropriate for an intermediate heat transfer course. Similarly, Section 10.9 Film Condensation on Radial Systems and Section 10.10 Condensation in Horizontal Tubes may be either covered as time permits or included in a subsequent heat transfer course.

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Preface

We recommend that all of Chapter 11 Heat Exchangers be covered in a first heat transfer course. A distinguishing feature of the text, from its inception, is the in-depth coverage of radiation heat transfer in Chapter 12 Radiation: Processes and Properties. The content of the chapter is perhaps more relevant today than ever, with applications ranging from advanced manufacturing, to radiation detection and monitoring, to environmental issues related to global climate change. Although Chapter 12 has been reorganized to accommodate instructors who may wish to skip ahead to Chapter 13 after Section 12.4, we encourage instructors to cover Chapter 12 in its entirety. Chapter 13 Radiation Exchange Between Surfaces may be covered as time permits or in an intermediate heat transfer course.

Acknowledgments We wish to acknowledge and thank many of our colleagues in the heat transfer community. In particular, we would like to express our appreciation to Diana Borca-Tasciuc of the Rensselaer Polytechnic Institute and David Cahill of the University of Illinois UrbanaChampaign for their assistance in developing the periodic heating material of Chapter 5. We thank John Abraham of the University of St. Thomas for recommendations that have led to an improved treatment of flow over noncircular tubes in Chapter 7. We are very grateful to Ken Smith, Clark Colton, and William Dalzell of the Massachusetts Institute of Technology for the stimulating and detailed discussion of thermal entry effects in Chapter 8. We acknowledge Amir Faghri of the University of Connecticut for his advice regarding the treatment of condensation in Chapter 10. We extend our gratitude to Ralph Grief of the University of California, Berkeley for his many constructive suggestions pertaining to material throughout the text. Finally, we wish to thank the many students, instructors, and practicing engineers from around the globe who have offered countless interesting, valuable, and stimulating suggestions. In closing, we are deeply grateful to our spouses and children, Tricia, Nate, Tico, Greg, Elias, Jacob, Andrea, Terri, Donna, and Shaunna for their endless love and patience. We extend appreciation to Tricia Bergman who expertly processed solutions for the end-ofchapter problems. Theodore L. Bergman ([email protected]) Storrs, Connecticut Adrienne S. Lavine ([email protected]) Los Angeles, California Frank P. Incropera ([email protected]) Notre Dame, Indiana

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Supplemental and Web Site Material The companion web site for the texts is www.wiley.com/college/bergman. By selecting one of the two texts and clicking on the “student companion site” link, students may access the Answers to Selected Exercises and the Supplemental Sections of the text. Supplemental Sections are identified throughout the text with the icon shown in the margin to the left. Material available for instructors only may also be found by selecting one of the two texts at www.wiley.com/college/bergman and clicking on the “instructor companion site” link. The available content includes the Solutions Manual, PowerPoint Slides that can be used by instructors for lectures, and Electronic Versions of figures from the text for those wishing to prepare their own materials for electronic classroom presentation. The Instructor Solutions Manual is copyrighted material for use only by instructors who are requiring the text for their course.1 Interactive Heat Transfer 4.0/FEHT is available either with the text or as a separate purchase. As described by the authors in the Approach and Organization, this simple-to-use software tool provides modeling and computational features useful in solving many problems in the text, and it enables rapid what-if and exploratory analysis of many types of problems. Instructors interested in using this tool in their course can download the software from the book’s web site at www.wiley.com/college/bergman. Students can download the software by registering on the student companion site; for details, see the registration card provided in this book. The software is also available as a stand-alone purchase at the web site. Any questions can be directed to your local Wiley representative.

This mouse icon identifies Supplemental Sections and is used throughout the text. Excerpts from the Solutions Manual may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of the contents of the Solutions Manual beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. 1

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Contents

CHAPTER

Symbols

xxi

1 Introduction

1

1.1 1.2

1.3

1.4 1.5

What and How? Physical Origins and Rate Equations 1.2.1 Conduction 3 1.2.2 Convection 6 1.2.3 Radiation 8 1.2.4 The Thermal Resistance Concept 12 Relationship to Thermodynamics 1.3.1 Relationship to the First Law of Thermodynamics (Conservation of Energy) 13 1.3.2 Relationship to the Second Law of Thermodynamics and the Efficiency of Heat Engines 31 Units and Dimensions Analysis of Heat Transfer Problems: Methodology

2 3

12

36 38

xii

Contents

1.6 1.7

CHAPTER

2 Introduction to Conduction 2.1 2.2

2.3 2.4 2.5

CHAPTER

Relevance of Heat Transfer Summary References Problems

The Conduction Rate Equation The Thermal Properties of Matter 2.2.1 Thermal Conductivity 70 2.2.2 Other Relevant Properties 78 The Heat Diffusion Equation Boundary and Initial Conditions Summary References Problems

3 One-Dimensional, Steady-State Conduction 3.1

The Plane Wall 3.1.1 Temperature Distribution 112 3.1.2 Thermal Resistance 114 3.1.3 The Composite Wall 115 3.1.4 Contact Resistance 117 3.1.5 Porous Media 119 3.2 An Alternative Conduction Analysis 3.3 Radial Systems 3.3.1 The Cylinder 136 3.3.2 The Sphere 141 3.4 Summary of One-Dimensional Conduction Results 3.5 Conduction with Thermal Energy Generation 3.5.1 The Plane Wall 143 3.5.2 Radial Systems 149 3.5.3 Tabulated Solutions 150 3.5.4 Application of Resistance Concepts 150 3.6 Heat Transfer from Extended Surfaces 3.6.1 A General Conduction Analysis 156 3.6.2 Fins of Uniform Cross-Sectional Area 158 3.6.3 Fin Performance 164 3.6.4 Fins of Nonuniform Cross-Sectional Area 167 3.6.5 Overall Surface Efficiency 170 3.7 The Bioheat Equation 3.8 Thermoelectric Power Generation 3.9 Micro- and Nanoscale Conduction 3.9.1 Conduction Through Thin Gas Layers 189 3.9.2 Conduction Through Thin Solid Films 190 3.10 Summary References Problems

41 45 48 49

67 68 70

82 90 94 95 95

111 112

132 136

142 142

154

178 182 189

190 193 193

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Contents

CHAPTER

4 Two-Dimensional, Steady-State Conduction 4.1 4.2 4.3 4.4

4.5

4.6

229

Alternative Approaches The Method of Separation of Variables The Conduction Shape Factor and the Dimensionless Conduction Heat Rate Finite-Difference Equations 4.4.1 The Nodal Network 241 4.4.2 Finite-Difference Form of the Heat Equation 242 4.4.3 The Energy Balance Method 243 Solving the Finite-Difference Equations 4.5.1 Formulation as a Matrix Equation 250 4.5.2 Verifying the Accuracy of the Solution 251 Summary References Problems

4S.1 The Graphical Method 4S.1.1 Methodology of Constructing a Flux Plot W-1 4S.1.2 Determination of the Heat Transfer Rate W-2 4S.1.3 The Conduction Shape Factor W-3 4S.2 The Gauss–Seidel Method: Example of Usage References Problems CHAPTER

5.4 5.5

5.6

5.7 5.8

The Lumped Capacitance Method Validity of the Lumped Capacitance Method General Lumped Capacitance Analysis 5.3.1 Radiation Only 288 5.3.2 Negligible Radiation 288 5.3.3 Convection Only with Variable Convection Coefficient 5.3.4 Additional Considerations 289 Spatial Effects The Plane Wall with Convection 5.5.1 Exact Solution 300 5.5.2 Approximate Solution 300 5.5.3 Total Energy Transfer 302 5.5.4 Additional Considerations 302 Radial Systems with Convection 5.6.1 Exact Solutions 303 5.6.2 Approximate Solutions 304 5.6.3 Total Energy Transfer 304 5.6.4 Additional Considerations 305 The Semi-Infinite Solid Objects with Constant Surface Temperatures or Surface Heat Fluxes 5.8.1 Constant Temperature Boundary Conditions 317 5.8.2 Constant Heat Flux Boundary Conditions 319 5.8.3 Approximate Solutions 320

250

256 257 257 W-1

W-5 W-9 W-10

5 Transient Conduction 5.1 5.2 5.3

230 231 235 241

279 280 283 287

289 298 299

303

310 317

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5.9 Periodic Heating 5.10 Finite-Difference Methods 5.10.1 Discretization of the Heat Equation: The Explicit Method 5.10.2 Discretization of the Heat Equation: The Implicit Method 5.11 Summary References Problems

327 330 330 337 345 346 346

5S.1 Graphical Representation of One-Dimensional, Transient Conduction in the Plane Wall, Long Cylinder, and Sphere 5S.2 Analytical Solution of Multidimensional Effects References Problems CHAPTER

6 Introduction to Convection 6.1

6.2

6.3

6.4

6.5

6.6 6.7 6.8

377

The Convection Boundary Layers 6.1.1 The Velocity Boundary Layer 378 6.1.2 The Thermal Boundary Layer 379 6.1.3 Significance of the Boundary Layers 380 Local and Average Convection Coefficients 6.2.1 Heat Transfer 381 6.2.2 The Problem of Convection 382 Laminar and Turbulent Flow 6.3.1 Laminar and Turbulent Velocity Boundary Layers 383 6.3.2 Laminar and Turbulent Thermal Boundary Layers 385 The Boundary Layer Equations 6.4.1 Boundary Layer Equations for Laminar Flow 389 6.4.2 Compressible Flow 391 Boundary Layer Similarity: The Normalized Boundary Layer Equations 6.5.1 Boundary Layer Similarity Parameters 392 6.5.2 Functional Form of the Solutions 393 Physical Interpretation of the Dimensionless Parameters Momentum and Heat Transfer (Reynolds) Analogy Summary References Problems

6S.1 Derivation of the Convection Transfer Equations 6S.1.1 Conservation of Mass W-25 6S.1.2 Newton’s Second Law of Motion W-26 6S.1.3 Conservation of Energy W-29 References Problems CHAPTER

The Empirical Method The Flat Plate in Parallel Flow 7.2.1 Laminar Flow over an Isothermal Plate: A Similarity Solution 7.2.2 Turbulent Flow over an Isothermal Plate 424

378

381

383

388

392

400 402 404 405 405 W-25

W-35 W-35

7 External Flow 7.1 7.2

W-12 W-16 W-22 W-22

415 416 418 418

Contents

7.3 7.4

7.5 7.6 7.7

7.8 7.9

CHAPTER

7.2.3 Mixed Boundary Layer Conditions 425 7.2.4 Unheated Starting Length 426 7.2.5 Flat Plates with Constant Heat Flux Conditions 427 7.2.6 Limitations on Use of Convection Coefficients 427 Methodology for a Convection Calculation The Cylinder in Cross Flow 7.4.1 Flow Considerations 433 7.4.2 Convection Heat Transfer 436 The Sphere Flow Across Banks of Tubes Impinging Jets 7.7.1 Hydrodynamic and Geometric Considerations 456 7.7.2 Convection Heat Transfer 458 Packed Beds Summary References Problems

8 Internal Flow 8.1

8.2

8.3

8.4

8.5 8.6 8.7 8.8

8.9

Hydrodynamic Considerations 8.1.1 Flow Conditions 490 8.1.2 The Mean Velocity 491 8.1.3 Velocity Profile in the Fully Developed Region 492 8.1.4 Pressure Gradient and Friction Factor in Fully Developed Flow 494 Thermal Considerations 8.2.1 The Mean Temperature 496 8.2.2 Newton’s Law of Cooling 497 8.2.3 Fully Developed Conditions 497 The Energy Balance 8.3.1 General Considerations 501 8.3.2 Constant Surface Heat Flux 502 8.3.3 Constant Surface Temperature 505 Laminar Flow in Circular Tubes: Thermal Analysis and Convection Correlations 8.4.1 The Fully Developed Region 509 8.4.2 The Entry Region 514 8.4.3 Temperature-Dependent Properties 516 Convection Correlations: Turbulent Flow in Circular Tubes Convection Correlations: Noncircular Tubes and the Concentric Tube Annulus Heat Transfer Enhancement Flow in Small Channels 8.8.1 Microscale Convection in Gases (0.1 m ⱗ Dh ⱗ 100 m) 530 8.8.2 Microscale Convection in Liquids 531 8.8.3 Nanoscale Convection (Dh ⱗ 100 nm) 532 Summary References Problems

xv

428 433

443 447 455

461 462 464 465

489 490

495

501

509

516 524 527 530

535 537 538

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Contents

9 Free Convection 9.1 9.2 9.3 9.4 9.5 9.6

Physical Considerations The Governing Equations for Laminar Boundary Layers Similarity Considerations Laminar Free Convection on a Vertical Surface The Effects of Turbulence Empirical Correlations: External Free Convection Flows 9.6.1 The Vertical Plate 573 9.6.2 Inclined and Horizontal Plates 576 9.6.3 The Long Horizontal Cylinder 581 9.6.4 Spheres 585 9.7 Free Convection Within Parallel Plate Channels 9.7.1 Vertical Channels 587 9.7.2 Inclined Channels 589 9.8 Empirical Correlations: Enclosures 9.8.1 Rectangular Cavities 589 9.8.2 Concentric Cylinders 592 9.8.3 Concentric Spheres 593 9.9 Combined Free and Forced Convection 9.10 Summary References Problems CHAPTER

10 Boiling and Condensation 10.1 Dimensionless Parameters in Boiling and Condensation 10.2 Boiling Modes 10.3 Pool Boiling 10.3.1 The Boiling Curve 622 10.3.2 Modes of Pool Boiling 623 10.4 Pool Boiling Correlations 10.4.1 Nucleate Pool Boiling 626 10.4.2 Critical Heat Flux for Nucleate Pool Boiling 628 10.4.3 Minimum Heat Flux 629 10.4.4 Film Pool Boiling 629 10.4.5 Parametric Effects on Pool Boiling 630 10.5 Forced Convection Boiling 10.5.1 External Forced Convection Boiling 636 10.5.2 Two-Phase Flow 636 10.5.3 Two-Phase Flow in Microchannels 639 10.6 Condensation: Physical Mechanisms 10.7 Laminar Film Condensation on a Vertical Plate 10.8 Turbulent Film Condensation 10.9 Film Condensation on Radial Systems 10.10 Condensation in Horizontal Tubes 10.11 Dropwise Condensation

561 562 565 566 567 570 572

586

589

595 596 597 598

619 620 621 622

626

635

639 641 645 650 655 656

Contents

CHAPTER

10.12 Summary References Problems

657 657 659

11 Heat Exchangers

671

11.1 11.2 11.3

11.4

11.5 11.6 11.7

Heat Exchanger Types The Overall Heat Transfer Coefficient Heat Exchanger Analysis: Use of the Log Mean Temperature Difference 11.3.1 The Parallel-Flow Heat Exchanger 678 11.3.2 The Counterflow Heat Exchanger 680 11.3.3 Special Operating Conditions 681 Heat Exchanger Analysis: The Effectiveness–NTU Method 11.4.1 Definitions 688 11.4.2 Effectiveness–NTU Relations 689 Heat Exchanger Design and Performance Calculations Additional Considerations Summary References Problems

11S.1 Log Mean Temperature Difference Method for Multipass and Cross-Flow Heat Exchangers 11S.2 Compact Heat Exchangers References Problems CHAPTER

xvii

12 Radiation: Processes and Properties 12.1 12.2 12.3

12.4

12.5 12.6

Fundamental Concepts Radiation Heat Fluxes Radiation Intensity 12.3.1 Mathematical Definitions 739 12.3.2 Radiation Intensity and Its Relation to Emission 740 12.3.3 Relation to Irradiation 745 12.3.4 Relation to Radiosity for an Opaque Surface 747 12.3.5 Relation to the Net Radiative Flux for an Opaque Surface 748 Blackbody Radiation 12.4.1 The Planck Distribution 749 12.4.2 Wien’s Displacement Law 750 12.4.3 The Stefan–Boltzmann Law 750 12.4.4 Band Emission 751 Emission from Real Surfaces Absorption, Reflection, and Transmission by Real Surfaces 12.6.1 Absorptivity 768 12.6.2 Reflectivity 769

672 674 677

688

696 705 713 714 714 W-38 W-42 W-47 W-48

733 734 737 739

748

758 767

xviii

Contents

12.6.3 Transmissivity 771 12.6.4 Special Considerations 771 12.7 Kirchhoff’s Law 12.8 The Gray Surface 12.9 Environmental Radiation 12.9.1 Solar Radiation 785 12.9.2 The Atmospheric Radiation Balance 12.9.3 Terrestrial Solar Irradiation 789 12.10 Summary References Problems CHAPTER

776 778 784 787

13 Radiation Exchange Between Surfaces 13.1

13.2 13.3

13.4 13.5 13.6

13.7

The View Factor 13.1.1 The View Factor Integral 828 13.1.2 View Factor Relations 829 Blackbody Radiation Exchange Radiation Exchange Between Opaque, Diffuse, Gray Surfaces in an Enclosure 13.3.1 Net Radiation Exchange at a Surface 843 13.3.2 Radiation Exchange Between Surfaces 844 13.3.3 The Two-Surface Enclosure 850 13.3.4 Radiation Shields 852 13.3.5 The Reradiating Surface 854 Multimode Heat Transfer Implications of the Simplifying Assumptions Radiation Exchange with Participating Media 13.6.1 Volumetric Absorption 862 13.6.2 Gaseous Emission and Absorption 863 Summary References Problems

792 796 796

827 828

838 842

859 862 862

867 868 869

APPENDIX

A Thermophysical Properties of Matter

897

APPENDIX

B Mathematical Relations and Functions

927

APPENDIX

C Thermal Conditions Associated with Uniform Energy

Generation in One-Dimensional, Steady-State Systems APPENDIX

D The Gauss–Seidel Method

933

939

Contents

APPENDIX

E The Convection Transfer Equations E.1 Conservation of Mass E.2 Newton’s Second Law of Motion E.3 Conservation of Energy

APPENDIX

F Boundary Layer Equations for Turbulent Flow

APPENDIX

G An Integral Laminar Boundary Layer Solution for

xix 941 942 942 943

945

Parallel Flow over a Flat Plate

949

Index

953

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Symbols

A Ab Ac Ap Ar a Bi Bo CD Cf Ct Co c cp cv D Db Dh d E E tot Ec E˙ g E˙ in E˙ out E˙ st e F Fo Fr

area, m2 area of prime (unfinned) surface, m2 cross-sectional area, m2 fin profile area, m2 nozzle area ratio acceleration, m/s2; speed of sound, m/s Biot number Bond number drag coefficient friction coefficient thermal capacitance, J/K Confinement number specific heat, J/kg 䡠 K; speed of light, m/s specific heat at constant pressure, J/kg 䡠 K specific heat at constant volume, J/kg 䡠 K diameter, m bubble diameter, m hydraulic diameter, m diameter of gas molecule, nm thermal plus mechanical energy, J; electric potential, V; emissive power, W/m2 total energy, J Eckert number rate of energy generation, W rate of energy transfer into a control volume, W rate of energy transfer out of control volume, W rate of increase of energy stored within a control volume, W thermal internal energy per unit mass, J/kg; surface roughness, m force, N; fraction of blackbody radiation in a wavelength band; view factor Fourier number Froude number

f G Gr Gz g H h hfg h⬘fg hsf hrad I i J Ja jH k kB L M ˙ in M ˙ out M ˙ st M ᏹi Ma m m˙ N NL, NT Nu

friction factor; similarity variable irradiation, W/m2; mass velocity, kg/s 䡠 m2 Grashof number Graetz number gravitational acceleration, m/s2 nozzle height, m; Henry’s constant, bars convection heat transfer coefficient, W/m2 䡠 K; Planck’s constant, J 䡠 s latent heat of vaporization, J/kg modified heat of vaporization, J/kg latent heat of fusion, J/kg radiation heat transfer coefficient, W/m2 䡠 K electric current, A; radiation intensity, W/m2 䡠 sr electric current density, A/m2; enthalpy per unit mass, J/kg radiosity, W/m2 Jakob number Colburn j factor for heat transfer thermal conductivity, W/m 䡠 K Boltzmann’s constant, J/K length, m mass, kg rate at which mass enters a control volume, kg/s rate at which mass leaves a control volume, kg/s rate of increase of mass stored within a control volume, kg/s molecular weight of species i, kg/kmol Mach number mass, kg mass flow rate, kg/s integer number number of tubes in longitudinal and transverse directions Nusselt number

xxii

Symbols

NTU ᏺ P PL , PT Pe Pr p Q q q˙ q⬘ q⬙ q* R Ra Re Re Rf Rm,n Rt Rt,c Rt,f Rt,o ro r, , z r, , S

Sc SD, SL, ST St T t U u, v, w V v W ˙ W We X Xtt X, Y, Z x, y, z xc xfd,h xfd,t Z

number of transfer units Avogadro’s number power, W; perimeter, m dimensionless longitudinal and transverse pitch of a tube bank Peclet number Prandtl number pressure, N/m2 energy transfer, J heat transfer rate, W rate of energy generation per unit volume, W/m3 heat transfer rate per unit length, W/m heat flux, W/m2 dimensionless conduction heat rate cylinder radius, m; gas constant, J/kg 䡠 K universal gas constant, J/kmol 䡠 K Rayleigh number Reynolds number electric resistance, ⍀ fouling factor, m2 䡠 K/W residual for the m, n nodal point thermal resistance, K/W thermal contact resistance, K/W fin thermal resistance, K/W thermal resistance of fin array, K/W cylinder or sphere radius, m cylindrical coordinates spherical coordinates shape factor for two-dimensional conduction, m; nozzle pitch, m; plate spacing, m; Seebeck coefficient, V/K solar constant, W/m2 diagonal, longitudinal, and transverse pitch of a tube bank, m Stanton number temperature, K time, s overall heat transfer coefficient, W/m2 䡠 K; internal energy, J mass average fluid velocity components, m/s volume, m3; fluid velocity, m/s specific volume, m3/kg width of a slot nozzle, m rate at which work is performed, W Weber number vapor quality Martinelli parameter components of the body force per unit volume, N/m3 rectangular coordinates, m critical location for transition to turbulence, m hydrodynamic entry length, m thermal entry length, m thermoelectric material property, K⫺1

Greek Letters ␣ thermal diffusivity, m2/s; accommodation coefficient; absorptivity

 ⌫ ␥ ␦ ␦p ␦t f f o mfp e ⌽ Subscripts abs am atm b C c cr cond conv CF D e f fc fd g H h i L l lat lm

volumetric thermal expansion coefficient, K⫺1 mass flow rate per unit width in film condensation, kg/s 䡠 m ratio of specific heats hydrodynamic boundary layer thickness, m thermal penetration depth, m thermal boundary layer thickness, m emissivity; porosity; heat exchanger effectiveness fin effectiveness thermodynamic efficiency; similarity variable fin efficiency overall efficiency of fin array zenith angle, rad; temperature difference, K absorption coefficient, m⫺1 wavelength, m mean free path length, nm viscosity, kg/s 䡠 m kinematic viscosity, m2/s; frequency of radiation, s⫺1 mass density, kg/m3; reflectivity electric resistivity, ⍀/m Stefan–Boltzmann constant, W/m2 䡠 K4; electrical conductivity, 1/⍀ 䡠 m; normal viscous stress, N/m2; surface tension, N/m viscous dissipation function, s⫺2 volume fraction azimuthal angle, rad stream function, m2/s shear stress, N/m2; transmissivity solid angle, sr; perfusion rate, s⫺1

absorbed arithmetic mean atmospheric base of an extended surface; blackbody Carnot cross-sectional; cold fluid; critical critical insulation thickness conduction convection counterflow diameter; drag excess; emission; electron fluid properties; fin conditions; saturated liquid conditions forced convection fully developed conditions saturated vapor conditions heat transfer conditions hydrodynamic; hot fluid; helical inner surface of an annulus; initial condition; tube inlet condition; incident radiation based on characteristic length saturated liquid conditions latent energy log mean condition

xxiii

Symbols

m max o p ph R r, ref rad S s sat sens sky

mean value over a tube cross section maximum center or midplane condition; tube outlet condition; outer momentum phonon reradiating surface reflected radiation radiation solar conditions surface conditions; solid properties; saturated solid conditions saturated conditions sensible energy sky conditions

ss sur t tr v x 앝

steady state surroundings thermal transmitted saturated vapor conditions local conditions on a surface spectral free stream conditions

Superscripts * dimensionless quantity Overbar surface average conditions; time mean

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C H A P T E R

Introduction

1

2

Chapter 1

䊏

Introduction

F

rom the study of thermodynamics, you have learned that energy can be transferred by interactions of a system with its surroundings. These interactions are called work and heat. However, thermodynamics deals with the end states of the process during which an interaction occurs and provides no information concerning the nature of the interaction or the time rate at which it occurs. The objective of this text is to extend thermodynamic analysis through the study of the modes of heat transfer and through the development of relations to calculate heat transfer rates. In this chapter we lay the foundation for much of the material treated in the text. We do so by raising several questions: What is heat transfer? How is heat transferred? Why is it important? One objective is to develop an appreciation for the fundamental concepts and principles that underlie heat transfer processes. A second objective is to illustrate the manner in which a knowledge of heat transfer may be used with the first law of thermodynamics (conservation of energy) to solve problems relevant to technology and society.

1.1 What and How? A simple, yet general, definition provides sufficient response to the question: What is heat transfer? Heat transfer (or heat) is thermal energy in transit due to a spatial temperature difference.

Whenever a temperature difference exists in a medium or between media, heat transfer must occur. As shown in Figure 1.1, we refer to different types of heat transfer processes as modes. When a temperature gradient exists in a stationary medium, which may be a solid or a fluid, we use the term conduction to refer to the heat transfer that will occur across the medium. In contrast, the term convection refers to heat transfer that will occur between a surface and a moving fluid when they are at different temperatures. The third mode of heat transfer is termed thermal radiation. All surfaces of finite temperature emit energy in the form of electromagnetic waves. Hence, in the absence of an intervening medium, there is net heat transfer by radiation between two surfaces at different temperatures.

Conduction through a solid or a stationary fluid

T1

T1 > T2

T2

Convection from a surface to a moving fluid

Net radiation heat exchange between two surfaces

Ts > T∞

Surface, T1

Moving fluid, T∞

q"

Surface, T2

q"

q"1 Ts

FIGURE 1.1

q"2

Conduction, convection, and radiation heat transfer modes.

1.2

3

Physical Origins and Rate Equations

䊏

1.2 Physical Origins and Rate Equations As engineers, it is important that we understand the physical mechanisms which underlie the heat transfer modes and that we be able to use the rate equations that quantify the amount of energy being transferred per unit time.

1.2.1

Conduction

At mention of the word conduction, we should immediately conjure up concepts of atomic and molecular activity because processes at these levels sustain this mode of heat transfer. Conduction may be viewed as the transfer of energy from the more energetic to the less energetic particles of a substance due to interactions between the particles. The physical mechanism of conduction is most easily explained by considering a gas and using ideas familiar from your thermodynamics background. Consider a gas in which a temperature gradient exists, and assume that there is no bulk, or macroscopic, motion. The gas may occupy the space between two surfaces that are maintained at different temperatures, as shown in Figure 1.2. We associate the temperature at any point with the energy of gas molecules in proximity to the point. This energy is related to the random translational motion, as well as to the internal rotational and vibrational motions, of the molecules. Higher temperatures are associated with higher molecular energies. When neighboring molecules collide, as they are constantly doing, a transfer of energy from the more energetic to the less energetic molecules must occur. In the presence of a temperature gradient, energy transfer by conduction must then occur in the direction of decreasing temperature. This would be true even in the absence of collisions, as is evident from Figure 1.2. The hypothetical plane at xo is constantly being crossed by molecules from above and below due to their random motion. However, molecules from above are associated with a higher temperature than those from below, in which case there must be a net transfer of energy in the positive x-direction. Collisions between molecules enhance this energy transfer. We may speak of the net transfer of energy by random molecular motion as a diffusion of energy. The situation is much the same in liquids, although the molecules are more closely spaced and the molecular interactions are stronger and more frequent. Similarly, in a solid, conduction may be attributed to atomic activity in the form of lattice vibrations. The modern T

xo

x

q"x

T1 > T2

q"x

T2

FIGURE 1.2 Association of conduction heat transfer with diffusion of energy due to molecular activity.

4

Chapter 1

䊏

Introduction

T

T1 q"x

T(x) T2 L

x

FIGURE 1.3 One-dimensional heat transfer by conduction (diffusion of energy).

view is to ascribe the energy transfer to lattice waves induced by atomic motion. In an electrical nonconductor, the energy transfer is exclusively via these lattice waves; in a conductor, it is also due to the translational motion of the free electrons. We treat the important properties associated with conduction phenomena in Chapter 2 and in Appendix A. Examples of conduction heat transfer are legion. The exposed end of a metal spoon suddenly immersed in a cup of hot coffee is eventually warmed due to the conduction of energy through the spoon. On a winter day, there is significant energy loss from a heated room to the outside air. This loss is principally due to conduction heat transfer through the wall that separates the room air from the outside air. Heat transfer processes can be quantified in terms of appropriate rate equations. These equations may be used to compute the amount of energy being transferred per unit time. For heat conduction, the rate equation is known as Fourier’s law. For the one-dimensional plane wall shown in Figure 1.3, having a temperature distribution T(x), the rate equation is expressed as q⬙x ⫽ ⫺ k dT dx

(1.1)

The heat flux q⬙x (W/m2) is the heat transfer rate in the x-direction per unit area perpendicular to the direction of transfer, and it is proportional to the temperature gradient, dT/dx, in this direction. The parameter k is a transport property known as the thermal conductivity (W/m 䡠 K) and is a characteristic of the wall material. The minus sign is a consequence of the fact that heat is transferred in the direction of decreasing temperature. Under the steady-state conditions shown in Figure 1.3, where the temperature distribution is linear, the temperature gradient may be expressed as dT ⫽ T2 ⫺ T1 dx L and the heat flux is then q⬙x ⫽ ⫺k

T2 ⫺ T1 L

or q⬙x ⫽ k

T1 ⫺ T2 ⫽ k ⌬T L L

(1.2)

Note that this equation provides a heat flux, that is, the rate of heat transfer per unit area. The heat rate by conduction, qx (W), through a plane wall of area A is then the product of the flux and the area, qx ⫽ q⬙x 䡠 A.

1.2

䊏

5

Physical Origins and Rate Equations

* EXAMPLE 1.1 The wall of an industrial furnace is constructed from 0.15-m-thick fireclay brick having a thermal conductivity of 1.7 W/m 䡠 K. Measurements made during steady-state operation reveal temperatures of 1400 and 1150 K at the inner and outer surfaces, respectively. What is the rate of heat loss through a wall that is 0.5 m ⫻ 1.2 m on a side?

SOLUTION Known: Steady-state conditions with prescribed wall thickness, area, thermal conductivity, and surface temperatures. Find: Wall heat loss. Schematic: W = 1.2 m H = 0.5 m k = 1.7 W/m•K T2 = 1150 K

T1 = 1400 K

qx qx''

Wall area, A

x

L = 0.15 m

x

L

Assumptions: 1. Steady-state conditions. 2. One-dimensional conduction through the wall. 3. Constant thermal conductivity. Analysis: Since heat transfer through the wall is by conduction, the heat flux may be determined from Fourier’s law. Using Equation 1.2, we have q⬙x ⫽ k ⌬T ⫽ 1.7 W/m 䡠 K ⫻ 250 K ⫽ 2833 W/m2 L 0.15 m The heat flux represents the rate of heat transfer through a section of unit area, and it is uniform (invariant) across the surface of the wall. The heat loss through the wall of area A ⫽ H ⫻ W is then qx ⫽ (HW ) q⬙x ⫽ (0.5 m ⫻ 1.2 m) 2833 W/m2 ⫽1700 W

䉰

Comments: Note the direction of heat flow and the distinction between heat flux and heat rate. *This icon identifies examples that are available in tutorial form in the Interactive Heat Transfer (IHT) software that accompanies the text. Each tutorial is brief and illustrates a basic function of the software. IHT can be used to solve simultaneous equations, perform parameter sensitivity studies, and graph the results. Use of IHT will reduce the time spent solving more complex end-of-chapter problems.

6

Chapter 1

1.2.2

䊏

Introduction

Convection

The convection heat transfer mode is comprised of two mechanisms. In addition to energy transfer due to random molecular motion (diffusion), energy is also transferred by the bulk, or macroscopic, motion of the fluid. This fluid motion is associated with the fact that, at any instant, large numbers of molecules are moving collectively or as aggregates. Such motion, in the presence of a temperature gradient, contributes to heat transfer. Because the molecules in the aggregate retain their random motion, the total heat transfer is then due to a superposition of energy transport by the random motion of the molecules and by the bulk motion of the fluid. The term convection is customarily used when referring to this cumulative transport, and the term advection refers to transport due to bulk fluid motion. We are especially interested in convection heat transfer, which occurs between a fluid in motion and a bounding surface when the two are at different temperatures. Consider fluid flow over the heated surface of Figure 1.4. A consequence of the fluid–surface interaction is the development of a region in the fluid through which the velocity varies from zero at the surface to a finite value u앝 associated with the flow. This region of the fluid is known as the hydrodynamic, or velocity, boundary layer. Moreover, if the surface and flow temperatures differ, there will be a region of the fluid through which the temperature varies from Ts at y ⫽ 0 to T앝 in the outer flow. This region, called the thermal boundary layer, may be smaller, larger, or the same size as that through which the velocity varies. In any case, if Ts ⬎ T앝, convection heat transfer will occur from the surface to the outer flow. The convection heat transfer mode is sustained both by random molecular motion and by the bulk motion of the fluid within the boundary layer. The contribution due to random molecular motion (diffusion) dominates near the surface where the fluid velocity is low. In fact, at the interface between the surface and the fluid (y ⫽ 0), the fluid velocity is zero, and heat is transferred by this mechanism only. The contribution due to bulk fluid motion originates from the fact that the boundary layer grows as the flow progresses in the x-direction. In effect, the heat that is conducted into this layer is swept downstream and is eventually transferred to the fluid outside the boundary layer. Appreciation of boundary layer phenomena is essential to understanding convection heat transfer. For this reason, the discipline of fluid mechanics will play a vital role in our later analysis of convection. Convection heat transfer may be classified according to the nature of the flow. We speak of forced convection when the flow is caused by external means, such as by a fan, a pump, or atmospheric winds. As an example, consider the use of a fan to provide forced convection air cooling of hot electrical components on a stack of printed circuit boards (Figure 1.5a). In contrast, for free (or natural) convection, the flow is induced by buoyancy forces, which are due to density differences caused by temperature variations in the fluid. An example is the free convection heat transfer that occurs from hot components on a vertical array of circuit

y

y

Fluid

u∞

Velocity distribution u(y)

q"

T∞

Temperature distribution T(y) Ts x

u(y)

Heated surface

T(y)

FIGURE 1.4 Boundary layer development in convection heat transfer.

1.2

䊏

7

Physical Origins and Rate Equations

boards in air (Figure 1.5b). Air that makes contact with the components experiences an increase in temperature and hence a reduction in density. Since it is now lighter than the surrounding air, buoyancy forces induce a vertical motion for which warm air ascending from the boards is replaced by an inflow of cooler ambient air. While we have presumed pure forced convection in Figure 1.5a and pure natural convection in Figure 1.5b, conditions corresponding to mixed (combined) forced and natural convection may exist. For example, if velocities associated with the flow of Figure 1.5a are small and/or buoyancy forces are large, a secondary flow that is comparable to the imposed forced flow could be induced. In this case, the buoyancy-induced flow would be normal to the forced flow and could have a significant effect on convection heat transfer from the components. In Figure 1.5b, mixed convection would result if a fan were used to force air upward between the circuit boards, thereby assisting the buoyancy flow, or downward, thereby opposing the buoyancy flow. We have described the convection heat transfer mode as energy transfer occurring within a fluid due to the combined effects of conduction and bulk fluid motion. Typically, the energy that is being transferred is the sensible, or internal thermal, energy of the fluid. However, for some convection processes, there is, in addition, latent heat exchange. This latent heat exchange is generally associated with a phase change between the liquid and vapor states of the fluid. Two special cases of interest in this text are boiling and condensation. For example, convection heat transfer results from fluid motion induced by vapor bubbles generated at the bottom of a pan of boiling water (Figure 1.5c) or by the condensation of water vapor on the outer surface of a cold water pipe (Figure 1.5d).

Buoyancy-driven flow Forced flow

q'' Hot components on printed circuit boards

Air

q''

Air (a)

(b)

Moist air

q'' Cold water Vapor bubbles

q" Water

Hot plate (c)

(d)

FIGURE 1.5 Convection heat transfer processes. (a) Forced convection. (b) Natural convection. (c) Boiling. (d) Condensation.

Water droplets

8

Chapter 1

䊏

Introduction

TABLE 1.1 Typical values of the convection heat transfer coefficient h (W/m2 䡠 K)

Process Free convection Gases Liquids Forced convection Gases Liquids Convection with phase change Boiling or condensation

2–25 50–1000 25–250 100–20,000 2500–100,000

Regardless of the nature of the convection heat transfer process, the appropriate rate equation is of the form q⬙ ⫽ h(Ts ⫺ T앝)

(1.3a)

where q⬙, the convective heat flux (W/m2), is proportional to the difference between the surface and fluid temperatures, Ts and T앝, respectively. This expression is known as Newton’s law of cooling, and the parameter h (W/m2 䡠 K) is termed the convection heat transfer coefficient. This coefficient depends on conditions in the boundary layer, which are influenced by surface geometry, the nature of the fluid motion, and an assortment of fluid thermodynamic and transport properties. Any study of convection ultimately reduces to a study of the means by which h may be determined. Although consideration of these means is deferred to Chapter 6, convection heat transfer will frequently appear as a boundary condition in the solution of conduction problems (Chapters 2 through 5). In the solution of such problems we presume h to be known, using typical values given in Table 1.1. When Equation 1.3a is used, the convection heat flux is presumed to be positive if heat is transferred from the surface (Ts ⬎ T앝) and negative if heat is transferred to the surface (T앝 ⬎ Ts). However, nothing precludes us from expressing Newton’s law of cooling as q⬙ ⫽ h(T앝 ⫺ Ts)

(1.3b)

in which case heat transfer is positive if it is to the surface.

1.2.3

Radiation

Thermal radiation is energy emitted by matter that is at a nonzero temperature. Although we will focus on radiation from solid surfaces, emission may also occur from liquids and gases. Regardless of the form of matter, the emission may be attributed to changes in the electron configurations of the constituent atoms or molecules. The energy of the radiation field is transported by electromagnetic waves (or alternatively, photons). While the transfer of energy by conduction or convection requires the presence of a material medium, radiation does not. In fact, radiation transfer occurs most efficiently in a vacuum. Consider radiation transfer processes for the surface of Figure 1.6a. Radiation that is emitted by the surface originates from the thermal energy of matter bounded by the surface,

1.2

䊏

9

Physical Origins and Rate Equations

and the rate at which energy is released per unit area (W/m2) is termed the surface emissive power, E. There is an upper limit to the emissive power, which is prescribed by the Stefan–Boltzmann law Eb ⫽ T s4

(1.4)

where Ts is the absolute temperature (K) of the surface and is the Stefan– Boltzmann constant ( ⫽ 5.67 ⫻ 10⫺8 W/m2 䡠 K4). Such a surface is called an ideal radiator or blackbody. The heat flux emitted by a real surface is less than that of a blackbody at the same temperature and is given by E ⫽ T 4s

(1.5)

where is a radiative property of the surface termed the emissivity. With values in the range 0 ⱕ ⱕ 1, this property provides a measure of how efficiently a surface emits energy relative to a blackbody. It depends strongly on the surface material and finish, and representative values are provided in Appendix A. Radiation may also be incident on a surface from its surroundings. The radiation may originate from a special source, such as the sun, or from other surfaces to which the surface of interest is exposed. Irrespective of the source(s), we designate the rate at which all such radiation is incident on a unit area of the surface as the irradiation G (Figure 1.6a). A portion, or all, of the irradiation may be absorbed by the surface, thereby increasing the thermal energy of the material. The rate at which radiant energy is absorbed per unit surface area may be evaluated from knowledge of a surface radiative property termed the absorptivity ␣. That is, Gabs ⫽ ␣G (1.6) where 0 ⱕ ␣ ⱕ 1. If ␣ ⬍ 1 and the surface is opaque, portions of the irradiation are reflected. If the surface is semitransparent, portions of the irradiation may also be transmitted. However, whereas absorbed and emitted radiation increase and reduce, respectively, the thermal energy of matter, reflected and transmitted radiation have no effect on this energy. Note that the value of ␣ depends on the nature of the irradiation, as well as on the surface itself. For example, the absorptivity of a surface to solar radiation may differ from its absorptivity to radiation emitted by the walls of a furnace.

G

Gas

Gas

T, h

T, h

E q"conv

Surface of emissivity , absorptivity ␣, and temperature Ts (a)

Surroundings at Tsur

q"rad

Surface of emissivity = ␣ , area A, and temperature Ts

q"conv

Ts > Tsur, Ts > T

(b)

FIGURE 1.6 Radiation exchange: (a) at a surface and (b) between a surface and large surroundings.

10

Chapter 1

䊏

Introduction

In many engineering problems (a notable exception being problems involving solar radiation or radiation from other very high temperature sources), liquids can be considered opaque to radiation heat transfer, and gases can be considered transparent to it. Solids can be opaque (as is the case for metals) or semitransparent (as is the case for thin sheets of some polymers and some semiconducting materials). A special case that occurs frequently involves radiation exchange between a small surface at Ts and a much larger, isothermal surface that completely surrounds the smaller one (Figure 1.6b). The surroundings could, for example, be the walls of a room or a furnace whose temperature Tsur differs from that of an enclosed surface (Tsur ⫽ Ts). We will show in Chapter 12 that, for such a condition, the irradiation may be approximated by emission from 4 . If the surface is assumed to be one for which a blackbody at Tsur, in which case G ⫽ T sur ␣ ⫽ (a gray surface), the net rate of radiation heat transfer from the surface, expressed per unit area of the surface, is q⬙rad ⫽

q 4 ) ⫽ Eb(Ts ) ⫺ ␣G ⫽ (T 4s ⫺ Tsur A

(1.7)

This expression provides the difference between thermal energy that is released due to radiation emission and that gained due to radiation absorption. For many applications, it is convenient to express the net radiation heat exchange in the form qrad ⫽ hr A(Ts ⫺ Tsur) (1.8) where, from Equation 1.7, the radiation heat transfer coefficient hr is 2 hr (Ts ⫹ Tsur)(Ts2 ⫹ Tsur )

(1.9)

Here we have modeled the radiation mode in a manner similar to convection. In this sense we have linearized the radiation rate equation, making the heat rate proportional to a temperature difference rather than to the difference between two temperatures to the fourth power. Note, however, that hr depends strongly on temperature, whereas the temperature dependence of the convection heat transfer coefficient h is generally weak. The surfaces of Figure 1.6 may also simultaneously transfer heat by convection to an adjoining gas. For the conditions of Figure 1.6b, the total rate of heat transfer from the surface is then 4 ) q ⫽ qconv ⫹ qrad ⫽ hA(Ts ⫺ T앝) ⫹ A(Ts4 ⫺ Tsur (1.10)

EXAMPLE 1.2 An uninsulated steam pipe passes through a room in which the air and walls are at 25⬚C. The outside diameter of the pipe is 70 mm, and its surface temperature and emissivity are 200⬚C and 0.8, respectively. What are the surface emissive power and irradiation? If the coefficient associated with free convection heat transfer from the surface to the air is 15 W/m2 䡠 K, what is the rate of heat loss from the surface per unit length of pipe?

SOLUTION Known: Uninsulated pipe of prescribed diameter, emissivity, and surface temperature in a room with fixed wall and air temperatures.

1.2

䊏

11

Physical Origins and Rate Equations

Find: 1. Surface emissive power and irradiation. 2. Pipe heat loss per unit length, q⬘. Schematic:

Air

q'

T∞ = 25°C h = 15 W/m2•K

E L Ts = 200°C ε = 0.8 G Tsur = 25°C

D = 70 mm

Assumptions: 1. Steady-state conditions. 2. Radiation exchange between the pipe and the room is between a small surface and a much larger enclosure. 3. The surface emissivity and absorptivity are equal. Analysis: 1. The surface emissive power may be evaluated from Equation 1.5, while the irradiation 4 corresponds to G ⫽ Tsur . Hence E ⫽ Ts4 ⫽ 0.8(5.67 ⫻ 10⫺8 W/m2 䡠 K4)(473 K)4 ⫽ 2270 W/m2 G⫽

4 T sur

⫺8

⫽ 5.67 ⫻ 10

W/m 䡠 K (298 K) ⫽ 447 W/m 2

4

4

2

䉰 䉰

2. Heat loss from the pipe is by convection to the room air and by radiation exchange with the walls. Hence, q ⫽ qconv ⫹ qrad and from Equation 1.10, with A ⫽ DL, 4 ) q ⫽ h(DL)(Ts ⫺ T앝) ⫹ (DL)(T 4s ⫺ Tsur

The heat loss per unit length of pipe is then q⬘ ⫽

q ⫽ 15 W/m2 䡠 K( ⫻ 0.07 m)(200 ⫺ 25)⬚C L ⫹ 0.8( ⫻ 0.07 m) 5.67 ⫻ 10⫺8 W/m2 䡠 K4 (4734 ⫺ 2984) K4 q⬘ ⫽ 577 W/m ⫹ 421 W/m ⫽ 998 W/m

䉰

Comments: 1. Note that temperature may be expressed in units of ⬚C or K when evaluating the temperature difference for a convection (or conduction) heat transfer rate. However, temperature must be expressed in kelvins (K) when evaluating a radiation transfer rate.

12

Chapter 1

䊏

Introduction

2. The net rate of radiation heat transfer from the pipe may be expressed as q⬘rad ⫽ D (E ⫺ ␣G) q⬘rad ⫽ ⫻ 0.07 m (2270 ⫺ 0.8 ⫻ 447) W/m2 ⫽ 421 W/m 3. In this situation, the radiation and convection heat transfer rates are comparable because Ts is large compared to Tsur and the coefficient associated with free convection is small. For more moderate values of Ts and the larger values of h associated with forced convection, the effect of radiation may often be neglected. The radiation heat transfer coefficient may be computed from Equation 1.9. For the conditions of this problem, its value is hr ⫽ 11 W/m2 䡠 K.

1.2.4 The Thermal Resistance Concept The three modes of heat transfer were introduced in the preceding sections. As is evident from Equations 1.2, 1.3, and 1.8, the heat transfer rate can be expressed in the form q ⫽ q⬙A ⫽ ⌬T Rt

(1.11)

where ⌬T is a relevant temperature difference and A is the area normal to the direction of heat transfer. The quantity Rt is called a thermal resistance and takes different forms for the three different modes of heat transfer. For example, Equation 1.2 may be multiplied by the area A and rewritten as qx ⫽ ⌬T/Rt,c , where Rt,c ⫽ L /kA is a thermal resistance associated with conduction, having the units K/W. The thermal resistance concept will be considered in detail in Chapter 3 and will be seen to have great utility in solving complex heat transfer problems.

1.3 Relationship to Thermodynamics The subjects of heat transfer and thermodynamics are highly complementary and interrelated, but they also have fundamental differences. If you have taken a thermodynamics course, you are aware that heat exchange plays a vital role in the first and second laws of thermodynamics because it is one of the primary mechanisms for energy transfer between a system and its surroundings. While thermodynamics may be used to determine the amount of energy required in the form of heat for a system to pass from one state to another, it considers neither the mechanisms that provide for heat exchange nor the methods that exist for computing the rate of heat exchange. The discipline of heat transfer specifically seeks to quantify the rate at which heat is exchanged through the rate equations expressed, for example, by Equations 1.2, 1.3, and 1.7. Indeed, heat transfer principles often enable the engineer to implement the concepts of thermodynamics. For example, the actual size of a power plant to be constructed cannot be determined from thermodynamics alone; the principles of heat transfer must also be invoked at the design stage. The remainder of this section considers the relationship of heat transfer to thermodynamics. Since the first law of thermodynamics (the law of conservation of energy) provides a useful, often essential, starting point for the solution of heat transfer problems, Section 1.3.1 will provide a development of the general formulations of the first law. The ideal

1.3

䊏

13

Relationship to Thermodynamics

(Carnot) efficiency of a heat engine, as determined by the second law of thermodynamics will be reviewed in Section 1.3.2. It will be shown that a realistic description of the heat transfer between a heat engine and its surroundings further limits the actual efficiency of a heat engine.

1.3.1 Relationship to the First Law of Thermodynamics (Conservation of Energy) At its heart, the first law of thermodynamics is simply a statement that the total energy of a system is conserved, and therefore the only way that the amount of energy in a system can change is if energy crosses its boundaries. The first law also addresses the ways in which energy can cross the boundaries of a system. For a closed system (a region of fixed mass), there are only two ways: heat transfer through the boundaries and work done on or by the system. This leads to the following statement of the first law for a closed system, which is familiar if you have taken a course in thermodynamics: ⌬Esttot ⫽ Q ⫺ W

(1.12a)

where ⌬Esttot is the change in the total energy stored in the system, Q is the net heat transferred to the system, and W is the net work done by the system. This is schematically illustrated in Figure 1.7a. The first law can also be applied to a control volume (or open system), a region of space bounded by a control surface through which mass may pass. Mass entering and leaving the control volume carries energy with it; this process, termed energy advection, adds a third way in which energy can cross the boundaries of a control volume. To summarize, the first law of thermodynamics can be very simply stated as follows for both a control volume and a closed system. First Law of Thermodynamics over a Time Interval (⌬t) The increase in the amount of energy stored in a control volume must equal the amount of energy that enters the control volume, minus the amount of energy that leaves the control volume.

In applying this principle, it is recognized that energy can enter and leave the control volume due to heat transfer through the boundaries, work done on or by the control volume, and energy advection. The first law of thermodynamics addresses total energy, which consists of kinetic and potential energies (together known as mechanical energy) and internal energy. Internal energy can be further subdivided into thermal energy (which will be defined more carefully later) W Q •

tot ∆ Est

E in

•

•

E g, E st •

E out

(a)

(b)

FIGURE 1.7 Conservation of energy: (a) for a closed system over a time interval and (b) for a control volume at an instant.

14

Chapter 1

䊏

Introduction

and other forms of internal energy, such as chemical and nuclear energy. For the study of heat transfer, we wish to focus attention on the thermal and mechanical forms of energy. We must recognize that the sum of thermal and mechanical energy is not conserved, because conversion can occur between other forms of energy and thermal or mechanical energy. For example, if a chemical reaction occurs that decreases the amount of chemical energy in the system, it will result in an increase in the thermal energy of the system. If an electric motor operates within the system, it will cause conversion from electrical to mechanical energy. We can think of such energy conversions as resulting in thermal or mechanical energy generation (which can be either positive or negative). So a statement of the first law that is well suited for heat transfer analysis is: Thermal and Mechanical Energy Equation over a Time Interval (⌬t) The increase in the amount of thermal and mechanical energy stored in the control volume must equal the amount of thermal and mechanical energy that enters the control volume, minus the amount of thermal and mechanical energy that leaves the control volume, plus the amount of thermal and mechanical energy that is generated within the control volume.

This expression applies over a time interval ⌬t, and all the energy terms are measured in joules. Since the first law must be satisfied at each and every instant of time t, we can also formulate the law on a rate basis. That is, at any instant, there must be a balance between all energy rates, as measured in joules per second (W). In words, this is expressed as follows: Thermal and Mechanical Energy Equation at an Instant (t) The rate of increase of thermal and mechanical energy stored in the control volume must equal the rate at which thermal and mechanical energy enters the control volume, minus the rate at which thermal and mechanical energy leaves the control volume, plus the rate at which thermal and mechanical energy is generated within the control volume.

If the inflow and generation of thermal and mechanical energy exceed the outflow, the amount of thermal and mechanical energy stored (accumulated) in the control volume must increase. If the converse is true, thermal and mechanical energy storage must decrease. If the inflow and generation equal the outflow, a steady-state condition must prevail such that there will be no change in the amount of thermal and mechanical energy stored in the control volume. We will now define symbols for each of the energy terms so that the boxed statements can be rewritten as equations. We let E stand for the sum of thermal and mechanical energy (in contrast to the symbol Etot for total energy). Using the subscript st to denote energy stored in the control volume, the change in thermal and mechanical energy stored over the time interval ⌬t is then ⌬Est. The subscripts in and out refer to energy entering and leaving the control volume. Finally, thermal and mechanical energy generation is given the symbol Eg. Thus, the first boxed statement can be written as: ⌬Est ⫽ Ein ⫺ Eout ⫹ Eg

(1.12b)

Next, using a dot over a term to indicate a rate, the second boxed statement becomes: dE E˙ st st ⫽ E˙ in ⫺ E˙ out ⫹ E˙ g dt

(1.12c)

1.3

䊏

Relationship to Thermodynamics

15

This expression is illustrated schematically in Figure 1.7b. Equations 1.12b,c provide important and, in some cases, essential tools for solving heat transfer problems. Every application of the first law must begin with the identification of an appropriate control volume and its control surface, to which an analysis is subsequently applied. The first step is to indicate the control surface by drawing a dashed line. The second step is to decide whether to perform the analysis for a time interval ⌬t (Equation 1.12b) or on a rate basis (Equation 1.12c). This choice depends on the objective of the solution and on how information is given in the problem. The next step is to identify the energy terms that are relevant in the problem you are solving. To develop your confidence in taking this last step, the remainder of this section is devoted to clarifying the following energy terms: • Stored thermal and mechanical energy, Est. • Thermal and mechanical energy generation, Eg. • Thermal and mechanical energy transport across the control surfaces, that is, the inflow and outflow terms, Ein and Eout. In the statement of the first law (Equation 1.12a), the total energy, E tot, consists of kinetic energy (KE ⫽ 1⁄2mV 2, where m and V are mass and velocity, respectively), potential energy (PE ⫽ mgz, where g is the gravitational acceleration and z is the vertical coordinate), and internal energy (U). Mechanical energy is defined as the sum of kinetic and potential energy. Most often in heat transfer problems, the changes in kinetic and potential energy are small and can be neglected. The internal energy consists of a sensible component, which accounts for the translational, rotational, and/or vibrational motion of the atoms/molecules comprising the matter; a latent component, which relates to intermolecular forces influencing phase change between solid, liquid, and vapor states; a chemical component, which accounts for energy stored in the chemical bonds between atoms; and a nuclear component, which accounts for the binding forces in the nucleus. For the study of heat transfer, we focus attention on the sensible and latent components of the internal energy (Usens and Ulat, respectively), which are together referred to as thermal energy, Ut. The sensible energy is the portion that we associate mainly with changes in temperature (although it can also depend on pressure). The latent energy is the component we associate with changes in phase. For example, if the material in the control volume changes from solid to liquid (melting) or from liquid to vapor (vaporization, evaporation, boiling), the latent energy increases. Conversely, if the phase change is from vapor to liquid (condensation) or from liquid to solid (solidification, freezing), the latent energy decreases. Obviously, if no phase change is occurring, there is no change in latent energy, and this term can be neglected. Based on this discussion, the stored thermal and mechanical energy is given by Est ⫽ KE ⫹ PE ⫹ Ut, where Ut ⫽ Usens ⫹ Ulat. In many problems, the only relevant energy term will be the sensible energy, that is, Est ⫽ Usens. The energy generation term is associated with conversion from some other form of internal energy (chemical, electrical, electromagnetic, or nuclear) to thermal or mechanical energy. It is a volumetric phenomenon. That is, it occurs within the control volume and is generally proportional to the magnitude of this volume. For example, an exothermic chemical reaction may be occurring, converting chemical energy to thermal energy. The net effect is an increase in the thermal energy of the matter within the control volume. Another source of thermal energy is the conversion from electrical energy that occurs due to resistance heating when an electric current is passed through a conductor. That is, if an electric current I passes through a resistance R in the control volume, electrical energy is dissipated at a rate I2R, which corresponds to the rate at which thermal energy is generated (released)

16

Chapter 1

䊏

Introduction

within the volume. In all applications of interest in this text, if chemical, electrical, or nuclear effects exist, they are treated as sources (or sinks, which correspond to negative sources) of thermal or mechanical energy and hence are included in the generation terms of Equations 1.12b,c. The inflow and outflow terms are surface phenomena. That is, they are associated exclusively with processes occurring at the control surface and are generally proportional to the surface area. As discussed previously, the energy inflow and outflow terms include heat transfer (which can be by conduction, convection, and/or radiation) and work interactions occurring at the system boundaries (e.g., due to displacement of a boundary, a rotating shaft, and/or electromagnetic effects). For cases in which mass crosses the control volume boundary (e.g., for situations involving fluid flow), the inflow and outflow terms also include energy (thermal and mechanical) that is advected (carried) by mass entering and leaving the . control volume. For instance, if the mass flow rate entering through the boundary is m , then . the rate at which thermal and mechanical energy enters with the flow is m (ut ⫹ 1⁄2V 2 ⫹ gz), where ut is the thermal energy per unit mass. When the first law is applied to a control volume with fluid crossing its boundary, it is customary to divide the work term into two contributions. The first contribution, termed flow work, is associated with work done by pressure forces moving fluid through the boundary. For a unit mass, the amount of work is equivalent to the product of the pressure ˙ is traditionally used for the rate at and the specific volume of the fluid ( pv). The symbol W which the remaining work (not including flow work) is perfomed. If operation is under steady-state conditions (dEst /dt ⫽ 0) and if there is no thermal or mechanical energy generation, Equation 1.12c reduces to the following form of the steady-flow energy equation (see Figure 1.8), which will be familiar if you have taken a thermodynamics course: ˙ ⫽0 m˙ (ut ⫹ pv ⫹ 1⁄2 V 2 ⫹ gz)in ⫺ m˙ (ut ⫹ pv ⫹ 1⁄2 V 2 ⫹ gz)out ⫹ q ⫺ W

(1.12d)

Terms within the parentheses are expressed for a unit mass of fluid at the inflow and outflow locations. When multiplied by the mass flow rate m˙ , they yield the rate at which the corresponding form of the energy (thermal, flow work, kinetic, and potential) enters or leaves the control volume. The sum of thermal energy and flow work per unit mass may be replaced by the enthalpy per unit mass, i ⫽ ut ⫹ pv. In most open system applications of interest in this text, changes in latent energy between the inflow and outflow conditions of Equation 1.12d may be neglected, so the thermal energy reduces to only the sensible component. If the fluid is approximated as an ideal gas with constant specific heats, the difference in enthalpies (per unit mass) between the inlet and outlet flows may then be expressed as (iin ⫺ iout) ⫽ cp(Tin ⫺ Tout), where cp is

q zout

(ut , pv, V)in

zin

(ut , pv, V)out

•

W

Reference height

FIGURE 1.8 Conservation of energy for a steady-flow, open system.

1.3

䊏

17

Relationship to Thermodynamics

the specific heat at constant pressure and Tin and Tout are the inlet and outlet temperatures, respectively. If the fluid is an incompressible liquid, its specific heats at constant pressure and volume are equal, cp ⫽ cv c, and for Equation 1.12d the change in sensible energy (per unit mass) reduces to (ut,in ⫺ ut,out) ⫽ c(Tin ⫺ Tout). Unless the pressure drop is extremely large, the difference in flow work terms, (pv)in ⫺ (pv)out, is negligible for a liquid. Having already assumed steady-state conditions, no changes in latent energy, and no thermal or mechanical energy generation, there are at least four cases in which further assumptions can be made to reduce Equation 1.12d to the simplified steady-flow thermal energy equation: q ⫽ m˙ cp(Tout ⫺ Tin)

(1.12e)

The right-hand side of Equation 1.12e represents the net rate of outflow of enthalpy (thermal energy plus flow work) for an ideal gas or of thermal energy for an incompressible liquid. The first two cases for which Equation 1.12e holds can readily be verified by examining Equation 1.12d. They are: 1. An ideal gas with negligible kinetic and potential energy changes and negligible work (other than flow work). 2. An incompressible liquid with negligible kinetic and potential energy changes and negligible work, including flow work. As noted in the preceding discussion, flow work is negligible for an incompressible liquid provided the pressure variation is not too great. The second pair of cases cannot be directly derived from Equation 1.12d but require further knowledge of how mechanical energy is converted into thermal energy. These cases are: 3. An ideal gas with negligible viscous dissipation and negligible pressure variation. 4. An incompressible liquid with negligible viscous dissipation. Viscous dissipation is the conversion from mechanical energy to thermal energy associated with viscous forces acting in a fluid. It is important only in cases involving high-speed flow and/or highly viscous fluid. Since so many engineering applications satisfy one or more of the preceding four conditions, Equation 1.12e is commonly used for the analysis of heat transfer in moving fluids. It will be used in Chapter 8 in the study of convection heat transfer in internal flow. The mass flow rate m˙ of the fluid may be expressed as m˙ ⫽ VAc, where is the fluid density and Ac is the cross-sectional area of the channel through which the fluid flows. The volumetric flow rate is simply ᭙˙ ⫽ VAc ⫽ m˙ /.

EXAMPLE 1.3 The blades of a wind turbine turn a large shaft at a relatively slow speed. The rotational speed is increased by a gearbox that has an efficiency of gb ⫽ 0.93. In turn, the gearbox output shaft drives an electric generator with an efficiency of gen ⫽ 0.95. The cylindrical nacelle, which houses the gearbox, generator, and associated equipment, is of length L ⫽ 6 m and diameter D ⫽ 3 m. If the turbine produces P ⫽ 2.5 MW of electrical power, and the air and surroundings temperatures are T앝 ⫽ 25⬚C and Tsur ⫽ 20⬚C, respectively, determine the minimum possible operating temperature inside the nacelle. The emissivity of the nacelle is ⫽ 0.83,

18

Chapter 1

䊏

Introduction

and the convective heat transfer coefficient is h ⫽ 35 W/m2 䡠 K. The surface of the nacelle that is adjacent to the blade hub can be considered to be adiabatic, and solar irradiation may be neglected.

Tsur 20°C h 35 W/m2·K L6m Air

D3m

T∞ 25°C Ts ,ε 0.83 Generator, ηgen 0.95 Gearbox, ηgb 0.93 Nacelle

Hub

SOLUTION Known: Electrical power produced by a wind turbine. Gearbox and generator efficiencies, dimensions and emissivity of the nacelle, ambient and surrounding temperatures, and heat transfer coefficient. Find: Minimum possible temperature inside the enclosed nacelle. Schematic:

Tsur 20°C

Air T∞ 25°C h 35 W/m2·K

qrad qconv

L6m • Eg

D3m

Ts ε 0.83 ηgen 0.95 ηgb 0.93

Assumptions: 1. Steady-state conditions. 2. Large surroundings. 3. Surface of the nacelle that is adjacent to the hub is adiabatic.

1.3

䊏

19

Relationship to Thermodynamics

Analysis: The nacelle temperature represents the minimum possible temperature inside the nacelle, and the first law of thermodynamics may be used to determine this temperature. The first step is to perform an energy balance on the nacelle to determine the rate of heat transfer from the nacelle to the air and surroundings under steady-state conditions. This step can be accomplished using either conservation of total energy or conservation of thermal and mechanical energy; we will compare these two approaches. Conservation of Total Energy The first of the three boxed statements of the first law in Section 1.3 can be converted to a rate basis and expressed in equation form as follows: dEsttot ˙ tot ˙ tot ⫽ E in ⫺ Eout dt

(1)

˙ tot ˙ tot Under steady-state conditions, this reduces to E˙ tot in ⫺ Eout ⫽ 0. The E in term corresponds to tot the mechanical work entering the nacelle W˙ , and the E˙out term includes the electrical power output P and the rate of heat transfer leaving the nacelle q. Thus W˙ ⫺ P ⫺ q ⫽ 0

(2)

Conservation of Thermal and Mechanical Energy Alternatively, we can express conservation of thermal and mechanical energy, starting with Equation 1.12c. Under steady-state conditions, this reduces to E˙ in ⫺ E˙ out ⫹ E˙ g ⫽ 0

(3)

Here, E˙ in once again corresponds to the mechanical work W˙ . However, E˙ out now includes only the rate of heat transfer leaving the nacelle q. It does not include the electrical power, since E represents only the thermal and mechanical forms of energy. The electrical power appears in the generation term, because mechanical energy is converted to electrical energy in the generator, giving rise to a negative source of mechanical energy. That is, E˙g ⫽ ⫺P. Thus, Equation (3) becomes W˙ ⫺ q ⫺ P ⫽ 0

(4)

which is equivalent to Equation (2), as it must be. Regardless of the manner in which the first law of thermodynamics is applied, the following expression for the rate of heat transfer evolves: q ⫽ W˙ ⫺ P

(5)

The mechanical work and electrical power are related by the efficiencies of the gearbox and generator, P ⫽ W˙ gbgen

(6)

Equation (5) can therefore be written as

1 ⫺ 1 ⫽ 0.33 ⫻ 106 W q ⫽ P 1 ⫺ 1 ⫽ 2.5 ⫻ 106 W ⫻ gb gen 0.93 ⫻ 0.95

(7)

Application of the Rate Equations Heat transfer is due to convection and radiation from the exterior surface of the nacelle, governed by Equations 1.3a and 1.7, respectively. Thus

20

Chapter 1

䊏

Introduction

q ⫽ qrad ⫹ qconv⫽ A[q⬙rad ⫹ q⬙conv]

⫽ DL ⫹ D 4

[(T ⫺ T

2

4 s

4 sur)

⫹ h(Ts ⫺ T앝)] ⫽ 0.33 ⫻ 106 W

or

⫻3m⫻6m⫹

⫻ (3 m)2 4

⫻ [0.83 ⫻ 5.67 ⫻ 10⫺8 W/m2 䡠 K4 (Ts4 ⫺ (273 ⫹ 20)4)K4 ⫹ 35 W/m2 䡠 K (Ts ⫺ (273 ⫹ 25)K)] ⫽ 0.33 ⫻ 106 W The preceding equation does not have a closed-form solution, but the surface temperature can be easily determined by trial and error or by using a software package such as the Interactive Heat Transfer (IHT) software accompanying your text. Doing so yields Ts ⫽ 416 K ⫽ 143⬚C We know that the temperature inside the nacelle must be greater than the exterior surface temperature of the nacelle Ts, because the heat generated within the nacelle must be transferred from the interior of the nacelle to its surface, and from the surface to the air and surroundings. Therefore, Ts represents the minimum possible temperature inside the enclosed nacelle. 䉰

Comments: 1. The temperature inside the nacelle is very high. This would preclude, for example, performance of routine maintenance by a worker, as illustrated in the problem statement. Thermal management approaches involving fans or blowers must be employed to reduce the temperature to an acceptable level. 2. Improvements in the efficiencies of either the gearbox or the generator would not only provide more electrical power, but would also reduce the size and cost of the thermal management hardware. As such, improved efficiencies would increase revenue generated by the wind turbine and decrease both its capital and operating costs. 3. The heat transfer coefficient would not be a steady value but would vary periodically as the blades sweep past the nacelle. Therefore, the value of the heat transfer coefficient represents a time-averaged quantity.

EXAMPLE 1.4 A long conducting rod of diameter D and electrical resistance per unit length R⬘e is initially in thermal equilibrium with the ambient air and its surroundings. This equilibrium is disturbed when an electrical current I is passed through the rod. Develop an equation that could be used to compute the variation of the rod temperature with time during the passage of the current.

1.3

䊏

21

Relationship to Thermodynamics

SOLUTION Known: Temperature of a rod of prescribed diameter and electrical resistance changes with time due to passage of an electrical current. Find: Equation that governs temperature change with time for the rod. Schematic: Air

•

E out

T∞, h

Tsur

T

I

•

•

E g, E st

Diameter D

L

Assumptions: 1. At any time t, the temperature of the rod is uniform. 2. Constant properties (r, c, ⫽ a). 3. Radiation exchange between the outer surface of the rod and the surroundings is between a small surface and a large enclosure. Analysis: The first law of thermodynamics may often be used to determine an unknown temperature. In this case, there is no mechanical energy component. So relevant terms include heat transfer by convection and radiation from the surface, thermal energy generation due to ohmic heating within the conductor, and a change in thermal energy storage. Since we wish to determine the rate of change of the temperature, the first law should be applied at an instant of time. Hence, applying Equation 1.12c to a control volume of length L about the rod, it follows that E˙g ⫺ E˙out ⫽ E˙ st where thermal energy generation is due to the electric resistance heating, E˙ g ⫽ I 2R⬘e L Heating occurs uniformly within the control volume and could also be expressed in terms of a volumetric heat generation rate q˙(W/m3). The generation rate for the entire control volume is then E˙g ⫽ q˙V, where q˙ ⫽ I 2R⬘e /(D2/4). Energy outflow is due to convection and net radiation from the surface, Equations 1.3a and 1.7, respectively, 4 ) E˙out ⫽ h(DL)(T ⫺ T앝) ⫹ (DL)(T 4 ⫺ Tsur

and the change in energy storage is due to the temperature change, dU E˙st ⫽ t ⫽ d (VcT) dt dt The term E˙st is associated with the rate of change in the internal thermal energy of the rod, where and c are the mass density and the specific heat, respectively, of the rod material,

Chapter 1

䊏

Introduction

and V is the volume of the rod, V ⫽ (D2/4)L. Substituting the rate equations into the energy balance, it follows that

2 4 ) ⫽ c D L dT I 2R⬘e L ⫺ h(DL)(T ⫺ T앝) ⫺ (DL)(T 4 ⫺ T sur 4 dt

Hence 2 4 4 dT ⫽ I R⬘e ⫺ Dh(T ⫺ T앝) ⫺ D(T ⫺ Tsur) dt c(D2/4)

䉰

Comments: 1. The preceding equation could be solved for the time dependence of the rod temperature by integrating numerically. A steady-state condition would eventually be reached for which dT/dt ⫽ 0. The rod temperature is then determined by an algebraic equation of the form 4 Dh(T ⫺ T앝) ⫹ D(T 4 ⫺ Tsur ) ⫽ I 2R⬘e

2. For fixed environmental conditions (h, T앝, Tsur), as well as a rod of fixed geometry (D) and properties (, R⬘e), the steady-state temperature depends on the rate of thermal energy generation and hence on the value of the electric current. Consider an uninsulated copper wire (D ⫽ 1 mm, ⫽ 0.8, R⬘e ⫽ 0.4 ⍀/m) in a relatively large enclosure (Tsur ⫽ 300 K) through which cooling air is circulated (h ⫽ 100 W/m2 䡠 K, T앝 ⫽ 300 K). Substituting these values into the foregoing equation, the rod temperature has been computed for operating currents in the range 0 ⱕ I ⱕ 10 A, and the following results were obtained: 150 125 100

T (C)

22

75 60 50 25 0

0

2

4

5.2

6

8

10

I (amperes)

3. If a maximum operating temperature of T ⫽ 60⬚C is prescribed for safety reasons, the current should not exceed 5.2 A. At this temperature, heat transfer by radiation (0.6 W/m) is much less than heat transfer by convection (10.4 W/m). Hence, if one wished to operate at a larger current while maintaining the rod temperature within the safety limit, the convection coefficient would have to be increased by increasing the velocity of the circulating air. For h ⫽ 250 W/m2 䡠 K, the maximum allowable current could be increased to 8.1 A. 4. The IHT software is especially useful for solving equations, such as the energy balance in Comment 1, and generating the graphical results of Comment 2.

1.3

䊏

23

Relationship to Thermodynamics

EXAMPLE 1.5 A hydrogen-air Proton Exchange Membrane (PEM) fuel cell is illustrated below. It consists of an electrolytic membrane sandwiched between porous cathode and anode materials, forming a very thin, three-layer membrane electrode assembly (MEA). At the anode, protons and electrons are generated (2H2 l 4H⫹ ⫹ 4e⫺); at the cathode, the protons and electrons recombine to form water (O2 ⫹ 4e⫺ ⫹ 4H⫹ l 2H2O). The overall reaction is then 2H2 ⫹ O2 l 2H2O. The dual role of the electrolytic membrane is to transfer hydrogen ions and serve as a barrier to electron transfer, forcing the electrons to the electrical load that is external to the fuel cell. Ec

I

e

e

Tsur

e O2

H2 e

H2

H

•

Eg H2

O2

H2O e

e

Tc O2

q

q H

H2O

H2 e

e O2 H

H2O

Porous anode

Tsur

H2O O2

Porous cathode Electrolytic membrane

Air

h, T∞

The membrane must operate in a moist state in order to conduct ions. However, the presence of liquid water in the cathode material may block the oxygen from reaching the cathode reaction sites, resulting in the failure of the fuel cell. Therefore, it is critical to control the temperature of the fuel cell, Tc , so that the cathode side contains saturated water vapor. For a given set of H2 and air inlet flow rates and use of a 50 mm ⫻ 50 mm MEA, the fuel cell generates P ⫽ I 䡠 Ec ⫽ 9 W of electrical power. Saturated vapor conditions exist in the fuel cell, corresponding to Tc ⫽ Tsat ⫽ 56.4⬚C. The overall electrochemical reaction is exothermic, and the corresponding thermal generation rate of E˙g ⫽ 11.25 W must be removed from the fuel cell by convection and radiation. The ambient and surrounding

24

Chapter 1

䊏

Introduction

temperatures are T앝 ⫽ Tsur ⫽ 25⬚C, and the relationship between the cooling air velocity and the convection heat transfer coefficient h is h ⫽ 10.9 W 䡠 s0.8/m2.8 䡠 K ⫻ V 0.8 where V has units of m/s. The exterior surface of the fuel cell has an emissivity of ⫽ 0.88. Determine the value of the cooling air velocity needed to maintain steady-state operating conditions. Assume the edges of the fuel cell are well insulated.

SOLUTION Known: Ambient and surrounding temperatures, fuel cell output voltage and electrical current, heat generated by the overall electrochemical reaction, and the desired fuel cell operating temperature. Find: The required cooling air velocity V needed to maintain steady-state operation at Tc 56.4⬚C. Schematic:

W = 50 mm

H = 50 mm q

Tsur = 25C

•

Eg

Tc = 56.4C ε = 0.88

Air

T∞ = 25C h

Assumptions: 1. Steady-state conditions. 2. Negligible temperature variations within the fuel cell. 3. Fuel cell is placed in large surroundings. 4. Edges of the fuel cell are well insulated. 5. Negligible energy entering or leaving the control volume due to gas or liquid flows.

1.3

䊏

25

Relationship to Thermodynamics

Analysis: To determine the required cooling air velocity, we must first perform an energy balance on the fuel cell. Noting that there is no mechanical energy component, we see that E˙in ⫽ 0 and E˙out ⫽ E˙g. This yields qconv ⫹ qrad ⫽ E˙g ⫽ 11.25 W where 4 qrad ⫽ A(Tc4 ⫺ Tsur )

⫽ 0.88 ⫻ (2 ⫻ 0.05 m ⫻ 0.05 m) ⫻ 5.67 ⫻ 10⫺8 W/m2 䡠 K4 ⫻ (329.44 ⫺ 2984) K4 ⫽ 0.97 W Therefore, we may find qconv ⫽ 11.25 W ⫺ 0.97 W ⫽ 10.28 W ⫽ hA(Tc ⫺ T앝) ⫽ 10.9 W 䡠 s0.8/m2.8 䡠 K ⫻ V 0.8 A(Tc ⫺ T앝) which may be rearranged to yield V⫽

10.28 W 10.9 W . s0.8Ⲑm2.8 . K ⫻ (2 ⫻ 0.05 m ⫻ 0.05 m) ⫻ (56.4 ⫺ 25oC)

V ⫽ 9.4 m/s

1.25

䉰

Comments: 1. Temperature and humidity of the MEA will vary from location to location within the fuel cell. Prediction of the local conditions within the fuel cell would require a more detailed analysis. 2. The required cooling air velocity is quite high. Decreased cooling velocities could be used if heat transfer enhancement devices were added to the exterior of the fuel cell. 3. The convective heat rate is significantly greater than the radiation heat rate. 4. The chemical energy (20.25 W) of the hydrogen and oxygen is converted to electrical (9 W) and thermal (11.25 W) energy. This fuel cell operates at a conversion efficiency of (9 W)/(20.25 W) ⫻ 100 ⫽ 44%.

EXAMPLE 1.6 Large PEM fuel cells, such as those used in automotive applications, often require internal cooling using pure liquid water to maintain their temperature at a desired level (see Example 1.5). In cold climates, the cooling water must be drained from the fuel cell to an adjoining container when the automobile is turned off so that harmful freezing does not occur within the fuel cell. Consider a mass M of ice that was frozen while the automobile was not being operated. The ice is at the fusion temperature (Tf ⫽ 0⬚C) and is enclosed in a cubical container of width W on a side. The container wall is of thickness L and thermal

26

Chapter 1

䊏

Introduction

conductivity k. If the outer surface of the wall is heated to a temperature T1 > Tf to melt the ice, obtain an expression for the time needed to melt the entire mass of ice and, in turn, deliver cooling water to, and energize, the fuel cell.

SOLUTION Known: Mass and temperature of ice. Dimensions, thermal conductivity, and outer surface temperature of containing wall. Find: Expression for time needed to melt the ice. Schematic: Section A-A

A

A

T1

Ein

∆ Est

Ice-water mixture (Tf )

W

k

L

Assumptions: 1. Inner surface of wall is at Tf throughout the process. 2. Constant properties. 3. Steady-state, one-dimensional conduction through each wall. 4. Conduction area of one wall may be approximated as W 2 (L Ⰶ W). Analysis: Since we must determine the melting time tm, the first law should be applied over the time interval ⌬t ⫽ tm. Hence, applying Equation 1.12b to a control volume about the ice–water mixture, it follows that Ein ⫽ ⌬Est ⫽ ⌬Ulat where the increase in energy stored within the control volume is due exclusively to the change in latent energy associated with conversion from the solid to liquid state. Heat is transferred to the ice by means of conduction through the container wall. Since the temperature difference across the wall is assumed to remain at (T1 ⫺ Tf) throughout the melting process, the wall conduction rate is constant qcond ⫽ k(6W 2)

T1 ⫺ Tf L

and the amount of energy inflow is

Ein ⫽ k(6W 2)

T1 ⫺ Tf L

t

m

The amount of energy required to effect such a phase change per unit mass of solid is termed the latent heat of fusion hsf . Hence the increase in energy storage is ⌬Est ⫽ Mhsf

1.3

䊏

27

Relationship to Thermodynamics

By substituting into the first law expression, it follows that tm ⫽

Mhsf L 6W k(T1 ⫺ Tf) 2

䉰

Comments: 1. Several complications would arise if the ice were initially subcooled. The storage term would have to include the change in sensible (internal thermal) energy required to take the ice from the subcooled to the fusion temperature. During this process, temperature gradients would develop in the ice. 2. Consider a cavity of width W ⫽ 100 mm on a side, wall thickness L ⫽ 5 mm, and thermal conductivity k ⫽ 0.05 W/m 䡠 K. The mass of the ice in the cavity is M ⫽ s(W ⫺ 2L)3 ⫽ 920 kg/m3 ⫻ (0.100 ⫺ 0.01)3 m3 ⫽ 0.67 kg If the outer surface temperature is T1 ⫽ 30⬚C, the time required to melt the ice is tm ⫽

0.67 kg ⫻ 334,000 J/kg ⫻ 0.005 m ⫽ 12,430 s ⫽ 207 min 6(0.100 m)2 ⫻ 0.05 W/m 䡠 K (30 ⫺ 0)⬚C

The density and latent heat of fusion of the ice are s ⫽ 920 kg/m3 and hsf ⫽ 334 kJ/kg, respectively. 3. Note that the units of K and ⬚C cancel each other in the foregoing expression for tm. Such cancellation occurs frequently in heat transfer analysis and is due to both units appearing in the context of a temperature difference.

We will frequently have occasion to apply the conservation of energy requirement at the surface of a medium. In this special case, the control surfaces are located on either side of the physical boundary and enclose no mass or volume (see Figure 1.9). Accordingly, the generation and storage terms of the conservation

The Surface Energy Balance

Surroundings

Tsur q"rad T1

q"cond Fluid

q"conv T

u∞, T∞

T2 x

T∞ Control surfaces

FIGURE 1.9 The energy balance for conservation of energy at the surface of a medium.

28

Chapter 1

䊏

Introduction

expression, Equation 1.12c, are no longer relevant, and it is necessary to deal only with surface phenomena. For this case, the conservation requirement becomes E˙in ⫺ E˙out ⫽ 0

(1.13)

Even though energy generation may be occurring in the medium, the process would not affect the energy balance at the control surface. Moreover, this conservation requirement holds for both steady-state and transient conditions. In Figure 1.9, three heat transfer terms are shown for the control surface. On a unit area basis, they are conduction from the medium to the control surface (q⬙cond), convection from the surface to a fluid (q⬙conv), and net radiation exchange from the surface to the surroundings (q⬙rad). The energy balance then takes the form. q⬙cond ⫺ q⬙conv ⫺ q⬙rad ⫽ 0

(1.14)

and we can express each of the terms using the appropriate rate equations, Equations 1.2, 1.3a, and 1.7.

EXAMPLE 1.7 Humans are able to control their heat production rate and heat loss rate to maintain a nearly constant core temperature of Tc ⫽ 37⬚C under a wide range of environmental conditions. This process is called thermoregulation. From the perspective of calculating heat transfer between a human body and its surroundings, we focus on a layer of skin and fat, with its outer surface exposed to the environment and its inner surface at a temperature slightly less than the core temperature, Ti ⫽ 35⬚C ⫽ 308 K. Consider a person with a skin/fat layer of thickness L ⫽ 3 mm and effective thermal conductivity k ⫽ 0.3 W/m 䡠 K. The person has a surface area A ⫽ 1.8 m2 and is dressed in a bathing suit. The emissivity of the skin is ⫽ 0.95. 1. When the person is in still air at T앝 ⫽ 297 K, what is the skin surface temperature and rate of heat loss to the environment? Convection heat transfer to the air is characterized by a free convection coefficient of h ⫽ 2 W/m2 䡠 K. 2. When the person is in water at T앝 ⫽ 297 K, what is the skin surface temperature and heat loss rate? Heat transfer to the water is characterized by a convection coefficient of h ⫽ 200 W/m2 䡠 K.

SOLUTION Known: Inner surface temperature of a skin/fat layer of known thickness, thermal conductivity, emissivity, and surface area. Ambient conditions. Find: Skin surface temperature and heat loss rate for the person in air and the person in water.

1.3

䊏

29

Relationship to Thermodynamics

Schematic: Ti = 308 K

Skin/fat

Ts ε = 0.95

Tsur = 297 K

q"rad q"cond q"conv

T∞ = 297 K h = 2 W/m2•K (Air) h = 200 W/m2•K (Water)

k = 0.3 W/m•K L = 3 mm Air or water

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer by conduction through the skin/fat layer. 3. Thermal conductivity is uniform. 4. Radiation exchange between the skin surface and the surroundings is between a small surface and a large enclosure at the air temperature. 5. Liquid water is opaque to thermal radiation. 6. Bathing suit has no effect on heat loss from body. 7. Solar radiation is negligible. 8. Body is completely immersed in water in part 2. Analysis: 1. The skin surface temperature may be obtained by performing an energy balance at the skin surface. From Equation 1.13, E˙ in ⫺ E˙ out ⫽ 0 It follows that, on a unit area basis, q⬙cond ⫺ q⬙conv ⫺ q⬙rad ⫽ 0 or, rearranging and substituting from Equations 1.2, 1.3a, and 1.7, Ti ⫺ Ts 4 ) ⫽ h(Ts ⫺ T앝) ⫹ (T s4 ⫺ Tsur L The only unknown is Ts, but we cannot solve for it explicitly because of the fourth-power dependence of the radiation term. Therefore, we must solve the equation iteratively, which can be done by hand or by using IHT or some other equation solver. To expedite a hand solution, we write the radiation heat flux in terms of the radiation heat transfer coefficient, using Equations 1.8 and 1.9: T ⫺ Ts k i ⫽ h(Ts ⫺ T앝) ⫹ hr (Ts ⫺ Tsur) L Solving for Ts, with Tsur ⫽ T앝, we have k

kTi ⫹ (h ⫹ hr)T앝 Ts ⫽ L k ⫹ (h ⫹ h ) r L

30

Chapter 1

䊏

Introduction

We estimate hr using Equation 1.9 with a guessed value of Ts ⫽ 305 K and T앝 ⫽ 297 K, to yield hr ⫽ 5.9 W/m2 䡠 K. Then, substituting numerical values into the preceding equation, we find 0.3 W/m 䡠 K ⫻ 308 K ⫹ (2 ⫹ 5.9) W/m2 䡠 K ⫻ 297 K 3 ⫻ 10⫺3 m Ts ⫽ ⫽ 307.2 K 0.3 W/m 䡠 K ⫹ (2 ⫹ 5.9) W/m2 䡠 K 3 ⫻ 10⫺3 m With this new value of Ts, we can recalculate hr and Ts, which are unchanged. Thus the skin temperature is 307.2 K 34⬚C. 䉰 The rate of heat loss can be found by evaluating the conduction through the skin/fat layer: T ⫺ Ts (308 ⫺ 307.2) K ⫽ 146 W ⫽ 0.3 W/m 䡠 K ⫻ 1.8 m2 ⫻ qs ⫽ kA i 䉰 L 3 ⫻ 10⫺3 m 2. Since liquid water is opaque to thermal radiation, heat loss from the skin surface is by convection only. Using the previous expression with hr ⫽ 0, we find 0.3 W/m 䡠 K ⫻ 308 K ⫹ 200 W/m2 䡠 K ⫻ 297 K 3 ⫻ 10⫺3 m ⫽ 300.7 K Ts ⫽ 0.3 W/m 䡠 K ⫹ 200 W/m2 䡠 K 3 ⫻ 10⫺3 m

䉰

and qs ⫽ kA

Ti ⫺ Ts (308 ⫺ 300.7) K ⫽ 1320 W ⫽ 0.3 W/m 䡠 K ⫻ 1.8 m2 ⫻ L 3 ⫻ 10⫺3 m

䉰

Comments: 1. When using energy balances involving radiation exchange, the temperatures appearing in the radiation terms must be expressed in kelvins, and it is good practice to use kelvins in all terms to avoid confusion. 2. In part 1, heat losses due to convection and radiation are 37 W and 109 W, respectively. Thus, it would not have been reasonable to neglect radiation. Care must be taken to include radiation when the heat transfer coefficient is small (as it often is for natural convection to a gas), even if the problem statement does not give any indication of its importance. 3. A typical rate of metabolic heat generation is 100 W. If the person stayed in the water too long, the core body temperature would begin to fall. The large heat loss in water is due to the higher heat transfer coefficient, which in turn is due to the much larger thermal conductivity of water compared to air. 4. The skin temperature of 34⬚C in part 1 is comfortable, but the skin temperature of 28⬚C in part 2 is uncomfortably cold.

1.3

䊏

Relationship to Thermodynamics

31

In addition to being familiar with the transport rate equations described in Section 1.2, the heat transfer analyst must be able to work with the energy conservation requirements of Equations 1.12 and 1.13. The application of these balances is simplified if a few basic rules are followed.

Application of the Conservation Laws: Methodology

1. The appropriate control volume must be defined, with the control surfaces represented by a dashed line or lines. 2. The appropriate time basis must be identified. 3. The relevant energy processes must be identified, and each process should be shown on the control volume by an appropriately labeled arrow. 4. The conservation equation must then be written, and appropriate rate expressions must be substituted for the relevant terms in the equation. Note that the energy conservation requirement may be applied to a finite control volume or a differential (infinitesimal) control volume. In the first case, the resulting expression governs overall system behavior. In the second case, a differential equation is obtained that can be solved for conditions at each point in the system. Differential control volumes are introduced in Chapter 2, and both types of control volumes are used extensively throughout the text.

1.3.2 Relationship to the Second Law of Thermodynamics and the Efficiency of Heat Engines In this section, we are interested in the efficiency of heat engines. The discussion builds on your knowledge of thermodynamics and shows how heat transfer plays a crucial role in managing and promoting the efficiency of a broad range of energy conversion devices. Recall that a heat engine is any device that operates continuously or cyclically and that converts heat to work. Examples include internal combustion engines, power plants, and thermoelectric devices (to be discussed in Section 3.8). Improving the efficiency of heat engines is a subject of extreme importance; for example, more efficient combustion engines consume less fuel to produce a given amount of work and reduce the corresponding emissions of pollutants and carbon dioxide. More efficient thermoelectric devices can generate more electricity from waste heat. Regardless of the energy conversion device, its size, weight, and cost can all be reduced through improvements in its energy conversion efficiency. The second law of thermodynamics is often invoked when efficiency is of concern and can be expressed in a variety of different but equivalent ways. The Kelvin–Planck statement is particularly relevant to the operation of heat engines [1]. It states: It is impossible for any system to operate in a thermodynamic cycle and deliver a net amount of work to its surroundings while receiving energy by heat transfer from a single thermal reservoir.

Recall that a thermodynamic cycle is a process for which the initial and final states of the system are identical. Consequently, the energy stored in the system does not change between the initial and final states, and the first law of thermodynamics (Equation 1.12a) reduces to W ⫽ Q. A consequence of the Kelvin–Planck statement is that a heat engine must exchange heat with two (or more) reservoirs, gaining thermal energy from the higher-temperature

32

Chapter 1

䊏

Introduction

reservoir and rejecting thermal energy to the lower-temperature reservoir. Thus, converting all of the input heat to work is impossible, and W ⫽ Qin – Qout, where Qin and Qout are both defined to be positive. That is, Qin is the heat transferred from the high temperature source to the heat engine, and Qout is the heat transferred from the heat engine to the low temperature sink. The efficiency of a heat engine is defined as the fraction of heat transferred into the system that is converted to work, namely Qin ⫺ Qout Qout W ⫽ ⫽1⫺ Qin Qin Qin

(1.15)

The second law also tells us that, for a reversible process, the ratio Qout/Qin is equal to the ratio of the absolute temperatures of the respective reservoirs [1]. Thus, the efficiency of a heat engine undergoing a reversible process, called the Carnot efficiency C, is given by C ⫽ 1 ⫺

Tc Th

(1.16)

where Tc and Th are the absolute temperatures of the low- and high-temperature reservoirs, respectively. The Carnot efficiency is the maximum possible efficiency that any heat engine can achieve operating between those two temperatures. Any real heat engine, which will necessarily undergo an irreversible process, will have a lower efficiency. From our knowledge of thermodynamics, we know that, for heat transfer to take place reversibly, it must occur through an infinitesimal temperature difference between the reservoir and heat engine. However, from our newly acquired knowledge of heat transfer mechanisms, as embodied, for example, in Equations 1.2, 1.3, and 1.7, we now realize that, for heat transfer to occur, there must be a nonzero temperature difference between the reservoir and the heat engine. This reality introduces irreversibility and reduces the efficiency. With the concepts of the preceding paragraph in mind, we now consider a more realistic model of a heat engine [2–5] in which heat is transferred into the engine through a thermal resistance Rt,h , while heat is extracted from the engine through a second thermal resistance Rt,c (Figure 1.10). The subscripts h and c refer to the hot and cold sides of the heat engine, respectively. As discussed in Section 1.2.4, these thermal resistances are associated with heat transfer between the heat engine and the reservoirs across a nonzero temperature difference, by way of the mechanisms of conduction, convection, and/or radiation. For example, the resistances could represent conduction through the walls separating the heat engine from the two reservoirs. Note that the reservoir temperatures are still Th and Tc but that the temperatures seen by the heat engine are Th,i ⬍ Th and Tc,i ⬎ Tc , as shown in the diagram. The heat engine is still assumed to be internally reversible, and its efficiency is still the Carnot efficiency. However,

High-temperature side resistance

High-temperature reservoir Q

Th

in

Th,i Heat engine walls

Internally reversible heat engine

W Tc,i

Low-temperature side resistance

Qout Low-temperature reservoir

Tc

FIGURE 1.10 Internally reversible heat engine exchanging heat with high- and low-temperature reservoirs through thermal resistances.

1.3

䊏

33

Relationship to Thermodynamics

the Carnot efficiency is now based on the internal temperatures Th,i and Tc,i. Therefore, a modified efficiency that accounts for realistic (irreversible) heat transfer processes m is m ⫽ 1 ⫺

Tc,i Qout q ⫽ 1 ⫺ qout ⫽ 1 ⫺ Qin Th,i in

(1.17)

where the ratio of heat flows over a time interval, Qout /Qin, has been replaced by the corresponding ratio of heat rates, qout /qin. This replacement is based on applying energy conservation at an instant in time,1 as discussed in Section 1.3.1. Utilizing the definition of a thermal resistance, the heat transfer rates into and out of the heat engine are given by qin ⫽ (Th ⫺ Th,i)/Rt,h

(1.18a)

qout ⫽ (Tc,i ⫺ Tc)/Rt,c

(1.18b)

Equations 1.18 can be solved for the internal temperatures, to yield Th,i ⫽ Th ⫺ qin Rt,h

(1.19a)

Tc,i ⫽ Tc ⫹ qoutRt,c ⫽ Tc ⫹ qin(1 ⫺ m)Rt,c

(1.19b)

In Equation 1.19b, qout has been related to qin and m, using Equation 1.17. The more realistic, modified efficiency can then be expressed as m ⫽ 1 ⫺

Tc,i Tc ⫹ qin(1 ⫺ m)Rt,c ⫽1 ⫺ Th,i Th ⫺ qinRt,h

(1.20)

Solving for m results in m ⫽ 1 ⫺

Tc Th ⫺ qin Rtot

(1.21)

where Rtot ⫽ Rt,h ⫹ Rt,c. It is readily evident that m ⫽ C only if the thermal resistances Rt,h and Rt,c could somehow be made infinitesimally small (or if qin ⫽ 0). For realistic (nonzero) values of Rtot , m ⬍ C , and m further deteriorates as either Rtot or qin increases. As an extreme case, note that m ⫽ 0 when Th ⫽ Tc ⫹ qin Rtot , meaning that no power could be produced even though the Carnot efficiency, as expressed in Equation 1.16, is nonzero. In addition to the efficiency, another important parameter to consider is the power output of the heat engine, given by

W˙ ⫽ qinm ⫽ qin 1 ⫺

Tc Th ⫺ qin Rtot

(1.22)

It has already been noted in our discussion of Equation 1.21 that the efficiency is equal to the maximum Carnot efficiency (m ⫽ C) if qin ⫽ 0. However, under these circumstances

1

The heat engine is assumed to undergo a continuous, steady-flow process, so that all heat and work processes are occurring simultaneously, and the corresponding terms would be expressed in watts (W). For a heat engine undergoing a cyclic process with sequential heat and work processes occurring over different time intervals, we would need to introduce the time intervals for each process, and each term would be expressed in joules (J).

34

Chapter 1

䊏

Introduction

˙ is zero according to Equation 1.22. To increase W˙ , qin must be the power output W increased at the expense of decreased efficiency. In any real application, a balance must be struck between maximizing the efficiency and maximizing the power output. If provision of the heat input is inexpensive (for example, if waste heat is converted to power), a case could be made for sacrificing efficiency to maximize power output. In contrast, if fuel is expensive or emissions are detrimental (such as for a conventional fossil fuel power plant), the efficiency of the energy conversion may be as or more important than the power output. In any case, heat transfer and thermodyamic principles should be used to determine the actual efficiency and power output of a heat engine. Although we have limited our discussion of the second law to heat engines, the preceding analysis shows how the principles of thermodynamics and heat transfer can be combined to address significant problems of contemporary interest.

EXAMPLE 1.8 In a large steam power plant, the combustion of coal provides a heat rate of qin ⫽ 2500 MW at a flame temperature of Th ⫽ 1000 K. Heat is rejected from the plant to a river flowing at Tc ⫽ 300 K. Heat is transferred from the combustion products to the exterior of large tubes in the boiler by way of radiation and convection, through the boiler tubes by conduction, and then from the interior tube surface to the working fluid (water) by convection. On the cold side, heat is extracted from the power plant by condensation of steam on the exterior condenser tube surfaces, through the condenser tube walls by conduction, and from the interior of the condenser tubes to the river water by convection. Hot and cold side thermal resistances account for the combined effects of conduction, convection, and radiation and, under design conditions, they are Rt,h ⫽ 8 ⫻ 10⫺8 K/W and Rt,c ⫽ 2 ⫻ 10⫺8 K/W, respectively. 1. Determine the efficiency and power output of the power plant, accounting for heat transfer effects to and from the cold and hot reservoirs. Treat the power plant as an internally reversible heat engine. 2. Over time, coal slag will accumulate on the combustion side of the boiler tubes. This fouling process increases the hot side resistance to Rt,h ⫽ 9 ⫻ 10⫺8 K/W. Concurrently, biological matter can accumulate on the river water side of the condenser tubes, increasing the cold side resistance to Rt,c ⫽ 2.2 ⫻ 10⫺8 K/W. Find the efficiency and power output of the plant under fouled conditions.

SOLUTION Known: Source and sink temperatures and heat input rate for an internally reversible heat engine. Thermal resistances separating heat engine from source and sink under clean and fouled conditions. Find: 1. Efficiency and power output for clean conditions. 2. Efficiency and power output under fouled conditions.

1.3

䊏

35

Relationship to Thermodynamics

Schematic: Products of combustion qin 2500 MW

8

Th 1000 K

Rt,h 8 10 K/W (clean) 8 Rt,h 9 10 K/W (fouled)

Th,i Power plant

Tc,i

8

Rt,c 2 10 K/W (clean) 8 Rt,c 2.2 10 K/W (fouled)

•

W

qout Cooling water

Tc 300 K

Assumptions: 1. Steady-state conditions. 2. Power plant behaves as an internally reversible heat engine, so its efficiency is the modified efficiency. Analysis: 1. The modified efficiency of the internally reversible power plant, considering realistic heat transfer effects on the hot and cold side of the power plant, is given by Equation 1.21: m ⫽ 1 ⫺

Tc Th ⫺ qinRtot

where, for clean conditions Rtot ⫽ Rt,h ⫹ Rt,c ⫽ 8 ⫻ 10⫺8 K/W ⫹ 2 ⫻ 10⫺8 K/W ⫽ 1.0 ⫻ 10⫺7 K/W Thus m ⫽ 1 ⫺

Tc 300 K ⫽1⫺ ⫽ 0.60 ⫽ 60% 䉰 Th ⫺ qin Rtot 1000 K ⫺ 2500 ⫻ 106 W ⫻ 1.0 ⫻ 10⫺7 K/W

The power output is given by W˙ ⫽ qinm ⫽ 2500 MW ⫻ 0.60 ⫽ 1500 MW

䉰

2. Under fouled conditions, the preceding calculations are repeated to find m ⫽ 0.583 ⫽ 58.3% and W˙ ⫽ 1460 MW

䉰

Comments: 1. The actual efficiency and power output of a power plant operating between these temperatures would be much less than the foregoing values, since there would be other irreversibilities internal to the power plant. Even if these irreversibilities

36

Chapter 1

䊏

Introduction

were considered in a more comprehensive analysis, fouling effects would still reduce the plant efficiency and power output. 2. The Carnot efficiency is C ⫽ 1 ⫺ Tc /Th ⫽ 1 ⫺ 300 K/1000 K ⫽ 70%. The corresponding power output would be W˙ ⫽ qinC ⫽ 2500 MW ⫻ 0.70 ⫽ 1750 MW. Thus, if the effect of irreversible heat transfer from and to the hot and cold reservoirs, respectively, were neglected, the power output of the plant would be significantly overpredicted. 3. Fouling reduces the power output of the plant by ⌬P ⫽ 40 MW. If the plant owner sells the electricity at a price of $0.08/kW ⭈ h, the daily lost revenue associated with operating the fouled plant would be C ⫽ 40,000 kW ⫻ $0.08/kW 䡠 h ⫻ 24 h/day ⫽ $76,800/day.

1.4 Units and Dimensions The physical quantities of heat transfer are specified in terms of dimensions, which are measured in terms of units. Four basic dimensions are required for the development of heat transfer: length (L), mass (M), time (t), and temperature (T). All other physical quantities of interest may be related to these four basic dimensions. In the United States, dimensions have been customarily measured in terms of the English system of units, for which the base units are: Dimension Length (L) Mass (M) Time (t) Temperature (T)

Unit l l l l

foot (ft) pound mass (lbm) second (s) degree Fahrenheit (⬚F)

The units required to specify other physical quantities may then be inferred from this group.

In recent years, there has been a strong trend toward the global usage of a standard set of units. In 1960, the SI (Système International d’Unités) system of units was defined by the Eleventh General Conference on Weights and Measures and recommended as a worldwide standard. In response to this trend, the American Society of Mechanical Engineers (ASME) has required the use of SI units in all of its publications since 1974. For this reason and because SI units are operationally more convenient than the English system, the SI system is used for calculations of this text. However, because for some time to come, engineers might also have to work with results expressed in the English system, you should be able to convert from one system to the other. For your convenience, conversion factors are provided on the inside back cover of the text. The SI base units required for this text are summarized in Table 1.2. With regard to these units, note that 1 mol is the amount of substance that has as many atoms or molecules as there are atoms in 12 g of carbon-12 (12C); this is the gram-mole (mol). Although the mole has been recommended as the unit quantity of matter for the SI system, it is more consistent to work with the kilogram-mol (kmol, kg-mol). One kmol is simply the amount of substance that has as many atoms or molecules as there are atoms in 12 kg of 12C. As long as the use is consistent within a given problem, no difficulties arise in using either mol or kmol. The molecular weight of a substance is the mass associated with a mole or

1.4

䊏

37

Units and Dimensions

kilogram-mole. For oxygen, as an example, the molecular weight ᏹ is 16 g/mol or 16 kg/kmol. Although the SI unit of temperature is the kelvin, use of the Celsius temperature scale remains widespread. Zero on the Celsius scale (0⬚C) is equivalent to 273.15 K on the thermodynamic scale,2 in which case T (K) ⫽ T (⬚C) ⫹ 273.15 However, temperature differences are equivalent for the two scales and may be denoted as ⬚C or K. Also, although the SI unit of time is the second, other units of time (minute, hour, and day) are so common that their use with the SI system is generally accepted. The SI units comprise a coherent form of the metric system. That is, all remaining units may be derived from the base units using formulas that do not involve any numerical factors. Derived units for selected quantities are listed in Table 1.3. Note that force is measured in newtons, where a 1-N force will accelerate a 1-kg mass at 1 m/s2. Hence 1 N ⫽ 1 kg 䡠 m/s2. The unit of pressure (N/m2) is often referred to as the pascal. In the SI system, there is one unit of energy (thermal, mechanical, or electrical) called the joule (J); 1 J ⫽ 1 N 䡠 m. The unit for energy rate, or power, is then J/s, where one joule per second is equivalent to one watt (1 J/s ⫽ 1 W). Since working with extremely large or small numbers is frequently necessary, a set of standard prefixes has been introduced to simplify matters (Table 1.4). For example, 1 megawatt (MW) ⫽ 106 W, and 1 micrometer (m) ⫽ 10⫺6 m.

TABLE 1.2

SI base and supplementary units

Quantity and Symbol

Unit and Symbol

Length (L) Mass (M) Amount of substance Time (t) Electric current (I) Thermodynamic temperature (T) Plane anglea () Solid anglea ()

meter (m) kilogram (kg) mole (mol) second (s) ampere (A) kelvin (K) radian (rad) steradian (sr)

a

Supplementary unit.

TABLE 1.3

SI derived units for selected quantities

Quantity

Name and Symbol

Formula

Expression in SI Base Units

Force Pressure and stress Energy Power

newton (N) pascal (Pa) joule (J) watt (W)

m 䡠 kg/s2 N/m2 N䡠m J/s

m 䡠 kg/s2 kg/m 䡠 s2 m2 䡠 kg/s2 m2 䡠 kg/s3

2

The degree symbol is retained for designating the Celsius temperature (⬚C) to avoid confusion with the use of C for the unit of electrical charge (coulomb).

38

Chapter 1

䊏

Introduction

TABLE 1.4

Multiplying prefixes

Prefix

Abbreviation

Multiplier

femto pico nano micro milli centi hecto kilo mega giga tera peta exa

f p n m c h k M G T P E

10⫺15 10⫺12 10⫺9 10⫺6 10⫺3 10⫺2 102 103 106 109 1012 1015 1018

1.5 Analysis of Heat Transfer Problems: Methodology A major objective of this text is to prepare you to solve engineering problems that involve heat transfer processes. To this end, numerous problems are provided at the end of each chapter. In working these problems you will gain a deeper appreciation for the fundamentals of the subject, and you will gain confidence in your ability to apply these fundamentals to the solution of engineering problems. In solving problems, we advocate the use of a systematic procedure characterized by a prescribed format. We consistently employ this procedure in our examples, and we require our students to use it in their problem solutions. It consists of the following steps: 1. Known: After carefully reading the problem, state briefly and concisely what is known about the problem. Do not repeat the problem statement. 2. Find: State briefly and concisely what must be found. 3. Schematic: Draw a schematic of the physical system. If application of the conservation laws is anticipated, represent the required control surface or surfaces by dashed lines on the schematic. Identify relevant heat transfer processes by appropriately labeled arrows on the schematic. 4. Assumptions: List all pertinent simplifying assumptions. 5. Properties: Compile property values needed for subsequent calculations and identify the source from which they are obtained. 6. Analysis: Begin your analysis by applying appropriate conservation laws, and introduce rate equations as needed. Develop the analysis as completely as possible before substituting numerical values. Perform the calculations needed to obtain the desired results. 7. Comments: Discuss your results. Such a discussion may include a summary of key conclusions, a critique of the original assumptions, and an inference of trends obtained by performing additional what-if and parameter sensitivity calculations.

1.5

䊏

39

Analysis of Heat Tranfer Problems: Methodology

The importance of following steps 1 through 4 should not be underestimated. They provide a useful guide to thinking about a problem before effecting its solution. In step 7, we hope you will take the initiative to gain additional insights by performing calculations that may be computer based. The software accompanying this text provides a suitable tool for effecting such calculations.

EXAMPLE 1.9 The coating on a plate is cured by exposure to an infrared lamp providing a uniform irradiation of 2000 W/m2. It absorbs 80% of the irradiation and has an emissivity of 0.50. It is also exposed to an airflow and large surroundings for which temperatures are 20⬚C and 30⬚C, respectively. 1. If the convection coefficient between the plate and the ambient air is 15 W/m2 䡠 K, what is the cure temperature of the plate? 2. The final characteristics of the coating, including wear and durability, are known to depend on the temperature at which curing occurs. An airflow system is able to control the air velocity, and hence the convection coefficient, on the cured surface, but the process engineer needs to know how the temperature depends on the convection coefficient. Provide the desired information by computing and plotting the surface temperature as a function of h for 2 ⱕ h ⱕ 200 W/m2 䡠 K. What value of h would provide a cure temperature of 50⬚C?

SOLUTION Known: Coating with prescribed radiation properties is cured by irradiation from an infrared lamp. Heat transfer from the coating is by convection to ambient air and radiation exchange with the surroundings. Find: 1. Cure temperature for h ⫽ 15 W/m2 䡠 K. 2. Effect of airflow on the cure temperature for 2 ⱕ h ⱕ 200 W/m2 䡠 K. Value of h for which the cure temperature is 50⬚C. Schematic:

Tsur = 30°C Glamp = 2000 W/m2 T∞ = 20°C 2 ≤ h ≤ 200 W/m2•K

q"conv

Air T

Coating, α = 0.8, ε = 0.5

q"rad

α Glamp

Chapter 1

䊏

Introduction

Assumptions: 1. Steady-state conditions. 2. Negligible heat loss from back surface of plate. 3. Plate is small object in large surroundings, and coating has an absorptivity of ␣sur ⫽ ⫽ 0.5 with respect to irradiation from the surroundings. Analysis: 1. Since the process corresponds to steady-state conditions and there is no heat transfer at the back surface, the plate must be isothermal (Ts ⫽ T). Hence the desired temperature may be determined by placing a control surface about the exposed surface and applying Equation 1.13 or by placing the control surface about the entire plate and applying Equation 1.12c. Adopting the latter approach and recognizing that there is no energy generation (E˙ g ⫽ 0), Equation 1.12c reduces to E˙ in ⫺ E˙ out ⫽ 0 where E˙ st ⫽ 0 for steady-state conditions. With energy inflow due to absorption of the lamp irradiation by the coating and outflow due to convection and net radiation transfer to the surroundings, it follows that (␣G)lamp ⫺ q⬙conv ⫺ q⬙rad ⫽ 0 Substituting from Equations 1.3a and 1.7, we obtain 4 (␣G)lamp ⫺ h(T ⫺ T앝) ⫺ (T 4 ⫺ Tsur )⫽0

Substituting numerical values 0.8 ⫻ 2000 W/m2 ⫺ 15 W/m2 䡠 K (T ⫺ 293) K ⫺ 0.5 ⫻ 5.67 ⫻ 10⫺8 W/m2 䡠 K4 (T 4 ⫺ 3034) K4 ⫽ 0 and solving by trial-and-error, we obtain T ⫽ 377 K ⫽ 104⬚C

䉰

2. Solving the foregoing energy balance for selected values of h in the prescribed range and plotting the results, we obtain 240 200 160

T (C)

40

120 80 50 40 0

0

20

40 51 60 h (W/m2•K)

80

100

If a cure temperature of 50⬚C is desired, the airflow must provide a convection coefficient of h(T ⫽ 50⬚C) ⫽ 51.0 W/m2 䡠 K 䉰

1.6

䊏

Relevance of Heat Tranfer

41

Comments: 1. The coating (plate) temperature may be reduced by decreasing T앝 and Tsur, as well as by increasing the air velocity and hence the convection coefficient. 2. The relative contributions of convection and radiation to heat transfer from the plate vary greatly with h. For h ⫽ 2 W/m2 䡠 K, T ⫽ 204⬚C and radiation dominates (q⬙rad 1232 W/m2, q⬙conv 368 W/m2). Conversely, for h ⫽ 200 W/m2 䡠 K, T ⫽ 28⬚C and convection dominates (q⬙conv 1606 W/m2, q⬙rad ⫺6 W/m2). In fact, for this condition the plate temperature is slightly less than that of the surroundings and net radiation exchange is to the plate.

1.6 Relevance of Heat Transfer We will devote much time to acquiring an understanding of heat transfer effects and to developing the skills needed to predict heat transfer rates and temperatures that evolve in certain situations. What is the value of this knowledge? To what problems may it be applied? A few examples will serve to illustrate the rich breadth of applications in which heat transfer plays a critical role. The challenge of providing sufficient amounts of energy for humankind is well known. Adequate supplies of energy are needed not only to fuel industrial productivity, but also to supply safe drinking water and food for much of the world’s population and to provide the sanitation necessary to control life-threatening diseases. To appreciate the role heat transfer plays in the energy challenge, consider a flow chart that represents energy use in the United States, as shown in Figure 1.11a. Currently, about 58% of the nearly 110 EJ of energy that is consumed annually in the United States is wasted in the form of heat. Nearly 70% of the energy used to generate electricity is lost in the form of heat. The transportation sector, which relies almost exclusively on petroleumbased fuels, utilizes only 21.5% of the energy it consumes; the remaining 78.5% is released in the form of heat. Although the industrial and residential/commercial use of energy is relatively more efficient, opportunities for energy conservation abound. Creative thermal engineering, utilizing the tools of thermodynamics and heat transfer, can lead to new ways to (1) increase the efficiency by which energy is generated and converted, (2) reduce energy losses, and (3) harvest a large portion of the waste heat. As evident in Figure 1.11a, fossil fuels (petroleum, natural gas, and coal) dominate the energy portfolio in many countries, such as the United States. The combustion of fossil fuels produces massive amounts of carbon dioxide; the amount of CO2 released in the United States on an annual basis due to combustion is currently 5.99 Eg (5.99 ⫻ 1015 kg). As more CO2 is pumped into the atmosphere, mechanisms of radiation heat transfer within the atmosphere are modified, resulting in potential changes in global temperatures. In a country like the United States, electricity generation and transportation are responsible for nearly 75% of the total CO2 released into the atmosphere due to energy use (Figure 1.11b). What are some of the ways engineers are applying the principles of heat transfer to address issues of energy and environmental sustainability? The efficiency of a gas turbine engine can be significantly increased by increasing its operating temperature. Today, the temperatures of the combustion gases inside these

42

Chapter 1

䊏

Introduction

Nuclear power 8.3%

Alternative sources 6.8%

Petroleum 39.3%

Electricity generation 35.4%

68.6%

Natural gas 23.3%

Transportation 25.4%

19.9% 19.9% Waste heat 57.6%

Coal 22.9%

Industrial 21.7%

78.5%

Residential/ commercial 17.4%

Useful power 42.4%

(a)

Petroleum 43.2%

Electricity generation 40.6%

Natural gas 20.7%

Transportation 33.5%

Coal 36.1%

Industrial 16.5%

Residential/ commercial 9.4%

(b)

FIGURE 1.11 Flow charts for energy consumption and associated CO2 emissions in the United States in 2007. (a) Energy production and consumption. (b) Carbon dioxide by source of fossil fuel and end-use application. Arrow widths represent relative magnitudes of the flow streams. (Credit: U.S. Department of Energy and the Lawrence Livermore National Laboratory.)

engines far exceed the melting point of the exotic alloys used to manufacture the turbine blades and vanes. Safe operation is typically achieved by three means. First, relatively cool gases are injected through small holes at the leading edge of a turbine blade (Figure 1.12). These gases hug the blade as they are carried downstream and help insulate the blade from the hot combustion gases. Second, thin layers of a very low thermal conductivity, ceramic thermal barrier coating are applied to the blades and vanes to provide an extra layer of insulation. These coatings are produced by spraying molten ceramic powders onto the engine components using extremely high temperature sources such as plasma spray guns

1.6

䊏

43

Relevance of Heat Tranfer

(a)

(b)

FIGURE 1.12 Gas turbine blade. (a) External view showing holes for injection of cooling gases. (b) X ray view showing internal cooling passages. (Credit: Images courtesy of FarField Technology, Ltd., Christchurch, New Zealand.)

that can operate in excess of 10,000 kelvins. Third, the blades and vanes are designed with intricate, internal cooling passages, all carefully configured by the heat transfer engineer to allow the gas turbine engine to operate under such extreme conditions. Alternative sources constitute a small fraction of the energy portfolio of many nations, as illustrated in the flow chart of Figure 1.11a for the United States. The intermittent nature of the power generated by sources such as the wind and solar irradiation limits their widespread utilization, and creative ways to store excess energy for use during low-power generation periods are urgently needed. Emerging energy conversion devices such as fuel cells could be used to (1) combine excess electricity that is generated during the day (in a solar power station, for example) with liquid water to produce hydrogen, and (2) subsequently convert the stored hydrogen at night by recombining it with oxygen to produce electricity and water. Roadblocks hindering the widespread use of hydrogen fuel cells are their size, weight, and limited durability. As with the gas turbine engine, the efficiency of a fuel cell increases with temperature, but high operating temperatures and large temperature gradients can cause the delicate polymeric materials within a hydrogen fuel cell to fail. More challenging is the fact that water exists inside any hydrogen fuel cell. If this water should freeze, the polymeric materials within the fuel cell would be destroyed, and the fuel cell would cease operation. Because of the necessity to utilize very pure water in a hydrogen fuel cell, common remedies such as antifreeze cannot be used. What heat transfer mechanisms must be controlled to avoid freezing of pure water within a fuel cell located at a wind farm or solar energy station in a cold climate? How might your developing knowledge of internal forced convection, evaporation, or condensation be applied to control the operating temperatures and enhance the durability of a fuel cell, in turn promoting more widespread use of solar and wind power? Due to the information technology revolution of the last two decades, strong industrial productivity growth has brought an improved quality of life worldwide. Many information technology breakthroughs have been enabled by advances in heat transfer engineering that have ensured the precise control of temperatures of systems ranging in size from nanoscale integrated circuits, to microscale storage media including compact discs, to large data centers filled with heat-generating equipment. As electronic devices become faster and incorporate

44

Chapter 1

䊏

Introduction

greater functionality, they generate more thermal energy. Simultaneously, the devices have become smaller. Inevitably, heat fluxes (W/m2) and volumetric energy generation rates (W/m3) keep increasing, but the operating temperatures of the devices must be held to reasonably low values to ensure their reliability. For personal computers, cooling fins (also known as heat sinks) are fabricated of a high thermal conductivity material (usually aluminum) and attached to the microprocessors to reduce their operating temperatures, as shown in Figure 1.13. Small fans are used to induce forced convection over the fins. The cumulative energy that is consumed worldwide, just to (1) power the small fans that provide the airflow over the fins and (2) manufacture the heat sinks for personal computers, is estimated to be over 109 kW 䡠 h per year [6]. How might your knowledge of conduction, convection, and radiation be used to, for example, eliminate the fan and minimize the size of the heat sink? Further improvements in microprocessor technology are currently limited by our ability to cool these tiny devices. Policy makers have voiced concern about our ability to continually reduce the cost of computing and, in turn as a society, continue the growth in productivity that has marked the last 30 years, specifically citing the need to enhance heat transfer in electronics cooling [7]. How might your knowledge of heat transfer help ensure continued industrial productivity into the future? Heat transfer is important not only in engineered systems but also in nature. Temperature regulates and triggers biological responses in all living systems and ultimately marks the boundary between sickness and health. Two common examples include hypothermia, which results from excessive cooling of the human body, and heat stroke, which is triggered in warm, humid environments. Both are deadly, and both are associated with core temperatures of the body exceeding physiological limits. Both are directly linked to the convection, radiation, and evaporation processes occurring at the surface of the body, the transport of heat within the body, and the metabolic energy generated volumetrically within the body. Recent advances in biomedical engineering, such as laser surgery, have been enabled by successfully applying fundamental heat transfer principles [8, 9]. While increased temperatures resulting from contact with hot objects may cause thermal burns, beneficial hyperthermal treatments are used to purposely destroy, for example, cancerous lesions. In a

Exploded view

FIGURE 1.13 A finned heat sink and fan assembly (left) and microprocessor (right).

1.7

䊏

45

Summary

Keratin Epidermal layer Epidermis Basal cell layer

Sebaceous gland Sensory receptor

Dermis

Sweat gland Nerve fiber Hair follicle

Subcutaneous layer Vein Artery

FIGURE 1.14 Morphology of human skin.

similar manner, very low temperatures might induce frostbite, but purposeful localized freezing can selectively destroy diseased tissue during cryosurgery. Many medical therapies and devices therefore operate by destructively heating or cooling diseased tissue, while leaving the surrounding healthy tissue unaffected. The ability to design many medical devices and to develop the appropriate protocol for their use hinges on the engineer’s ability to predict and control the distribution of temperatures during thermal treatment and the distribution of chemical species in chemotherapies. The treatment of mammalian tissue is made complicated by its morphology, as shown in Figure 1.14. The flow of blood within the venular and capillary structure of a thermally treated area affects heat transfer through advection processes. Larger veins and arteries, which commonly exist in pairs throughout the body, carry blood at different temperatures and advect thermal energy at different rates. Therefore, the veins and arteries exist in a counterflow heat exchange arrangement with warm, arteriolar blood exchanging thermal energy with the cooler, venular blood through the intervening solid tissue. Networks of smaller capillaries can also affect local temperatures by perfusing blood through the treated area. In subsequent chapters, example and homework problems will deal with the analysis of these and many other thermal systems.

1.7 Summary Although much of the material of this chapter will be discussed in greater detail, you should now have a reasonable overview of heat transfer. You should be aware of the

46 TABLE 1.5

Mode Conduction

Convection

Radiation

Chapter 1

䊏

Introduction

Summary of heat transfer processes

Mechanism(s) Diffusion of energy due to random molecular motion Diffusion of energy due to random molecular motion plus energy transfer due to bulk motion (advection) Energy transfer by electromagnetic waves

Rate Equation q⬙x (W/m2) ⫽ ⫺k

dT dx

q⬙(W/m2) ⫽ h(Ts ⫺ T앝)

4 q⬙(W/m2) ⫽ (Ts4 ⫺ Tsur ) or q (W) ⫽ hr A(Ts ⫺ Tsur)

Equation Number

Transport Property or Coefficient

(1.1)

k (W/m 䡠 K)

(1.3a)

h (W/m2 䡠 K)

(1.7) (1.8)

hr (W/m2 䡠 K)

several modes of transfer and their physical origins. You will be devoting much time to acquiring the tools needed to calculate heat transfer phenomena. However, before you can use these tools effectively, you must have the intuition to determine what is happening physically. Specifically, given a physical situation, you must be able to identify the relevant transport phenomena; the importance of developing this facility must not be underestimated. The example and problems at the end of this chapter will launch you on the road to developing this intuition. You should also appreciate the significance of the rate equations and feel comfortable in using them to compute transport rates. These equations, summarized in Table 1.5, should be committed to memory. You must also recognize the importance of the conservation laws and the need to carefully identify control volumes. With the rate equations, the conservation laws may be used to solve numerous heat transfer problems. Lastly, you should have begun to acquire an appreciation for the terminology and physical concepts that underpin the subject of heat transfer. Test your understanding of the important terms and concepts introduced in this chapter by addressing the following questions: • What are the physical mechanisms associated with heat transfer by conduction, convection, and radiation? • What is the driving potential for heat transfer? What are analogs to this potential and to heat transfer itself for the transport of electric charge? • What is the difference between a heat flux and a heat rate? What are their units? • What is a temperature gradient? What are its units? What is the relationship of heat flow to a temperature gradient? • What is the thermal conductivity? What are its units? What role does it play in heat transfer? • What is Fourier’s law? Can you write the equation from memory? • If heat transfer by conduction through a medium occurs under steady-state conditions, will the temperature at a particular instant vary with location in the medium? Will the temperature at a particular location vary with time?

1.7

䊏

Summary

47

• What is the difference between natural convection and forced convection? • What conditions are necessary for the development of a hydrodynamic boundary layer? A thermal boundary layer? What varies across a hydrodynamic boundary layer? Across a thermal boundary layer? • If convection heat transfer for flow of a liquid or a vapor is not characterized by liquid/vapor phase change, what is the nature of the energy being transferred? What is it if there is such a phase change? • What is Newton’s law of cooling? Can you write the equation from memory? • What role is played by the convection heat transfer coefficient in Newton’s law of cooling? What are its units? • What effect does convection heat transfer from or to a surface have on the solid bounded by the surface? • What is predicted by the Stefan–Boltzmann law, and what unit of temperature must be used with the law? Can you write the equation from memory? • What is the emissivity, and what role does it play in characterizing radiation transfer at a surface? • What is irradiation? What are its units? • What two outcomes characterize the response of an opaque surface to incident radiation? Which outcome affects the thermal energy of the medium bounded by the surface and how? What property characterizes this outcome? • What conditions are associated with use of the radiation heat transfer coefficient? • Can you write the equation used to express net radiation exchange between a small isothermal surface and a large isothermal enclosure? • Consider the surface of a solid that is at an elevated temperature and exposed to cooler surroundings. By what mode(s) is heat transferred from the surface if (1) it is in intimate (perfect) contact with another solid, (2) it is exposed to the flow of a liquid, (3) it is exposed to the flow of a gas, and (4) it is in an evacuated chamber? • What is the inherent difference between the application of conservation of energy over a time interval and at an instant of time? • What is thermal energy storage? How does it differ from thermal energy generation? What role do the terms play in a surface energy balance?

EXAMPLE 1.10 A closed container filled with hot coffee is in a room whose air and walls are at a fixed temperature. Identify all heat transfer processes that contribute to the cooling of the coffee. Comment on features that would contribute to a superior container design.

SOLUTION Known: Hot coffee is separated from its cooler surroundings by a plastic flask, an air space, and a plastic cover. Find: Relevant heat transfer processes.

48

Chapter 1

䊏

Introduction

Schematic:

q8

q5

Hot coffee

q1

q2

q6 q3

Coffee Cover

Plastic flask

q7

q4

Air space

Room air Cover

Surroundings

Air space Plastic flask

Pathways for energy transfer from the coffee are as follows: q1: free convection from the coffee to the flask. q2: conduction through the flask. q3: free convection from the flask to the air. q4: free convection from the air to the cover. q5: net radiation exchange between the outer surface of the flask and the inner surface of the cover. q6: conduction through the cover. q7: free convection from the cover to the room air. q8: net radiation exchange between the outer surface of the cover and the surroundings.

Comments: Design improvements are associated with (1) use of aluminized (lowemissivity) surfaces for the flask and cover to reduce net radiation, and (2) evacuating the air space or using a filler material to retard free convection.

References 1. Moran, M. J., and H. N. Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, Hoboken, NJ, 2004. 2. Curzon, F. L., and B. Ahlborn, American J. Physics, 43, 22, 1975. 3. Novikov, I. I., J. Nuclear Energy II, 7, 125, 1958. 4. Callen, H. B., Thermodynamics and an Introduction to Thermostatistics, Wiley, Hoboken, NJ, 1985. 5. Bejan, A., American J. Physics, 64, 1054, 1996.

6. Bar-Cohen, A., and I. Madhusudan, IEEE Trans. Components and Packaging Tech., 25, 584, 2002. 7. Miller, R., Business Week, November 11, 2004. 8. Diller, K. R., and T. P. Ryan, J. Heat Transfer, 120, 810, 1998. 9. Datta, A.K., Biological and Bioenvironmental Heat and Mass Transfer, Marcel Dekker, New York, 2002.

䊏

49

Problems

Problems Conduction 1.1 The thermal conductivity of a sheet of rigid, extruded insulation is reported to be k ⫽ 0.029 W/m 䡠 K. The measured temperature difference across a 20-mm-thick sheet of the material is T1 ⫺ T2 ⫽ 10⬚C. (a) What is the heat flux through a 2 m ⫻ 2 m sheet of the insulation? (b) What is the rate of heat transfer through the sheet of insulation? 1.2 The heat flux that is applied to the left face of a plane wall is q⬙ ⫽ 20 W/m2. The wall is of thickness L ⫽ 10 mm and of thermal conductivity k ⫽ 12 W/m 䡠 K. If the surface temperatures of the wall are measured to be 50⬚C on the left side and 30⬚C on the right side, do steady-state conditions exist? 1.3 A concrete wall, which has a surface area of 20 m2 and is 0.30 m thick, separates conditioned room air from ambient air. The temperature of the inner surface of the wall is maintained at 25⬚C, and the thermal conductivity of the concrete is 1 W/m 䡠 K. (a) Determine the heat loss through the wall for outer surface temperatures ranging from ⫺15⬚C to 38⬚C, which correspond to winter and summer extremes, respectively. Display your results graphically. (b) On your graph, also plot the heat loss as a function of the outer surface temperature for wall materials having thermal conductivities of 0.75 and 1.25 W/m 䡠 K. Explain the family of curves you have obtained. 1.4 The concrete slab of a basement is 11 m long, 8 m wide, and 0.20 m thick. During the winter, temperatures are nominally 17⬚C and 10⬚C at the top and bottom surfaces, respectively. If the concrete has a thermal conductivity of 1.4 W/m 䡠 K, what is the rate of heat loss through the slab? If the basement is heated by a gas furnace operating at an efficiency of f ⫽ 0.90 and natural gas is priced at Cg ⫽ $0.02/MJ, what is the daily cost of the heat loss? 1.5 Consider Figure 1.3. The heat flux in the x-direction is q⬙x ⫽ 10 W/m2, the thermal conductivity and wall thickness are k ⫽ 2.3 W/m 䡠 K and L ⫽ 20 mm, respectively, and steady-state conditions exist. Determine the value of the temperature gradient in units of K/m. What is the value of the temperature gradient in units of ⬚C/m? 1.6 The heat flux through a wood slab 50 mm thick, whose inner and outer surface temperatures are 40 and 20⬚C, respectively, has been determined to be 40 W/m2. What is the thermal conductivity of the wood?

1.7 The inner and outer surface temperatures of a glass window 5 mm thick are 15 and 5⬚C. What is the heat loss through a 1 m ⫻ 3 m window? The thermal conductivity of glass is 1.4 W/m 䡠 K. 1.8 A thermodynamic analysis of a proposed Brayton cycle gas turbine yields P ⫽ 5 MW of net power production. The compressor, at an average temperature of Tc ⫽ 400⬚C, is driven by the turbine at an average temperature of Th ⫽ 1000⬚C by way of an L ⫽ 1-m-long, d ⫽ 70-mmdiameter shaft of thermal conductivity k ⫽ 40 W/m 䡠 K. Combustion chamber

Turbine

Compressor

d

Tc Shaft

Th P

m• in

L • m out

(a) Compare the steady-state conduction rate through the shaft connecting the hot turbine to the warm compressor to the net power predicted by the thermodynamics-based analysis. (b) A research team proposes to scale down the gas turbine of part (a), keeping all dimensions in the same proportions. The team assumes that the same hot and cold temperatures exist as in part (a) and that the net power output of the gas turbine is proportional to the overall volume of the device. Plot the ratio of the conduction through the shaft to the net power output of the turbine over the range 0.005 m ⱕ L ⱕ 1 m. Is a scaled-down device with L ⫽ 0.005 m feasible? 1.9 A glass window of width W ⫽ 1 m and height H ⫽ 2 m is 5 mm thick and has a thermal conductivity of kg ⫽ 1.4 W/m 䡠 K. If the inner and outer surface temperatures of the glass are 15⬚C and ⫺20⬚C, respectively, on a cold winter day, what is the rate of heat loss through the glass? To reduce heat loss through windows, it is customary to use a double pane construction in which adjoining panes are separated by an air space. If the spacing is 10 mm and the glass surfaces in contact with the air have temperatures of 10⬚C and ⫺15⬚C, what is the rate of heat loss from a 1 m ⫻ 2 m window? The thermal conductivity of air is ka ⫽ 0.024 W/m 䡠 K. 1.10 A freezer compartment consists of a cubical cavity that is 2 m on a side. Assume the bottom to be perfectly

50

Chapter 1

䊏

Introduction

insulated. What is the minimum thickness of styrofoam insulation (k ⫽ 0.030 W/m 䡠 K) that must be applied to the top and side walls to ensure a heat load of less than 500 W, when the inner and outer surfaces are ⫺10 and 35⬚C? 1.11 The heat flux that is applied to one face of a plane wall is q⬙ ⫽ 20 W/m2. The opposite face is exposed to air at temperature 30⬚C, with a convection heat transfer coefficient of 20 W/m2 䡠 K. The surface temperature of the wall exposed to air is measured and found to be 50⬚C. Do steady-state conditions exist? If not, is the temperature of the wall increasing or decreasing with time? 1.12 An inexpensive food and beverage container is fabricated from 25-mm-thick polystyrene (k ⫽ 0.023 W/m 䡠 K) and has interior dimensions of 0.8 m ⫻ 0.6 m ⫻ 0.6 m. Under conditions for which an inner surface temperature of approximately 2⬚C is maintained by an ice-water mixture and an outer surface temperature of 20⬚C is maintained by the ambient, what is the heat flux through the container wall? Assuming negligible heat gain through the 0.8 m ⫻ 0.6 m base of the cooler, what is the total heat load for the prescribed conditions? 1.13 What is the thickness required of a masonry wall having thermal conductivity 0.75 W/m 䡠 K if the heat rate is to be 80% of the heat rate through a composite structural wall having a thermal conductivity of 0.25 W/m 䡠 K and a thickness of 100 mm? Both walls are subjected to the same surface temperature difference. 1.14 A wall is made from an inhomogeneous (nonuniform) material for which the thermal conductivity varies through the thickness according to k ⫽ ax ⫹ b, where a and b are constants. The heat flux is known to be constant. Determine expressions for the temperature gradient and the temperature distribution when the surface at x ⫽ 0 is at temperature T1. 1.15 The 5-mm-thick bottom of a 200-mm-diameter pan may be made from aluminum (k ⫽ 240 W/m 䡠 K) or copper (k ⫽ 390 W/m 䡠 K). When used to boil water, the surface of the bottom exposed to the water is nominally at 110⬚C. If heat is transferred from the stove to the pan at a rate of 600 W, what is the temperature of the surface in contact with the stove for each of the two materials? 1.16 A square silicon chip (k ⫽ 150 W/m 䡠 K) is of width w ⫽ 5 mm on a side and of thickness t ⫽ 1 mm. The chip is mounted in a substrate such that its side and back surfaces are insulated, while the front surface is exposed to a coolant. If 4 W are being dissipated in circuits mounted to the back surface of the chip, what is the steady-state temperature difference between back and front surfaces?

Coolant w Chip

Circuits

t

Convection 1.17 For a boiling process such as shown in Figure 1.5c, the ambient temperature T앝 in Newton’s law of cooling is replaced by the saturation temperature of the fluid Tsat. Consider a situation where the heat flux from the hot plate is q⬙ ⫽ 20 ⫻ 105 W/m2. If the fluid is water at atmospheric pressure and the convection heat transfer coefficient is hw ⫽ 20 ⫻ 103 W/m2 䡠 K, determine the upper surface temperature of the plate, Ts,w. In an effort to minimize the surface temperature, a technician proposes replacing the water with a dielectric fluid whose saturation temperature is Tsat,d ⫽ 52⬚C. If the heat transfer coefficient associated with the dielectric fluid is hd ⫽ 3 ⫻ 103 W/m2 䡠 K, will the technician’s plan work? 1.18 You’ve experienced convection cooling if you’ve ever extended your hand out the window of a moving vehicle or into a flowing water stream. With the surface of your hand at a temperature of 30⬚C, determine the convection heat flux for (a) a vehicle speed of 35 km/h in air at ⫺5⬚C with a convection coefficient of 40 W/m2 䡠 K and (b) a velocity of 0.2 m/s in a water stream at 10⬚C with a convection coefficient of 900 W/m2 䡠 K. Which condition would feel colder? Contrast these results with a heat loss of approximately 30 W/m2 under normal room conditions. 1.19 Air at 40⬚C flows over a long, 25-mm-diameter cylinder with an embedded electrical heater. In a series of tests, measurements were made of the power per unit length, P⬘, required to maintain the cylinder surface temperature at 300⬚C for different free stream velocities V of the air. The results are as follows: Air velocity, V (m/s) Power, P⬘ (W/m)

1 450

2 658

4 983

8 1507

12 1963

(a) Determine the convection coefficient for each velocity, and display your results graphically. (b) Assuming the dependence of the convection coefficient on the velocity to be of the form h ⫽ CV n, determine the parameters C and n from the results of part (a).

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51

Problems

1.20 A wall has inner and outer surface temperatures of 16 and 6⬚C, respectively. The interior and exterior air temperatures are 20 and 5⬚C, respectively. The inner and outer convection heat transfer coefficients are 5 and 20 W/m2 䡠 K, respectively. Calculate the heat flux from the interior air to the wall, from the wall to the exterior air, and from the wall to the interior air. Is the wall under steady-state conditions? 1.21 An electric resistance heater is embedded in a long cylinder of diameter 30 mm. When water with a temperature of 25⬚C and velocity of 1 m/s flows crosswise over the cylinder, the power per unit length required to maintain the surface at a uniform temperature of 90⬚C is 28 kW/m. When air, also at 25⬚C, but with a velocity of 10 m/s is flowing, the power per unit length required to maintain the same surface temperature is 400 W/m. Calculate and compare the convection coefficients for the flows of water and air. 1.22 The free convection heat transfer coefficient on a thin hot vertical plate suspended in still air can be determined from observations of the change in plate temperature with time as it cools. Assuming the plate is isothermal and radiation exchange with its surroundings is negligible, evaluate the convection coefficient at the instant of time when the plate temperature is 225⬚C and the change in plate temperature with time (dT/dt) is ⫺0.022 K/s. The ambient air temperature is 25⬚C and the plate measures 0.3 ⫻ 0.3 m with a mass of 3.75 kg and a specific heat of 2770 J/kg 䡠 K. 1.23 A transmission case measures W ⫽ 0.30 m on a side and receives a power input of Pi ⫽ 150 hp from the engine.

1.24 A cartridge electrical heater is shaped as a cylinder of length L ⫽ 200 mm and outer diameter D ⫽ 20 mm. Under normal operating conditions, the heater dissipates 2 kW while submerged in a water flow that is at 20⬚C and provides a convection heat transfer coefficient of h ⫽ 5000 W/m2 䡠 K. Neglecting heat transfer from the ends of the heater, determine its surface temperature Ts. If the water flow is inadvertently terminated while the heater continues to operate, the heater surface is exposed to air that is also at 20⬚C but for which h ⫽ 50 W/m2 䡠 K. What is the corresponding surface temperature? What are the consequences of such an event? 1.25 A common procedure for measuring the velocity of an airstream involves the insertion of an electrically heated wire (called a hot-wire anemometer) into the airflow, with the axis of the wire oriented perpendicular to the flow direction. The electrical energy dissipated in the wire is assumed to be transferred to the air by forced convection. Hence, for a prescribed electrical power, the temperature of the wire depends on the convection coefficient, which, in turn, depends on the velocity of the air. Consider a wire of length L ⫽ 20 mm and diameter D ⫽ 0.5 mm, for which a calibration of the form V ⫽ 6.25 ⫻ 10⫺5 h2 has been determined. The velocity V and the convection coefficient h have units of m/s and W/m2 䡠 K, respectively. In an application involving air at a temperature of T앝 ⫽ 25⬚C, the surface temperature of the anemometer is maintained at Ts ⫽ 75⬚C with a voltage drop of 5 V and an electric current of 0.1 A. What is the velocity of the air? 1.26 A square isothermal chip is of width w ⫽ 5 mm on a side and is mounted in a substrate such that its side and back surfaces are well insulated; the front surface is exposed to the flow of a coolant at T앝 ⫽ 15⬚C. From reliability considerations, the chip temperature must not exceed T ⫽ 85⬚C. Coolant

T∞, h w

Transmission case, η, Ts

Air

Chip

T∞, h Pi

W

If the transmission efficiency is ⫽ 0.93 and airflow over the case corresponds to T앝 ⫽ 30⬚C and h ⫽ 200 W/m2 䡠 K, what is the surface temperature of the transmission?

If the coolant is air and the corresponding convection coefficient is h ⫽ 200 W/m2 䡠 K, what is the maximum allowable chip power? If the coolant is a dielectric liquid for which h ⫽ 3000 W/m2 䡠 K, what is the maximum allowable power? 1.27 The temperature controller for a clothes dryer consists of a bimetallic switch mounted on an electrical heater attached to a wall-mounted insulation pad.

52

Chapter 1

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Introduction

Dryer wall

Pe

Insulation pad Air T∞, h

Tset = 70°C

Electrical heater Bimetallic switch

The switch is set to open at 70⬚C, the maximum dryer air temperature. To operate the dryer at a lower air temperature, sufficient power is supplied to the heater such that the switch reaches 70⬚C (Tset) when the air temperature T is less than Tset. If the convection heat transfer coefficient between the air and the exposed switch surface of 30 mm2 is 25 W/m2 䡠 K, how much heater power Pe is required when the desired dryer air temperature is T앝 ⫽ 50⬚C?

Radiation 1.28 An overhead 25-m-long, uninsulated industrial steam pipe of 100-mm diameter is routed through a building whose walls and air are at 25⬚C. Pressurized steam maintains a pipe surface temperature of 150⬚C, and the coefficient associated with natural convection is h ⫽ 10 W/m2 䡠 K. The surface emissivity is ⫽ 0.8. (a) What is the rate of heat loss from the steam line? (b) If the steam is generated in a gas-fired boiler operating at an efficiency of f ⫽ 0.90 and natural gas is priced at Cg ⫽ $0.02 per MJ, what is the annual cost of heat loss from the line? 1.29 Under conditions for which the same room temperature is maintained by a heating or cooling system, it is not uncommon for a person to feel chilled in the winter but comfortable in the summer. Provide a plausible explanation for this situation (with supporting calculations) by considering a room whose air temperature is maintained at 20⬚C throughout the year, while the walls of the room are nominally at 27⬚C and 14⬚C in the summer and winter, respectively. The exposed surface of a person in the room may be assumed to be at a temperature of 32⬚C throughout the year and to have an emissivity of 0.90. The coefficient associated with heat transfer by natural convection between the person and the room air is approximately 2 W/m2 䡠 K.

range 40 ⱕ T ⱕ 85⬚C, what is the range of acceptable power dissipation for the package? Display your results graphically, showing also the effect of variations in the emissivity by considering values of 0.20 and 0.30. 1.32 Consider the conditions of Problem 1.22. However, now the plate is in a vacuum with a surrounding temperature of 25⬚C. What is the emissivity of the plate? What is the rate at which radiation is emitted by the surface? 1.33 If Ts Tsur in Equation 1.9, the radiation heat transfer coefficient may be approximated as hr,a ⫽ 4T 3 where T (Ts ⫹ Tsur)/2. We wish to assess the validity of this approximation by comparing values of hr and hr,a for the following conditions. In each case, represent your results graphically and comment on the validity of the approximation. (a) Consider a surface of either polished aluminum ( ⫽ 0.05) or black paint ( ⫽ 0.9), whose temperature may exceed that of the surroundings (Tsur ⫽ 25⬚C) by 10 to 100°C. Also compare your results with values of the coefficient associated with free convection in air (T앝 ⫽ Tsur), where h(W/m2 䡠 K) ⫽ 0.98 ⌬T 1/3. (b) Consider initial conditions associated with placing a workpiece at Ts ⫽ 25⬚C in a large furnace whose wall temperature may be varied over the range 100 ⱕ Tsur ⱕ 1000⬚C. According to the surface finish or coating, its emissivity may assume values of 0.05, 0.2, and 0.9. For each emissivity, plot the relative error, (hr ⫺ hr,a )/hr , as a function of the furnace temperature. 1.34 A vacuum system, as used in sputtering electrically conducting thin films on microcircuits, is comprised of a baseplate maintained by an electrical heater at 300 K and a shroud within the enclosure maintained at 77 K by a liquid-nitrogen coolant loop. The circular baseplate, insulated on the lower side, is 0.3 m in diameter and has an emissivity of 0.25.

Vacuum enclosure

1.30 A spherical interplanetary probe of 0.5-m diameter contains electronics that dissipate 150 W. If the probe surface has an emissivity of 0.8 and the probe does not receive radiation from other surfaces, as, for example, from the sun, what is its surface temperature? 1.31 An instrumentation package has a spherical outer surface of diameter D ⫽ 100 mm and emissivity ⫽ 0.25. The package is placed in a large space simulation chamber whose walls are maintained at 77 K. If operation of the electronic components is restricted to the temperature

Liquid-nitrogen filled shroud

LN2

Electrical heater Baseplate

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53

Problems

(a) How much electrical power must be provided to the baseplate heater? (b) At what rate must liquid nitrogen be supplied to the shroud if its heat of vaporization is 125 kJ/kg? (c) To reduce the liquid nitrogen consumption, it is proposed to bond a thin sheet of aluminum foil ( ⫽ 0.09) to the baseplate. Will this have the desired effect?

Relationship to Thermodynamics 1.35 An electrical resistor is connected to a battery, as shown schematically. After a brief transient, the resistor assumes a nearly uniform, steady-state temperature of 95⬚C, while the battery and lead wires remain at the ambient temperature of 25⬚C. Neglect the electrical resistance of the lead wires. I = 6A

Resistor

Battery

1.37 Consider the tube and inlet conditions of Problem 1.36. Heat transfer at a rate of q ⫽ 3.89 MW is delivered to the tube. For an exit pressure of p ⫽ 8 bar, determine (a) the temperature of the water at the outlet as well as the change in (b) combined thermal and flow work, (c) mechanical energy, and (d) total energy of the water from the inlet to the outlet of the tube. Hint: As a first estimate, neglect the change in mechanical energy in solving part (a). Relevant properties may be obtained from a thermodynamics text. 1.38 An internally reversible refrigerator has a modified coefficient of performance accounting for realistic heat transfer processes of COPm ⫽

where qin is the refrigerator cooling rate, qout is the heat ˙ is the power input. Show that COPm rejection rate, and W can be expressed in terms of the reservoir temperatures Tc and Th, the cold and hot thermal resistances Rt,c and Rt,h, and qin, as

Air

V = 24 V

COPm ⫽

T• = 25C

Lead wire

(c) Neglecting radiation from the resistor, what is the convection coefficient? 1.36 Pressurized water (pin ⫽ 10 bar, Tin ⫽ 110⬚C) enters the bottom of an L ⫽ 10-m-long vertical tube of diameter D ⫽ 100 mm at a mass flow rate of m˙ ⫽ 1.5 kg/s. The tube is located inside a combustion chamber, resulting in heat transfer to the tube. Superheated steam exits the top of the tube at pout ⫽ 7 bar, Tout ⫽ 600⬚C. Determine the change in the rate at which the following quantities enter and exit the tube: (a) the combined thermal and flow work, (b) the mechanical energy, and (c) the total energy of the water. Also, (d) determine the heat transfer rate, q. Hint: Relevant properties may be obtained from a thermodynamics text.

Tc ⫺ qin Rtot Th ⫺ Tc ⫹ qin Rtot

where Rtot ⫽ Rt,c ⫹ Rt,h. Also, show that the power input may be expressed as Th ⫺ Tc ⫹ qin Rtot W˙ ⫽ qin Tc ⫺ qin Rtot

(a) Consider the resistor as a system about which a control surface is placed and Equation 1.12c is applied. Determine the corresponding values of E˙ in(W), E˙ g(W), E˙ out(W), and E˙ st(W). If a control surface is placed about the entire system, what are the values of E˙ in, E˙ g, E˙ out, and E˙ st? (b) If electrical energy is dissipated uniformly within the resistor, which is a cylinder of diameter D ⫽ 60 mm and length L ⫽ 250 mm, what is the volumetric heat generation rate, q˙ (W/m3)?

Tc,i qin qin ⫽q ⫺q ⫽ out in T h,i ⫺ Tc,i W˙

High-temperature reservoir Q

Th

out

Th,i W

Internally reversible refrigerator

Tc,i Qin Low-temperature reservoir

High-temperature side resistance Low-temperature side resistance

Tc

1.39 A household refrigerator operates with cold- and hot-temperature reservoirs of Tc ⫽ 5⬚C and Th ⫽ 25⬚C, respectively. When new, the cold and hot side resistances are Rc,n ⫽ 0.05 K/W and Rh,n ⫽ 0.04 K/W, respectively. Over time, dust accumulates on the refrigerator’s condenser coil, which is located behind the refrigerator, increasing the hot side resistance to Rh,d ⫽ 0.1 K/W. It is desired to have a refrigerator cooling rate of qin ⫽ 750 W. Using the results of Problem 1.38, determine the modified coefficient of performance and the required power input ˙ under (a) clean and (b) dusty coil conditions. W

54

Chapter 1

䊏

Introduction exposed surface is h ⫽ 8 W/m2 䡠 K, and the surface is characterized by an emissivity of s ⫽ 0.9. The solid silicon powder is at Tsi,i ⫽ 298 K, and the solid silicon sheet exits the chamber at Tsi,o ⫽ 420 K. Both the surroundings and ambient temperatures are T앝 ⫽ Tsur ⫽ 298 K.

Energy Balance and Multimode Effects 1.40 Chips of width L ⫽ 15 mm on a side are mounted to a substrate that is installed in an enclosure whose walls and air are maintained at a temperature of Tsur ⫽ 25⬚C. The chips have an emissivity of ⫽ 0.60 and a maximum allowable temperature of Ts ⫽ 85⬚C.

Solid silicon powder

Enclosure, Tsur

Vsi Ts,o

Tsur

Ts, εs

Substrate

Air T∞, h

tsi

Molten silicon String

Pelec

Chip (Ts, ε)

Solid silicon sheet

Solid silicon sheet

H

Air T∞, h

Vsi

• •

•

Molten silicon Crucible D

L

(a) If heat is rejected from the chips by radiation and natural convection, what is the maximum operating power of each chip? The convection coefficient depends on the chip-to-air temperature difference and may be approximated as h ⫽ C(Ts ⫺ T앝)1/4, where C ⫽ 4.2 W/m2 䡠 K5/4. (b) If a fan is used to maintain airflow through the enclosure and heat transfer is by forced convection, with h ⫽ 250 W/m2 䡠 K, what is the maximum operating power? 1.41 Consider the transmission case of Problem 1.23, but now allow for radiation exchange with the ground/ chassis, which may be approximated as large surroundings at Tsur ⫽ 30⬚C. If the emissivity of the case is ⫽ 0.80, what is the surface temperature? 1.42 One method for growing thin silicon sheets for photovoltaic solar panels is to pass two thin strings of high melting temperature material upward through a bath of molten silicon. The silicon solidifies on the strings near the surface of the molten pool, and the solid silicon sheet is pulled slowly upward out of the pool. The silicon is replenished by supplying the molten pool with solid silicon powder. Consider a silicon sheet that is Wsi ⫽ 85 mm wide and tsi ⫽ 150 m thick that is pulled at a velocity of Vsi ⫽ 20 mm/min. The silicon is melted by supplying electric power to the cylindrical growth chamber of height H ⫽ 350 mm and diameter D ⫽ 300 mm. The exposed surfaces of the growth chamber are at Ts ⫽ 320 K, the corresponding convection coefficient at the

(a) Determine the electric power, Pelec, needed to operate the system at steady state. (b) If the photovoltaic panel absorbs a time-averaged solar flux of q⬙sol ⫽ 180 W/m2 and the panel has a conversion efficiency (the ratio of solar power absorbed to electric power produced) of ⫽ 0.20, how long must the solar panel be operated to produce enough electric energy to offset the electric energy that was consumed in its manufacture? 1.43 Heat is transferred by radiation and convection between the inner surface of the nacelle of the wind turbine of Example 1.3 and the outer surfaces of the gearbox and generator. The convection heat flux associated with the gearbox and the generator may be described by q⬙conv,gb ⫽ h(Tgb ⫺ T앝) and q⬙conv,gen ⫽ h(Tgen ⫺ T앝), respectively, where the ambient temperature T앝 Ts (which is the nacelle temperature) and h ⫽ 40 W/m2 䡠 K. The outer surfaces of both the gearbox and the generator are characterized by an emissivity of ⫽ 0.9. If the surface areas of the gearbox and generator are Agb ⫽ 6 m2 and Agen ⫽ 4 m2, respectively, determine their surface temperatures. 1.44 Radioactive wastes are packed in a long, thin-walled cylindrical container. The wastes generate thermal energy nonuniformly according to the relation q˙ ⫽ q˙o[1 ⫺ (r/ro)2], where q˙ is the local rate of energy generation per unit volume, q˙o is a constant, and ro is the radius of the container. Steady-state conditions are maintained by submerging the container in a liquid that is at T앝 and provides a uniform convection coefficient h.

䊏

55

Problems

estimate the magnitudes of kinetic and potential energy changes. Assume the blood’s properties are similar to those of water.

ro

T∞, h q• = q•o [1 – (r/ro)2]

Obtain an expression for the total rate at which energy is generated in a unit length of the container. Use this result to obtain an expression for the temperature Ts of the container wall. 1.45 An aluminum plate 4 mm thick is mounted in a horizontal position, and its bottom surface is well insulated. A special, thin coating is applied to the top surface such that it absorbs 80% of any incident solar radiation, while having an emissivity of 0.25. The density and specific heat c of aluminum are known to be 2700 kg/m3 and 900 J/kg 䡠 K, respectively. (a) Consider conditions for which the plate is at a temperature of 25⬚C and its top surface is suddenly exposed to ambient air at T앝 ⫽ 20⬚C and to solar radiation that provides an incident flux of 900 W/m2. The convection heat transfer coefficient between the surface and the air is h ⫽ 20 W/m2 䡠 K. What is the initial rate of change of the plate temperature? (b) What will be the equilibrium temperature of the plate when steady-state conditions are reached? (c) The surface radiative properties depend on the specific nature of the applied coating. Compute and plot the steady-state temperature as a function of the emissivity for 0.05 ⱕ ⱕ 1, with all other conditions remaining as prescribed. Repeat your calculations for values of ␣S ⫽ 0.5 and 1.0, and plot the results with those obtained for ␣S ⫽ 0.8. If the intent is to maximize the plate temperature, what is the most desirable combination of the plate emissivity and its absorptivity to solar radiation? 1.46 A blood warmer is to be used during the transfusion of blood to a patient. This device is to heat blood taken from the blood bank at 10⬚C to 37⬚C at a flow rate of 200 ml/min. The blood passes through tubing of length 2 m, with a rectangular cross section 6.4 mm ⫻ 1.6 mm At what rate must heat be added to the blood to accomplish the required temperature increase? If the fluid originates from a large tank with nearly zero velocity and flows vertically downward for its 2-m length,

1.47 Consider a carton of milk that is refrigerated at a temperature of Tm ⫽ 5⬚C. The kitchen temperature on a hot summer day is T앝 ⫽ 30⬚C. If the four sides of the carton are of height and width L ⫽ 200 mm and w ⫽ 100 mm, respectively, determine the heat transferred to the milk carton as it sits on the kitchen counter for durations of t ⫽ 10 s, 60 s, and 300 s before it is returned to the refrigerator. The convection coefficient associated with natural convection on the sides of the carton is h ⫽ 10 W/m2 䡠 K. The surface emissivity is 0.90. Assume the milk carton temperature remains at 5⬚C during the process. Your parents have taught you the importance of refrigerating certain foods from the food safety perspective. Comment on the importance of quickly returning the milk carton to the refrigerator from an energy conservation point of view. 1.48 The energy consumption associated with a home water heater has two components: (i) the energy that must be supplied to bring the temperature of groundwater to the heater storage temperature, as it is introduced to replace hot water that has been used; (ii) the energy needed to compensate for heat losses incurred while the water is stored at the prescribed temperature. In this problem, we will evaluate the first of these components for a family of four, whose daily hot water consumption is approximately 100 gal. If groundwater is available at 15⬚C, what is the annual energy consumption associated with heating the water to a storage temperature of 55⬚C? For a unit electrical power cost of $0.18/kW 䡠 h, what is the annual cost associated with supplying hot water by means of (a) electric resistance heating or (b) a heat pump having a COP of 3. 1.49 Liquid oxygen, which has a boiling point of 90 K and a latent heat of vaporization of 214 kJ/kg, is stored in a spherical container whose outer surface is of 500-mm diameter and at a temperature of ⫺10⬚C. The container is housed in a laboratory whose air and walls are at 25⬚C. (a) If the surface emissivity is 0.20 and the heat transfer coefficient associated with free convection at the outer surface of the container is 10 W/m2 䡠 K, what is the rate, in kg/s, at which oxygen vapor must be vented from the system? (b) Moisture in the ambient air will result in frost formation on the container, causing the surface emissivity to increase. Assuming the surface temperature and convection coefficient to remain at ⫺10⬚C and

56

Chapter 1

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Introduction

10 W/m2 䡠 K, respectively, compute the oxygen evaporation rate (kg/s) as a function of surface emissivity over the range 0.2 ⱕ ⱕ 0.94. 1.50 The emissivity of galvanized steel sheet, a common roofing material, is ⫽ 0.13 at temperatures around 300 K, while its absorptivity for solar irradiation is ␣S ⫽ 0.65. Would the neighborhood cat be comfortable walking on a roof constructed of the material on a day when GS ⫽ 750 W/m2, T앝 ⫽ 16⬚C, and h ⫽ 7 W/m2 䡠 K? Assume the bottom surface of the steel is insulated. 1.51 Three electric resistance heaters of length L ⫽ 250 mm and diameter D ⫽ 25 mm are submerged in a 10-gal tank of water, which is initially at 295 K. The water may be assumed to have a density and specific heat of ⫽ 990 kg/m3 and c ⫽ 4180 J/kg 䡠 K. (a) If the heaters are activated, each dissipating q1 ⫽ 500 W, estimate the time required to bring the water to a temperature of 335 K. (b) If the natural convection coefficient is given by an expression of the form h ⫽ 370 (Ts ⫺ T)1/3, where Ts and T are temperatures of the heater surface and water, respectively, what is the temperature of each heater shortly after activation and just before deactivation? Units of h and (Ts ⫺ T) are W/m2 ⭈ K and K, respectively. (c) If the heaters are inadvertently activated when the tank is empty, the natural convection coefficient associated with heat transfer to the ambient air at T앝 ⫽ 300 K may be approximated as h ⫽ 0.70 (Ts ⫺ T앝)1/3. If the temperature of the tank walls is also 300 K and the emissivity of the heater surface is ⫽ 0.85, what is the surface temperature of each heater under steady-state conditions? 1.52 A hair dryer may be idealized as a circular duct through which a small fan draws ambient air and within which the air is heated as it flows over a coiled electric resistance wire.

(a) If a dryer is designed to operate with an electric power consumption of Pelec ⫽ 500 W and to heat air from an ambient temperature of Ti ⫽ 20⬚C to a discharge temperature of To ⫽ 45⬚C, at what volu˙ should the fan operate? Heat loss metric flow rate ᭙ from the casing to the ambient air and the surroundings may be neglected. If the duct has a diameter of D ⫽ 70 mm, what is the discharge velocity Vo of the air? The density and specific heat of the air may be approximated as ⫽ 1.10 kg/m3 and cp ⫽ 1007 J/kg 䡠 K, respectively. (b) Consider a dryer duct length of L ⫽ 150 mm and a surface emissivity of ⫽ 0.8. If the coefficient associated with heat transfer by natural convection from the casing to the ambient air is h ⫽ 4 W/m2 䡠 K and the temperature of the air and the surroundings is T앝 ⫽ Tsur ⫽ 20⬚C, confirm that the heat loss from the casing is, in fact, negligible. The casing may be assumed to have an average surface temperature of Ts ⫽ 40⬚C. 1.53 In one stage of an annealing process, 304 stainless steel sheet is taken from 300 K to 1250 K as it passes through an electrically heated oven at a speed of Vs ⫽ 10 mm/s. The sheet thickness and width are ts ⫽ 8 mm and Ws ⫽ 2 m, respectively, while the height, width, and length of the oven are Ho ⫽ 2 m, Wo ⫽ 2.4 m, and Lo ⫽ 25 m, respectively. The top and four sides of the oven are exposed to ambient air and large surroundings, each at 300 K, and the corresponding surface temperature, convection coefficient, and emissivity are Ts ⫽ 350 K, h ⫽ 10 W/m2 䡠 K, and s ⫽ 0.8. The bottom surface of the oven is also at 350 K and rests on a 0.5-m-thick concrete pad whose base is at 300 K. Estimate the required electric power input, Pelec, to the oven. Tsur Pelec

Air

T∞, h

Ts, εs Lo Steel sheet

ts Vs

Surroundings, Tsur

Ts

Air T∞, h Electric resistor

Discharge

To, Vo

Concrete pad

Fan •

Inlet, ∀, Ti

D Pelec

Tb

Dryer, Ts, ε

1.54 Convection ovens operate on the principle of inducing forced convection inside the oven chamber with a fan. A small cake is to be baked in an oven when the convection feature is disabled. For this situation, the free convection coefficient associated with the cake and its

䊏

57

Problems

pan is hfr ⫽ 3 W/m2 䡠 K. The oven air and wall are at temperatures T앝 ⫽ Tsur ⫽ 180⬚C. Determine the heat flux delivered to the cake pan and cake batter when they are initially inserted into the oven and are at a temperature of Ti ⫽ 24⬚C. If the convection feature is activated, the forced convection heat transfer coefficient is hfo ⫽ 27 W/m2 䡠 K. What is the heat flux at the batter or pan surface when the oven is operated in the convection mode? Assume a value of 0.97 for the emissivity of the cake batter and pan. 1.55 Annealing, an important step in semiconductor materials processing, can be accomplished by rapidly heating the silicon wafer to a high temperature for a short period of time. The schematic shows a method involving the use of a hot plate operating at an elevated temperature Th. The wafer, initially at a temperature of Tw,i, is suddenly positioned at a gap separation distance L from the hot plate. The purpose of the analysis is to compare the heat fluxes by conduction through the gas within the gap and by radiation exchange between the hot plate and the cool wafer. The initial time rate of change in the temperature of the wafer, (dTw /dt)i, is also of interest. Approximating the surfaces of the hot plate and the wafer as blackbodies and assuming their diameter D to be much larger than the spacing L, the radiative heat flux may be expressed as q⬙rad ⫽ (Th4 ⫺ Tw4). The silicon wafer has a thickness of d ⫽ 0.78 mm, a density of 2700 kg/m3, and a specific heat of 875 J/kg 䡠 K. The thermal conductivity of the gas in the gap is 0.0436 W/m 䡠 K. D Hot plate, Th Stagnant gas, k Silicon wafer, Tw, i Gap, L

d

Positioner motion

(a) For Th ⫽ 600⬚C and Tw,i ⫽ 20⬚C, calculate the radiative heat flux and the heat flux by conduction across a gap distance of L ⫽ 0.2 mm. Also determine the value of (dTw /dt)i, resulting from each of the heating modes. (b) For gap distances of 0.2, 0.5, and 1.0 mm, determine the heat fluxes and temperature-time change as a function of the hot plate temperature for 300 ⱕ Th ⱕ 1300⬚C. Display your results graphically. Comment on the relative importance of the two heat

transfer modes and the effect of the gap distance on the heating process. Under what conditions could a wafer be heated to 900⬚C in less than 10 s? 1.56 In the thermal processing of semiconductor materials, annealing is accomplished by heating a silicon wafer according to a temperature-time recipe and then maintaining a fixed elevated temperature for a prescribed period of time. For the process tool arrangement shown as follows, the wafer is in an evacuated chamber whose walls are maintained at 27⬚C and within which heating lamps maintain a radiant flux q⬙s at its upper surface. The wafer is 0.78 mm thick, has a thermal conductivity of 30 W/m 䡠 K, and an emissivity that equals its absorptivity to the radiant flux ( ⫽ ␣l ⫽ 0.65). For q⬙s ⫽ 3.0 ⫻ 105 W/m2, the temperature on its lower surface is measured by a radiation thermometer and found to have a value of Tw,l ⫽ 997⬚C.

Heating lamps

Tsur = 27°C

qs'' = 3 × 105 W/m2 Wafer, k, ε , αl

L = 0.78 mm

Tw, l = 997°C

To avoid warping the wafer and inducing slip planes in the crystal structure, the temperature difference across the thickness of the wafer must be less than 2⬚C. Is this condition being met? 1.57 A furnace for processing semiconductor materials is formed by a silicon carbide chamber that is zone-heated on the top section and cooled on the lower section. With the elevator in the lowest position, a robot arm inserts the silicon wafer on the mounting pins. In a production operation, the wafer is rapidly moved toward the hot zone to achieve the temperature-time history required for the process recipe. In this position, the top and bottom surfaces of the wafer exchange radiation with the hot and cool zones, respectively, of the chamber. The zone temperatures are Th ⫽ 1500 K and Tc ⫽ 330 K, and the emissivity and thickness of the wafer are ⫽ 0.65 and d ⫽ 0.78 mm, respectively. With the ambient gas at T앝 ⫽ 700 K, convection coefficients at the upper and lower surfaces of the wafer are 8 and 4 W/m2 䡠 K, respectively. The silicon wafer has a density of 2700 kg/m3 and a specific heat of 875 J/kg 䡠 K.

58

Chapter 1

䊏

Introduction

Lstack SiC chamber

Gas, T•

Estack

e

Heating zone Wafer, Tw, ε hu

hl

Mounting pin holder

Hot zone, Th = 1500 K Cool zone, Tc = 330 K

Elevator Water channel

Bipolar plate

Hydrogen flow channel

Airflow channel

Membrane

(a) For an initial condition corresponding to a wafer temperature of Tw,i ⫽ 300 K and the position of the wafer shown schematically, determine the corresponding time rate of change of the wafer temperature, (dTw /dt)i. (b) Determine the steady-state temperature reached by the wafer if it remains in this position. How significant is convection heat transfer for this situation? Sketch how you would expect the wafer temperature to vary as a function of vertical distance. 1.58 Single fuel cells such as the one of Example 1.5 can be scaled up by arranging them into a fuel cell stack. A stack consists of multiple electrolytic membranes that are sandwiched between electrically conducting bipolar plates. Air and hydrogen are fed to each membrane through flow channels within each bipolar plate, as shown in the sketch. With this stack arrangement, the individual fuel cells are connected in series, electrically, producing a stack voltage of Estack ⫽ N ⫻ Ec, where Ec is the voltage produced across each membrane and N is the number of membranes in the stack. The electrical current is the same for each membrane. The cell voltage, Ec, as well as the cell efficiency, increases with temperature (the air and hydrogen fed to the stack are humidified to allow operation at temperatures greater than in Example 1.5), but the membranes will fail at temperatures exceeding T 85⬚C. Consider L ⫻ w membranes, where L ⫽ w ⫽ 100 mm, of thickness tm ⫽ 0.43 mm, that each produce Ec ⫽ 0.6 V at I ⫽ 60 A, and E˙ c,g ⫽ 45 W of thermal energy when operating at T ⫽ 80⬚C. The external surfaces of the stack are exposed to air at T앝 ⫽ 25⬚C and surroundings at Tsur ⫽ 30⬚C, with ⫽ 0.88 and h ⫽ 150 W/m2 䡠 K.

(a) Find the electrical power produced by a stack that is Lstack ⫽ 200 mm long, for bipolar plate thickness in the range 1 mm ⬍ tbp ⬍ 10 mm. Determine the total thermal energy generated by the stack. (b) Calculate the surface temperature and explain whether the stack needs to be internally heated or cooled to operate at the optimal internal temperature of 80⬚C for various bipolar plate thicknesses. (c) Identify how the internal stack operating temperature might be lowered or raised for a given bipolar plate thickness, and discuss design changes that would promote a more uniform temperature distribution within the stack. How would changes in the external air and surroundings temperature affect your answer? Which membrane in the stack is most likely to fail due to high operating temperature? 1.59 Consider the wind turbine of Example 1.3. To reduce the nacelle temperature to Ts ⫽ 30⬚C, the nacelle is vented and a fan is installed to force ambient air into and out of the nacelle enclosure. What is the minimum mass flow rate of air required if the air temperature increases to the nacelle surface temperature before exiting the nacelle? The specific heat of air is 1007 J/kg 䡠 K. 1.60 Consider the conducting rod of Example 1.4 under steady-state conditions. As suggested in Comment 3, the temperature of the rod may be controlled by varying the speed of airflow over the rod, which, in turn, alters the convection heat transfer coefficient. To consider the effect of the convection coefficient, generate plots of T versus I for values of h ⫽ 50, 100, and 250 W/m2 䡠 K. Would variations in the surface emissivity have a significant effect on the rod temperature?

䊏

59

Problems

1.61 A long bus bar (cylindrical rod used for making electrical connections) of diameter D is installed in a large conduit having a surface temperature of 30⬚C and in which the ambient air temperature is T앝 ⫽ 30⬚C. The electrical resistivity, e(⍀ 䡠 m), of the bar material is a function of temperature, e,o ⫽ e [1 ⫹ ␣ (T ⫺ To)], where e,o ⫽ 0.0171 ⍀ 䡠 m, To ⫽ 25⬚C, and ␣ ⫽ 0.00396 K⫺1. The bar experiences free convection in the ambient air, and the convection coefficient depends on the bar diameter, as well as on the difference between the surface and ambient temperatures. The governing relation is of the form, h ⫽ CD⫺0.25 (T ⫺ T앝)0.25, where C ⫽ 1.21 W 䡠 m⫺1.75 䡠 K⫺1.25. The emissivity of the bar surface is ⫽ 0.85. (a) Recognizing that the electrical resistance per unit length of the bar is R⬘e ⫽ e /Ac, where Ac is its cross-sectional area, calculate the current-carrying capacity of a 20-mm-diameter bus bar if its temperature is not to exceed 65⬚C. Compare the relative importance of heat transfer by free convection and radiation exchange. (b) To assess the trade-off between current-carrying capacity, operating temperature, and bar diameter, for diameters of 10, 20, and 40 mm, plot the bar temperature T as a function of current for the range 100 ⱕ I ⱕ 5000 A. Also plot the ratio of the heat transfer by convection to the total heat transfer. 1.62 A small sphere of reference-grade iron with a specific heat of 447 J/kg 䡠 K and a mass of 0.515 kg is suddenly immersed in a water–ice mixture. Fine thermocouple wires suspend the sphere, and the temperature is observed to change from 15 to 14⬚C in 6.35 s. The experiment is repeated with a metallic sphere of the same diameter, but of unknown composition with a mass of 1.263 kg. If the same observed temperature change occurs in 4.59 s, what is the specific heat of the unknown material? 1.63 A 50 mm ⫻ 45 mm ⫻ 20 mm cell phone charger has a surface temperature of Ts ⫽ 33⬚C when plugged into an electrical wall outlet but not in use. The surface of the charger is of emissivity ⫽ 0.92 and is subject to a free convection heat transfer coefficient of h ⫽ 4.5 W/m2 䡠 K. The room air and wall temperatures are T앝 ⫽ 22⬚C and Tsur ⫽ 20⬚C, respectively. If electricity costs C ⫽ $0.18/kW 䡠 h, determine the daily cost of leaving the charger plugged in when not in use.

Tsur w 20 mm

L 50 mm

Wall Charger

Air T∞, h

1.64 A spherical, stainless steel (AISI 302) canister is used to store reacting chemicals that provide for a uniform heat flux q⬙i to its inner surface. The canister is suddenly submerged in a liquid bath of temperature T앝 ⬍ Ti, where Ti is the initial temperature of the canister wall. Canister

Reacting chemicals

ro = 0.6 m Ti = 500 K

3 ρ = 8055 kg/m c = 510 J/kg•K

p

T∞ = 300 K h = 500 W/m2•K

q"i Bath

ri = 0.5 m

(a) Assuming negligible temperature gradients in the canister wall and a constant heat flux q⬙i , develop an equation that governs the variation of the wall temperature with time during the transient process. What is the initial rate of change of the wall temperature if q⬙i ⫽ 105 W/m2? (b) What is the steady-state temperature of the wall? (c) The convection coefficient depends on the velocity associated with fluid flow over the canister and whether the wall temperature is large enough to induce boiling in the liquid. Compute and plot the steady-state temperature as a function of h for the range 100 ⱕ h ⱕ 10,000 W/m2 䡠 K. Is there a value of h below which operation would be unacceptable? 1.65 A freezer compartment is covered with a 2-mm-thick layer of frost at the time it malfunctions. If the compartment is in ambient air at 20⬚C and a coefficient of h ⫽ 2 W/m2 䡠 K characterizes heat transfer by natural convection from the exposed surface of the layer, estimate the time required to completely melt the frost. The frost may be assumed to have a mass density of 700 kg/m3 and a latent heat of fusion of 334 kJ/kg.

60

Chapter 1

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Introduction

1.66 A vertical slab of Wood’s metal is joined to a substrate on one surface and is melted as it is uniformly irradiated by a laser source on the opposite surface. The metal is initially at its fusion temperature of Tf ⫽ 72⬚C, and the melt runs off by gravity as soon as it is formed. The absorptivity of the metal to the laser radiation is ␣1 ⫽ 0.4, and its latent heat of fusion is hsf ⫽ 33 kJ/kg. (a) Neglecting heat transfer from the irradiated surface by convection or radiation exchange with the surroundings, determine the instantaneous rate of melting in kg/s 䡠 m2 if the laser irradiation is 5 kW/m2. How much material is removed if irradiation is maintained for a period of 2 s? (b) Allowing for convection to ambient air, with T앝 ⫽ 20⬚C and h ⫽ 15 W/m2 䡠 K, and radiation exchange with large surroundings ( ⫽ 0.4, Tsur ⫽ 20⬚C), determine the instantaneous rate of melting during irradiation. 1.67 A photovoltaic panel of dimension 2 m ⫻ 4 m is installed on the roof of a home. The panel is irradiated with a solar flux of GS ⫽ 700 W/m2, oriented normal to the top panel surface. The absorptivity of the panel to the solar irradiation is ␣S ⫽ 0.83, and the efficiency of conversion of the absorbed flux to electrical power is ⫽ P/␣S GS A ⫽ 0.553 ⫺ 0.001 K⫺1Tp, where Tp is the panel temperature expressed in kelvins and A is the solar panel area. Determine the electrical power generated for (a) a still summer day, in which Tsur ⫽ T앝 ⫽ 35⬚C, h ⫽ 10 W/m2 䡠 K, and (b) a breezy winter day, for which Tsur ⫽ T앝 ⫽ ⫺15⬚C, h ⫽ 30 W/m2 䡠 K. The panel emissivity is ⫽ 0.90.

GS

Air T∞ , h

Tsur

Bank of infrared radiant heaters Gas-fired furnace Carton

Conveyor

The chief engineer of your plant will approve the purchase of the heaters if they can reduce the water content by 10% of the total mass. Would you recommend the purchase? Assume the heat of vaporization of water is hfg ⫽ 2400 kJ/kg. 1.69 Electronic power devices are mounted to a heat sink having an exposed surface area of 0.045 m2 and an emissivity of 0.80. When the devices dissipate a total power of 20 W and the air and surroundings are at 27⬚C, the average sink temperature is 42⬚C. What average temperature will the heat sink reach when the devices dissipate 30 W for the same environmental condition? Power device

Tsur = 27°C

Heat sink, Ts A s, ε

Air

T∞ = 27°C

1.70 A computer consists of an array of five printed circuit boards (PCBs), each dissipating Pb ⫽ 20 W of power. Cooling of the electronic components on a board is provided by the forced flow of air, equally distributed in passages formed by adjoining boards, and the convection coefficient associated with heat transfer from the components to the air is approximately h ⫽ 200 W/m2 䡠 K. Air enters the computer console at a temperature of Ti ⫽ 20⬚C, and flow is driven by a fan whose power consumption is Pf ⫽ 25 W. •

Outlet air ∀, To

P

Photovoltaic panel, Tp

1.68 Following the hot vacuum forming of a paper-pulp mixture, the product, an egg carton, is transported on a conveyor for 18 s toward the entrance of a gas-fired oven where it is dried to a desired final water content. Very little water evaporates during the travel time. So, to increase the productivity of the line, it is proposed that a bank of infrared radiation heaters, which provide a uniform radiant flux of 5000 W/m2, be installed over the conveyor. The carton has an exposed area of 0.0625 m2 and a mass of 0.220 kg, 75% of which is water after the forming process.

PCB, Pb

•

Inlet air ∀, Ti

Fan, Pf

䊏

61

Problems

(a) If the temperature rise of the airflow, (To ⫺ Ti), is not to exceed 15⬚C, what is the minimum allowable volu˙ of the air? The density and specific metric flow rate ᭙ heat of the air may be approximated as ⫽ 1.161 kg/m3 and cp ⫽ 1007 J/kg 䡠 K, respectively. (b) The component that is most susceptible to thermal failure dissipates 1 W/cm2 of surface area. To minimize the potential for thermal failure, where should the component be installed on a PCB? What is its surface temperature at this location? 1.71 Consider a surface-mount type transistor on a circuit board whose temperature is maintained at 35⬚C. Air at 20⬚C flows over the upper surface of dimensions 4 mm ⫻ 8 mm with a convection coefficient of 50 W/m2 䡠 K. Three wire leads, each of cross section 1 mm ⫻ 0.25 mm and length 4 mm, conduct heat from the case to the circuit board. The gap between the case and the board is 0.2 mm. Air

Transistor case Wire lead Circuit board

Gap

(a) Assuming the case is isothermal and neglecting radiation, estimate the case temperature when 150 mW is dissipated by the transistor and (i) stagnant air or (ii) a conductive paste fills the gap. The thermal conductivities of the wire leads, air, and conductive paste are 25, 0.0263, and 0.12 W/m 䡠 K, respectively. (b) Using the conductive paste to fill the gap, we wish to determine the extent to which increased heat dissipation may be accommodated, subject to the constraint that the case temperature not exceed 40⬚C. Options include increasing the air speed to achieve a larger convection coefficient h and/or changing the lead wire material to one of larger thermal conductivity. Independently considering leads fabricated from materials with thermal conductivities of 200 and 400 W/m 䡠 K, compute and plot the maximum allowable heat dissipation for variations in h over the range 50 ⱕ h ⱕ 250 W/m2 䡠 K. 1.72 The roof of a car in a parking lot absorbs a solar radiant flux of 800 W/m2, and the underside is perfectly insulated. The convection coefficient between the roof and the ambient air is 12 W/m2 䡠 K. (a) Neglecting radiation exchange with the surroundings, calculate the temperature of the roof under steadystate conditions if the ambient air temperature is 20⬚C.

(b) For the same ambient air temperature, calculate the temperature of the roof if its surface emissivity is 0.8. (c) The convection coefficient depends on airflow conditions over the roof, increasing with increasing air speed. Compute and plot the roof temperature as a function of h for 2 ⱕ h ⱕ 200 W/m2 䡠 K. 1.73 Consider the conditions of Problem 1.22, but the surroundings temperature is 25⬚C and radiation exchange with the surroundings is not negligible. If the convection coefficient is 6.4 W/m2 䡠 K and the emissivity of the plate is ⫽ 0.42, determine the time rate of change of the plate temperature, dT/dt, when the plate temperature is 225⬚C. Evaluate the heat loss by convection and the heat loss by radiation. 1.74 Most of the energy we consume as food is converted to thermal energy in the process of performing all our bodily functions and is ultimately lost as heat from our bodies. Consider a person who consumes 2100 kcal per day (note that what are commonly referred to as food calories are actually kilocalories), of which 2000 kcal is converted to thermal energy. (The remaining 100 kcal is used to do work on the environment.) The person has a surface area of 1.8 m2 and is dressed in a bathing suit. (a) The person is in a room at 20⬚C, with a convection heat transfer coefficient of 3 W/m2 䡠 K. At this air temperature, the person is not perspiring much. Estimate the person’s average skin temperature. (b) If the temperature of the environment were 33⬚C, what rate of perspiration would be needed to maintain a comfortable skin temperature of 33⬚C? 1.75 Consider Problem 1.1. (a) If the exposed cold surface of the insulation is at T2 ⫽ 20⬚C, what is the value of the convection heat transfer coefficient on the cold side of the insulation if the surroundings temperature is Tsur ⫽ 320 K, the ambient temperature is T앝 ⫽ 5⬚C, and the emissivity is ⫽ 0.95? Express your results in units of W/m2 䡠 K and W/m2 䡠 ⬚C. (b) Using the convective heat transfer coefficient you calculated in part (a), determine the surface temperature, T2, as the emissivity of the surface is varied over the range 0.05 ⱕ ⱕ 0.95. The hot wall temperature of the insulation remains fixed at T1 ⫽ 30⬚C. Display your results graphically. 1.76 The wall of an oven used to cure plastic parts is of thickness L ⫽ 0.05 m and is exposed to large surroundings and air at its outer surface. The air and the surroundings are at 300 K. (a) If the temperature of the outer surface is 400 K and its convection coefficient and emissivity are

62

Chapter 1

䊏

Introduction

h ⫽ 20 W/m2 䡠 K and ⫽ 0.8, respectively, what is the temperature of the inner surface if the wall has a thermal conductivity of k ⫽ 0.7 W/m2 䡠 K? (b) Consider conditions for which the temperature of the inner surface is maintained at 600 K, while the air and large surroundings to which the outer surface is exposed are maintained at 300 K. Explore the effects of variations in k, h, and on (i) the temperature of the outer surface, (ii) the heat flux through the wall, and (iii) the heat fluxes associated with convection and radiation heat transfer from the outer surface. Specifically, compute and plot the foregoing dependent variables for parametric variations about baseline conditions of k ⫽ 10 W/m 䡠 K, h ⫽ 20 W/m2 䡠 K, and ⫽ 0.5. The suggested ranges of the independent variables are 0.1 ⱕ k ⱕ 400 W/m 䡠 K, 2 ⱕ h ⱕ 200 W/m2 䡠 K, and 0.05 ⱕ ⱕ 1. Discuss the physical implications of your results. Under what conditions will the temperature of the outer surface be less than 45⬚C, which is a reasonable upper limit to avoid burn injuries if contact is made? 1.77 An experiment to determine the convection coefficient associated with airflow over the surface of a thick stainless steel casting involves the insertion of thermocouples into the casting at distances of 10 and 20 mm from the surface along a hypothetical line normal to the surface. The steel has a thermal conductivity of 15 W/m 䡠 K. If the thermocouples measure temperatures of 50 and 40⬚C in the steel when the air temperature is 100⬚C, what is the convection coefficient? 1.78 A thin electrical heating element provides a uniform heat flux q⬙o to the outer surface of a duct through which airflows. The duct wall has a thickness of 10 mm and a thermal conductivity of 20 W/m 䡠 K. Air

Duct

Air

Ti Duct wall

To Electrical heater Insulation

(a) At a particular location, the air temperature is 30⬚C and the convection heat transfer coefficient between the air and inner surface of the duct is 100 W/m2 䡠 K. What heat flux q⬙o is required to maintain the inner surface of the duct at Ti ⫽ 85⬚C?

(b) For the conditions of part (a), what is the temperature (To ) of the duct surface next to the heater? (c) With Ti ⫽ 85⬚C, compute and plot q⬙o and To as a function of the air-side convection coefficient h for the range 10 ⱕ h ⱕ 200 W/m2 䡠 K. Briefly discuss your results. 1.79 A rectangular forced air heating duct is suspended from the ceiling of a basement whose air and walls are at a temperature of T앝 ⫽ Tsur ⫽ 5⬚C. The duct is 15 m long, and its cross section is 350 mm ⫻ 200 mm. (a) For an uninsulated duct whose average surface temperature is 50⬚C, estimate the rate of heat loss from the duct. The surface emissivity and convection coefficient are approximately 0.5 and 4 W/m2 䡠 K, respectively. (b) If heated air enters the duct at 58⬚C and a velocity of 4 m/s and the heat loss corresponds to the result of part (a), what is the outlet temperature? The density and specific heat of the air may be assumed to be ⫽ 1.10 kg/m3 and c ⫽ 1008 J/kg 䡠 K, respectively. 1.80 Consider the steam pipe of Example 1.2. The facilities manager wants you to recommend methods for reducing the heat loss to the room, and two options are proposed. The first option would restrict air movement around the outer surface of the pipe and thereby reduce the convection coefficient by a factor of two. The second option would coat the outer surface of the pipe with a low emissivity ( ⫽ 0.4) paint. (a) Which of the foregoing options would you recommend? (b) To prepare for a presentation of your recommendation to management, generate a graph of the heat loss q⬘ as a function of the convection coefficient for 2 ⱕ h ⱕ 20 W/m 2 䡠 K and emissivities of 0.2, 0.4, and 0.8. Comment on the relative efficacy of reducing heat losses associated with convection and radiation. 1.81 During its manufacture, plate glass at 600⬚C is cooled by passing air over its surface such that the convection heat transfer coefficient is h ⫽ 5 W/m2 䡠 K. To prevent cracking, it is known that the temperature gradient must not exceed 15⬚C/mm at any point in the glass during the cooling process. If the thermal conductivity of the glass is 1.4 W/m 䡠 K and its surface emissivity is 0.8, what is the lowest temperature of the air that can initially be used for the cooling? Assume that the temperature of the air equals that of the surroundings. 1.82 The curing process of Example 1.9 involves exposure of the plate to irradiation from an infrared lamp and attendant cooling by convection and radiation exchange

䊏

63

Problems

with the surroundings. Alternatively, in lieu of the lamp, heating may be achieved by inserting the plate in an oven whose walls (the surroundings) are maintained at an elevated temperature. (a) Consider conditions for which the oven walls are at 200⬚C, airflow over the plate is characterized by T앝 ⫽ 20⬚C and h ⫽ 15 W/m2 䡠 K, and the coating has an emissivity of ⫽ 0.5. What is the temperature of the plate? (b) For ambient air temperatures of 20, 40, and 60⬚C, determine the plate temperature as a function of the oven wall temperature over the range from 150 to 250⬚C. Plot your results, and identify conditions for which acceptable curing temperatures between 100 and 110⬚C may be maintained.

1.85 A solar flux of 700 W/m2 is incident on a flat-plate solar collector used to heat water. The area of the collector is 3 m2, and 90% of the solar radiation passes through the cover glass and is absorbed by the absorber plate. The remaining 10% is reflected away from the collector. Water flows through the tube passages on the back side of the absorber plate and is heated from an inlet temperature Ti to an outlet temperature To. The cover glass, operating at a temperature of 30⬚C, has an emissivity of 0.94 and experiences radiation exchange with the sky at ⫺10⬚C. The convection coefficient between the cover glass and the ambient air at 25⬚C is 10 W/m2 䡠 K. GS Cover glass

1.83 The diameter and surface emissivity of an electrically heated plate are D ⫽ 300 mm and ⫽ 0.80, respectively. (a) Estimate the power needed to maintain a surface temperature of 200⬚C in a room for which the air and the walls are at 25⬚C. The coefficient characterizing heat transfer by natural convection depends on the surface temperature and, in units of W/m2 䡠 K, may be approximated by an expression of the form h ⫽ 0.80(Ts ⫺ T앝)1/3. (b) Assess the effect of surface temperature on the power requirement, as well as on the relative contributions of convection and radiation to heat transfer from the surface. 1.84 Bus bars proposed for use in a power transmission station have a rectangular cross section of height H ⫽ 600 mm and width W ⫽ 200 mm. The electrical resistivity, e(⍀ 䡠 m), of the bar material is a function of temperature, e ⫽ e,o[1 ⫹ ␣(T ⫺ To)], where e,o ⫽ 0.0828 ⍀ 䡠 m, To ⫽ 25⬚C, and ␣ ⫽ 0.0040 K⫺1. The emissivity of the bar’s painted surface is 0.8, and the temperature of the surroundings is 30⬚C. The convection coefficient between the bar and the ambient air at 30⬚C is 10 W/m2 䡠 K. (a) Assuming the bar has a uniform temperature T, calculate the steady-state temperature when a current of 60,000 A passes through the bar. (b) Compute and plot the steady-state temperature of the bar as a function of the convection coefficient for 10 ⱕ h ⱕ 100 W/m2 䡠 K. What minimum convection coefficient is required to maintain a safe-operating temperature below 120⬚C? Will increasing the emissivity significantly affect this result?

Air space Absorber plate Water tubing Insulation

(a) Perform an overall energy balance on the collector to obtain an expression for the rate at which useful heat is collected per unit area of the collector, q⬙u. Determine the value of q⬙u. (b) Calculate the temperature rise of the water, To ⫺ Ti, if the flow rate is 0.01 kg/s. Assume the specific heat of the water to be 4179 J/kg 䡠 K. (c) The collector efficiency is defined as the ratio of the useful heat collected to the rate at which solar energy is incident on the collector. What is the value of ?

Process Identification 1.86 In analyzing the performance of a thermal system, the engineer must be able to identify the relevant heat transfer processes. Only then can the system behavior be properly quantified. For the following systems, identify the pertinent processes, designating them by appropriately labeled arrows on a sketch of the system. Answer additional questions that appear in the problem statement. (a) Identify the heat transfer processes that determine the temperature of an asphalt pavement on a summer day. Write an energy balance for the surface of the pavement.

64

Chapter 1

䊏

Introduction

(b) Microwave radiation is known to be transmitted by plastics, glass, and ceramics but to be absorbed by materials having polar molecules such as water. Water molecules exposed to microwave radiation align and reverse alignment with the microwave radiation at frequencies up to 109 s⫺1, causing heat to be generated. Contrast cooking in a microwave oven with cooking in a conventional radiant or convection oven. In each case, what is the physical mechanism responsible for heating the food? Which oven has the greater energy utilization efficiency? Why? Microwave heating is being considered for drying clothes. How would the operation of a microwave clothes dryer differ from a conventional dryer? Which is likely to have the greater energy utilization efficiency? Why? (c) To prevent freezing of the liquid water inside the fuel cell of an automobile, the water is drained to an onboard storage tank when the automobile is not in use. (The water is transferred from the tank back to the fuel cell when the automobile is turned on.) Consider a fuel cell–powered automobile that is parked outside on a very cold evening with T앝 ⫽ ⫺20⬚C. The storage tank is initially empty at Ti,t ⫽ ⫺20⬚C, when liquid water, at atmospheric pressure and temperature Ti,w ⫽ 50⬚C, is introduced into the tank. The tank has a wall thickness tt and is blanketed with insulation of thickness tins. Identify the heat transfer processes that will promote freezing of the water. Will the likelihood of freezing change as the insulation thickness is modified? Will the likelihood of freezing depend on the tank wall’s thickness and material? Would freezing of the water be more likely if plastic (low thermal conductivity) or stainless steel (moderate thermal conductivity) tubing is used to transfer the water to and from the tank? Is there an optimal tank shape that would minimize the probability of the water freezing? Would freezing be more likely or less likely to occur if a thin sheet of aluminum foil (high thermal conductivity, low emissivity) is applied to the outside of the insulation? To fuel cell Transfer tubing

Tsur Water

tt tins h, T∞

(d) Your grandmother is concerned about reducing her winter heating bills. Her strategy is to loosely fit rigid polystyrene sheets of insulation over her double-pane windows right after the first freezing weather arrives in the autumn. Identify the relevant heat transfer processes on a cold winter night when the foamed insulation sheet is placed (i) on the inner surface and (ii) on the outer surface of her window. To avoid condensation damage, which configuration is preferred? Condensation on the window pane does not occur when the foamed insulation is not in place.

Cold, dry night air

Warm, moist room air

Exterior pane Air gap Interior pane Insulation

Insulation on inner surface

Cold, dry night air

Warm, moist room air

Exterior pane Air gap Interior pane Insulation

Insulation on outer surface

(e) There is considerable interest in developing building materials with improved insulating qualities. The development of such materials would do much to enhance energy conservation by reducing space heating requirements. It has been suggested that superior structural and insulating qualities could be obtained by using the composite shown. The material consists of a honeycomb, with cells of square cross section, sandwiched between solid slabs. The cells are filled with air, and the slabs, as well as the honeycomb matrix, are fabricated from plastics of low thermal conductivity. For heat transfer normal to the slabs, identify all heat transfer processes pertinent to the performance of the composite. Suggest ways in which this performance could be enhanced.

䊏

65

Problems

Surface slabs

(h) A thermocouple junction is used to measure the temperature of a solid material. The junction is inserted into a small circular hole and is held in place by epoxy. Identify the heat transfer processes associated with the junction. Will the junction sense a temperature less than, equal to, or greater than the solid temperature? How will the thermal conductivity of the epoxy affect the junction temperature? Hot solid

Cellular air spaces

(f) A thermocouple junction (bead) is used to measure the temperature of a hot gas stream flowing through a channel by inserting the junction into the mainstream of the gas. The surface of the channel is cooled such that its temperature is well below that of the gas. Identify the heat transfer processes associated with the junction surface. Will the junction sense a temperature that is less than, equal to, or greater than the gas temperature? A radiation shield is a small, openended tube that encloses the thermocouple junction, yet allows for passage of the gas through the tube. How does use of such a shield improve the accuracy of the temperature measurement? Cool channel Shield Hot gases

Thermocouple bead

(g) A double-glazed, glass fire screen is inserted between a wood-burning fireplace and the interior of a room. The screen consists of two vertical glass plates that are separated by a space through which room air may flow (the space is open at the top and bottom). Identify the heat transfer processes associated with the fire screen. Air channel Glass plate

Air

Thermocouple bead

Cool gases

Epoxy

1.87 In considering the following problems involving heat transfer in the natural environment (outdoors), recognize that solar radiation is comprised of long and short wavelength components. If this radiation is incident on a semitransparent medium, such as water or glass, two things will happen to the nonreflected portion of the radiation. The long wavelength component will be absorbed at the surface of the medium, whereas the short wavelength component will be transmitted by the surface. (a) The number of panes in a window can strongly influence the heat loss from a heated room to the outside ambient air. Compare the single- and double-paned units shown by identifying relevant heat transfer processes for each case.

Double pane Ambient air

Room air Single pane

(b) In a typical flat-plate solar collector, energy is collected by a working fluid that is circulated through tubes that are in good contact with the back face of an absorber plate. The back face is insulated from

66

Chapter 1

䊏

Introduction

the surroundings, and the absorber plate receives solar radiation on its front face, which is typically covered by one or more transparent plates. Identify the relevant heat transfer processes, first for the absorber plate with no cover plate and then for the absorber plate with a single cover plate. (c) The solar energy collector design shown in the schematic has been used for agricultural applications. Air is blown through a long duct whose cross section is in the form of an equilateral triangle. One side of the triangle is comprised of a double-paned, semitransparent cover; the other two sides are constructed from aluminum sheets painted flat black on the inside and covered on the outside with a layer of styrofoam insulation. During sunny periods, air entering the system is heated for delivery to either a greenhouse, grain drying unit, or storage system.

Identify all heat transfer processes associated with the cover plates, the absorber plate(s), and the air. (d) Evacuated-tube solar collectors are capable of improved performance relative to flat-plate collectors. The design consists of an inner tube enclosed in an outer tube that is transparent to solar radiation. The annular space between the tubes is evacuated. The outer, opaque surface of the inner tube absorbs solar radiation, and a working fluid is passed through the tube to collect the solar energy. The collector design generally consists of a row of such tubes arranged in front of a reflecting panel. Identify all heat transfer processes relevant to the performance of this device.

Solar radiation Evacuated tubes Reflecting panel

Doublepaned cover

Working fluid

Styrofoam

Transparent outer tube

Absorber plates Evacuated space

Inner tube

C H A P T E R

Introduction to Conduction

2

68

Chapter 2

䊏

Introduction to Conduction

R

ecall that conduction is the transport of energy in a medium due to a temperature gradient, and the physical mechanism is one of random atomic or molecular activity. In Chapter 1 we learned that conduction heat transfer is governed by Fourier’s law and that use of the law to determine the heat flux depends on knowledge of the manner in which temperature varies within the medium (the temperature distribution). By way of introduction, we restricted our attention to simplified conditions (one-dimensional, steady-state conduction in a plane wall). However, Fourier’s law is applicable to transient, multidimensional conduction in complex geometries. The objectives of this chapter are twofold. First, we wish to develop a deeper understanding of Fourier’s law. What are its origins? What form does it take for different geometries? How does its proportionality constant (the thermal conductivity) depend on the physical nature of the medium? Our second objective is to develop, from basic principles, the general equation, termed the heat equation, which governs the temperature distribution in a medium. The solution to this equation provides knowledge of the temperature distribution, which may then be used with Fourier’s law to determine the heat flux.

2.1 The Conduction Rate Equation Although the conduction rate equation, Fourier’s law, was introduced in Section 1.2, it is now appropriate to consider its origin. Fourier’s law is phenomenological; that is, it is developed from observed phenomena rather than being derived from first principles. Hence, we view the rate equation as a generalization based on much experimental evidence. For example, consider the steady-state conduction experiment of Figure 2.1. A cylindrical rod of known material is insulated on its lateral surface, while its end faces are maintained at different temperatures, with T1 T2. The temperature difference causes conduction heat transfer in the positive x-direction. We are able to measure the heat transfer rate qx, and we seek to determine how qx depends on the following variables: T, the temperature difference; x, the rod length; and A, the cross-sectional area. We might imagine first holding T and x constant and varying A. If we do so, we find that qx is directly proportional to A. Similarly, holding T and A constant, we observe that qx varies inversely with x. Finally, holding A and x constant, we find that qx is directly proportional to T. The collective effect is then qx A T x In changing the material (e.g., from a metal to a plastic), we would find that this proportionality remains valid. However, we would also find that, for equal values of A, x, and T, ∆T = T1 – T2

A, T1

T2

qx

x

∆x

FIGURE 2.1 Steady-state heat conduction experiment.

2.1

䊏

69

The Conduction Rate Equation

the value of qx would be smaller for the plastic than for the metal. This suggests that the proportionality may be converted to an equality by introducing a coefficient that is a measure of the material behavior. Hence, we write qx kA T x where k, the thermal conductivity (W/m 䡠 K) is an important property of the material. Evaluating this expression in the limit as x l 0, we obtain for the heat rate qx kA dT dx

(2.1)

or for the heat flux qx

qx k dT A dx

(2.2)

Recall that the minus sign is necessary because heat is always transferred in the direction of decreasing temperature. Fourier’s law, as written in Equation 2.2, implies that the heat flux is a directional quantity. In particular, the direction of qx is normal to the cross-sectional area A. Or, more generally, the direction of heat flow will always be normal to a surface of constant temperature, called an isothermal surface. Figure 2.2 illustrates the direction of heat flow qx in a plane wall for which the temperature gradient dT/dx is negative. From Equation 2.2, it follows that qx is positive. Note that the isothermal surfaces are planes normal to the x-direction. Recognizing that the heat flux is a vector quantity, we can write a more general statement of the conduction rate equation (Fourier’s law) as follows:

冢 ⭸T⭸x j ⭸T⭸y k ⭸T⭸z 冣

q kT k i

(2.3)

where is the three-dimensional del operator and T(x, y, z) is the scalar temperature field. It is implicit in Equation 2.3 that the heat flux vector is in a direction perpendicular to the isothermal surfaces. An alternative form of Fourier’s law is therefore q qn n k

⭸T n ⭸n

T(x)

T1 q''x T2 x

FIGURE 2.2 The relationship between coordinate system, heat flow direction, and temperature gradient in one dimension.

(2.4)

70

Chapter 2

䊏

Introduction to Conduction

qy''

qn''

qx'' y

n Isotherm

x

FIGURE 2.3 The heat flux vector normal to an isotherm in a two-dimensional coordinate system.

where qn is the heat flux in a direction n, which is normal to an isotherm, and n is the unit normal vector in that direction. This is illustrated for the two-dimensional case in Figure 2.3. The heat transfer is sustained by a temperature gradient along n. Note also that the heat flux vector can be resolved into components such that, in Cartesian coordinates, the general expression for q is q iqx jqy kqz

(2.5)

where, from Equation 2.3, it follows that qx k

⭸T ⭸x

qy k

⭸T ⭸y

qz k

⭸T ⭸z

(2.6)

Each of these expressions relates the heat flux across a surface to the temperature gradient in a direction perpendicular to the surface. It is also implicit in Equation 2.3 that the medium in which the conduction occurs is isotropic. For such a medium, the value of the thermal conductivity is independent of the coordinate direction. Fourier’s law is the cornerstone of conduction heat transfer, and its key features are summarized as follows. It is not an expression that may be derived from first principles; it is instead a generalization based on experimental evidence. It is an expression that defines an important material property, the thermal conductivity. In addition, Fourier’s law is a vector expression indicating that the heat flux is normal to an isotherm and in the direction of decreasing temperature. Finally, note that Fourier’s law applies for all matter, regardless of its state (solid, liquid, or gas).

2.2 The Thermal Properties of Matter To use Fourier’s law, the thermal conductivity of the material must be known. This property, which is referred to as a transport property, provides an indication of the rate at which energy is transferred by the diffusion process. It depends on the physical structure of matter, atomic and molecular, which is related to the state of the matter. In this section we consider various forms of matter, identifying important aspects of their behavior and presenting typical property values.

2.2.1

Thermal Conductivity

From Fourier’s law, Equation 2.6, the thermal conductivity associated with conduction in the x-direction is defined as qx kx ⬅ (⭸T/⭸x)

2.2

䊏

71

The Thermal Properties of Matter

Similar definitions are associated with thermal conductivities in the y- and z-directions (ky, kz), but for an isotropic material the thermal conductivity is independent of the direction of transfer, kx ky kz ⬅ k. From the foregoing equation, it follows that, for a prescribed temperature gradient, the conduction heat flux increases with increasing thermal conductivity. In general, the thermal conductivity of a solid is larger than that of a liquid, which is larger than that of a gas. As illustrated in Figure 2.4, the thermal conductivity of a solid may be more than four orders of magnitude larger than that of a gas. This trend is due largely to differences in intermolecular spacing for the two states. In the modern view of materials, a solid may be comprised of free electrons and atoms bound in a periodic arrangement called the lattice. Accordingly, transport of thermal energy may be due to two effects: the migration of free electrons and lattice vibrational waves. When viewed as a particle-like phenomenon, the lattice vibration quanta are termed phonons. In pure metals, the electron contribution to conduction heat transfer dominates, whereas in nonconductors and semiconductors, the phonon contribution is dominant. Kinetic theory yields the following expression for the thermal conductivity [1]:

The Solid State

k 1 C c mfp 3

(2.7)

For conducting materials such as metals, C ⬅ Ce is the electron specific heat per unit volume, c is the mean electron velocity, and mfp ⬅ e is the electron mean free path, which is defined as the average distance traveled by an electron before it collides with either an imperfection in the material or with a phonon. In nonconducting solids, C ⬅ Cph is the phonon specific heat, c is the average speed of sound, and mfp ⬅ ph is the phonon mean free path, which again is determined by collisions with imperfections or other phonons. In all cases, the thermal conductivity increases as the mean free path of the energy carriers (electrons or phonons) is increased.

Zinc Silver PURE METALS Nickel Aluminum ALLOYS Plastics Ice Oxides NONMETALLIC SOLIDS Foams Fibers INSULATION SYSTEMS Oils Water Mercury LIQUIDS Carbon Hydrogen dioxide GASES

0.01

0.1

1 10 Thermal conductivity (W/m•K)

100

1000

FIGURE 2.4 Range of thermal conductivity for various states of matter at normal temperatures and pressure.

Chapter 2

䊏

Introduction to Conduction

When electrons and phonons carry thermal energy leading to conduction heat transfer in a solid, the thermal conductivity may be expressed as k ke kph

(2.8)

To a first approximation, ke is inversely proportional to the electrical resistivity, e. For pure metals, which are of low e, ke is much larger than kph. In contrast, for alloys, which are of substantially larger e, the contribution of kph to k is no longer negligible. For nonmetallic solids, k is determined primarily by kph, which increases as the frequency of interactions between the atoms and the lattice decreases. The regularity of the lattice arrangement has an important effect on kph, with crystalline (well-ordered) materials like quartz having a higher thermal conductivity than amorphous materials like glass. In fact, for crystalline, nonmetallic solids such as diamond and beryllium oxide, kph can be quite large, exceeding values of k associated with good conductors, such as aluminum. The temperature dependence of k is shown in Figure 2.5 for representative metallic and nonmetallic solids. Values for selected materials of technical importance are also provided in Table A.1 (metallic solids) and Tables A.2 and A.3 (nonmetallic solids). More detailed treatments of thermal conductivity are available in the literature [2]. In the preceding discussion, the bulk thermal conductivity is described, and the thermal conductivity values listed in Tables A.1 through A.3 are appropriate for use when the physical dimensions of the material of interest are relatively large. This is the case in many commonplace engineering problems. However, in several

The Solid State: Micro- and Nanoscale Effects

500 400

Silver Copper

300

Gold Aluminum Aluminum alloy 2024 Tungsten

200

100 Thermal conductivity (W/m•K)

72

Platinum 50 Iron

20

Stainless steel, AISI 304

10

Aluminum oxide

5 Pyroceram

2 Fused quartz 1 100

300

500 1000 Temperature (K)

2000

4000

FIGURE 2.5 The temperature dependence of the thermal conductivity of selected solids.

2.2

䊏

73

The Thermal Properties of Matter

areas of technology, such as microelectronics, the material’s characteristic dimensions can be on the order of micrometers or nanometers, in which case care must be taken to account for the possible modifications of k that can occur as the physical dimensions become small. Cross sections of films of the same material having thicknesses L1 and L2 are shown in Figure 2.6. Electrons or phonons that are associated with conduction of thermal energy are also shown qualitatively. Note that the physical boundaries of the film act to scatter the energy carriers and redirect their propagation. For large L/mfp1 (Figure 2.6a), the effect of the boundaries on reducing the average energy carrier path length is minor, and conduction heat transfer occurs as described for bulk materials. However, as the film becomes thinner, the physical boundaries of the material can decrease the average net distance traveled by the energy carriers, as shown in Figure 2.6b. Moreover, electrons and phonons moving in the thin x-direction (representing conduction in the x-direction) are affected by the boundaries to a more significant degree than energy carriers moving in the y-direction. As such, for films characterized by small L/mfp, we find that kx ky k, where k is the bulk thermal conductivity of the film material. For L/mfp 1, the predicted values of kx and ky may be estimated to within 20% from the following expression [1]: mfp kx 1 k 3L ky k

1

(2.9a)

2mfp 3L

(2.9b)

Equations 2.9a, b reveal that the values of kx and ky are within approximately 5% of the bulk thermal conductivity if L/mfp 7 (for kx ) and L/mfp 4.5 (for ky). Values of the mean free path as well as critical film thicknesses below which microscale effects must be considered, Lcrit, are included in Table 2.1 for several materials at T ⬇ 300 K. For films with mfp L Lcrit, kx and ky are reduced from the bulk value as indicated in Equations 2.9a,b.

y L1

L2 < L1

(a)

(b)

x

FIGURE 2.6 Electron or phonon trajectories in (a) a relatively thick film and (b) a relatively thin film with boundary effects.

The quantity mfp/L is a dimensionless parameter known as the Knudsen number. Large Knudsen numbers (small L/mfp) suggest potentially significant nano- or microscale effects. 1

Chapter 2

䊏

Introduction to Conduction

No general guidelines exist for predicting values of the thermal conductivities for L/mfp 1. Note that, in solids, the value of mfp decreases as the temperature increases. In addition to scattering from physical boundaries, as in the case of Figure 2.6b, energy carriers may be redirected by chemical dopants embedded within a material or by grain boundaries that separate individual clusters of material in otherwise homogeneous matter. Nanostructured materials are chemically identical to their conventional counterparts but are processed to provide very small grain sizes. This feature impacts heat transfer by increasing the scattering and reflection of energy carriers at grain boundaries. Measured values of the thermal conductivity of a bulk, nanostructured yttria-stabilized zirconia material are shown in Figure 2.7. This particular ceramic is widely used for insulation purposes in high-temperature combustion devices. Conduction is dominated by phonon transfer, and the mean free path of the phonon energy carriers is, from Table 2.1, mfp 25 nm at 300 K. As the grain sizes are reduced to characteristic dimensions less than 25 nm (and more grain boundaries are introduced in the material per unit volume), significant reduction of the thermal conductivity occurs. Extrapolation of the results of Figure 2.7 to higher temperatures is not recommended, since the mean free path decreases with increasing temperature (mfp ⬇ 4 nm at T ⬇ 1525 K ) and grains of the material may coalesce, merge, and enlarge at elevated temperatures. Therefore, L/mfp becomes larger at high temperatures, and

TABLE 2.1 Mean free path and critical film thickness for various materials at T 艐 300 K [3,4] Material Aluminum oxide Diamond (IIa) Gallium arsenide Gold Silicon Silicon dioxide Yttria-stabilized zirconia

mfp (nm)

Lcrit, x (nm)

Lcrit,y (nm)

5.08 315 23 31 43 0.6 25

36 2200 160 220 290 4 170

22 1400 100 140 180 3 110

2.5 L = 98 nm 2 Thermal conductivity (W/m•K)

74

L = 55 nm L = 32 nm

1.5 L = 23 nm

1

L = 10 nm

0.5 λmfp (T = 300 K) = 25 nm 0

0

100

200

300

Temperature (K)

400

500

FIGURE 2.7 Measured thermal conductivity of yttria-stabilized zirconia as a function of temperature and mean grain size, L [3].

2.2

75

The Thermal Properties of Matter

䊏

reduction of k due to nanoscale effects becomes less pronounced. Research on heat transfer in nanostructured materials continues to reveal novel ways engineers can manipulate the nanostructure to reduce or increase thermal conductivity [5]. Potentially important consequences include applications such as gas turbine engine technology [6], microelectronics [7], and renewable energy [8]. The Fluid State The fluid state includes both liquids and gases. Because the intermolecular spacing is much larger and the motion of the molecules is more random for the fluid state than for the solid state, thermal energy transport is less effective. The thermal conductivity of gases and liquids is therefore generally smaller than that of solids. The effect of temperature, pressure, and chemical species on the thermal conductivity of a gas may be explained in terms of the kinetic theory of gases [9]. From this theory it is known that the thermal conductivity is directly proportional to the density of the gas, the mean molecular speed c, and the mean free path mfp, which is the average distance traveled by an energy carrier (a molecule) before experiencing a collision.

k 艐 1 cv c mfp 3

(2.10)

For an ideal gas, the mean free path may be expressed as mfp

k BT 兹2d 2p

(2.11)

where kB is Boltzmann’s constant, kB 1.381 1023 J/K, d is the diameter of the gas molecule, representative values of which are included in Figure 2.8, and p is the pressure. 0.3 Hydrogen

= 2.016, d ⫽ 0.274

Thermal conductivity (W/m•K)

Helium 4.003, 0.219

0.2

Water (steam, 1 atm) 18.02, 0.458

0.1

Carbon dioxide 44.01, 0.464

Air 28.97, 0.372

0

0

200

400 600 Temperature (K)

800

1000

FIGURE 2.8 The temperature dependence of the thermal conductivity of selected gases at normal pressures. Molecular diameters (d) are in nm [10]. Molecular weights (ᏹ) of the gases are also shown.

Chapter 2

䊏

Introduction to Conduction

As expected, the mean free path is small for high pressure or low temperature, which causes densely packed molecules. The mean free path also depends on the diameter of the molecule, with larger molecules more likely to experience collisions than small molecules; in the limiting case of an infinitesimally small molecule, the molecules cannot collide, resulting in an infinite mean free path. The mean molecular speed, c, can be determined from the kinetic theory of gases, and Equation 2.10 may ultimately be expressed as k

9␥ 5 cv 4 d 2

冪 ᏺ

ᏹkBT

(2.12)

where the parameter ␥ is the ratio of specific heats, ␥ ⬅ cp /cv, and ᏺ is Avogadro’s number, ᏺ 6.022 1023 molecules per mol. Equation 2.12 can be used to estimate the thermal conductivity of gas, although more accurate models have been developed [10]. It is important to note that the thermal conductivity is independent of pressure except in extreme cases as, for example, when conditions approach that of a perfect vacuum. Therefore, the assumption that k is independent of gas pressure for large volumes of gas is appropriate for the pressures of interest in this text. Accordingly, although the values of k presented in Table A.4 pertain to atmospheric pressure or the saturation pressure corresponding to the prescribed temperature, they may be used over a much wider pressure range. Molecular conditions associated with the liquid state are more difficult to describe, and physical mechanisms for explaining the thermal conductivity are not well understood [11]. The thermal conductivity of nonmetallic liquids generally decreases with increasing temperature. As shown in Figure 2.9, water, glycerine, and engine oil are notable exceptions. The thermal conductivity of liquids is usually insensitive to pressure except near the critical point. Also, thermal conductivity generally decreases with increasing molecular weight. Values of

0.8

Water

0.6 Thermal conductivity (W/m•K)

76

Ammonia

0.4 Glycerine

0.2 Engine oil Freon 12 0 200

300

400 Temperature (K)

500

FIGURE 2.9 The temperature dependence of the thermal conductivity of selected nonmetallic liquids under saturated conditions.

2.2

䊏

The Thermal Properties of Matter

77

the thermal conductivity are often tabulated as a function of temperature for the saturated state of the liquid. Tables A.5 and A.6 present such data for several common liquids. Liquid metals are commonly used in high heat flux applications, such as occur in nuclear power plants. The thermal conductivity of such liquids is given in Table A.7. Note that the values are much larger than those of the nonmetallic liquids [12]. The Fluid State: Micro- and Nanoscale Effects As for the solid state, the bulk thermal conductivity of a fluid may be modified when the characteristic dimension of the system becomes small, in particular for small values of L/mfp. Similar to the situation of a thin solid film shown in Figure 2.6b, the molecular mean free path is restricted when a fluid is constrained by a small physical dimension, affecting conduction across a thin fluid layer. Mixtures of fluids and solids can also be formulated to tailor the transport properties of the resulting suspension. For example, nanofluids are base liquids that are seeded with nanometer-sized solid particles. Their very small size allows the solid particles to remain suspended within the base liquid for a long time. From the heat transfer perspective, a nanofluid exploits the high thermal conductivity that is characteristic of most solids, as is evident in Figure 2.5, to boost the relatively low thermal conductivity of base liquids, typical values of which are shown in Figure 2.9. Typical nanofluids involve liquid water seeded with nominally spherical nanoparticles of Al2O3 or CuO. Insulation Systems Thermal insulations consist of low thermal conductivity materials combined to achieve an even lower system thermal conductivity. In conventional fiber-, powder-, and flake-type insulations, the solid material is finely dispersed throughout an air space. Such systems are characterized by an effective thermal conductivity, which depends on the thermal conductivity and surface radiative properties of the solid material, as well as the nature and volumetric fraction of the air or void space. A special parameter of the system is its bulk density (solid mass/total volume), which depends strongly on the manner in which the material is packed. If small voids or hollow spaces are formed by bonding or fusing portions of the solid material, a rigid matrix is created. When these spaces are sealed from each other, the system is referred to as a cellular insulation. Examples of such rigid insulations are foamed systems, particularly those made from plastic and glass materials. Reflective insulations are composed of multilayered, parallel, thin sheets or foils of high reflectivity, which are spaced to reflect radiant energy back to its source. The spacing between the foils is designed to restrict the motion of air, and in high-performance insulations, the space is evacuated. In all types of insulation, evacuation of the air in the void space will reduce the effective thermal conductivity of the system. Heat transfer through any of these insulation systems may include several modes: conduction through the solid materials; conduction or convection through the air in the void spaces; and radiation exchange between the surfaces of the solid matrix. The effective thermal conductivity accounts for all of these processes, and values for selected insulation systems are summarized in Table A.3. Additional background information and data are available in the literature [13, 14]. As with thin films, micro- and nanoscale effects can influence the effective thermal conductivity of insulating materials. The value of k for a nanostructured silica aerogel material that is composed of approximately 5% by volume solid material and 95% by volume air that is trapped within pores of L ⬇ 20 nm is shown in Figure 2.10. Note that at T ⬇ 300 K, the mean free path for air at atmospheric pressure is approximately 80 nm. As the gas pressure is reduced, mfp would increase for an unconfined gas, but the molecular

Chapter 2

䊏

Introduction to Conduction

0.014 Effective thermal conductivity (W/m•K)

78

0.012 0.01 0.008 0.006 0.004 0.002 0 10ⴚ3

10ⴚ2

10ⴚ1

100

Pressure (atm)

FIGURE 2.10 Measured thermal conductivity of carbon-doped silica aerogel as a function of pressure at T 艐 300 K [15].

motion of the trapped air is restricted by the walls of the small pores and k is reduced to extremely small values relative to the thermal conductivities of conventional matter reported in Figure 2.4.

2.2.2

Other Relevant Properties

In our analysis of heat transfer problems, it will be necessary to use several properties of matter. These properties are generally referred to as thermophysical properties and include two distinct categories, transport and thermodynamic properties. The transport properties include the diffusion rate coefficients such as k, the thermal conductivity (for heat transfer), and , the kinematic viscosity (for momentum transfer). Thermodynamic properties, on the other hand, pertain to the equilibrium state of a system. Density () and specific heat (cp) are two such properties used extensively in thermodynamic analysis. The product cp (J/m3 䡠 K), commonly termed the volumetric heat capacity, measures the ability of a material to store thermal energy. Because substances of large density are typically characterized by small specific heats, many solids and liquids, which are very good energy storage media, have comparable heat capacities (cp 1 MJ/m3 䡠 K). Because of their very small densities, however, gases are poorly suited for thermal energy storage (cp ⬇ 1 kJ/m3 䡠 K). Densities and specific heats are provided in the tables of Appendix A for a wide range of solids, liquids, and gases. In heat transfer analysis, the ratio of the thermal conductivity to the heat capacity is an important property termed the thermal diffusivity ␣, which has units of m2/s: k ␣ rc

p

It measures the ability of a material to conduct thermal energy relative to its ability to store thermal energy. Materials of large ␣ will respond quickly to changes in their thermal environment, whereas materials of small ␣ will respond more sluggishly, taking longer to reach a new equilibrium condition. The accuracy of engineering calculations depends on the accuracy with which the thermophysical properties are known [16–18]. Numerous examples could be cited of flaws

2.2

䊏

79

The Thermal Properties of Matter

in equipment and process design or failure to meet performance specifications that were attributable to misinformation associated with the selection of key property values used in the initial system analysis. Selection of reliable property data is an integral part of any careful engineering analysis. The casual use of data that have not been well characterized or evaluated, as may be found in some literature or handbooks, is to be avoided. Recommended data values for many thermophysical properties can be obtained from Reference 19. This reference, available in most institutional libraries, was prepared by the Thermophysical Properties Research Center (TPRC) at Purdue University.

EXAMPLE 2.1 The thermal diffusivity ␣ is the controlling transport property for transient conduction. Using appropriate values of k, , and cp from Appendix A, calculate ␣ for the following materials at the prescribed temperatures: pure aluminum, 300 and 700 K; silicon carbide, 1000 K; paraffin, 300 K.

SOLUTION Known: Definition of the thermal diffusivity ␣. Find: Numerical values of ␣ for selected materials and temperatures. Properties: Table A.1, pure aluminum (300 K):

冧

2702 kg/m3 k 237 W/m 䡠 K cp 903 J/kg 䡠 K ␣ c 3 p 2702 kg/m 903 J/kg 䡠 K k 237 W/m 䡠 K 97.1 106 m2/s

䉰

Table A.1, pure aluminum (700 K): 2702 kg/m3 cp 1090 J/kg 䡠 K k 225 W/m 䡠 K

at 300 K at 700 K (by linear interpolation) at 700 K (by linear interpolation)

Hence

␣ kc p

225 W/m 䡠 K 76 106 m2/s 2702 kg/m3 1090 J/kg 䡠 K

䉰

Table A.2, silicon carbide (1000 K):

3160 kg/m3 cp 1195 J/kg 䡠 K k 87 W/m 䡠 K

冧

at 300 K 87 W/m 䡠 K at 1000 K ␣ 3160 kg/m3 1195 J/kg 䡠 K at 1000 K 23 106 m2/s

䉰

80

Chapter 2

䊏

Introduction to Conduction

Table A.3, paraffin (300 K):

冧

900 kg/m3 0.24 W/m 䡠 K cp 2890 J/kg 䡠 K ␣ kc p 900 kg/m3 2890 J/kg 䡠 K k 0.24 W/m 䡠 K 9.2 108 m2/s

䉰

Comments: 1. Note the temperature dependence of the thermophysical properties of aluminum and silicon carbide. For example, for silicon carbide, ␣(1000 K) ⬇ 0.1 ␣(300 K); hence properties of this material have a strong temperature dependence. 2. The physical interpretation of ␣ is that it provides a measure of heat transport (k) relative to energy storage (cp). In general, metallic solids have higher ␣, whereas nonmetallics (e.g., paraffin) have lower values of ␣. 3. Linear interpolation of property values is generally acceptable for engineering calculations. 4. Use of the low-temperature (300 K) density at higher temperatures ignores thermal expansion effects but is also acceptable for engineering calculations. 5. The IHT software provides a library of thermophysical properties for selected solids, liquids, and gases that can be accessed from the toolbar button, Properties. See Example 2.1 in IHT.

EXAMPLE 2.2 The bulk thermal conductivity of a nanofluid containing uniformly dispersed, noncontacting spherical nanoparticles may be approximated by knf

kp 2kbf 2(kp kbf)

冤 k 2k p

bf

(kp kbf)

冥k

bf

where is the volume fraction of the nanoparticles, and kbf, kp, and knf are the thermal conductivities of the base fluid, particle, and nanofluid, respectively. Likewise, the dynamic viscosity may be approximated as [20] nf bf (1 2.5) Determine the values of knf, nf, cp,nf, nf, and ␣nf for a mixture of water and Al2O3 nanoparticles at a temperature of T 300 K and a particle volume fraction of 0.05. The thermophysical properties of the particle are kp 36.0 W/m 䡠 K, p 3970 kg/m3, and cp,p 0.765 kJ/kg 䡠 K.

SOLUTION Known: Expressions for the bulk thermal conductivity and viscosity of a nanofluid with spherical nanoparticles. Nanoparticle properties.

2.2

䊏

81

The Thermal Properties of Matter

Find: Values of the nanofluid thermal conductivity, density, specific heat, dynamic viscosity, and thermal diffusivity. Schematic: Water Nanoparticle kp ⫽ 36.0 W/m·K ρp ⫽ 3970 kg/m3 cp,p ⫽ 0.765 kJ/kg·K

Assumptions: 1. Constant properties. 2. Density and specific heat are not affected by nanoscale phenomena. 3. Isothermal conditions. Properties: Table A.6 (T 300 K): Water; kbf 0.613 W/m K, bf 997 kg/m3, cp,bf 4.179 kJ/kg K, bf 855 106 N s/m2. Analysis: From the problem statement, knf

kp 2kbf 2(kp kbf)

冤 k 2k p

bf

(kp kbf)

冥k

bf

䡠 K 2 0.613 W/m 䡠 K 2 0.05(36.0 0.613) W/m 䡠 K 冤36.036.0W/m W/m 䡠 K 2 0.613 W/m 䡠 K 0.05(36.0 0.613) W/m 䡠 K 冥 0.613 W/m 䡠 K

0.705 W/m K

䉰

Consider the control volume shown in the schematic to be of total volume V. Then the conservation of mass principle yields nfV bfV(1 ) pV or, after dividing by the volume V, nf 997 kg/m3 (1 0.05) 3970 kg/m3 0.05 1146 kg/m3

䉰

Similarly, the conservation of energy principle yields, nfVcp,nf T bfV(1 )cp,bf T pVcp,p T Dividing by the volume V, temperature T, and nanofluid density nf yields cp,nf

bf cp,bf (1 ) pcp,p nf

997 kg/m3 4.179 kJ/kg 䡠 K (1 0.05) 3970 kg/m3 0.765 kJ/kg 䡠 K (0.05) 1146 kg/m3 3.587 kJ/kg 䡠 K 䉰

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From the problem statement, the dynamic viscosity of the nanofluid is nf 855 106 N 䡠 s/m2 (1 2.5 0.05) 962 106 N 䡠 s/m2 The nanofluid’s thermal diffusivity is k 0.705 W/m 䡠 K 171 109 m2/s ␣nf cnf 3 nf p,nf 1146 kg/m 3587 J/kg 䡠 K

䉰

䉰

Comments: 1. Ratios of the properties of the nanofluid to the properties of water are as follows. knf 0.705 W/m 䡠 K 1.150 kbf 0.613 W/m 䡠 K cp,nf 3587 J/kg 䡠 K cp,bf 4179 J/kg 䡠 K 0.858

nf 1146 kg/m3 bf 997 kg/m3 1.149 nf 962 106 N 䡠 s/m2 bf 855 106 N 䡠 s/m2 1.130

␣nf 171 109 m2 /s ␣bf 147 109 m2/s 1.166 The relatively large thermal conductivity and thermal diffusivity of the nanofluid enhance heat transfer rates in some applications. However, all of the thermophysical properties are affected by the addition of the nanoparticles, and, as will become evident in Chapters 6 through 9, properties such as the viscosity and specific heat are adversely affected. This condition can degrade thermal performance when the use of nanofluids involves convection heat transfer. 2. The expression for the nanofluid’s thermal conductivity (and viscosity) is limited to dilute mixtures of noncontacting, spherical particles. In some cases, the particles do not remain separated but can agglomerate into long chains, providing effective paths for heat conduction through the fluid and larger bulk thermal conductivities. Hence, the expression for the thermal conductivity represents the minimum possible enhancement of the thermal conductivity by spherical nanoparticles. An expression for the maximum possible isotropic thermal conductivity of a nanofluid, corresponding to agglomeration of the spherical particles, is available [21], as are expressions for dilute suspensions of nonspherical particles [22]. Note that these expressions can also be applied to nanostructured composite materials consisting of a particulate phase interspersed within a host binding medium, as will be discussed in more detail in Chapter 3. 3. The nanofluid’s density and specific heat are determined by applying the principles of mass and energy conservation, respectively. As such, these properties do not depend on the manner in which the nanoparticles are dispersed within the base liquid.

2.3 The Heat Diffusion Equation A major objective in a conduction analysis is to determine the temperature field in a medium resulting from conditions imposed on its boundaries. That is, we wish to know the temperature distribution, which represents how temperature varies with position in the medium. Once this distribution is known, the conduction heat flux at any point in the medium or on its surface may be computed from Fourier’s law. Other important

2.3

䊏

83

The Heat Diffusion Equation

quantities of interest may also be determined. For a solid, knowledge of the temperature distribution could be used to ascertain structural integrity through determination of thermal stresses, expansions, and deflections. The temperature distribution could also be used to optimize the thickness of an insulating material or to determine the compatibility of special coatings or adhesives used with the material. We now consider the manner in which the temperature distribution can be determined. The approach follows the methodology described in Section 1.3.1 of applying the energy conservation requirement. In this case, we define a differential control volume, identify the relevant energy transfer processes, and introduce the appropriate rate equations. The result is a differential equation whose solution, for prescribed boundary conditions, provides the temperature distribution in the medium. Consider a homogeneous medium within which there is no bulk motion (advection) and the temperature distribution T(x, y, z) is expressed in Cartesian coordinates. Following the methodology of applying conservation of energy (Section 1.3.1), we first define an infinitesimally small (differential) control volume, dx 䡠 dy 䡠 dz, as shown in Figure 2.11. Choosing to formulate the first law at an instant of time, the second step is to consider the energy processes that are relevant to this control volume. In the absence of motion (or with uniform motion), there are no changes in mechanical energy and no work being done on the system. Only thermal forms of energy need be considered. Specifically, if there are temperature gradients, conduction heat transfer will occur across each of the control surfaces. The conduction heat rates perpendicular to each of the control surfaces at the x-, y-, and z-coordinate locations are indicated by the terms qx, qy, and qz, respectively. The conduction heat rates at the opposite surfaces can then be expressed as a Taylor series expansion where, neglecting higher-order terms, ⭸qx dx ⭸x ⭸qy qy dy qy dy ⭸y qx dx qx

qz dz qz

(2.13a) (2.13b)

⭸qz dz ⭸z

(2.13c)

T(x, y, z)

qz + dz qy + dy

dz •

Eg

qx

•

qx + dx

E st

z y x

dy

qy dx qz

FIGURE 2.11 Differential control volume, dx dy dz, for conduction analysis in Cartesian coordinates.

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Introduction to Conduction

In words, Equation 2.13a simply states that the x-component of the heat transfer rate at x dx is equal to the value of this component at x plus the amount by which it changes with respect to x times dx. Within the medium there may also be an energy source term associated with the rate of thermal energy generation. This term is represented as E˙ g q˙ dx dy dz

(2.14)

where q˙ is the rate at which energy is generated per unit volume of the medium (W/m3). In addition, changes may occur in the amount of the internal thermal energy stored by the material in the control volume. If the material is not experiencing a change in phase, latent energy effects are not pertinent, and the energy storage term may be expressed as ⭸T E˙ st cp dx dy dz (2.15) ⭸t where cp ⭸T/⭸t is the time rate of change of the sensible (thermal) energy of the medium per unit volume. Once again it is important to note that the terms E˙ g and E˙ st represent different physical processes. The energy generation term E˙ g is a manifestation of some energy conversion process involving thermal energy on one hand and some other form of energy, such as chemical, electrical, or nuclear, on the other. The term is positive (a source) if thermal energy is being generated in the material at the expense of some other energy form; it is negative (a sink) if thermal energy is being consumed. In contrast, the energy storage term E˙ st refers to the rate of change of thermal energy stored by the matter. The last step in the methodology outlined in Section 1.3.1 is to express conservation of energy using the foregoing rate equations. On a rate basis, the general form of the conservation of energy requirement is E˙in E˙g E˙out E˙st

(1.12c)

Hence, recognizing that the conduction rates constitute the energy inflow E˙ in and outflow E˙ out, and substituting Equations 2.14 and 2.15, we obtain ⭸T qx qy qz q˙ dx dy dz qxdx qydy qzdz cp dx dy dz (2.16) ⭸t Substituting from Equations 2.13, it follows that

⭸qy ⭸qz ⭸qx ⭸T dx dy dz q˙ dx dy dz cp dx dy dz ⭸x ⭸y ⭸z ⭸t

(2.17)

The conduction heat rates in an isotropic material may be evaluated from Fourier’s law, ⭸T ⭸x ⭸T qy k dx dz ⭸y ⭸T qz k dx dy ⭸z

qx k dy dz

(2.18a) (2.18b) (2.18c)

where each heat flux component of Equation 2.6 has been multiplied by the appropriate control surface (differential) area to obtain the heat transfer rate. Substituting

2.3

䊏

85

The Heat Diffusion Equation

Equations 2.18 into Equation 2.17 and dividing out the dimensions of the control volume (dx dy dz), we obtain

冢 冣

冢 冣

冢 冣

⭸ ⭸T ⭸T ⭸T ⭸T ⭸ ⭸ k k k q˙ cp ⭸x ⭸x ⭸y ⭸y ⭸z ⭸z ⭸t

(2.19)

Equation 2.19 is the general form, in Cartesian coordinates, of the heat diffusion equation. This equation, often referred to as the heat equation, provides the basic tool for heat conduction analysis. From its solution, we can obtain the temperature distribution T(x, y, z) as a function of time. The apparent complexity of this expression should not obscure the fact that it describes an important physical condition, that is, conservation of energy. You should have a clear understanding of the physical significance of each term appearing in the equation. For example, the term ⭸(k⭸T/⭸x)/⭸x is related to the net conduction heat flux into the control volume for the x-coordinate direction. That is, multiplying by dx,

冢 冣

⭸ ⭸T k dx qx qxdx ⭸x ⭸x

(2.20)

with similar expressions applying for the fluxes in the y- and z-directions. In words, the heat equation, Equation 2.19, therefore states that at any point in the medium the net rate of energy transfer by conduction into a unit volume plus the volumetric rate of thermal energy generation must equal the rate of change of thermal energy stored within the volume. It is often possible to work with simplified versions of Equation 2.19. For example, if the thermal conductivity is constant, the heat equation is ⭸2T ⭸2T ⭸2T q˙ 1 ⭸T ⭸x2 ⭸y2 ⭸z2 k ␣ ⭸t

(2.21)

where ␣ k/cp is the thermal diffusivity. Additional simplifications of the general form of the heat equation are often possible. For example, under steady-state conditions, there can be no change in the amount of energy storage; hence Equation 2.19 reduces to

冢 冣

冢 冣

冢 冣

⭸ ⭸T ⭸T ⭸T ⭸ ⭸ k k k q˙ 0 ⭸x ⭸x ⭸y ⭸y ⭸z ⭸z

(2.22)

Moreover, if the heat transfer is one-dimensional (e.g., in the x-direction) and there is no energy generation, Equation 2.22 reduces to

冢 冣

d k dT 0 dx dx

(2.23)

The important implication of this result is that, under steady-state, one-dimensional conditions with no energy generation, the heat flux is a constant in the direction of transfer (dqx /dx 0). The heat equation may also be expressed in cylindrical and spherical coordinates. The differential control volumes for these two coordinate systems are shown in Figures 2.12 and 2.13.

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Chapter 2

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Introduction to Conduction

qz + dz

rdφ

qr

qφ + dφ dz z r T(r,φ ,z)

qφ qr + dr

dr

y

r

φ

x qz

FIGURE 2.12 Differential control volume, dr 䡠 r d 䡠 dz, for conduction analysis in cylindrical coordinates (r, , z).

When the del operator of Equation 2.3 is expressed in cylindrical coordinates, the general form of the heat flux vector and hence of Fourier’s law is

Cylindrical Coordinates

⭸T ⭸T k 冣 冢 ⭸T⭸r j 1r ⭸ ⭸z

q kT k i

(2.24)

where ⭸T ⭸r

qr k

⭸T q kr ⭸

qz k

⭸T ⭸z

qθ + dθ r sin θ dφ qr

qφ + dφ rdθ

qφ

z θ

r φ

T(r, φ , θ)

qr + dr

dr

y qθ

x

FIGURE 2.13 Differential control volume, dr 䡠 r sin d 䡠 r d, for conduction analysis in spherical coordinates (r, , ).

(2.25)

2.3

䊏

87

The Heat Diffusion Equation

are heat flux components in the radial, circumferential, and axial directions, respectively. Applying an energy balance to the differential control volume of Figure 2.12, the following general form of the heat equation is obtained:

冢 冣

冢 冣

冢 冣

1 ⭸ kr ⭸T 1 ⭸ k ⭸T ⭸ k ⭸T q˙ c ⭸T p r ⭸r ⭸r ⭸z ⭸z ⭸t r 2 ⭸ ⭸ Spherical Coordinates

(2.26)

In spherical coordinates, the general form of the heat flux vector

and Fourier’s law is 1 ⭸T 冢 ⭸T⭸r j 1r ⭸T⭸ k r sin ⭸冣

q kT k i

(2.27)

where qr k

⭸T ⭸r

⭸T q kr ⭸

q

k ⭸T r sin ⭸

(2.28)

are heat flux components in the radial, polar, and azimuthal directions, respectively. Applying an energy balance to the differential control volume of Figure 2.13, the following general form of the heat equation is obtained:

冢

冣

冢 冣

⭸ ⭸T 1 1 ⭸ kr 2 ⭸T k 2 ⭸r 2 2 ⭸ ⭸r ⭸ r r sin

冢

冣

⭸ ⭸T ⭸T 1 k sin q˙ cp ⭸t ⭸ r 2 sin ⭸

(2.29)

You should attempt to derive Equation 2.26 or 2.29 to gain experience in applying conservation principles to differential control volumes (see Problems 2.35 and 2.36). Note that the temperature gradient in Fourier’s law must have units of K/m. Hence, when evaluating the gradient for an angular coordinate, it must be expressed in terms of the differential change in arc length. For example, the heat flux component in the circumferential direction of a cylindrical coordinate system is q (k/r)(⭸T/⭸), not q k(⭸T/⭸).

EXAMPLE 2.3 The temperature distribution across a wall 1 m thick at a certain instant of time is given as T(x) a bx cx2 where T is in degrees Celsius and x is in meters, while a 900 C, b 300 C/m, and . c 50 C/m2. A uniform heat generation, q 1000 W/m3, is present in the wall of area 2 3 10 m having the properties 1600 kg/m , k 40 W/m 䡠 K, and cp 4 kJ/kg 䡠 K.

88

Chapter 2

䊏

Introduction to Conduction

1. Determine the rate of heat transfer entering the wall (x 0) and leaving the wall (x 1 m). 2. Determine the rate of change of energy storage in the wall. 3. Determine the time rate of temperature change at x 0, 0.25, and 0.5 m.

SOLUTION Known: Temperature distribution T(x) at an instant of time t in a one-dimensional wall with uniform heat generation. Find: 1. Heat rates entering, qin (x 0), and leaving, qout (x 1 m), the wall. 2. Rate of change of energy storage in the wall, E˙ st. 3. Time rate of temperature change at x 0, 0.25, and 0.5 m. Schematic: q• = 1000 W/m3 k = 40 W/m•K ρ = 1600 kg/m3 cp = 4 kJ/kg•K

A = 10 m2

T(x) = a + bx + cx2 •

Eg •

E st qin

qout

L=1m x

Assumptions: 1. One-dimensional conduction in the x-direction. 2. Isotropic medium with constant properties. . 3. Uniform internal heat generation, q (W/m3). Analysis: 1. Recall that once the temperature distribution is known for a medium, it is a simple matter to determine the conduction heat transfer rate at any point in the medium or at its surfaces by using Fourier’s law. Hence the desired heat rates may be determined by using the prescribed temperature distribution with Equation 2.1. Accordingly, qin qx(0) kA

⭸T 兩 kA(b 2cx)x0 ⭸x x0

qin bkA 300 C/m 40 W/m 䡠 K 10 m2 120 kW

䉰

2.3

䊏

89

The Heat Diffusion Equation

Similarly, qout qx(L) kA

⭸T 兩 kA(b 2cx)xL ⭸x xL

qout (b 2cL)kA [300 C/m 2(50 C/m2) 1 m] 40 W/m 䡠 K 10 m2 160 kW

䉰

2. The rate of change of energy storage in the wall E˙ st may be determined by applying an overall energy balance to the wall. Using Equation 1.12c for a control volume about the wall, E˙ in E˙ g E˙ out E˙ st where E˙ g q˙AL, it follows that E˙ st E˙ in E˙ g E˙ out qin q˙AL qout E˙ st 120 kW 1000 W/m3 10 m2 1 m 160 kW E˙ st 30 kW

䉰

3. The time rate of change of the temperature at any point in the medium may be determined from the heat equation, Equation 2.21, rewritten as ⭸T k ⭸2T q˙ c cp ⭸t p ⭸x2 From the prescribed temperature distribution, it follows that

冢 冣

⭸2T ⭸ ⭸T 2 ⭸x ⭸x ⭸x

⭸ (b 2cx) 2c 2(50 C/m2) 100 C/m2 ⭸x

Note that this derivative is independent of position in the medium. Hence the time rate of temperature change is also independent of position and is given by ⭸T 40 W/m 䡠 K (100 C/m2) ⭸t 1600 kg/m3 4 kJ/kg 䡠 K

1000 W/m3 1600 kg/m3 4 kJ/kg 䡠 K

⭸T 6.25 104 C/s 1.56 104 C/s ⭸t 4.69 104 C/s

䉰

90

Chapter 2

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Introduction to Conduction

Comments: 1. From this result, it is evident that the temperature at every point within the wall is decreasing with time. 2. Fourier’s law can always be used to compute the conduction heat rate from knowledge of the temperature distribution, even for unsteady conditions with internal heat generation.

Microscale Effects For most practical situations, the heat diffusion equations generated in this text may be used with confidence. However, these equations are based on Fourier’s law, which does not account for the finite speed at which thermal information is propagated within the medium by the various energy carriers. The consequences of the finite propagation speed may be neglected if the heat transfer events of interest occur over a sufficiently long time scale, t, such that mfp 1 (2.30) ct The heat diffusion equations of this text are likewise invalid for problems where boundary scattering must be explicitly considered. For example, the temperature distribution within the thin film of Figure 2.6b cannot be determined by applying the foregoing heat diffusion equations. Additional discussion of micro- and nanoscale heat transfer applications and analysis methods is available in the literature [1, 5, 10, 23].

2.4 Boundary and Initial Conditions To determine the temperature distribution in a medium, it is necessary to solve the appropriate form of the heat equation. However, such a solution depends on the physical conditions existing at the boundaries of the medium and, if the situation is time dependent, on conditions existing in the medium at some initial time. With regard to the boundary conditions, there are several common possibilities that are simply expressed in mathematical form. Because the heat equation is second order in the spatial coordinates, two boundary conditions must be expressed for each coordinate needed to describe the system. Because the equation is first order in time, however, only one condition, termed the initial condition, must be specified. Three kinds of boundary conditions commonly encountered in heat transfer are summarized in Table 2.2. The conditions are specified at the surface x 0 for a one-dimensional system. Heat transfer is in the positive x-direction with the temperature distribution, which may be time dependent, designated as T(x, t). The first condition corresponds to a situation for which the surface is maintained at a fixed temperature Ts. It is commonly termed a Dirichlet condition, or a boundary condition of the first kind. It is closely approximated, for example, when the surface is in contact with a melting solid or a boiling liquid. In both cases, there is heat transfer at the surface, while the surface remains at the temperature of the phase change process. The second condition corresponds to the existence of a fixed or constant heat flux qs at the surface. This heat flux is related to the temperature gradient at the surface by Fourier’s law, Equation 2.6, which may be expressed as qx (0) k

⭸T 兩 qs ⭸x x0

2.4

䊏

91

Boundary and Initial Conditions

TABLE 2.2 Boundary conditions for the heat diffusion equation at the surface (x 0) 1.

Ts

Constant surface temperature T(0, t) Ts

(2.31)

T(x, t) x

2.

Constant surface heat flux (a) Finite heat flux ⭸T k 兩 qs ⭸x x0

qs'' T(x, t)

(2.32) x

(b) Adiabatic or insulated surface ⭸T 兩 0 ⭸x x0

T(x, t)

(2.33) x

3.

Convection surface condition ⭸T k 兩 h[T앝 T(0, t)] ⭸x x0

T(0, t)

(2.34)

T∞, h x

T(x, t)

It is termed a Neumann condition, or a boundary condition of the second kind, and may be realized by bonding a thin film electric heater to the surface. A special case of this condition corresponds to the perfectly insulated, or adiabatic, surface for which ⭸T/⭸x冷 x0 0. The boundary condition of the third kind corresponds to the existence of convection heating (or cooling) at the surface and is obtained from the surface energy balance discussed in Section 1.3.1.

EXAMPLE 2.4 A long copper bar of rectangular cross section, whose width w is much greater than its thickness L, is maintained in contact with a heat sink at its lower surface, and the temperature throughout the bar is approximately equal to that of the sink, To. Suddenly, an electric current is passed through the bar and an airstream of temperature T앝 is passed over the top surface, while the bottom surface continues to be maintained at To. Obtain the differential equation and the boundary and initial conditions that could be solved to determine the temperature as a function of position and time in the bar.

SOLUTION Known: Copper bar initially in thermal equilibrium with a heat sink is suddenly heated by passage of an electric current.

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Introduction to Conduction

Find: Differential equation and boundary and initial conditions needed to determine temperature as a function of position and time within the bar. Schematic: Copper bar (k, α) T(x, y, z, t) T(x, t)

y

Air

T∞, h

x

Air

w

T∞, h

L

T(L, t) •

q

z

L

I

Heat sink

To

x

To = T(0, t)

Assumptions: 1. Since the bar is long and w L, end and side effects are negligible and heat transfer within the bar is primarily one dimensional in the x-direction. 2. Uniform volumetric heat generation, q˙. 3. Constant properties. Analysis: The temperature distribution is governed by the heat equation (Equation 2.19), which, for the one-dimensional and constant property conditions of the present problem, reduces to ⭸2T q˙ 1 ⭸T ⭸x2 k ␣ ⭸t

(1)

䉰

where the temperature is a function of position and time, T(x, t). Since this differential equation is second order in the spatial coordinate x and first order in time t, there must be two boundary conditions for the x-direction and one condition, termed the initial condition, for time. The boundary condition at the bottom surface corresponds to case 1 of Table 2.2. In particular, since the temperature of this surface is maintained at a value, To, which is fixed with time, it follows that T(0, t) To

(2)

䉰

The convection surface condition, case 3 of Table 2.2, is appropriate for the top surface. Hence k

⭸T 兩 h[T(L, t) T앝] ⭸x xL

(3)

䉰

The initial condition is inferred from recognition that, before the change in conditions, the bar is at a uniform temperature To. Hence T(x, 0) To

(4)

䉰

2.4

䊏

93

Boundary and Initial Conditions

. If To, T앝, q, and h are known, Equations 1 through 4 may be solved to obtain the time-varying temperature distribution T(x, t) following imposition of the electric current.

Comments: 1. The heat sink at x 0 could be maintained by exposing the surface to an ice bath or by attaching it to a cold plate. A cold plate contains coolant channels machined in a solid of large thermal conductivity (usually copper). By circulating a liquid (usually water) through the channels, the plate and hence the surface to which it is attached may be maintained at a nearly uniform temperature. 2. The temperature of the top surface T(L, t) will change with time. This temperature is an unknown and may be obtained after finding T(x, t). 3. We may use our physical intuition to sketch temperature distributions in the bar at selected times from the beginning to the end of the transient process. If we assume that T앝 To and that the electric current is sufficiently large to heat the bar to temperatures in excess of T⬁, the following distributions would correspond to the initial condition (t 0), the final (steady-state) condition (t l 앝), and two intermediate times.

T(x, t)

T(x, ∞), Steady-state condition T∞

T∞

b

a T(x, 0), Initial condition

To L

0 Distance, x

Note how the distributions comply with the initial and boundary conditions. What is a special feature of the distribution labeled (b)? 4. Our intuition may also be used to infer the manner in which the heat flux varies with time at the surfaces (x 0, L) of the bar. On qx t coordinates, the transient variations are as follows. +

q"x (x, t)

q"x (L, t)

0

q"x (0, t)

– 0

Time, t

Convince yourself that the foregoing variations are consistent with the temperature distributions of Comment 3. For t l 앝, how are qx (0) and qx (L) related to the volumetric rate of energy generation?

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Chapter 2

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Introduction to Conduction

2.5 Summary Despite the relative brevity of this chapter, its importance must not be underestimated. Understanding the conduction rate equation, Fourier’s law, is essential. You must be cognizant of the importance of thermophysical properties; over time, you will develop a sense of the magnitudes of the properties of many real materials. Likewise, you must recognize that the heat equation is derived by applying the conservation of energy principle to a differential control volume and that it is used to determine temperature distributions within matter. From knowledge of the distribution, Fourier’s law can be used to determine the corresponding conduction heat rates. A firm grasp of the various types of thermal boundary conditions that are used in conjunction with the heat equation is vital. Indeed, Chapter 2 is the foundation on which Chapters 3 through 5 are based, and you are encouraged to revisit this chapter often. You may test your understanding of various concepts by addressing the following questions. • In the general formulation of Fourier’s law (applicable to any geometry), what are the vector and scalar quantities? Why is there a minus sign on the right-hand side of the equation? • What is an isothermal surface? What can be said about the heat flux at any location on this surface? • What form does Fourier’s law take for each of the orthogonal directions of Cartesian, cylindrical, and spherical coordinate systems? In each case, what are the units of the temperature gradient? Can you write each equation from memory? • An important property of matter is defined by Fourier’s law. What is it? What is its physical significance? What are its units? • What is an isotropic material? • Why is the thermal conductivity of a solid generally larger than that of a liquid? Why is the thermal conductivity of a liquid larger than that of a gas? • Why is the thermal conductivity of an electrically conducting solid generally larger than that of a nonconductor? Why are materials such as beryllium oxide, diamond, and silicon carbide (see Table A.2) exceptions to this rule? • Is the effective thermal conductivity of an insulation system a true manifestation of the efficacy with which heat is transferred through the system by conduction alone? • Why does the thermal conductivity of a gas increase with increasing temperature? Why is it approximately independent of pressure? • What is the physical significance of the thermal diffusivity? How is it defined and what are its units? • What is the physical significance of each term appearing in the heat equation? • Cite some examples of thermal energy generation. If the rate at which thermal energy is generated per unit volume, q˙, varies with location in a medium of volume V, how can the rate of energy generation for the entire medium, E˙ g, be determined from knowledge of q˙(x, y, z)? • For a chemically reacting medium, what kind of reaction provides a source of thermal energy (q˙ 0)? What kind of reaction provides a sink for thermal energy (q˙ 0)? • To solve the heat equation for the temperature distribution in a medium, boundary conditions must be prescribed at the surfaces of the medium. What physical conditions are commonly suitable for this purpose?

䊏

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Problems

References 1. Flik, M. I., B.-I. Choi, and K. E. Goodson, J. Heat Transfer, 114, 666, 1992. 2. Klemens, P. G., “Theory of the Thermal Conductivity of Solids,” in R. P. Tye, Ed., Thermal Conductivity, Vol. 1, Academic Press, London, 1969. 3. Yang, H.-S., G.-R. Bai, L. J. Thompson, and J. A. Eastman, Acta Materialia, 50, 2309, 2002. 4. Chen, G., J. Heat Transfer, 118, 539, 1996. 5. Carey, V. P., G. Chen, C. Grigoropoulos, M. Kaviany, and A. Majumdar, Nano. and Micro. Thermophys. Engng. 12, 1, 2008. 6. Padture, N. P., M. Gell, and E. H. Jordan, Science, 296, 280, 2002. 7. Schelling, P. K., L. Shi, and K. E. Goodson, Mat. Today, 8, 30, 2005. 8. Baxter, J., Z. Bian, G. Chen, D. Danielson, M. S. Dresselhaus, A. G. Federov, T. S. Fisher, C. W. Jones, E. Maginn, W. Kortshagen, A. Manthiram, A. Nozik, D. R. Rolison, T. Sands, L. Shi, D. Sholl, and Y. Wu, Energy and Environ. Sci., 2, 559, 2009. 9. Vincenti, W. G., and C. H. Kruger Jr., Introduction to Physical Gas Dynamics, Wiley, New York, 1986. 10. Zhang, Z. M., Nano/Microscale Heat Transfer, McGrawHill, New York, 2007. 11. McLaughlin, E., “Theory of the Thermal Conductivity of Fluids,” in R. P. Tye, Ed., Thermal Conductivity, Vol. 2, Academic Press, London, 1969. 12. Foust, O. J., Ed., “Sodium Chemistry and Physical Properties,” in Sodium-NaK Engineering Handbook, Vol. 1, Gordon & Breach, New York, 1972.

13. Mallory, J. F., Thermal Insulation, Reinhold Book Corp., New York, 1969. 14. American Society of Heating, Refrigeration and Air Conditioning Engineers, Handbook of Fundamentals, Chapters 23–25 and 31, ASHRAE, New York, 2001. 15. Zeng, S. Q., A. Hunt, and R. Greif, J. Heat Transfer, 117, 1055, 1995. 16. Sengers, J. V., and M. Klein, Eds., The Technical Importance of Accurate Thermophysical Property Information, National Bureau of Standards Technical Note No. 590, 1980. 17. Najjar, M. S., K. J. Bell, and R. N. Maddox, Heat Transfer Eng., 2, 27, 1981. 18. Hanley, H. J. M., and M. E. Baltatu, Mech. Eng., 105, 68, 1983. 19. Touloukian, Y. S., and C. Y. Ho, Eds., Thermophysical Properties of Matter, The TPRC Data Series (13 volumes on thermophysical properties: thermal conductivity, specific heat, thermal radiative, thermal diffusivity, and thermal linear expansion), Plenum Press, New York, 1970 through 1977. 20. Chow, T. S., Phys. Rev. E, 48, 1977, 1993. 21. Keblinski, P., R. Prasher, and J. Eapen, J. Nanopart. Res., 10, 1089, 2008. 22. Hamilton, R. L., and O. K. Crosser, I&EC Fundam. 1, 187, 1962. 23. Cahill, D. G., W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar, H. J. Maris, R. Merlin, and S. R. Phillpot, App. Phys. Rev., 93, 793, 2003.

Problems Fourier’s Law 2.1 Assume steady-state, one-dimensional heat conduction through the axisymmetric shape shown below. T1

T2

2.2 Assume steady-state, one-dimensional conduction in the axisymmetric object below, which is insulated around its perimeter. T1

T2 T1 > T2

T1 > T2 x x

L

Assuming constant properties and no internal heat generation, sketch the temperature distribution on T x coordinates. Briefly explain the shape of your curve.

L

If the properties remain constant and no internal heat generation occurs, sketch the heat flux distribution, qx (x), and the temperature distribution, T(x). Explain the shapes of your curves. How do your curves depend on the thermal conductivity of the material?

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2.3 A hot water pipe with outside radius r1 has a temperature T1. A thick insulation, applied to reduce the heat loss, has an outer radius r2 and temperature T2. On T r coordinates, sketch the temperature distribution in the insulation for one-dimensional, steady-state heat transfer with constant properties. Give a brief explanation, justifying the shape of your curve.

T1, A1

r

x

2.4 A spherical shell with inner radius r1 and outer radius r2 has surface temperatures T1 and T2, respectively, where T1 T2. Sketch the temperature distribution on T r coordinates assuming steady-state, one-dimensional conduction with constant properties. Briefly justify the shape of your curve. 2.5 Assume steady-state, one-dimensional heat conduction through the symmetric shape shown.

T2 < T1 A2 > A1

The thermal conductivity of the solid depends on temperature according to the relation k k0 aT, where a is a positive constant, and the sides of the cone are well insulated. Do the following quantities increase, decrease, or remain the same with increasing x: the heat transfer rate qx , the heat flux qx , the thermal conductivity k, and the temperature gradient dT/dx? 2.8 To determine the effect of the temperature dependence of the thermal conductivity on the temperature distribution in a solid, consider a material for which this dependence may be represented as

qx

k ko aT x

Assuming that there is no internal heat generation, derive an expression for the thermal conductivity k(x) for these conditions: A(x) (1 x), T(x) 300 (1 2x x3), and q 6000 W, where A is in square meters, T in kelvins, and x in meters. 2.6 A composite rod consists of two different materials, A and B, each of length 0.5L. T1

T2

T1 < T 2

A

x

0.5 L

B

L

The thermal conductivity of Material A is half that of Material B, that is, kA/kB 0.5. Sketch the steady-state temperature and heat flux distributions, T(x) and qx , respectively. Assume constant properties and no internal heat generation in either material. 2.7 A solid, truncated cone serves as a support for a system that maintains the top (truncated) face of the cone at a temperature T1, while the base of the cone is at a temperature T2 T1.

where ko is a positive constant and a is a coefficient that may be positive or negative. Sketch the steady-state temperature distribution associated with heat transfer in a plane wall for three cases corresponding to a 0, a 0, and a 0. 2.9 A young engineer is asked to design a thermal protection barrier for a sensitive electronic device that might be exposed to irradiation from a high-powered infrared laser. Having learned as a student that a low thermal conductivity material provides good insulating characteristics, the engineer specifies use of a nanostructured aerogel, characterized by a thermal conductivity of ka 0.005 W/m 䡠 K, for the protective barrier. The engineer’s boss questions the wisdom of selecting the aerogel because it has a low thermal conductivity. Consider the sudden laser irradiation of (a) pure aluminum, (b) glass, and (c) aerogel. The laser provides irradiation of G 10 106 W/m2. The absorptivities of the materials are ␣ 0.2, 0.9, and 0.8 for the aluminum, glass, and aerogel, respectively, and the initial temperature of the barrier is Ti 300 K. Explain why the boss is concerned. Hint: All materials experience thermal expansion (or contraction), and local stresses that develop within a material are, to a first approximation, proportional to the local temperature gradient. 2.10 A one-dimensional plane wall of thickness 2L 100 mm experiences uniform thermal energy generation of q˙ 1000 W/m3 and is convectively cooled at x 50 mm by an ambient fluid characterized by T앝 20 C. If the steady-state temperature distribution

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within the wall is T(x) a(L2 x2) b where a 10 C/m2 and b 30 C, what is the thermal conductivity of the wall? What is the value of the convection heat transfer coefficient, h?

Insulation 1m

k = 10 W/m•K y

2.11 Consider steady-state conditions for one-dimensional conduction in a plane wall having a thermal conductivity k 50 W/m 䡠 K and a thickness L 0.25 m, with no internal heat generation.

T1 L

Determine the heat flux and the unknown quantity for each case and sketch the temperature distribution, indicating the direction of the heat flux. Case

T1(ⴗC)

T2(ⴗC)

1 2 3 4 5

50 30 70

20 10 Ao

160 80 200

T(x)

qx(x)

T(x) T2

T2

T1

T1 x (a)

2.16 Steady-state, one-dimensional conduction occurs in a rod of constant thermal conductivity k and variable crosssectional area Ax(x) Aoeax, where Ao and a are constants. The lateral surface of the rod is well insulated. Ax(x) = Aoeax

2.12 Consider a plane wall 100 mm thick and of thermal conductivity 100 W/m 䡠 K. Steady-state conditions are known to exist with T1 400 K and T2 600 K. Determine the heat flux qx and the temperature gradient dT/dx for the coordinate systems shown.

T2

x A, TA = 0°C

dT/dx (K/m)

40 30

T(x)

2m

2.15 Consider the geometry of Problem 2.14 for the case where the thermal conductivity varies with temperature as k ko aT, where ko 10 W/m 䡠 K, a 103 W/m 䡠 K2, and T is in kelvins. The gradient at surface B is ⭸T/⭸x 30 K/m. What is ⭸T/⭸y at surface A?

T2

x

B, TB = 100°C

x

L

(a) Write an expression for the conduction heat rate, qx(x). Use this expression to determine the temperature distribution T(x) and qualitatively sketch the distribution for T(0) T(L). (b) Now consider conditions for which thermal energy is generated in the rod at a volumetric rate q˙ q˙o exp(ax), where q˙o is a constant. Obtain an expression for qx(x) when the left face (x 0) is well insulated.

T1

x

x (b)

(c)

2.13 A cylinder of radius ro, length L, and thermal conductivity k is immersed in a fluid of convection coefficient h and unknown temperature T앝. At a certain instant the temperature distribution in the cylinder is T(r) a br2, where a and b are constants. Obtain expressions for the heat transfer rate at ro and the fluid temperature. 2.14 In the two-dimensional body illustrated, the gradient at surface A is found to be ⭸T/⭸y 30 K/m. What are ⭸T/⭸y and ⭸T/⭸x at surface B?

Thermophysical Properties 2.17 An apparatus for measuring thermal conductivity employs an electrical heater sandwiched between two identical samples of diameter 30 mm and length 60 mm, which are pressed between plates maintained at a uniform temperature To 77 C by a circulating fluid. A conducting grease is placed between all the surfaces to ensure good thermal contact. Differential thermocouples are imbedded in the samples with a spacing of 15 mm. The lateral sides of the samples are insulated to ensure onedimensional heat transfer through the samples.

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Sample Heater leads

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Introduction to Conduction

Plate, To

(a) Explain why the apparatus of Problem 2.17 cannot be used to obtain an accurate measurement of the aerogel’s thermal conductivity.

∆T1

(b) The engineer designs a new apparatus for which an electric heater of diameter D 150 mm is sandwiched between two thin plates of aluminum. The steady-state temperatures of the 5-mm-thick aluminum plates, T1 and T2, are measured with thermocouples. Aerogel sheets of thickness t 5 mm are placed outside the aluminum plates, while a coolant with an inlet temperature of Tc,i 25 C maintains the exterior surfaces of the aerogel at a low temperature. The circular aerogel sheets are formed so that they encase the heater and aluminum sheets, providing insulation to minimize radial heat losses. At steady state, T1 T2 55 C, and the heater draws 125 mA at 10 V. Determine the value of the aerogel thermal conductivity ka.

Insulation ∆T2

Sample

Plate, To

(a) With two samples of SS316 in the apparatus, the heater draws 0.353 A at 100 V, and the differential thermocouples indicate T1 T2 25.0 C. What is the thermal conductivity of the stainless steel sample material? What is the average temperature of the samples? Compare your result with the thermal conductivity value reported for this material in Table A.1. (b) By mistake, an Armco iron sample is placed in the lower position of the apparatus with one of the SS316 samples from part (a) in the upper portion. For this situation, the heater draws 0.601 A at 100 V, and the differential thermocouples indicate T1 T2 15.0 C. What are the thermal conductivity and average temperature of the Armco iron sample? (c) What is the advantage in constructing the apparatus with two identical samples sandwiching the heater rather than with a single heater–sample combination? When would heat leakage out of the lateral surfaces of the samples become significant? Under what conditions would you expect T1 T2 ? 2.18 An engineer desires to measure the thermal conductivity of an aerogel material. It is expected that the aerogel will have an extremely small thermal conductivity. Heater leads

Tc,i

Coolant in

(c) Calculate the temperature difference across the thickness of the 5-mm-thick aluminum plates. Comment on whether it is important to know the axial locations at which the temperatures of the aluminum plates are measured. (d) If liquid water is used as the coolant with a total flow rate of m˙ 1 kg/min (0.5 kg/min for each of the two streams), calculate the outlet temperature of the water, Tc,o. 2.19 Consider a 300 mm 300 mm window in an aircraft. For a temperature difference of 80 C from the inner to the outer surface of the window, calculate the heat loss through L 10-mm-thick polycarbonate, soda lime glass, and aerogel windows, respectively. The thermal conductivities of the aerogel and polycarbonate are kag 0.014 W/m 䡠 K and kpc 0.21 W/m 䡠 K, respectively. Evaluate the thermal conductivity of the soda lime glass at 300 K. If the aircraft has 130 windows and the cost to heat the cabin air is $1/kW 䡠 h, compare the costs associated with the heat loss through the windows for an 8-hour intercontinental flight. 2.20 Consider a small but known volume of metal that has a large thermal conductivity.

t

Aerogel sample

D

Heater x Aluminum plate T 2 T1

(a) Since the thermal conductivity is large, spatial temperature gradients that develop within the metal in response to mild heating are small. Neglecting spatial temperature gradients, derive a differential equation that could be solved for the temperature of the metal versus time T(t) if the metal is subjected to a fixed surface heat rate q supplied by an electric heater. (b) A student proposes to identify the unknown metal by comparing measured and predicted thermal

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99

Problems

responses. Once a match is made, relevant thermophysical properties might be determined, and, in turn, the metal may be identified by comparison to published property data. Will this approach work? Consider aluminum, gold, and silver as the candidate metals.

Sample 1, D, L, ρ

To(t)

Heater leads Sample 2, D, L, ρ

2.21 Use IHT to perform the following tasks. (a) Graph the thermal conductivity of pure copper, 2024 aluminum, and AISI 302 stainless steel over the temperature range 300 T 600 K. Include all data on a single graph, and comment on the trends you observe. (b) Graph the thermal conductivity of helium and air over the temperature range 300 T 800 K. Include the data on a single graph, and comment on the trends you observe. (c) Graph the kinematic viscosity of engine oil, ethylene glycol, and liquid water over the temperature range 300 T 360 K. Include all data on a single graph, and comment on the trends you observe. (d) Graph the thermal conductivity of a water-Al2O3 nanofluid at T 300 K over the volume fraction range 0 0.08. See Example 2.2. 2.22 Calculate the thermal conductivity of air, hydrogen, and carbon dioxide at 300 K, assuming ideal gas behavior. Compare your calculated values to values from Table A.4. 2.23 A method for determining the thermal conductivity k and the specific heat cp of a material is illustrated in the sketch. Initially the two identical samples of diameter D 60 mm and thickness L 10 mm and the thin heater are at a uniform temperature of Ti 23.00 C, while surrounded by an insulating powder. Suddenly the heater is energized to provide a uniform heat flux qo on each of the sample interfaces, and the heat flux is maintained constant for a period of time, to. A short time after sudden heating is initiated, the temperature at this interface To is related to the heat flux as

冢ct k冣

1/ 2

To(t) Ti 2qo

p

For a particular test run, the electrical heater dissipates 15.0 W for a period of to 120 s, and the temperature at the interface is To(30 s) 24.57 C after 30 s of heating. A long time after the heater is deenergized, t t0, the samples reach the uniform temperature of To(앝) 33.50 C. The density of the sample materials, determined by measurement of volume and mass, is 3965 kg/m3.

Determine the specific heat and thermal conductivity of the test material. By looking at values of the thermophysical properties in Table A.1 or A.2, identify the test sample material. 2.24 Compare and contrast the heat capacity cp of common brick, plain carbon steel, engine oil, water, and soil. Which material provides the greatest amount of thermal energy storage per unit volume? Which material would you expect to have the lowest cost per unit heat capacity? Evaluate properties at 300 K. 2.25 A cylindrical rod of stainless steel is insulated on its exterior surface except for the ends. The steady-state temperature distribution is T(x) a bx/L, where a 305 K and b 10 K. The diameter and length of the rod are D 20 mm and L 100 mm, respectively. Determine the heat flux along the rod, qx . Hint: The mass of the rod is M 0.248 kg.

The Heat Equation 2.26 At a given instant of time, the temperature distribution within an infinite homogeneous body is given by the function T(x, y, z) x2 2y2 z2 xy 2yz Assuming constant properties and no internal heat generation, determine the regions where the temperature changes with time. 2.27 A pan is used to boil water by placing it on a stove, from which heat is transferred at a fixed rate qo. There are two stages to the process. In Stage 1, the water is taken from its initial (room) temperature Ti to the boiling point, as heat is transferred from the pan by natural convection. During this stage, a constant value of the convection coefficient h may be assumed, while the bulk temperature of the water increases with time, T앝 T앝(t). In Stage 2, the water has come to a boil, and its temperature remains at a fixed value, T앝 Tb, as heating continues. Consider a pan bottom of thickness L and diameter D, with a coordinate system corresponding to x 0 and x L for the surfaces in contact with the stove and water, respectively. (a) Write the form of the heat equation and the boundary/ initial conditions that determine the variation of

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Chapter 2

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Introduction to Conduction

temperature with position and time, T(x, t), in the pan bottom during Stage 1. Express your result in terms of the parameters qo, D, L, h, and T앝, as well as appropriate properties of the pan material. (b) During Stage 2, the surface of the pan in contact with the water is at a fixed temperature, T(L, t) TL Tb. Write the form of the heat equation and boundary conditions that determine the temperature distribution T(x) in the pan bottom. Express your result in terms of the parameters qo, D, L, and TL, as well as appropriate properties of the pan material. 2.28 Uniform internal heat generation at q˙ 5 107 W/m3 is occurring in a cylindrical nuclear reactor fuel rod of 50-mm diameter, and under steady-state conditions the temperature distribution is of the form T(r) a br2, where T is in degrees Celsius and r is in meters, while a 800 C and b 4.167 105 C/m2. The fuel rod properties are k 30 W/m 䡠 K, 1100 kg/m3, and cp 800 J/kg K. (a) What is the rate of heat transfer per unit length of the rod at r 0 (the centerline) and at r 25 mm (the surface)? (b) If the reactor power level is suddenly increased to . q2 108 W/m3, what is the initial time rate of temperature change at r 0 and r 25 mm? 2.29 Consider a one-dimensional plane wall with constant properties and uniform internal generation q˙. The left face is insulated, and the right face is held at a uniform temperature.

thickness 50 mm is observed to be T( C) a bx2, where a 200 C, b 2000 C/m2, and x is in meters. (a) What is the heat generation rate q˙ in the wall? (b) Determine the heat fluxes at the two wall faces. In what manner are these heat fluxes related to the heat generation rate? 2.31 The temperature distribution across a wall 0.3 m thick at a certain instant of time is T(x) a bx cx2, where T is in degrees Celsius and x is in meters, a 200 C, b 200 C/m, and c 30 C/m2. The wall has a thermal conductivity of 1 W/m 䡠 K. (a) On a unit surface area basis, determine the rate of heat transfer into and out of the wall and the rate of change of energy stored by the wall. (b) If the cold surface is exposed to a fluid at 100 C, what is the convection coefficient? 2.32 A plane wall of thickness 2L 40 mm and thermal conductivity k 5 W/m 䡠 K experiences uniform volumetric . heat generation at a rate q, while convection heat transfer occurs at both of its surfaces (x L, L), each of which is exposed to a fluid of temperature T앝 20 C. Under steady-state conditions, the temperature distribution in the wall is of the form T(x) a bx cx2 where a 82.0 C, b 210 C/m, c 2 104 C/m2, and x is in meters. The origin of the x-coordinate is at the midplane of the wall. (a) Sketch the temperature distribution and identify significant physical features. (b) What is the volumetric rate of heat generation q˙ in the wall?

ξ Tc

•

q

(c) Determine the surface heat fluxes, qx(L) and qx(L). How are these fluxes related to the heat generation rate? (d) What are the convection coefficients for the surfaces at x L and x L?

x

(e) Obtain an expression for the heat flux distribution qx(x). Is the heat flux zero at any location? Explain any significant features of the distribution.

(a) Using the appropriate form of the heat equation, derive an expression for the x-dependence of the steady-state heat flux q(x).

(f) If the source of the heat generation is suddenly deactivated (q˙ 0), what is the rate of change of energy stored in the wall at this instant?

(b) Using a finite volume spanning the range 0 x , derive an expression for q() and compare the expression to your result for part (a).

(g) What temperature will the wall eventually reach with q˙ 0? How much energy must be removed by the fluid per unit area of the wall (J/m2) to reach this state? The density and specific heat of the wall material are 2600 kg/m3 and 800 J/kg 䡠 K, respectively.

2.30 The steady-state temperature distribution in a onedimensional wall of thermal conductivity 50 W/m 䡠 K and

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Problems

2.33 Temperature distributions within a series of onedimensional plane walls at an initial time, at steady state, and at several intermediate times are as shown. t→∞

t⫽0

t→∞

t⫽0 x

L

x (b)

(a)

x

L

t→∞

t→∞

t⫽0

t⫽0

L

x

(c)

L

(d)

For each case, write the appropriate form of the heat diffusion equation. Also write the equations for the initial condition and the boundary conditions that are applied at x 0 and x L. If volumetric generation occurs, it is uniform throughout the wall. The properties are constant. 2.34 One-dimensional, steady-state conduction with uniform internal energy generation occurs in a plane wall with a thickness of 50 mm and a constant thermal conductivity of 5 W/m 䡠 K. For these conditions, the temperature distribution has the form T(x) a bx cx2. The surface at x 0 has a temperature of T(0) ⬅ To 120 C and experiences convection with a fluid for which T앝 20 C and h 500 W/m2 䡠 K. The surface at x L is well insulated.

(a) Applying an overall energy balance to the wall, calculate the volumetric energy generation rate q˙. (b) Determine the coefficients a, b, and c by applying the boundary conditions to the prescribed temperature distribution. Use the results to calculate and plot the temperature distribution. (c) Consider conditions for which the convection coefficient is halved, but the volumetric energy generation rate remains unchanged. Determine the new values of a, b, and c, and use the results to plot the temperature distribution. Hint: recognize that T(0) is no longer 120 C. (d) Under conditions for which the volumetric energy generation rate is doubled, and the convection coefficient remains unchanged (h 500 W/m2 䡠 K), determine the new values of a, b, and c and plot the corresponding temperature distribution. Referring to the results of parts (b), (c), and (d) as Cases 1, 2, and 3, respectively, compare the temperature distributions for the three cases and discuss the effects of h and q˙ on the distributions. 2.35 Derive the heat diffusion equation, Equation 2.26, for cylindrical coordinates beginning with the differential control volume shown in Figure 2.12. 2.36 Derive the heat diffusion equation, Equation 2.29, for spherical coordinates beginning with the differential control volume shown in Figure 2.13. 2.37 The steady-state temperature distribution in a semitransparent material of thermal conductivity k and thickness L exposed to laser irradiation is of the form T(x)

A ax e Bx C ka2

where A, a, B, and C are known constants. For this situation, radiation absorption in the material is manifested by a distributed heat generation term, q˙(x). Laser irradiation

x To = 120°C

T(x)

L Semitransparent medium, T(x)

T∞ = 20°C h = 500 W/m2•K •

q , k = 5 W/m•K

Fluid

x

L = 50 mm

(a) Obtain expressions for the conduction heat fluxes at the front and rear surfaces. (b) Derive an expression for q˙(x). (c) Derive an expression for the rate at which radiation is absorbed in the entire material, per unit surface

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Chapter 2

䊏

Introduction to Conduction (b) With the temperature at x 0 and the fluid temperature fixed at T(0) 0 C and T앝 20 C, respectively, compute and plot the temperature at x L, T(L), as a function of h for 10 h 100 W/m2 䡠 K. Briefly explain your results.

area. Express your result in terms of the known constants for the temperature distribution, the thermal conductivity of the material, and its thickness. 2.38 One-dimensional, steady-state conduction with no energy generation is occurring in a cylindrical shell of inner radius r1 and outer radius r2. Under what condition is the linear temperature distribution shown possible? T(r) T(r1)

T(r2) r1

r

r2

2.39 One-dimensional, steady-state conduction with no energy generation is occurring in a spherical shell of inner radius r1 and outer radius r2. Under what condition is the linear temperature distribution shown in Problem 2.38 possible? 2.40 The steady-state temperature distribution in a onedimensional wall of thermal conductivity k and thickness L is of the form T ax3 bx2 cx d. Derive expressions for the heat generation rate per unit volume in the wall and the heat fluxes at the two wall faces (x 0, L). 2.41 One-dimensional, steady-state conduction with no energy generation is occurring in a plane wall of constant thermal conductivity. 120

Ambient air T∞, h

GS

Ts

L Coal, k, q•

x

(a) Write the steady-state form of the heat diffusion equation for the layer of coal. Verify that this equation is satisfied by a temperature distribution of the form T(x) Ts

冢

q˙L2 x2 1 2 2k L

冣

From this distribution, what can you say about conditions at the bottom surface (x 0)? Sketch the temperature distribution and label key features.

100 80

T(⬚C)

2.42 A plane layer of coal of thickness L 1 m experiences uniform volumetric generation at a rate of q˙ 20 W/m3 due to slow oxidation of the coal particles. Averaged over a daily period, the top surface of the layer transfers heat by convection to ambient air for which h 5 W/m2 䡠 K and T앝 25 C, while receiving solar irradiation in the amount GS 400 W/m2. Irradiation from the atmosphere may be neglected. The solar absorptivity and emissivity of the surface are each ␣S 0.95.

(b) Obtain an expression for the rate of heat transfer by conduction per unit area at x L. Applying an energy balance to a control surface about the top surface of the layer, obtain an expression for Ts. Evaluate Ts and T(0) for the prescribed conditions.

60 40 20 0

x •

q = 0, k = 4.5 W/m•K T∞ = 20°C h = 30 W/m2•K 0.18 m Air

(a) Is the prescribed temperature distribution possible? Briefly explain your reasoning.

(c) Daily average values of GS and h depend on a number of factors, such as time of year, cloud cover, and wind conditions. For h 5 W/m2 䡠 K, compute and plot TS and T(0) as a function of GS for 50 GS 500 W/m2. For GS 400 W/m2, compute and plot TS and T(0) as a function of h for 5 h 50 W/m2 䡠 K. 2.43 The cylindrical system illustrated has negligible variation of temperature in the r- and z-directions. Assume

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Problems

that r ro ri is small compared to ri, and denote the length in the z-direction, normal to the page, as L. Insulation

φ

2.48 Passage of an electric current through a long conducting rod of radius ri and thermal conductivity kr results in uniform volumetric heating at a rate of q˙. The conducting rod is wrapped in an electrically nonconducting cladding material of outer radius ro and thermal conductivity kc, and convection cooling is provided by an adjoining fluid.

ri r o

T2

T1

(a) Beginning with a properly defined control volume and considering energy generation and storage effects, derive the differential equation that prescribes the variation in temperature with the angular coordinate . Compare your result with Equation 2.26. (b) For steady-state conditions with no internal heat generation and constant properties, determine the temperature distribution T() in terms of the constants T1, T2, ri, and ro. Is this distribution linear in ? (c) For the conditions of part (b) write the expression for the heat rate q. 2.44 Beginning with a differential control volume in the form of a cylindrical shell, derive the heat diffusion equation for a one-dimensional, cylindrical, radial coordinate system with internal heat generation. Compare your result with Equation 2.26. 2.45 Beginning with a differential control volume in the form of a spherical shell, derive the heat diffusion equation for a one-dimensional, spherical, radial coordinate system with internal heat generation. Compare your result with Equation 2.29. 2.46 A steam pipe is wrapped with insulation of inner and outer radii ri and ro, respectively. At a particular instant the temperature distribution in the insulation is known to be of the form

冢冣

r T(r) C1 ln r C2 o Are conditions steady-state or transient? How do the heat flux and heat rate vary with radius? 2.47 For a long circular tube of inner and outer radii r1 and r2, respectively, uniform temperatures T1 and T2 are maintained at the inner and outer surfaces, while thermal energy generation is occurring within the tube wall (r1 r r2). Consider steady-state conditions for which T1 T2. Is it possible to maintain a linear radial temperature distribution in the wall? If so, what special conditions must exist?

Conducting rod, q•, kr

ri

T∞, h ro Cladding, kc

For steady-state conditions, write appropriate forms of the heat equations for the rod and cladding. Express appropriate boundary conditions for the solution of these equations. 2.49 Two-dimensional, steady-state conduction occurs in a hollow cylindrical solid of thermal conductivity k 16 W/m 䡠 K, outer radius r o 1 m and overall length 2zo 5 m, where the origin of the coordinate system is located at the midpoint of the center line. The inner surface of the cylinder is insulated, and the temperature distribution within the cylinder has the form T(r, z) a br2 clnr dz2, where a 20 C, b 150 C/m2, c 12 C, d 300 C/m2 and r and z are in meters. (a) Determine the inner radius ri of the cylinder. (b) Obtain an expression for the volumetric rate of heat generation, q˙(W/m3). (c) Determine the axial distribution of the heat flux at the outer surface, qr(ro, z). What is the heat rate at the outer surface? Is it into or out of the cylinder? (d) Determine the radial distribution of the heat flux at the end faces of the cylinder, qr (r, zo) and qr (r, zo). What are the corresponding heat rates? Are they into or out of the cylinder? (e) Verify that your results are consistent with an overall energy balance on the cylinder. 2.50 An electric cable of radius r1 and thermal conductivity kc is enclosed by an insulating sleeve whose outer surface is of radius r2 and experiences convection heat transfer and radiation exchange with the adjoining air and large surroundings, respectively. When electric

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Introduction to Conduction

current passes through the cable, thermal energy is generated within the cable at a volumetric rate q˙. Tsur

2.51 A spherical shell of inner and outer radii ri and ro, respectively, contains heat-dissipating components, and at a particular instant the temperature distribution in the shell is known to be of the form C1 T(r) r C2

Electrical cable Insulation

Ts, 1 r1

Ambient air T∞, h

r2 Ts, 2

(a) Write the steady-state forms of the heat diffusion equation for the insulation and the cable. Verify that these equations are satisfied by the following temperature distributions: Insulation: T(r) Ts,2 (Ts,1 Ts,2) Cable: T(r) Ts,1

冢

q˙ r 21 r2 1 2 4kc r1

ln(r/r2) ln(r1/r2)

冣

Sketch the temperature distribution, T(r), in the cable and the sleeve, labeling key features. (b) Applying Fourier’s law, show that the rate of conduction heat transfer per unit length through the sleeve may be expressed as qr

Are conditions steady-state or transient? How do the heat flux and heat rate vary with radius? 2.52 A chemically reacting mixture is stored in a thin-walled spherical container of radius r1 200 mm, and the exothermic reaction generates heat at a uniform, but temperaturedependent volumetric rate of q˙ q˙o exp(A/To), where q˙o 5000 W/m3, A 75 K, and To is the mixture temperature in kelvins. The vessel is enclosed by an insulating material of outer radius r2, thermal conductivity k, and emissivity . The outer surface of the insulation experiences convection heat transfer and net radiation exchange with the adjoining air and large surroundings, respectively.

Tsur Chemical reaction, q• (To) Ambient air

T∞, h Insulation, k, ε

2pks(Ts,1 Ts,2) ln(r2/r1)

Applying an energy balance to a control surface placed around the cable, obtain an alternative expression for qr , expressing your result in terms of q˙ and r1. (c) Applying an energy balance to a control surface placed around the outer surface of the sleeve, obtain an expression from which Ts,2 may be determined as a function of q˙, r1, h, T앝, , and Tsur. (d) Consider conditions for which 250 A are passing through a cable having an electric resistance per unit length of Re 0.005 /m, a radius of r1 15 mm, and a thermal conductivity of kc 200 W/m 䡠 K. For ks 15 W/m 䡠 K, r2 15.5 mm, h 25 W/m2 K, 0.9, T앝 25 C, and Tsur 35 C, evaluate the surface temperatures, Ts,1 and Ts,2, as well as the temperature To at the centerline of the cable. (e) With all other conditions remaining the same, compute and plot To, Ts,1, and Ts,2 as a function of r2 for 15.5 r2 20 mm.

r1

r2

(a) Write the steady-state form of the heat diffusion equation for the insulation. Verify that this equation is satisfied by the temperature distribution T(r) Ts,1 (Ts,1 Ts,2)

冤11(r(r /r/r))冥 1

1

2

Sketch the temperature distribution, T(r), labeling key features. (b) Applying Fourier’s law, show that the rate of heat transfer by conduction through the insulation may be expressed as qr

4k(Ts,1 Ts,2) (1/r1) (1/r2)

Applying an energy balance to a control surface about the container, obtain an alternative expression for qr, expressing your result in terms of q˙ and r1.

䊏

105

Problems

(c) With the system operating as described in part (b), the surface x L also experiences a sudden loss of coolant. This dangerous situation goes undetected for 15 min, at which time the power to the heater is deactivated. Assuming no heat losses from the surfaces of the plates, what is the eventual (t l 앝), uniform, steady-state temperature distribution in the plates? Show this distribution as Case 3 on your sketch, and explain its key features. Hint: Apply the conservation of energy requirement on a time-interval basis, Eq. 1.12b, for the initial and final conditions corresponding to Case 2 and Case 3, respectively.

(c) Applying an energy balance to a control surface placed around the outer surface of the insulation, obtain an expression from which Ts,2 may be determined as a function of q˙, r1, h, T앝, , and Tsur. (d) The process engineer wishes to maintain a reactor temperature of To T(r1) 95 C under conditions for which k 0.05 W/m 䡠 K, r2 208 mm, h 5 W/m2 䡠 K, 0.9, T앝 25 C, and Tsur 35 C. What is the actual reactor temperature and the outer surface temperature Ts,2 of the insulation? (e) Compute and plot the variation of Ts,2 with r2 for 201 r2 210 mm. The engineer is concerned about potential burn injuries to personnel who may come into contact with the exposed surface of the insulation. Is increasing the insulation thickness a practical solution to maintaining Ts,2 45 C? What other parameter could be varied to reduce Ts,2?

Graphical Representations 2.53 A thin electrical heater dissipating 4000 W/m2 is sandwiched between two 25-mm-thick plates whose exposed surfaces experience convection with a fluid for which T앝 20 C and h 400 W/m2 䡠 K. The thermophysical properties of the plate material are 2500 kg/m3, c 700 J/kg 䡠 K, and k 5 W/m 䡠 K. Electric heater, q"o

(d) On T t coordinates, sketch the temperature history at the plate locations x 0, L during the transient period between the distributions for Cases 2 and 3. Where and when will the temperature in the system achieve a maximum value? 2.54 The one-dimensional system of mass M with constant properties and no internal heat generation shown in the figure is initially at a uniform temperature Ti. The electrical heater is suddenly energized, providing a uniform heat flux qo at the surface x 0. The boundaries at x L and elsewhere are perfectly insulated. Insulation

L x

System, mass M Electrical heater

ρ , c, k Fluid

Fluid

T∞, h

T∞, h

–L

0

+L

x

(a) On T x coordinates, sketch the steady-state temperature distribution for L x L. Calculate values of the temperatures at the surfaces, x L, and the midpoint, x 0. Label this distribution as Case 1, and explain its salient features. (b) Consider conditions for which there is a loss of coolant and existence of a nearly adiabatic condition on the x L surface. On the T x coordinates used for part (a), sketch the corresponding steady-state temperature distribution and indicate the temperatures at x 0, L. Label the distribution as Case 2, and explain its key features.

(a) Write the differential equation, and identify the boundary and initial conditions that could be used to determine the temperature as a function of position and time in the system. (b) On T x coordinates, sketch the temperature distributions for the initial condition (t 0) and for several times after the heater is energized. Will a steady-state temperature distribution ever be reached? (c) On qx t coordinates, sketch the heat flux qx (x, t) at the planes x 0, x L/2, and x L as a function of time. (d) After a period of time te has elapsed, the heater power is switched off. Assuming that the insulation is perfect, the system will eventually reach a final uniform temperature Tf. Derive an expression that can be used to determine Tf as a function of the parameters qo , te, Ti, and the system characteristics M, cp, and As (the heater surface area). 2.55 Consider a one-dimensional plane wall of thickness 2L. The surface at x L is subjected to convective conditions characterized by T앝,1, h1, while the surface

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Introduction to Conduction

at x L is subjected to conditions T앝,2, h2. The initial temperature of the wall is To (T앝,1 T앝,2)/2 where T앝,1 T앝,2.

To h1, T∞,1

h2, T∞,2

T∞,1 T∞,2 2L

x

(a) Write the differential equation, and identify the boundary and initial conditions that could be used to determine the temperature distribution T(x, t) as a function of position and time. (b) On T x coordinates, sketch the temperature distributions for the initial condition, the steady-state condition, and for two intermediate times for the case h1 h2. (c) On qx t coordinates, sketch the heat flux qx (x, t) at the planes x 0, L, and L. (d) The value of h1 is now doubled with all other conditions being identical as in parts (a) through (c). On T x coordinates drawn to the same scale as used in part (b), sketch the temperature distributions for the initial condition, the steady-state condition, and for two intermediate times. Compare the sketch to that of part (b). (e) Using the doubled value of h1, sketch the heat flux qx(x, t) at the planes x 0, L, and L on the same plot you prepared for part (c). Compare the two responses. 2.56 A large plate of thickness 2L is at a uniform temperature of Ti 200 C, when it is suddenly quenched by dipping it in a liquid bath of temperature T앝 20 C. Heat transfer to the liquid is characterized by the convection coefficient h. (a) If x 0 corresponds to the midplane of the wall, on T x coordinates, sketch the temperature distributions for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and two intermediate times. (b) On qx t coordinates, sketch the variation with time of the heat flux at x L.

(c) If h 100 W/m2 䡠 K, what is the heat flux at x L and t 0? If the wall has a thermal conductivity of k 50 W/m 䡠 K what is the corresponding temperature gradient at x L? (d) Consider a plate of thickness 2L 20 mm with a density of 2770 kg/m3 and a specific heat cp 875 J/kg 䡠 K. By performing an energy balance on the plate, determine the amount of energy per unit surface area of the plate (J/m2) that is transferred to the bath over the time required to reach steady-state conditions. (e) From other considerations, it is known that, during the quenching process, the heat flux at x L and x L decays exponentially with time according to the relation, qx A exp(Bt), where t is in seconds, A 1.80 104 W/m2, and B 4.126 103 s1. Use this information to determine the energy per unit surface area of the plate that is transferred to the fluid during the quenching process. 2.57 The plane wall with constant properties and no internal heat generation shown in the figure is initially at a uniform temperature Ti. Suddenly the surface at x L is heated by a fluid at T앝 having a convection heat transfer coefficient h. The boundary at x 0 is perfectly insulated.

T∞, h Insulation

x

L

(a) Write the differential equation, and identify the boundary and initial conditions that could be used to determine the temperature as a function of position and time in the wall. (b) On T x coordinates, sketch the temperature distributions for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and two intermediate times. (c) On qx t coordinates, sketch the heat flux at the locations x 0, x L. That is, show qualitatively how qx (0, t) and qx (L, t) vary with time. (d) Write an expression for the total energy transferred to the wall per unit volume of the wall (J/m3). 2.58 Consider the steady-state temperature distributions within a composite wall composed of Material A and Material B for the two cases shown. There is no

䊏

107

Problems

internal generation, and the conduction process is onedimensional. T(x)

T(x)

(b) On qx x coordinates, sketch the heat flux corresponding to the four temperature distributions of part (a). (c) On qx t coordinates, sketch the heat flux at the locations x 0 and x L. That is, show qualitatively how qx (0, t) and qx (L, t) vary with time. (d) Derive an expression for the steady-state temperature at the heater surface, T(0, 앝), in terms of qo , T앝, k, h, and L.

LA

LB

kA

LA kB

LB

kA

x

kB x

Case 1

Case 2

Answer the following questions for each case. Which material has the higher thermal conductivity? Does the thermal conductivity vary significantly with temperature? If so, how? Describe the heat flux distribution qx(x) through the composite wall. If the thickness and thermal conductivity of each material were both doubled and the boundary temperatures remained the same, what would be the effect on the heat flux distribution? Case 1. Linear temperature distributions exist in both materials, as shown. Case 2. Nonlinear temperature distributions exist in both materials, as shown. 2.59 A plane wall has constant properties, no internal heat generation, and is initially at a uniform temperature Ti. Suddenly, the surface at x L is heated by a fluid at T앝 having a convection coefficient h. At the same instant, the electrical heater is energized, providing a constant heat flux qo at x 0.

T∞, h

Heater

2.60 A plane wall with constant properties is initially at a uniform temperature To. Suddenly, the surface at x L is exposed to a convection process with a fluid at T앝 (To) having a convection coefficient h. Also, suddenly the wall experiences a uniform internal volumetric heating q˙ that is sufficiently large to induce a maximum steadystate temperature within the wall, which exceeds that of the fluid. The boundary at x 0 remains at To.

k, q• (t ≥ 0) To T∞, h

L x

(a) On T x coordinates, sketch the temperature distributions for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and for two intermediate times. Show also the distribution for the special condition when there is no heat flow at the x L boundary. (b) On qx t coordinates, sketch the heat flux for the locations x 0 and x L, that is, qx(0, t) and qx(L, t), respectively. 2.61 Consider the conditions associated with Problem 2.60, but now with a convection process for which T앝 To.

Insulation

L x

(a) On T x coordinates, sketch the temperature distributions for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and for two intermediate times.

(a) On T x coordinates, sketch the temperature distributions for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and for two intermediate times. Identify key features of the distributions, especially the location of the maximum temperature and the temperature gradient at x L. (b) On qx t coordinates, sketch the heat flux for the locations x 0 and x L, that is, qx(0, t) and qx(L, t), respectively. Identify key features of the flux histories.

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䊏

2.62 Consider the steady-state temperature distribution within a composite wall composed of Materials A and B.

determine a relationship between the temperature gradient dT/dr and the local radius r, for r1 r r2. (c) On T r coordinates, sketch the temperature distribution over the range 0 r r2.

T(x)

LA

LB

kA

kB x

The conduction process is one-dimensional. Within which material does uniform volumetric generation occur? What is the boundary condition at x LA? How would the temperature distribution change if the thermal conductivity of Material A were doubled? How would the temperature distribution change if the thermal conductivity of Material B were doubled? Does a contact resistance exist at the interface between the two materials? Sketch the heat flux distribution qx(x) through the composite wall. 2.63 A spherical particle of radius r1 experiences uniform ther. mal generation at a rate of q. The particle is encapsulated by a spherical shell of outside radius r2 that is cooled by ambient air. The thermal conductivities of the particle and shell are k1 and k2, respectively, where k1 2k2.

2.64 A long cylindrical rod, initially at a uniform temperature Ti, is suddenly immersed in a large container of liquid at T앝 Ti. Sketch the temperature distribution within the rod, T(r), at the initial time, at steady state, and at two intermediate times. On the same graph, carefully sketch the temperature distributions that would occur at the same times within a second rod that is the same size as the first rod. The densities and specific heats of the two rods are identical, but the thermal conductivity of the second rod is very large. Which rod will approach steady-state conditions sooner? Write the appropriate boundary conditions that would be applied at r 0 and r D/2 for either rod. 2.65 A plane wall of thickness L 0.1 m experiences uniform . volumetric heating at a rate q. One surface of the wall (x 0) is insulated, and the other surface is exposed to a fluid at T앝 20 C, with convection heat transfer characterized by h 1000 W/m2 䡠 K. Initially, the temperature distribution in the wall is T(x, 0) a bx2, where a 300 C, b 1.0 104 C/m2, and x is in meters. Suddenly, the volumetric heat generation is deactivated . (q 0 for t 0), while convection heat transfer continues to occur at x L. The properties of the wall are 7000 kg/m3, cp 450 J/kg 䡠 K, and k 90 W/m 䡠 K. •

k, ρ , cp, q (t < – 0)

Chemical reaction •

q T∞, h

x r1

Ambient air T∞, h

r2 Control volume B Control volume A

(a) By applying the conservation of energy principle to spherical control volume A, which is placed at an arbitrary location within the sphere, determine a relationship between the temperature gradient dT/dr and the local radius r, for 0 r r1. (b) By applying the conservation of energy principle to spherical control volume B, which is placed at an arbitrary location within the spherical shell,

L

(a) Determine the magnitude of the volumetric energy . generation rate q associated with the initial condition (t 0). (b) On T x coordinates, sketch the temperature distribution for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and two intermediate conditions. (c) On qx t coordinates, sketch the variation with time of the heat flux at the boundary exposed to the convection process, qx(L, t). Calculate the corresponding value of the heat flux at t 0, qx(L, 0). (d) Calculate the amount of energy removed from the wall per unit area (J/m2) by the fluid stream

䊏

109

Problems

as the wall cools from its initial to steady-state condition. 2.66 A plane wall that is insulated on one side (x 0) is initially at a uniform temperature Ti, when its exposed surface at x L is suddenly raised to a temperature Ts. (a) Verify that the following equation satisfies the heat equation and boundary conditions:

冢

冣 冢 冣

T(x, t) Ts 2 ␣t x C1 exp cos Ti Ts 4 L2 2L

where C1 is a constant and ␣ is the thermal diffusivity. (b) Obtain expressions for the heat flux at x 0 and x L. (c) Sketch the temperature distribution T(x) at t 0, at t l 앝, and at an intermediate time. Sketch the variation with time of the heat flux at x L, qL(t). (d) What effect does ␣ have on the thermal response of the material to a change in surface temperature? 2.67 A composite one-dimensional plane wall is of overall thickness 2L. Material A spans the domain L x 0 and experiences an exothermic chemical reaction leading . to a uniform volumetric generation rate of qA. Material B spans the domain 0 x L and undergoes an endothermic chemical reaction corresponding to a uniform . . volumetric generation rate of qB qA. The surfaces at x L are insulated. Sketch the steady-state temperature and heat flux distributions T(x) and qx(x), respectively, over the domain L x L for kA kB, kA 0.5kB, and kA 2kB. Point out the important features of the distributions you have drawn. If q˙B 2q˙A, can you sketch the steady-state temperature distribution? 2.68 Typically, air is heated in a hair dryer by blowing it across a coiled wire through which an electric current is passed. Thermal energy is generated by electric resistance heating within the wire and is transferred by convection from the surface of the wire to the air. Consider conditions for which the wire is initially at room temperature, Ti, and resistance heating is concurrently initiated with airflow at t 0.

Coiled wire (ro, L, k, ρ , cp)

•

q

Airflow

Air

Pelec

T∞, h ro

r

(a) For a wire radius ro, an air temperature T앝, and a convection coefficient h, write the form of the heat equation and the boundary/initial conditions that govern the transient thermal response, T(r, t), of the wire. (b) If the length and radius of the wire are 500 mm and 1 mm, respectively, what is the volumetric rate of thermal energy generation for a power consumption of Pelec 500 W? What is the convection heat flux under steady-state conditions? (c) On T r coordinates, sketch the temperature distributions for the following conditions: initial condition (t 0), steady-state condition (t l 앝), and for two intermediate times. (d) On qr t coordinates, sketch the variation of the heat flux with time for locations at r 0 and r ro. 2.69 The steady-state temperature distribution in a composite plane wall of three different materials, each of constant thermal conductivity, is shown. 1

2

3

4

T A

B

C

q"2

q"3

q"4 x

(a) Comment on the relative magnitudes of q2 and q3 , and of q3 and q4 . (b) Comment on the relative magnitudes of kA and kB, and of kB and kC. (c) Sketch the heat flux as a function of x.

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C H A P T E R

One-Dimensional, Steady-State Conduction

3

112

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

I

n this chapter we treat situations for which heat is transferred by diffusion under onedimensional, steady-state conditions. The term one-dimensional refers to the fact that only one coordinate is needed to describe the spatial variation of the dependent variables. Hence, in a one-dimensional system, temperature gradients exist along only a single coordinate direction, and heat transfer occurs exclusively in that direction. The system is characterized by steady-state conditions if the temperature at each point is independent of time. Despite their inherent simplicity, one-dimensional, steady-state models may be used to accurately represent numerous engineering systems. We begin our consideration of one-dimensional, steady-state conduction by discussing heat transfer with no internal generation of thermal energy (Sections 3.1 through 3.4). The objective is to determine expressions for the temperature distribution and heat transfer rate in common (planar, cylindrical, and spherical) geometries. For such geometries, an additional objective is to introduce the concept of thermal resistance and to show how thermal circuits may be used to model heat flow, much as electrical circuits are used for current flow. The effect of internal heat generation is treated in Section 3.5, and again our objective is to obtain expressions for determining temperature distributions and heat transfer rates. In Section 3.6, we consider the special case of one-dimensional, steady-state conduction for extended surfaces. In their most common form, these surfaces are termed fins and are used to enhance heat transfer by convection to an adjoining fluid. In addition to determining related temperature distributions and heat rates, our objective is to introduce performance parameters that may be used to determine their efficacy. Finally, in Sections 3.7 through 3.9 we apply heat transfer and thermal resistance concepts to the human body, including the effects of metabolic heat generation and perfusion; to thermoelectric power generation driven by the Seebeck effect; and to micro- and nanoscale conduction in thin gas layers and thin solid films.

3.1 The Plane Wall For one-dimensional conduction in a plane wall, temperature is a function of the x-coordinate only and heat is transferred exclusively in this direction. In Figure 3.1a, a plane wall separates two fluids of different temperatures. Heat transfer occurs by convection from the hot fluid at T앝,1 to one surface of the wall at Ts,1, by conduction through the wall, and by convection from the other surface of the wall at Ts,2 to the cold fluid at T앝,2. We begin by considering conditions within the wall. We first determine the temperature distribution, from which we can then obtain the conduction heat transfer rate.

3.1.1

Temperature Distribution

The temperature distribution in the wall can be determined by solving the heat equation with the proper boundary conditions. For steady-state conditions with no distributed source or sink of energy within the wall, the appropriate form of the heat equation is Equation 2.23

冢 冣

d k dT 0 dx dx

(3.1)

3.1

䊏

113

The Plane Wall

T∞,1 Ts,1

Ts,2 T∞,2 qx Hot fluid

T∞,1, h1 x

x=L Cold fluid

T∞,2, h2

(a)

T∞,1 qx

Ts,1

T∞,2

Ts,2

1 ____

L ____

h1A

kA

FIGURE 3.1 Heat transfer through a plane wall. (a) Temperature distribution. (b) Equivalent thermal circuit.

1 ____

h2A

(b)

Hence, from Equation 2.2, it follows that, for one-dimensional, steady-state conduction in a plane wall with no heat generation, the heat flux is a constant, independent of x. If the thermal conductivity of the wall material is assumed to be constant, the equation may be integrated twice to obtain the general solution T(x) C1x C2

(3.2)

To obtain the constants of integration, C1 and C2, boundary conditions must be introduced. We choose to apply conditions of the first kind at x 0 and x L, in which case T(0) Ts,1

and

T(L) Ts,2

Applying the condition at x 0 to the general solution, it follows that Ts,1 C2 Similarly, at x L, Ts,2 C1L C2 C1L Ts,1 in which case Ts,2 Ts,1 C1 L Substituting into the general solution, the temperature distribution is then T(x) (Ts,2 Ts,1) x Ts,1 L

(3.3)

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Chapter 3

䊏

One-Dimensional, Steady-State Conduction

From this result it is evident that, for one-dimensional, steady-state conduction in a plane wall with no heat generation and constant thermal conductivity, the temperature varies linearly with x. Now that we have the temperature distribution, we may use Fourier’s law, Equation 2.1, to determine the conduction heat transfer rate. That is, qx kA dT kA (Ts,1 Ts,2) dx L

(3.4)

Note that A is the area of the wall normal to the direction of heat transfer and, for the plane wall, it is a constant independent of x. The heat flux is then qx

qx k (T Ts,2) A L s,1

(3.5)

Equations 3.4 and 3.5 indicate that both the heat rate qx and heat flux qx are constants, independent of x. In the foregoing paragraphs we have used the standard approach to solving conduction problems. That is, the general solution for the temperature distribution is first obtained by solving the appropriate form of the heat equation. The boundary conditions are then applied to obtain the particular solution, which is used with Fourier’s law to determine the heat transfer rate. Note that we have opted to prescribe surface temperatures at x 0 and x L as boundary conditions, even though it is the fluid temperatures, not the surface temperatures, that are typically known. However, since adjoining fluid and surface temperatures are easily related through a surface energy balance (see Section 1.3.1), it is a simple matter to express Equations 3.3 through 3.5 in terms of fluid, rather than surface, temperatures. Alternatively, equivalent results could be obtained directly by using the surface energy balances as boundary conditions of the third kind in evaluating the constants of Equation 3.2 (see Problem 3.1).

3.1.2

Thermal Resistance

At this point we note that, for the special case of one-dimensional heat transfer with no internal energy generation and with constant properties, a very important concept is suggested by Equation 3.4. In particular, an analogy exists between the diffusion of heat and electrical charge. Just as an electrical resistance is associated with the conduction of electricity, a thermal resistance may be associated with the conduction of heat. Defining resistance as the ratio of a driving potential to the corresponding transfer rate, it follows from Equation 3.4 that the thermal resistance for conduction in a plane wall is Rt,cond ⬅

Ts,1 Ts,2 L qx kA

(3.6)

Similarly, for electrical conduction in the same system, Ohm’s law provides an electrical resistance of the form Re

Es,1 Es,2 L I A

(3.7)

3.1

䊏

115

The Plane Wall

The analogy between Equations 3.6 and 3.7 is obvious. A thermal resistance may also be associated with heat transfer by convection at a surface. From Newton’s law of cooling, q hA(Ts T앝)

(3.8)

The thermal resistance for convection is then Rt,conv ⬅

Ts T앝 1 q hA

(3.9)

Circuit representations provide a useful tool for both conceptualizing and quantifying heat transfer problems. The equivalent thermal circuit for the plane wall with convection surface conditions is shown in Figure 3.1b. The heat transfer rate may be determined from separate consideration of each element in the network. Since qx is constant throughout the network, it follows that qx

T앝,1 Ts,1 Ts,1 Ts,2 Ts,2 T앝,2 1/h1A L/kA 1/h2A

(3.10)

In terms of the overall temperature difference, T앝,1 T앝,2, and the total thermal resistance, Rtot, the heat transfer rate may also be expressed as qx

T앝,1 T앝,2 Rtot

(3.11)

Because the conduction and convection resistances are in series and may be summed, it follows that Rtot 1 L 1 h1A kA h2A

(3.12)

Radiation exchange between the surface and surroundings may also be important if the convection heat transfer coefficient is small (as it often is for natural convection in a gas). A thermal resistance for radiation may be defined by reference to Equation 1.8: Rt,rad

Ts Tsur 1 qrad hr A

(3.13)

For radiation between a surface and large surroundings, hr is determined from Equation 1.9. Surface radiation and convection resistances act in parallel, and if T앝 Tsur, they may be combined to obtain a single, effective surface resistance.

3.1.3

The Composite Wall

Equivalent thermal circuits may also be used for more complex systems, such as composite walls. Such walls may involve any number of series and parallel thermal resistances due to layers of different materials. Consider the series composite wall of Figure 3.2. The onedimensional heat transfer rate for this system may be expressed as qx

T앝,1 T앝,4 Rt

(3.14)

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Chapter 3

䊏

One-Dimensional, Steady-State Conduction

T∞,1 Ts,1

T2

T3 Ts,4

Hot fluid

LA

LB

LC

kA

kB

kC

A

B

C

T∞,4

T∞,1, h1

x

qx

Cold fluid

1 ____

LA ____

LB ____

LC ____

1 ____

h1A

kA A

kB A

kC A

h4 A

T∞,1

Ts,1

T2

T3

Ts,4

T∞,4, h4

T∞,4

FIGURE 3.2 Equivalent thermal circuit for a series composite wall.

where T앝,1 T앝,4 is the overall temperature difference, and the summation includes all thermal resistances. Hence qx

T앝,1 T앝,4 [(1/h1A) (LA /kAA) (LB /kBA) (LC /kC A) (1/h4A)]

(3.15)

Alternatively, the heat transfer rate can be related to the temperature difference and resistance associated with each element. For example, qx

T앝,1 Ts,1 Ts,1 T2 T T3 … 2 (1/h1A) (LA/kAA) (LB /kBA)

(3.16)

With composite systems, it is often convenient to work with an overall heat transfer coefficient U, which is defined by an expression analogous to Newton’s law of cooling. Accordingly, qx ⬅ UA T

(3.17)

where T is the overall temperature difference. The overall heat transfer coefficient is related to the total thermal resistance, and from Equations 3.14 and 3.17 we see that UA 1/Rtot. Hence, for the composite wall of Figure 3.2, 1 U 1 Rtot A [(1/h1) (LA /kA) (LB /kB) (LC /kC) (1/h4)]

(3.18)

In general, we may write Rtot

1 兺R T q UA t

(3.19)

3.1

䊏

117

The Plane Wall

LE

LF = LG

Area, A

LH

kF

F

T1

T2 kE

kG

kH

E

G

H

x LF ________ kF(A/2) LE ____

LH ____ kHA

kEA qx

LG ________ kG(A/2)

T1

T2

(a)

qx

LE ________ kE(A/2)

LF ________ kF(A/2)

LH ________ kH(A/2)

T1 L E ________ kE(A/2)

LG ________ kG(A/2)

LH ________ kH(A/2)

T2

FIGURE 3.3 Equivalent thermal circuits for a series–parallel composite wall.

(b)

Composite walls may also be characterized by series–parallel configurations, such as that shown in Figure 3.3. Although the heat flow is now multidimensional, it is often reasonable to assume one-dimensional conditions. Subject to this assumption, two different thermal circuits may be used. For case (a) it is presumed that surfaces normal to the x-direction are isothermal, whereas for case (b) it is assumed that surfaces parallel to the x-direction are adiabatic. Different results are obtained for Rtot, and the corresponding values of q bracket the actual heat transfer rate. These differences increase with increasing 冨kF kG冨, as multidimensional effects become more significant.

3.1.4

Contact Resistance

Although neglected until now, it is important to recognize that, in composite systems, the temperature drop across the interface between materials may be appreciable. This temperature change is attributed to what is known as the thermal contact resistance, Rt,c. The effect is shown in Figure 3.4, and for a unit area of the interface, the resistance is defined as Rt,c

TA TB qx

(3.20)

The existence of a finite contact resistance is due principally to surface roughness effects. Contact spots are interspersed with gaps that are, in most instances, air filled. Heat transfer is therefore due to conduction across the actual contact area and to conduction and/or radiation across the gaps. The contact resistance may be viewed as two parallel resistances: that due to

118

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

q"x qc"ontact q"x

TA ∆T

A

B

TB

T

qg"ap A

B

FIGURE 3.4 Temperature drop due to thermal contact resistance.

x

the contact spots and that due to the gaps. The contact area is typically small, and, especially for rough surfaces, the major contribution to the resistance is made by the gaps. For solids whose thermal conductivities exceed that of the interfacial fluid, the contact resistance may be reduced by increasing the area of the contact spots. Such an increase may be effected by increasing the joint pressure and/or by reducing the roughness of the mating surfaces. The contact resistance may also be reduced by selecting an interfacial fluid of large thermal conductivity. In this respect, no fluid (an evacuated interface) eliminates conduction across the gap, thereby increasing the contact resistance. Likewise, if the characteristic gap width L becomes small (as, for example, in the case of very smooth surfaces in contact), L/mfp can approach values for which the thermal conductivity of the interfacial gas is reduced by microscale effects, as discussed in Section 2.2. Although theories have been developed for predicting Rt,c, the most reliable results are those that have been obtained experimentally. The effect of loading on metallic interfaces can be seen in Table 3.1a, which presents an approximate range of thermal resistances under vacuum conditions. The effect of interfacial fluid on the thermal resistance of an aluminum interface is shown in Table 3.1b. Contrary to the results of Table 3.1, many applications involve contact between dissimilar solids and/or a wide range of possible interstitial (filler) materials (Table 3.2). Any interstitial substance that fills the gap between contacting surfaces and whose thermal conductivity exceeds that of air will decrease the contact resistance. Two classes of materials that are well suited for this purpose are soft metals and thermal greases. The metals, which include

TABLE 3.1 Thermal contact resistance for (a) metallic interfaces under vacuum conditions and (b) aluminum interface (10-m surface roughness, 105 N/m2) with different interfacial fluids [1] Thermal Resistance, Rⴖt, c ⫻ 104 (m2 䡠 K/W) (a) Vacuum Interface Contact pressure 100 kN/m2 Stainless steel 6–25 Copper 1–10 Magnesium 1.5–3.5 Aluminum 1.5–5.0

2

10,000 kN/m 0.7–4.0 0.1–0.5 0.2–0.4 0.2–0.4

(b) Interfacial Fluid Air 2.75 Helium 1.05 Hydrogen 0.720 Silicone oil 0.525 Glycerine 0.265

3.1

䊏

119

The Plane Wall

TABLE 3.2

Thermal resistance of representative solid/solid interfaces

Interface Silicon chip/lapped aluminum in air (27–500 kN/m2) Aluminum/aluminum with indium foil filler (⬃100 kN/m2) Stainless/stainless with indium foil filler (⬃3500 kN/m2) Aluminum/aluminum with metallic (Pb) coating Aluminum/aluminum with Dow Corning 340 grease (⬃100 kN/m2) Stainless/stainless with Dow Corning 340 grease (⬃3500 kN/m2) Silicon chip/aluminum with 0.02-mm epoxy Brass/brass with 15-m tin solder

Rⴖt,c ⫻ 104 (m2 䡠 K/W)

Source

0.3–0.6

[2]

⬃0.07

[1, 3]

⬃0.04

[1, 3]

0.01–0.1

[4]

⬃0.07

[1, 3]

⬃0.04

[1, 3]

0.2–0.9

[5]

0.025–0.14

[6]

indium, lead, tin, and silver, may be inserted as a thin foil or applied as a thin coating to one of the parent materials. Silicon-based thermal greases are attractive on the basis of their ability to completely fill the interstices with a material whose thermal conductivity is as much as 50 times that of air. Unlike the foregoing interfaces, which are not permanent, many interfaces involve permanently bonded joints. The joint could be formed from an epoxy, a soft solder rich in lead, or a hard solder such as a gold/tin alloy. Due to interface resistances between the parent and bonding materials, the actual thermal resistance of the joint exceeds the theoretical value (L/k) computed from the thickness L and thermal conductivity k of the joint material. The thermal resistance of epoxied and soldered joints is also adversely affected by voids and cracks, which may form during manufacture or as a result of thermal cycling during normal operation. Comprehensive reviews of thermal contact resistance results and models are provided by Snaith et al. [3], Madhusudana and Fletcher [7], and Yovanovich [8].

3.1.5

Porous Media

In many applications, heat transfer occurs within porous media that are combinations of a stationary solid and a fluid. When the fluid is either a gas or a liquid, the resulting porous medium is said to be saturated. In contrast, all three phases coexist in an unsaturated porous medium. Examples of porous media include beds of powder with a fluid occupying the interstitial regions between individual granules, as well as the insulation systems and nanofluids of Section 2.2.1. A saturated porous medium that consists of a stationary solid phase through which a fluid flows is referred to as a packed bed and is discussed in Section 7.8. Consider a saturated porous medium that is subjected to surface temperatures T1 at x 0 and T2 at x L, as shown in Figure 3.5a. After steady-state conditions are reached and if T1 T2, the heat rate may be expressed as qx

keff A (T1 T2) L

(3.21)

120

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

where keff is an effective thermal conductivity. Equation 3.21 is valid if fluid motion, as well as radiation heat transfer within the medium, are negligible. The effective thermal conductivity varies with the porosity or void fraction of the medium which is defined as the volume of fluid relative to the total volume (solid and fluid). In addition, keff depends on the thermal conductivities of each of the phases and, in this discussion, it is assumed that ks kf. The detailed solid phase geometry, for example the size distribution and packing arrangement of individual powder particles, also affects the value of keff. Contact resistances that might evolve at interfaces between adjacent solid particles can impact the value of keff. As discussed in Section 2.2.1, nanoscale phenomena might also influence the effective thermal conductivity. Hence, prediction of keff can be difficult and, in general, requires detailed knowledge of parameters that might not be readily available. Despite the complexity of the situation, the value of the effective thermal conductivity may be bracketed by considering the composite walls of Figures 3.5b and 3.5c. In Figure 3.5b, the medium is modeled as an equivalent, series composite wall consisting of a fluid region of length L and a solid region of length (1 – )L. Applying Equations 3.17 and 3.18 to this model for which there is no convection (h1 h2 0) and only two conduction terms, it follows that qx

A T (1 )L /ks L /kf

(3.22)

Equating this result to Equation 3.21, we then obtain keff,min

1 (1 )/ks /kf

(3.23)

Alternatively, the medium of Figure 3.5a could be described by the equivalent, parallel composite wall consisting of a fluid region of width w and a solid region of width (1 – )w, as shown in Figure 3.5c. Combining Equation 3.21 with an expression for the equivalent resistance of two resistors in parallel gives keff,max kf (1 )ks

(3.24)

L

(1 − )L

L

L

Area A , ks, kf, keff

T1

Area A ks

T1

kf

Area A T1

ks

qx.s

A w

x

x

T1

L

T2

qx

T1

keff A (a)

qx.f

kf

x

q

T2

qx

T2

qx

(b)

(1 − )L ks A

(1 − )w

T2

w

x

L kf A

T2

qx T1

(c)

L ks(1 − )A

T2

L kf A

FIGURE 3.5 A porous medium. (a) The medium and its properties. (b) Series thermal resistance representation. (c) Parallel resistance representation.

3.1

䊏

121

The Plane Wall

While Equations 3.23 and 3.24 provide the minimum and maximum possible values of keff, more accurate expressions have been derived for specific composite systems within which nanoscale effects are negligible. Maxwell [9] derived an expression for the effective electrical conductivity of a solid matrix interspersed with uniformly distributed, noncontacting spherical inclusions. Noting the analogy between Equations 3.6 and 3.7, Maxwell’s result may be used to determine the effective thermal conductivity of a saturated porous medium consisting of an interconnected solid phase within which a dilute distribution of spherical fluid regions exists, resulting in an expression of the form [10] keff

kf 2ks 2(ks kf)

冤 k 2k (k k ) 冥k

s

f

s

s

(3.25)

f

Equation 3.25 is valid for relatively small porosities ( 0.25) as shown schematically in Figure 3.5a [11]. It is equivalent to the expression introduced in Example 2.2 for a fluid that contains a dilute mixture of solid particles, but with reversal of the fluid and solid. When analyzing conduction within porous media, it is important to consider the potential directional dependence of the effective thermal conductivity. For example, the media represented in Figure 3.5b or Figure 3.5c would not be characterized by isotropic properties, since the effective thermal conductivity in the x-direction is clearly different from values of keff in the vertical direction. Hence, although Equations 3.23 and 3.24 can be used to bracket the actual value of the effective thermal conductivity, they will generally overpredict the possible range of keff for isotropic media. For isotropic media, expressions have been developed to determine the minimum and maximum possible effective thermal conductivities based solely on knowledge of the porosity and the thermal conductivities of the solid and fluid. Specifically, the maximum possible value of keff in an isotropic porous medium is given by Equation 3.25, which corresponds to an interconnected, high thermal conductivity solid phase. The minimum possible value of keff for an isotropic medium corresponds to the case where the fluid phase forms long, randomly oriented fingers within the medium [12]. Additional information regarding conduction in saturated porous media is available [13].

EXAMPLE 3.1 In Example 1.7, we calculated the heat loss rate from a human body in air and water environments. Now we consider the same conditions except that the surroundings (air or water) are at 10 C. To reduce the heat loss rate, the person wears special sporting gear (snow suit and wet suit) made from a nanostructured silica aerogel insulation with an extremely low thermal conductivity of 0.014 W/m K. The emissivity of the outer surface of the snow and wet suits is 0.95. What thickness of aerogel insulation is needed to reduce the heat loss rate to 100 W (a typical metabolic heat generation rate) in air and water? What are the resulting skin temperatures?

SOLUTION Known: Inner surface temperature of a skin/fat layer of known thickness, thermal conductivity, and surface area. Thermal conductivity and emissivity of snow and wet suits. Ambient conditions.

122

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Find: Insulation thickness needed to reduce heat loss rate to 100 W and corresponding skin temperature. Schematic: Ti = 35°C

Ts

ε = 0.95

Tsur = 10°C

Insulation

Skin/fat

kins = 0.014 W/m•K

ksf = 0.3 W/m•K

T∞ = 10°C h = 2 W/m2•K (Air) h = 200 W/m2•K (Water) Lins

Lsf = 3 mm

Air or water

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer by conduction through the skin/fat and insulation layers. 3. Contact resistance is negligible. 4. Thermal conductivities are uniform. 5. Radiation exchange between the skin surface and the surroundings is between a small surface and a large enclosure at the air temperature. 6. Liquid water is opaque to thermal radiation. 7. Solar radiation is negligible. 8. Body is completely immersed in water in part 2. Analysis: The thermal circuit can be constructed by recognizing that resistance to heat flow is associated with conduction through the skin/fat and insulation layers and convection and radiation at the outer surface. Accordingly, the circuit and the resistances are of the following form (with hr 0 for water): 1 ____

Lsf ____ ksf A q

Ti

Lins kins A

hr A Tsur Tsur = T∞

Ts

T∞ 1 ____

hA

The total thermal resistance needed to achieve the desired heat loss rate is found from Equation 3.19, Rtot

Ti T앝 (35 10) K 0.25 K/ W q 100 W

The total thermal resistance between the inside of the skin/fat layer and the cold surroundings includes conduction resistances for the skin/fat and insulation layers and an

3.1

123

The Plane Wall

䊏

effective resistance associated with convection and radiation, which act in parallel. Hence, Rtot

冢

Lsf L 1 1 ins ksf A kins A 1/hA 1/hr A

冣

1

1 A

冢Lk

sf

sf

L ins 1 kins h hr

冣

This equation can be solved for the insulation thickness.

Air The radiation heat transfer coefficient is approximated as having the same value as in Example 1.7: hr = 5.9 W/m2 䡠 K.

冤

Lins kins ARtot

冥

Lsf 1 ksf h hr

冤

冥

3 1 0.014 W/m 䡠 K 1.8 m2 0.25 K/W 3 10 m 0.3 W/m 䡠 K (2 5.9) W/m2 䡠 K

0.0044 m 4.4 mm

䉰

Water

冤

Lins kins ARtot

冥

Lsf 1 ksf h

冤

冥

3 1 0.014 W/m 䡠 K 1.8 m2 0.25 K/W 3 10 m 0.3 W/m 䡠 K 200 W/m2 䡠 K

䉰

0.0061 m 6.1 mm

These required thicknesses of insulation material can easily be incorporated into the snow and wet suits. The skin temperature can be calculated by considering conduction through the skin/fat layer: q

ksf A(Ti Ts) Lsf

or solving for Ts, Ts Ti

3 qLsf 35 C 100 W 3 10 m2 34.4 C ksfA 0.3 W/m 䡠 K 1.8 m

䉰

The skin temperature is the same in both cases because the heat loss rate and skin/fat properties are the same.

Comments: 1. The nanostructured silica aerogel is a porous material that is only about 5% solid. Its thermal conductivity is less than the thermal conductivity of the gas that fills its pores. As explained in Section 2.2, the reason for this seemingly impossible result is that the pore size is only around 20 nm, which reduces the mean free path of the gas and hence decreases its thermal conductivity.

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

2. By reducing the heat loss rate to 100 W, a person could remain in the cold environments indefinitely without becoming chilled. The skin temperature of 34.4 C would feel comfortable. 3. In the water case, the thermal resistance of the insulation dominates and all other resistances can be neglected. 4. The convection heat transfer coefficient associated with the air depends on the wind conditions, and it can vary over a broad range. As it changes, so will the outer surface temperature of the insulation layer. Since the radiation heat transfer coefficient depends on this temperature, it will also vary. We can perform a more complete analysis that takes this into account. The radiation heat transfer coefficient is given by Equation 1.9: 2 2 hr (Ts,o Tsur)(Ts,o Tsur )

(1)

Here Ts,o is the outer surface temperature of the insulation layer, which can be calculated from

冤kL A kL A冥

Ts,o Ti q

sf

ins

sf

(2)

ins

Since this depends on the insulation thickness, we also need the previous equation for Lins:

冢

Lins kins ARtot

Lsf 1 ksf h hr

冣

(3)

With all other values known, these three equations can be solved for the required insulation thickness. Using all the values from above, these equations have been solved for values of h in the range 0 h 100 W/m2 K, and the results are represented graphically. 7

6

Lins (mm)

124

5

4

3

0

10

20

30

40

50

60

70

80

90

100

h (W/m2•K)

Increasing h reduces the corresponding convection resistance, which then requires additional insulation to maintain the heat transfer rate at 100 W. Once the heat transfer coefficient exceeds approximately 60 W/m2 䡠 K, the convection resistance is negligible and further increases in h have little effect on the required insulation thickness.

3.1

䊏

125

The Plane Wall

The outer surface temperature and radiation heat transfer coefficient can also be calculated. As h increases from 0 to 100 W/m2 䡠 K, Ts,o decreases from 294 to 284 K, while hr decreases from 5.2 to 4.9 W/m2 䡠 K. The initial estimate of hr 5.9 W/m2 䡠 K was not highly accurate. Using this more complete model of the radiation heat transfer, with h 2 W/m2 䡠 K, the radiation heat transfer coefficient is 5.1 W/m2 K, and the required insulation thickness is 4.2 mm, close to the value calculated in the first part of the problem. 5. See Example 3.1 in IHT. This problem can also be solved using the thermal resistance network builder, Models/Resistance Networks, available in IHT.

EXAMPLE 3.2 A thin silicon chip and an 8-mm-thick aluminum substrate are separated by a 0.02-mm-thick epoxy joint. The chip and substrate are each 10 mm on a side, and their exposed surfaces are cooled by air, which is at a temperature of 25 C and provides a convection coefficient of 100 W/m2 䡠 K. If the chip dissipates 104 W/m2 under normal conditions, will it operate below a maximum allowable temperature of 85 C?

SOLUTION Known: Dimensions, heat dissipation, and maximum allowable temperature of a silicon chip. Thickness of aluminum substrate and epoxy joint. Convection conditions at exposed chip and substrate surfaces. Find:

Whether maximum allowable temperature is exceeded.

Schematic: Air

Silicon chip

q1"

T∞ = 25°C h = 100 W/m2•K q1"

q2" L = 8 mm

Aluminum substrate

T∞

h qc"

q"c Epoxy joint (0.02 mm)

_1_

Insulation

Tc R"t,c _L_

k _1_

h Air

T∞ = 25°C h = 100 W/m2•K

T∞ q2"

Assumptions: 1. Steady-state conditions. 2. One-dimensional conduction (negligible heat transfer from sides of composite). 3. Negligible chip thermal resistance (an isothermal chip). 4. Constant properties. 5. Negligible radiation exchange with surroundings.

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Properties: Table A.1, pure aluminum (T ⬃ 350 K): k 239 W/m 䡠 K. Analysis: Heat dissipated in the chip is transferred to the air directly from the exposed surface and indirectly through the joint and substrate. Performing an energy balance on a control surface about the chip, it follows that, on the basis of a unit surface area, qc q1 q2 or qc

Tc T앝 Tc T앝 (L/k) (1/h) (1/h) Rt,c

To conservatively estimate Tc, the maximum possible value of Rt,c 0.9 104 m2 䡠 K/W is obtained from Table 3.2. Hence

冤

Tc T앝 qc h

1 Rt,c (L/k) (1/h)

冥

1

or Tc 25 C 104 W/m2

冤

100

冥

1 (0.9 0.33 100) 104

1

m2 䡠 K/W

Tc 25 C 50.3 C 75.3 C

䉰

Hence the chip will operate below its maximum allowable temperature.

Comments: 1. The joint and substrate thermal resistances are much less than the convection resistance. The joint resistance would have to increase to the unrealistically large value of 50 104 m2 䡠 K/W, before the maximum allowable chip temperature would be exceeded. 2. The allowable power dissipation may be increased by increasing the convection coefficients, either by increasing the air velocity and/or by replacing the air with a more effective heat transfer fluid. Exploring this option for 100 h 2000 W/m2 䡠 K with Tc 85 C, the following results are obtained. 2.5

Tc = 85°C

2.0

q"c × 10–5 (W/m2)

126

1.5 1.0 0.5 0

0

500

1000

1500

2000

h (W/m2•K)

As h l 앝, q2 l 0 and virtually all of the chip power is transferred directly to the fluid stream.

3.1

䊏

127

The Plane Wall

3. As calculated, the difference between the air temperature (T앝 25 C) and the chip temperature (Tc 75.3 C) is 50.3 K. Keep in mind that this is a temperature difference and therefore is the same as 50.3 C. 4. Consider conditions for which airflow over the chip (upper) or substrate (lower) surface ceases due to a blockage in the air supply channel. If heat transfer from either surface is negligible, what are the resulting chip temperatures for qc 104 W/m2? [Answer, 126 C or 125 C]

EXAMPLE 3.3 A photovoltaic panel consists of (top to bottom) a 3-mm-thick ceria-doped glass (kg 1.4 W/m 䡠 K), a 0.1-mm-thick optical grade adhesive (ka 145 W/m 䡠 K), a very thin layer of silicon within which solar energy is converted to electrical energy, a 0.1-mm-thick solder layer (ksdr 50 W/m 䡠 K), and a 2-mm-thick aluminum nitride substrate (kan 120 W/m 䡠 K). The solar-to-electrical conversion efficiency within the silicon layer decreases with increasing silicon temperature, Tsi, and is described by the expression a – bTsi, where a 0.553 and b 0.001 K1. The temperature T is expressed in kelvins over the range 300 K Tsi 525 K. Of the incident solar irradiation, G 700 W/m2, 7% is reflected from the top surface of the glass, 10% is absorbed at the top surface of the glass, and 83% is transmitted to and absorbed within the silicon layer. Part of the solar irradiation absorbed in the silicon is converted to thermal energy, and the remainder is converted to electrical energy. The glass has an emissivity of 0.90, and the bottom as well as the sides of the panel are insulated. Determine the electric power P produced by an L 1-m-long, w 0.1-m-wide solar panel for conditions characterized by h 35 W/m2 䡠 K and T앝 Tsur 20 C.

Air

T∞ = 20°C h = 35 W/m2 •K

Tsur = 20°C G = 700 W/m2

Glass Adhesive Electric power to grid, P

Solder

Silicon layer

Substrate

Lg = 3 mm La = 0.1 mm Lan = 2 mm

Lsdr = 0.1 mm

L=1m

SOLUTION Known: Dimensions and materials of a photovoltaic solar panel. Material properties, solar irradiation, convection coefficient and ambient temperature, emissivity of top panel surface and surroundings temperature. Partitioning of the solar irradiation, and expression for the solar-to-electrical conversion efficiency. Find:

Electric power produced by the photovoltaic panel.

128

Chapter 3

One-Dimensional, Steady-State Conduction

䊏

Schematic: qrad

qconv Tsur

Solar irradiation G = 700 W/m2

Air

T∞ = 20°C h = 35 W/m2·K

Tsur = 20°C

1 hLw Tg,top Lg kg Lw Tg,bot La ka Lw 0.83ηGLw

0.10GLw

0.07G (reflected) 0.10G (absorbed at surface)

Glass

T

1 hr Lw

Lg = 3 mm

Adhesive Silicon layer

0.83GLw

0.83G (absorbed in silicon) La = 0.1 mm

Tsi

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer. 3. Constant properties. 4. Negligible thermal contact resistances. 5. Negligible temperature differences within the silicon layer. Analysis: Recognize that there is no heat transfer to the bottom insulated surface of the solar panel. Hence, the solder layer and aluminum nitride substrate do not affect the solution, and all of the solar energy absorbed by the panel must ultimately leave the panel in the form of radiation and convection heat transfer from the top surface of the glass, and electric power to the grid, P 0.83 GLw. Performing an energy balance on the node associated with the silicon layer yields 0.83 GLw 0.83 GLw

Tsi Tg,top Lg La kaLw kg Lw

Substituting the expression for the solar-to-electrical conversion efficiency and simplifying leads to 0.83 G(1 a bTsi)

Tsi Tg,top La L g ka kg

(1)

Performing a second energy balance on the node associated with the top surface of the glass gives 4 4 Tsur ) 0.83 GLw(1 ) 0.1 GLw hLw(Tg,top T앝) Lw(Tg,top

Substituting the expression for the solar-to-electrical conversion efficiency into the preceding equation and simplifying provides 4 4 Tsur ) 0.83 G(1 a bTsi) 0.1 G h(Tg,top T앝) (Tg,top

(2)

3.1

䊏

The Plane Wall

129

Finally, substituting known values into Equations 1 and 2 and solving simultaneously yields Tsi 307 K 34 C, providing a solar-to-electrical conversion efficiency of 0.553 – 0.001 K1 307 K 0.247. Hence, the power produced by the photovoltaic panel is P 0.83 GLw 0.247 0.83 700 W/m2 1 m 0.1 m 14.3 W

䉰

Comments: 1. The correct application of the conservation of energy requirement is crucial to determining the silicon temperature and the electric power. Note that solar energy is converted to both thermal and electrical energy, and the thermal circuit is used to quantify only the thermal energy transfer. 2. Because of the thermally insulated boundary condition, it is not necessary to include the solder or substrate layers in the analysis. This is because there is no conduction through these materials and, from Fourier’s law, there can be no temperature gradients within these materials. At steady state, Tsdr Tan Tsi. 3. As the convection coefficient increases, the temperature of the silicon decreases. This leads to a higher solar-to-electrical conversion efficiency and increased electric power output. Similarly, higher silicon temperatures and less power production are associated with smaller convection coefficients. For example, P 13.6 W and 14.6 W for h 15 W/m2 䡠 K and 55 W/m2 䡠 K, respectively. 4. The cost of a photovoltaic system can be reduced significantly by concentrating the solar energy onto the relatively expensive photovoltaic panel using inexpensive focusing mirrors or lenses. However, good thermal management then becomes even more important. For example, if the irradiation supplied to the panel were increased to G 7,000 W/m2 through concentration, the conversion efficiency drops to 0.160 as the silicon temperature increases to Tsi 119 C, even for h 55 W/m2 䡠 K. A key to reducing the cost of photovoltaic power generation is developing innovative cooling technologies for use in concentrating photovoltaic systems. 5. The simultaneous solution of Equations 1 and 2 may be achieved by using IHT, another commercial code, or a handheld calculator. A trial-and-error solution could also be obtained, but with considerable effort. Equations 1 and 2 could be combined to write a single transcendental expression for the silicon temperature, but the equation must still be solved numerically or by trial-and-error.

EXAMPLE 3.4 The thermal conductivity of a D 14-nm-diameter carbon nanotube is measured with an instrument that is fabricated of a wafer of silicon nitride at a temperature of T앝 300 K. The 20-m-long nanotube rests on two 0.5-m-thick, 10 m 10 m square islands that are separated by a distance s 5 m. A thin layer of platinum is used as an electrical resistor on the heated island (at temperature Th) to dissipate q 11.3 W of electrical power. On the sensing island, a similar layer of platinum is used to determine its temperature, Ts. The platinum’s electrical resistance, R(Ts) E/I, is found by measuring the voltage drop and electrical current across the platinum layer. The temperature of the sensing island, Ts, is then determined from the relationship of the platinum electrical resistance to its temperature.

130

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Each island is suspended by two Lsn 250-m-long silicon nitride beams that are wsn 3 m wide and tsn 0.5 m thick. A platinum line of width wpt 1 m and thickness tpt 0.2 m is deposited within each silicon nitride beam to power the heated island or to detect the voltage drop associated with the determination of Ts. The entire experiment is performed in a vacuum with Tsur 300 K and at steady state, Ts 308.4 K. Estimate the thermal conductivity of the carbon nanotube.

SOLUTION Known: Dimensions, heat dissipated at the heated island, and temperatures of the sensing island and surrounding silicon nitride wafer. Find:

The thermal conductivity of the carbon nanotube.

Schematic:

Tsur = 300 K

Carbon nanotube

D = 14 nm

Sensing island

Heated island

s = 5 µm

Sensing island Ts = 308.4 K Heated island

Th

s = 5 µm Lsn = 250 µm

10 µm 10 µm

tpt = 0.2 µm wpt = 1 µm

tsn = 0.5 µm wsn = 3 µm Silicon nitride block

T∞ = 300 K

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer. 3. The heated and sensing islands are isothermal. 4. Radiation exchange between the surfaces and the surroundings is negligible. 5. Negligible convection losses.

3.1

䊏

131

The Plane Wall

6. Ohmic heating in the platinum signal lines is negligible. 7. Constant properties. 8. Negligible contact resistance between the nanotube and the islands.

Properties: Table A.1, platinum (325 K, assumed): kpt 71.6 W/m 䡠 K. Table A.2, silicon nitride (325 K, assumed): ksn 15.5 W/m 䡠 K. Analysis: Energy that is dissipated at the heated island is transferred to the silicon nitride block through the support beams of the heated island, the carbon nanotube, and subsequently through the support beams of the sensing island. Therefore, the thermal circuit may be constructed as follows qh /2

qs /2

T∞

T∞

Rt,sup

q

Rt,sup

Th

s kcn Acn

Rt,sup

Ts Rt,sup

T∞

T∞

qh /2

qs /2

where each supporting beam provides a thermal resistance Rt,sup that is composed of a resistance due to the silicon nitride (sn) in parallel with a resistance due to the platinum (pt) line. The cross-sectional areas of the materials in the support beams are Apt wpttpt (1 106 m) (0.2 106 m) 2 1013 m2 Asn wsntsn Apt (3 106 m) (0.5 106 m) 2 1013 m2 1.3 1012 m2 while the cross-sectional area of the carbon nanotube is Acn D2/4 (14 109 m)2/4 1.54 1016 m2 The thermal resistance of each support is

冤 L k LA 冥 冤71.6 W/m 䡠 K 2 10 250 10 m

Rt,sup

kpt Apt pt

sn

sn

1

sn

6

13

冥

m2 15.5 W/m 䡠 K 1.3 1012 m2 250 106 m

7.25 106 K/W The combined heat loss through both sensing island supports is qs 2(Ts T )/Rt,sup 2 (308.4 K 300 K)/(7.25 106 K/W) 2.32 106 W 2.32 W

1

132

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

It follows that qh q qs 11.3 W 2.32 W 8.98 W and Th attains a value of Th T 1 qh Rt,sup 300 K 2

8.98 106 W 7.25 106 K/W 332.6 K 2

For the portion of the thermal circuit connecting Th and Ts, qs

Th Ts s/(kcn Acn)

from which kcn

qss 2.32 106 W 5 106 m Acn(Th Ts) 1.54 1016 m2 (332.6 K 308.4 K)

kcn 3113 W/m 䡠 K

䉰

Comments: 1. The measured thermal conductivity is extremely large, as evident by comparing its value to the thermal conductivities of pure metals shown in Figure 2.4. Carbon nanotubes might be used to dope otherwise low thermal conductivity materials to improve heat transfer. 2. Contact resistances between the carbon nanotube and the heated and sensing islands were neglected because little is known about such resistances at the nanoscale. However, if a contact resistance were included in the analysis, the measured thermal conductivity of the carbon nanotube would be even higher than the predicted value. 3. The significance of radiation heat transfer may be estimated by approximating the heated island as a blackbody radiating to Tsur from both its top and bottom surfaces. Hence, qrad,b ⬇ 5.67 108 W/m2 䡠 K4 2 (10 106 m)2 (332.64 3004)K4 4.7 108 W 0.047 W, and radiation is negligible.

3.2 An Alternative Conduction Analysis The conduction analysis of Section 3.1 was performed using the standard approach. That is, the heat equation was solved to obtain the temperature distribution, Equation 3.3, and Fourier’s law was then applied to obtain the heat transfer rate, Equation 3.4. However, an alternative approach may be used for the conditions presently of interest. Considering conduction in the system of Figure 3.6, we recognize that, for steady-state conditions with no heat generation and no heat loss from the sides, the heat transfer rate qx must be a constant independent of x. That is, for any differential element dx, qx qxdx. This condition is, of course, a consequence of the energy conservation requirement, and it must apply even if the area varies with position A(x) and the thermal conductivity varies with temperature k(T). Moreover, even though the temperature distribution may be two-dimensional, varying with x and y, it is often reasonable to neglect the y-variation and to assume a one-dimensional distribution in x. For the above conditions it is possible to work exclusively with Fourier’s law when performing a conduction analysis. In particular, since the conduction rate is a constant, the

3.2

䊏

133

An Alternative Conduction Analysis

Insulation

qx Adiabatic surface

T1

T0, A(x) z y

qx+dx

x1 x

x

qx

dx

FIGURE 3.6 System with a constant conduction heat transfer rate.

x0

rate equation may be integrated, even though neither the rate nor the temperature distribution is known. Consider Fourier’s law, Equation 2.1, which may be applied to the system of Figure 3.6. Although we may have no knowledge of the value of qx or the form of T(x), we do know that qx is a constant. Hence we may express Fourier’s law in the integral form qx

dx 冕 A(x) 冕 k(T ) dT x

T

x0

T0

(3.26)

The cross-sectional area may be a known function of x, and the material thermal conductivity may vary with temperature in a known manner. If the integration is performed from a point x0 at which the temperature T0 is known, the resulting equation provides the functional form of T(x). Moreover, if the temperature T T1 at some x x1 is also known, integration between x0 and x1 provides an expression from which qx may be computed. Note that, if the area A is uniform and k is independent of temperature, Equation 3.26 reduces to qx x k T A

(3.27)

where x x1 x0 and T T1 – T0. We frequently elect to solve diffusion problems by working with integrated forms of the diffusion rate equations. However, the limiting conditions for which this may be done should be firmly fixed in our minds: steady-state and one-dimensional transfer with no heat generation.

EXAMPLE 3.5 The diagram shows a conical section fabricated from pyroceram. It is of circular cross section with the diameter D ax, where a 0.25. The small end is at x1 50 mm and the large end at x2 250 mm. The end temperatures are T1 400 K and T2 600 K, while the lateral surface is well insulated. T2 T1

x1 x2

x

134

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

1. Derive an expression for the temperature distribution T(x) in symbolic form, assuming one-dimensional conditions. Sketch the temperature distribution. 2. Calculate the heat rate qx through the cone.

SOLUTION Known: Conduction in a circular conical section having a diameter D ax, where a 0.25. Find: 1. Temperature distribution T(x). 2. Heat transfer rate qx. Schematic: T2 = 600 K T1 = 400 K qx

x1 = 0.05 m x2 = 0.25 m x Pyroceram

Assumptions: 1. Steady-state conditions. 2. One-dimensional conduction in the x-direction. 3. No internal heat generation. 4. Constant properties. Properties: Table A.2, pyroceram (500 K): k 3.46 W/m 䡠 K. Analysis: 1. Since heat conduction occurs under steady-state, one-dimensional conditions with no internal heat generation, the heat transfer rate qx is a constant independent of x. Accordingly, Fourier’s law, Equation 2.1, may be used to determine the temperature distribution qx kA dT dx where A D2/4 a2x2/4. Separating variables, 4qxdx kdT a2x2 Integrating from x1 to any x within the cone, and recalling that qx and k are constants, it follows that 4qx a2

冕 dxx k冕 dT x

x1

T

2

T1

3.2

䊏

135

An Alternative Conduction Analysis

Hence

冢

冣

4qx 1x x1 k(T T1) 1 a2 or solving for T T(x) T1

冢

4qx 1 1 a2k x1 x

冣

Although qx is a constant, it is as yet an unknown. However, it may be determined by evaluating the above expression at x x2, where T(x2) T2. Hence T2 T1

冢

4qx 1 1 a2k x1 x2

冣

and solving for qx qx

a2k(T1 T2) 4[(1/x1) (1/x2)]

Substituting for qx into the expression for T(x), the temperature distribution becomes T(x) T1 (T1 T2)

(1/x) (1/x ) 冤(1/x ) (1/x )冥 1

1

䉰

2

From this result, temperature may be calculated as a function of x and the distribution is as shown.

T(x)

T2

T1 x2

x1 x

Note that, since dT/dx – 4qx/ka2x2 from Fourier’s law, it follows that the temperature gradient and heat flux decrease with increasing x. 2. Substituting numerical values into the foregoing result for the heat transfer rate, it follows that qx

(0.25)2 3.46 W/m 䡠 K (400 600) K 2.12 W 4 (1/0.05 m 1/0.25 m)

䉰

Comments: When the parameter a increases, the cross-sectional area changes more rapidly with distance, causing the one-dimensional assumption to become less appropriate.

136

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

3.3 Radial Systems Cylindrical and spherical systems often experience temperature gradients in the radial direction only and may therefore be treated as one-dimensional. Moreover, under steady-state conditions with no heat generation, such systems may be analyzed by using the standard method, which begins with the appropriate form of the heat equation, or the alternative method, which begins with the appropriate form of Fourier’s law. In this section, the cylindrical system is analyzed by means of the standard method and the spherical system by means of the alternative method.

3.3.1

The Cylinder

A common example is the hollow cylinder whose inner and outer surfaces are exposed to fluids at different temperatures (Figure 3.7). For steady-state conditions with no heat generation, the appropriate form of the heat equation, Equation 2.26, is

冢

冣

1 d kr dT 0 r dr dr

(3.28)

where, for the moment, k is treated as a variable. The physical significance of this result becomes evident if we also consider the appropriate form of Fourier’s law. The rate at which energy is conducted across any cylindrical surface in the solid may be expressed as qr kA dT k(2rL) dT dr dr

(3.29)

where A 2rL is the area normal to the direction of heat transfer. Since Equation 3.28 dictates that the quantity kr(dT/dr) is independent of r, it follows from Equation 3.29 that the conduction heat transfer rate qr (not the heat flux qr ) is a constant in the radial direction. Hot fluid T∞,1, h1 Cold fluid T∞,2, h2

Ts,1 r

Ts,2 Ts,1

r1

r2 r

r1

L r2 qr

Ts,2

FIGURE 3.7

T∞,1

Ts,1

________ 1

h12 π r1L

Ts,2 In( r2/r1) ________ 2 π kL

Hollow cylinder with convective surface conditions.

T∞,2

________ 1

h22 π r2L

3.3

䊏

137

Radial Systems

We may determine the temperature distribution in the cylinder by solving Equation 3.28 and applying appropriate boundary conditions. Assuming the value of k to be constant, Equation 3.28 may be integrated twice to obtain the general solution T(r) C1 ln r C2

(3.30)

To obtain the constants of integration C1 and C2, we introduce the following boundary conditions: T(r1) Ts,1

and

T(r2) Ts,2

Applying these conditions to the general solution, we then obtain Ts,1 C1 ln r1 C2

and

Ts,2 C1 ln r2 C2

Solving for C1 and C2 and substituting into the general solution, we then obtain T(r)

冢冣

Ts,1 Ts,2 ln rr Ts,2 2 ln (r1 /r2)

(3.31)

Note that the temperature distribution associated with radial conduction through a cylindrical wall is logarithmic, not linear, as it is for the plane wall under the same conditions. The logarithmic distribution is sketched in the inset of Figure 3.7. If the temperature distribution, Equation 3.31, is now used with Fourier’s law, Equation 3.29, we obtain the following expression for the heat transfer rate: qr

2Lk(Ts,1 Ts,2) ln (r2 /r1)

(3.32)

From this result it is evident that, for radial conduction in a cylindrical wall, the thermal resistance is of the form Rt,cond

ln (r2 /r1) 2Lk

(3.33)

This resistance is shown in the series circuit of Figure 3.7. Note that since the value of qr is independent of r, the foregoing result could have been obtained by using the alternative method, that is, by integrating Equation 3.29. Consider now the composite system of Figure 3.8. Recalling how we treated the composite plane wall and neglecting the interfacial contact resistances, the heat transfer rate may be expressed as qr

T앝,1 T앝,4 ln (r2 /r1) ln (r3 /r2) ln (r4 /r3) 1 1 2r1Lh1 2kAL 2kBL 2kCL 2r4Lh4

(3.34)

The foregoing result may also be expressed in terms of an overall heat transfer coefficient. That is, qr

T앝,1 T앝,4 UA(T앝,1 T앝,4) Rtot

(3.35)

138

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Ts,4

T∞,4, h4

T∞,1, h1

T3 T2 Ts,1

r1

r2

r3 r4

L

T∞,1, h1

A

B

C

T∞,4, h4 T∞,1

Ts,1 T2 T3 Ts,4

qr

T∞,1

Ts,1 In(r2/r1) _________ 2π kAL

1 __________ h12 π r1L

FIGURE 3.8

T2

T3

In(r3/r2) _________ 2π kBL

Ts,4 In(r4/r3) _________ 2π kCL

T∞,4 T∞,4 1 __________ h42 π r4L

Temperature distribution for a composite cylindrical wall.

If U is defined in terms of the inside area, A1 2r1L, Equations 3.34 and 3.35 may be equated to yield U1

1 r r r 1 1 ln 2 1 ln r3 r1 ln r4 r1 1 h1 kA r1 kB r2 kC r3 r4 h4

(3.36)

This definition is arbitrary, and the overall coefficient may also be defined in terms of A4 or any of the intermediate areas. Note that U1A1 U2A2 U3A3 U4A4 (Rt)1

(3.37)

and the specific forms of U2, U3, and U4 may be inferred from Equations 3.34 and 3.35.

EXAMPLE 3.6 The possible existence of an optimum insulation thickness for radial systems is suggested by the presence of competing effects associated with an increase in this thickness. In particular, although the conduction resistance increases with the addition of insulation, the convection resistance decreases due to increasing outer surface area. Hence there may exist an insulation thickness that minimizes heat loss by maximizing the total resistance to heat transfer. Resolve this issue by considering the following system.

3.3

䊏

139

Radial Systems

1. A thin-walled copper tube of radius ri is used to transport a low-temperature refrigerant and is at a temperature Ti that is less than that of the ambient air at T앝 around the tube. Is there an optimum thickness associated with application of insulation to the tube? 2. Confirm the above result by computing the total thermal resistance per unit length of tube for a 10-mm-diameter tube having the following insulation thicknesses: 0, 2, 5, 10, 20, and 40 mm. The insulation is composed of cellular glass, and the outer surface convection coefficient is 5 W/m2 䡠 K.

SOLUTION Known: Radius ri and temperature Ti of a thin-walled copper tube to be insulated from the ambient air. Find: 1. Whether there exists an optimum insulation thickness that minimizes the heat transfer rate. 2. Thermal resistance associated with using cellular glass insulation of varying thickness. Schematic: T∞ h = 5 W/m2•K r ri Air

Ti Insulation, k

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer in the radial (cylindrical) direction. 3. Negligible tube wall thermal resistance. 4. Constant properties for insulation. 5. Negligible radiation exchange between insulation outer surface and surroundings. Properties: Table A.3, cellular glass (285 K, assumed): k 0.055 W/m 䡠 K. Analysis: 1. The resistance to heat transfer between the refrigerant and the air is dominated by conduction in the insulation and convection in the air. The thermal circuit is therefore q'

Ti

T∞ In(r/ri) ________ 2π k

1 _______ 2 π rh

where the conduction and convection resistances per unit length follow from Equations 3.33 and 3.9, respectively. The total thermal resistance per unit length of tube is then Rtot

ln (r/ri ) 1 2k 2rh

140

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

where the rate of heat transfer per unit length of tube is q

T앝 Ti Rtot

An optimum insulation thickness would be associated with the value of r that minimized q or maximized Rtot. Such a value could be obtained from the requirement that dRtot 0 dr Hence 1 1 0 2kr 2r 2h or rk h To determine whether the foregoing result maximizes or minimizes the total resistance, the second derivative must be evaluated. Hence d 2Rtot 1 2 13 dr 2 2kr r h or, at r k/h,

冢

冣

d 2Rtot 1 1 2 1 1 0 dr 2 (k /h) k 2k 2k 3/h2 Since this result is always positive, it follows that r k/h is the insulation radius for which the total resistance is a minimum, not a maximum. Hence an optimum insulation thickness does not exist. From the above result it makes more sense to think in terms of a critical insulation radius rcr ⬅ k h which maximizes heat transfer, that is, below which q increases with increasing r and above which q decreases with increasing r. 2. With h 5 W/m2 䡠 K and k 0.055 W/m 䡠 K, the critical radius is 䡠 K 0.011 m rcr 0.055 W/m 2 5 W/m 䡠 K Hence rcr ri and heat transfer will increase with the addition of insulation up to a thickness of rcr ri (0.011 0.005) m 0.006 m

3.3

䊏

141

Radial Systems

The thermal resistances corresponding to the prescribed insulation thicknesses may be calculated and are plotted as follows: 8

R'tot

R't (m•K/W)

6

R'cond

4

R'conv 2

0

0

6

10

20

30

40

50

r – ri (mm)

Comments: 1. The effect of the critical radius is revealed by the fact that, even for 20 mm of insulation, the total resistance is not as large as the value for no insulation. 2. If ri rcr, as it is in this case, the total resistance decreases and the heat rate therefore increases with the addition of insulation. This trend continues until the outer radius of the insulation corresponds to the critical radius. The trend is desirable for electrical current flow through a wire, since the addition of electrical insulation would aid in transferring heat dissipated in the wire to the surroundings. Conversely, if ri rcr, any addition of insulation would increase the total resistance and therefore decrease the heat loss. This behavior would be desirable for steam flow through a pipe, where insulation is added to reduce heat loss to the surroundings. 3. For radial systems, the problem of reducing the total resistance through the application of insulation exists only for small diameter wires or tubes and for small convection coefficients, such that rcr ri. For a typical insulation (k ⬇ 0.03 W/m 䡠 K) and free convection in air (h ⬇ 10 W/m2 䡠 K), rcr (k/h) ⬇ 0.003 m. Such a small value tells us that, normally, ri rcr and we need not be concerned with the effects of a critical radius. 4. The existence of a critical radius requires that the heat transfer area change in the direction of transfer, as for radial conduction in a cylinder (or a sphere). In a plane wall the area perpendicular to the direction of heat flow is constant and there is no critical insulation thickness (the total resistance always increases with increasing insulation thickness).

3.3.2

The Sphere

Now consider applying the alternative method to analyzing conduction in the hollow sphere of Figure 3.9. For the differential control volume of the figure, energy conservation requires that qr qrdr for steady-state, one-dimensional conditions with no heat generation. The appropriate form of Fourier’s law is qr kA dT k(4r 2) dT dr dr 2 where A 4r is the area normal to the direction of heat transfer.

(3.38)

142

Chapter 3

r1

䊏

One-Dimensional, Steady-State Conduction

r

qr

r2

qr + dr

Ts, 2 Ts, 1

dr

FIGURE 3.9 Conduction in a spherical shell.

Acknowledging that qr is a constant, independent of r, Equation 3.38 may be expressed in the integral form qr 4

冕

r2

r1

冕

dr r2

Ts,2

k(T) dT

(3.39)

Ts,1

Assuming constant k, we then obtain qr

4k(Ts,1 Ts,2) (1/r1) (1/r2)

(3.40)

Remembering that the thermal resistance is defined as the temperature difference divided by the heat transfer rate, we obtain

冢

Rt,cond 1 r1 r1 2 4k 1

冣

(3.41)

Note that the temperature distribution and Equations 3.40 and 3.41 could have been obtained by using the standard approach, which begins with the appropriate form of the heat equation. Spherical composites may be treated in much the same way as composite walls and cylinders, where appropriate forms of the total resistance and overall heat transfer coefficient may be determined.

3.4 Summary of One-Dimensional Conduction Results Many important problems are characterized by one-dimensional, steady-state conduction in plane, cylindrical, or spherical walls without thermal energy generation. Key results for these three geometries are summarized in Table 3.3, where T refers to the temperature difference, Ts,1 Ts,2, between the inner and outer surfaces identified in Figures 3.1, 3.7, and 3.9. In each case, beginning with the heat equation, you should be able to derive the corresponding expressions for the temperature distribution, heat flux, heat rate, and thermal resistance.

3.5 Conduction with Thermal Energy Generation In the preceding section we considered conduction problems for which the temperature distribution in a medium was determined solely by conditions at the boundaries of the medium. We now want to consider the additional effect on the temperature distribution of processes that may be occurring within the medium. In particular, we wish to consider situations for which thermal energy is being generated due to conversion from some other energy form.

3.5

䊏

143

Conduction with Thermal Energy Generation

TABLE 3.3 One-dimensional, steady-state solutions to the heat equation with no generation

Heat equation Temperature distribution

Plane Wall

Cylindrical Walla

Spherical Walla

d 2T 0 dx2

dT 1 d r dr r dr 0

冢 冣

1 d 2 dT r 0 dr r 2 dr

Ts,1 T

Ts, 2 T

T L

Heat flux (q⬙)

k

Heat rate (q)

kA

Thermal resistance (Rt,cond)

x L

T L

L kA

ln (r/r2) ln (r1/r2)

Ts,1

冢 冣 1 (r /r) T 冤 1 (r /r )冥 1

1

2

k T r ln (r2 /r1)

k T r 2[(1/r1) (1/r2)]

2Lk T ln (r2 /r1)

4k T (1/r1) (1/r2)

ln (r2 /r1) 2Lk

(1/r1) (1/r2) 4 k

The critical radius of insulation is rcr k/h for the cylinder and rcr 2k/h for the sphere.

a

A common thermal energy generation process involves the conversion from electrical to thermal energy in a current-carrying medium (Ohmic, or resistance, or Joule heating). The rate at which energy is generated by passing a current I through a medium of electrical resistance Re is E˙g I 2Re

(3.42)

If this power generation (W) occurs uniformly throughout the medium of volume V, the volumetric generation rate (W/m3) is then q˙ ⬅

E˙ g V

I 2Re V

(3.43)

Energy generation may also occur as a result of the deceleration and absorption of neutrons in the fuel element of a nuclear reactor or exothermic chemical reactions occurring within a medium. Endothermic reactions would, of course, have the inverse effect (a thermal energy sink) of converting thermal energy to chemical bonding energy. Finally, a conversion from electromagnetic to thermal energy may occur due to the absorption of radiation within the medium. The process occurs, for example, when gamma rays are absorbed in external nuclear reactor components (cladding, thermal shields, pressure vessels, etc.) or when visible radiation is absorbed in a semitransparent medium. Remember not to confuse energy generation with energy storage (Section 1.3.1).

3.5.1

The Plane Wall

Consider the plane wall of Figure 3.10a, in which there is uniform energy generation per unit volume (q˙ is constant) and the surfaces are maintained at Ts,1 and Ts,2. For constant thermal conductivity k, the appropriate form of the heat equation, Equation 2.22, is d 2T q˙ 0 dx2 k

(3.44)

144

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

The general solution is T

q˙ 2 x C1x C2 2k

(3.45)

where C1 and C2 are the constants of integration. For the prescribed boundary conditions, T(L) Ts,1

T(L) Ts,2

and

The constants may be evaluated and are of the form C1

Ts,2 Ts,1 2L

q˙ 2 Ts,1 Ts,2 L 2k 2

C2

and

in which case the temperature distribution is T(x)

冢

冣

2 Ts,2 Ts,1 x Ts,1 Ts,2 q˙L2 1 x2 2k 2 L 2 L

(3.46)

The heat flux at any point in the wall may, of course, be determined by using Equation 3.46 with Fourier’s law. Note, however, that with generation the heat flux is no longer independent of x. The preceding result simplifies when both surfaces are maintained at a common temperature, Ts,1 Ts,2 ⬅ Ts. The temperature distribution is then symmetrical about the midplane, Figure 3.10b, and is given by T(x)

冢

冣

2 q˙L2 1 x 2 Ts 2k L

(3.47)

x –L

x +L

–L

q•

+L

T0

T(x)

Ts,1

q• T(x)

Ts

Ts

T∞,1,h1

T∞ ,h (a)

q"conv

q"cond

Ts,2

T∞, h

T∞,2,h2

(b )

q• T0 T(x) Ts q"conv

q"cond

T∞, h (c)

FIGURE 3.10 Conduction in a plane wall with uniform heat generation. (a) Asymmetrical boundary conditions. (b) Symmetrical boundary conditions. (c) Adiabatic surface at midplane.

3.5

䊏

145

Conduction with Thermal Energy Generation

The maximum temperature exists at the midplane T(0) ⬅ T0

q˙L2 Ts 2k

(3.48)

in which case the temperature distribution, Equation 3.47, may be expressed as

冢冣

T(x) T0 x Ts T0 L

2

(3.49)

It is important to note that at the plane of symmetry in Figure 3.10b, the temperature gradient is zero, (dT/dx)x0 0. Accordingly, there is no heat transfer across this plane, and it may be represented by the adiabatic surface shown in Figure 3.10c. One implication of this result is that Equation 3.47 also applies to plane walls that are perfectly insulated on one side (x 0) and maintained at a fixed temperature Ts on the other side (x L). To use the foregoing results, the surface temperature(s) Ts must be known. However, a common situation is one for which it is the temperature of an adjoining fluid, T앝, and not Ts, which is known. It then becomes necessary to relate Ts to T앝. This relation may be developed by applying a surface energy balance. Consider the surface at x L for the symmetrical plane wall (Figure 3.10b) or the insulated plane wall (Figure 3.10c). Neglecting radiation and substituting the appropriate rate equations, the energy balance given by Equation 1.13 reduces to k dT dx

冏

xL

h(Ts T앝)

(3.50)

Substituting from Equation 3.47 to obtain the temperature gradient at x L, it follows that Ts T앝

q˙L h

(3.51)

Hence Ts may be computed from knowledge of T앝, q˙ , L, and h. Equation 3.51 may also be obtained by applying an overall energy balance to the plane wall of Figure 3.10b or 3.10c. For example, relative to a control surface about the wall of Figure 3.10c, the rate at which energy is generated within the wall must be balanced by the rate at which energy leaves via convection at the boundary. Equation 1.12c reduces to E˙g E˙out

(3.52)

q˙L h(Ts T앝)

(3.53)

or, for a unit surface area,

Solving for Ts, Equation 3.51 is obtained. Equation 3.51 may be combined with Equation 3.47 to eliminate Ts from the temperature distribution, which is then expressed in terms of the known quantities q˙, L, k, h, and T앝. The same result may be obtained directly by using Equation 3.50 as a boundary condition to evaluate the constants of integration appearing in Equation 3.45.

EXAMPLE 3.7 A plane wall is a composite of two materials, A and B. The wall of material A has uniform heat generation q˙ 1.5 106 W/m3, kA 75 W/m K, and thickness LA 50 mm. The

146

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

wall material B has no generation with kB 150 W/m 䡠 K and thickness LB 20 mm. The inner surface of material A is well insulated, while the outer surface of material B is cooled by a water stream with T앝 30 C and h 1000 W/m2 䡠 K. 1. Sketch the temperature distribution that exists in the composite under steady-state conditions. 2. Determine the temperature T0 of the insulated surface and the temperature T2 of the cooled surface.

SOLUTION Known: Plane wall of material A with internal heat generation is insulated on one side and bounded by a second wall of material B, which is without heat generation and is subjected to convection cooling. Find: 1. Sketch of steady-state temperature distribution in the composite. 2. Inner and outer surface temperatures of the composite. Schematic: T0

T1

T2 T∞ = 30°C h = 1000 W/m2•K

Insulation

q•A = 1.5 × 106 W/m3 kA = 75 W/m•K

q"

A

LA = 50 mm x

B

LB = 20 mm

Water

kB = 150 W/m•K q• B = 0

Assumptions: 1. Steady-state conditions. 2. One-dimensional conduction in x-direction. 3. Negligible contact resistance between walls. 4. Inner surface of A adiabatic. 5. Constant properties for materials A and B. Analysis: 1. From the prescribed physical conditions, the temperature distribution in the composite is known to have the following features, as shown: (a) Parabolic in material A. (b) Zero slope at insulated boundary. (c) Linear in material B. (d) Slope change kB/kA 2 at interface.

3.5

䊏

147

Conduction with Thermal Energy Generation

The temperature distribution in the water is characterized by (e) Large gradients near the surface. b

a

T(x)

T0

d

c

T1 T2

e A

T∞

B

LA

0

LA + LB x

2. The outer surface temperature T2 may be obtained by performing an energy balance on a control volume about material B. Since there is no generation in this material, it follows that, for steady-state conditions and a unit surface area, the heat flux into the material at x LA must equal the heat flux from the material due to convection at x LA LB. Hence q h(T2 T )

(1)

The heat flux q may be determined by performing a second energy balance on a control volume about material A. In particular, since the surface at x 0 is adiabatic, there is no inflow and the rate at which energy is generated must equal the outflow. Accordingly, for a unit surface area, q˙LA q

(2)

Combining Equations 1 and 2, the outer surface temperature is T2 T

q˙LA h

T2 30 C 1.5 10 W/m 2 0.05 m 105 C 1000 W/m 䡠 K 6

3

䉰

From Equation 3.48 the temperature at the insulated surface is T0

q˙L2A T1 2kA

(3)

where T1 may be obtained from the following thermal circuit: q''

T1

T2 Rcond, '' B

T∞ Rconv ''

That is, T1 T (Rcond,B Rconv) q where the resistances for a unit surface area are Rcond, B

LB kB

Rconv 1 h

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Hence, T1 30 C

m 1 冢1500.02 W/m 䡠 K 1000 W/m 䡠 K冣 2

1.5 106 W/m3 0.05 m T1 30 C 85 C 115 C Substituting into Equation 3, 1.5 106 W/m3 (0.05 m)2 115 C 2 75 W/m 䡠 K T0 25oC 115oC 140oC

T0

䉰

Comments: 1. Material A, having heat generation, cannot be represented by a thermal circuit element. 2. Since the resistance to heat transfer by convection is significantly larger than that due to conduction in material B, Rconv/Rcond 7.5, the surface-to-fluid temperature difference is much larger than the temperature drop across material B, (T2 – T앝)/(T1 – T2) 7.5. This result is consistent with the temperature distribution plotted in part 1. 3. The surface and interface temperatures (T0, T1, and T2) depend on the generation rate q˙, the thermal conductivities kA and kB, and the convection coefficient h. Each material will have a maximum allowable operating temperature, which must not be exceeded if thermal failure of the system is to be avoided. We explore the effect of one of these parameters by computing and plotting temperature distributions for values of h 200 and 1000 W/m2 䡠 K, which would be representative of air and liquid cooling, respectively. 450

440

h = 200 W/m2•K

T (°C)

430

420

410

400 0

10

20

30

40

50

60

70

60

70

x (mm) 150

140

h = 1000 W/m2•K

130

T (°C)

148

120

110

100 0

10

20

30

40

x (mm)

50

3.5

Conduction with Thermal Energy Generation

䊏

149

For h 200 W/m2 䡠 K, there is a significant increase in temperature throughout the system and, depending on the selection of materials, thermal failure could be a problem. Note the slight discontinuity in the temperature gradient, dT/dx, at x 50 mm. What is the physical basis for this discontinuity? We have assumed negligible contact resistance at this location. What would be the effect of such a resistance on the temperature distribution throughout the system? Sketch a representative distribution. What would be the effect on the temperature distribution of an increase in q˙, kA, or kB? Qualitatively sketch the effect of such changes on the temperature distribution. 4. This example is solved in the Advanced section of IHT.

3.5.2

Radial Systems

Heat generation may occur in a variety of radial geometries. Consider the long, solid cylinder of Figure 3.11, which could represent a current-carrying wire or a fuel element in a nuclear reactor. For steady-state conditions, the rate at which heat is generated within the cylinder must equal the rate at which heat is convected from the surface of the cylinder to a moving fluid. This condition allows the surface temperature to be maintained at a fixed value of Ts. To determine the temperature distribution in the cylinder, we begin with the appropriate form of the heat equation. For constant thermal conductivity k, Equation 2.26 reduces to

冢 冣

1 d r dT q˙ 0 r dr dr k

(3.54)

Separating variables and assuming uniform generation, this expression may be integrated to obtain q˙ r dT r2 C1 dr 2k

(3.55)

Repeating the procedure, the general solution for the temperature distribution becomes T(r)

q˙ 2 r C1 ln r C2 4k

(3.56)

Cold fluid

T∞, h

qr

Ts

q• L

r ro

FIGURE 3.11 Conduction in a solid cylinder with uniform heat generation.

150

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

To obtain the constants of integration C1 and C2, we apply the boundary conditions dT dr

冏

0

T(r0) Ts

and

r0

The first condition results from the symmetry of the situation. That is, for the solid cylinder the centerline is a line of symmetry for the temperature distribution and the temperature gradient must be zero. Recall that similar conditions existed at the midplane of a wall having symmetrical boundary conditions (Figure 3.10b). From the symmetry condition at r 0 and Equation 3.55, it is evident that C1 0. Using the surface boundary condition at r ro with Equation 3.56, we then obtain C2 Ts

q˙ 2 r 4k o

(3.57)

冣

(3.58)

The temperature distribution is therefore T(r)

冢

2 q˙ro2 1 r 2 Ts 4k ro

Evaluating Equation 3.58 at the centerline and dividing the result into Equation 3.58, we obtain the temperature distribution in nondimensional form,

冢冣

T(r) Ts 1 rr o To Ts

2

(3.59)

where To is the centerline temperature. The heat rate at any radius in the cylinder may, of course, be evaluated by using Equation 3.58 with Fourier’s law. To relate the surface temperature, Ts, to the temperature of the cold fluid T앝, either a surface energy balance or an overall energy balance may be used. Choosing the second approach, we obtain q˙(ro2L) h(2ro L)(Ts T앝) or Ts T앝

3.5.3

q˙ro 2h

(3.60)

Tabulated Solutions

Appendix C provides a convenient and systematic procedure for treating the different combinations of surface conditions that may be applied to one-dimensional planar and radial (cylindrical and spherical) geometries with uniform thermal energy generation. From the tabulated results of this appendix, it is a simple matter to obtain distributions of the temperature, heat flux, and heat rate for boundary conditions of the second kind (a uniform surface heat flux) and the third kind (a surface heat flux that is proportional to a convection coefficient h or the overall heat transfer coefficient U). You are encouraged to become familiar with the contents of the appendix.

3.5.4

Application of Resistance Concepts

We conclude our discussion of heat generation effects with a word of caution. In particular, when such effects are present, the heat transfer rate is not a constant, independent of the

3.5

䊏

151

Conduction with Thermal Energy Generation

spatial coordinate. Accordingly, it would be incorrect to use the conduction resistance concepts and the related heat rate equations developed in Sections 3.1 and 3.3.

EXAMPLE 3.8 Consider a long solid tube, insulated at the outer radius r2 and cooled at the inner radius r1, with uniform heat generation q˙ (W/m3) within the solid. 1. Obtain the general solution for the temperature distribution in the tube. 2. In a practical application a limit would be placed on the maximum temperature that is permissible at the insulated surface (r r2). Specifying this limit as Ts,2, identify appropriate boundary conditions that could be used to determine the arbitrary constants appearing in the general solution. Determine these constants and the corresponding form of the temperature distribution. 3. Determine the heat removal rate per unit length of tube. 4. If the coolant is available at a temperature T앝, obtain an expression for the convection coefficient that would have to be maintained at the inner surface to allow for operation at prescribed values of Ts,2 and q˙ .

SOLUTION Known: Solid tube with uniform heat generation is insulated at the outer surface and cooled at the inner surface. Find: 1. General solution for the temperature distribution T(r). 2. Appropriate boundary conditions and the corresponding form of the temperature distribution. 3. Heat removal rate for specified maximum temperature. 4. Corresponding required convection coefficient at the inner surface. Schematic:

Ts,2 Ts,1

q'conv

q'cond

r1 T∞, h

r2

q•, k

Coolant

T∞, h

Insulation

152

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Assumptions: 1. Steady-state conditions. 2. One-dimensional radial conduction. 3. Constant properties. 4. Uniform volumetric heat generation. 5. Outer surface adiabatic. Analysis: 1. To determine T(r), the appropriate form of the heat equation, Equation 2.26, must be solved. For the prescribed conditions, this expression reduces to Equation 3.54, and the general solution is given by Equation 3.56. Hence, this solution applies in a cylindrical shell, as well as in a solid cylinder (Figure 3.11). 2. Two boundary conditions are needed to evaluate C1 and C2, and in this problem it is appropriate to specify both conditions at r2. Invoking the prescribed temperature limit, T(r2) Ts,2

(1)

and applying Fourier’s law, Equation 3.29, at the adiabatic outer surface dT dr

冏

0

(2)

r2

Using Equations 3.56 and 1, it follows that Ts,2

q˙ 2 r C1 ln r2 C2 4k 2

(3)

Similarly, from Equations 3.55 and 2 0

q˙ 2 r C1 2k 2

(4)

q˙ 2 r 2k 2

(5)

Hence, from Equation 4, C1 and from Equation 3 C2 Ts,2

q˙ 2 q˙ 2 r r ln r2 4k 2 2k 2

(6)

Substituting Equations 5 and 6 into the general solution, Equation 3.56, it follows that T(r) Ts,2

r q˙ 2 q˙ (r r2) r 22 ln r2 4k 2 2k

(7)

3.5

䊏

153

Conduction with Thermal Energy Generation

3. The heat removal rate may be determined by obtaining the conduction rate at r1 or by evaluating the total generation rate for the tube. From Fourier’s law qr k2r dT dr Hence, substituting from Equation 7 and evaluating the result at r1,

冢

qr(r1) k2r1

冣

q˙ q˙ r 2 r1 r2 q˙(r22 r12) 2k 2k 1

(8)

Alternatively, because the tube is insulated at r2, the rate at which heat is generated in the tube must equal the rate of removal at r1. That is, for a control volume about the tube, the energy conservation requirement, Equation 1.12c, reduces to E˙ g E˙ out 0, where E˙ g q˙(r22 r12)L and E˙out qcond L qr(r1)L. Hence qr(r1) q˙(r22 r12)

(9)

4. Applying the energy conservation requirement, Equation 1.13, to the inner surface, it follows that qcond qconv or

q˙(r22 r12) h2r1(Ts,1 T ) Hence h

q˙(r22 r12) 2r1(Ts,1 T앝)

(10)

where Ts,1 may be obtained by evaluating Equation 7 at r r1.

Comments: 1. Note that, through application of Fourier’s law in part 3, the sign on qr(r1) was found to be negative, Equation 8, implying that heat flow is in the negative r-direction. However, in applying the energy balance, we acknowledged that heat flow was out of the wall. Hence we expressed qcond as qr(r1) and we expressed qconv in terms of (Ts,1 – T앝), rather than (T앝 – Ts,1). 2. Results of the foregoing analysis may be used to determine the convection coefficient required to maintain the maximum tube temperature Ts,2 below a prescribed value. Consider a tube of thermal conductivity k 5 W/m K and inner and outer radii of r1 20 mm and r2 25 mm, respectively, with a maximum allowable temperature of Ts,2 350 C. The tube experiences heat generation at a rate of q· 5 106 W/m3, and the coolant is at a temperature of T앝 80 C. Obtaining T(r1) Ts,1 336.5 C from Equation 7 and substituting into Equation 10, the required convection coefficient is found to be h 110 W/m2 䡠 K. Using the IHT Workspace, parametric calculations may be performed to determine the effects of the convection coefficient and the generation rate on the maximum tube temperature, and results are plotted as a function of h for three values of q·.

154

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Maximum tube temperature, Ts,2 (°C)

500 •

q × 10–6 (W/m3) 7.5 5.0 2.5

400

300

200

100 20

60 100 140 Convection coefficient, h (W/m2•K)

180

For each generation rate, the minimum value of h needed to maintain Ts,2 350 C may be determined from the figure. 3. The temperature distribution, Equation 7, may also be obtained by using the results of Appendix C. Applying a surface energy balance at r r1, with q(r) q˙(r22 r21)L, (Ts,2 Ts,1) may be determined from Equation C.8 and the result substituted into Equation C.2 to eliminate Ts,1 and obtain the desired expression.

3.6 Heat Transfer from Extended Surfaces The term extended surface is commonly used to depict an important special case involving heat transfer by conduction within a solid and heat transfer by convection (and/or radiation) from the boundaries of the solid. Until now, we have considered heat transfer from the boundaries of a solid to be in the same direction as heat transfer by conduction in the solid. In contrast, for an extended surface, the direction of heat transfer from the boundaries is perpendicular to the principal direction of heat transfer in the solid. Consider a strut that connects two walls at different temperatures and across which there is fluid flow (Figure 3.12). With T1 T2, temperature gradients in the x-direction sustain heat transfer by conduction in the strut. However, with T1 T2 T앝, there is concurrent heat

T2 qx, 2 L

x

qconv

Fluid

T∞, h

T1 T1

qx, 1

T2 T(x)

T1 > T2 > T∞

0

FIGURE 3.12 Combined conduction and convection in a structural element.

3.6

䊏

155

Heat Transfer from Extended Surfaces

transfer by convection to the fluid, causing qx, and hence the magnitude of the temperature gradient, 兩dT/dx兩, to decrease with increasing x. Although there are many different situations that involve such combined conduction– convection effects, the most frequent application is one in which an extended surface is used specifically to enhance heat transfer between a solid and an adjoining fluid. Such an extended surface is termed a fin. Consider the plane wall of Figure 3.13a . If Ts is fixed, there are two ways in which the heat transfer rate may be increased. The convection coefficient h could be increased by increasing the fluid velocity, and/or the fluid temperature T앝 could be reduced. However, there are many situations for which increasing h to the maximum possible value is either insufficient to obtain the desired heat transfer rate or the associated costs are prohibitive. Such costs are related to the blower or pump power requirements needed to increase h through increased fluid motion. Moreover, the second option of reducing T앝 is often impractical. Examining Figure 3.13b , however, we see that there exists a third option. That is, the heat transfer rate may be increased by increasing the surface area across which the convection occurs. This may be done by employing fins that extend from the wall into the surrounding fluid. The thermal conductivity of the fin material can have a strong effect on the temperature distribution along the fin and therefore influences the degree to which the heat transfer rate is enhanced. Ideally, the fin material should have a large thermal conductivity to minimize temperature variations from its base to its tip. In the limit of infinite thermal conductivity, the entire fin would be at the temperature of the base surface, thereby providing the maximum possible heat transfer enhancement. Examples of fin applications are easy to find. Consider the arrangement for cooling engine heads on motorcycles and lawn mowers or for cooling electric power transformers. Consider also the tubes with attached fins used to promote heat exchange between air and the working fluid of an air conditioner. Two common finned-tube arrangements are shown in Figure 3.14. Different fin configurations are illustrated in Figure 3.15. A straight fin is any extended surface that is attached to a plane wall. It may be of uniform cross-sectional area, or its cross-sectional area may vary with the distance x from the wall. An annular fin is one that is circumferentially attached to a cylinder, and its cross section varies with radius from the wall of the cylinder. The foregoing fin types have rectangular cross sections, whose area may be expressed as a product of the fin thickness t and the width w for straight fins or the circumference 2r for annular fins. In contrast a pin fin, or spine, is an extended surface of circular cross section. Pin fins may also be of uniform or nonuniform cross section. In any

T∞, h

T∞, h

A

q = hA(Ts – T∞)

Ts, A

Ts (a)

(b)

FIGURE 3.13 Use of fins to enhance heat transfer from a plane wall. (a) Bare surface. (b) Finned surface.

156

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Liquid flow Gas flow

Liquid flow Gas flow

FIGURE 3.14 Schematic of typical finned-tube heat exchangers.

application, selection of a particular fin configuration may depend on space, weight, manufacturing, and cost considerations, as well as on the extent to which the fins reduce the surface convection coefficient and increase the pressure drop associated with flow over the fins.

3.6.1

A General Conduction Analysis

As engineers we are primarily interested in knowing the extent to which particular extended surfaces or fin arrangements could improve heat transfer from a surface to the surrounding fluid. To determine the heat transfer rate associated with a fin, we must first obtain the temperature distribution along the fin. As we have done for previous systems, we begin by performing an energy balance on an appropriate differential element. Consider the extended surface of Figure 3.16. The analysis is simplified if certain assumptions are made. We choose to assume one-dimensional conditions in the longitudinal (x-) direction, even though conduction within the fin is actually two-dimensional. The rate at which energy is convected to the fluid from any point on the fin surface must be balanced by the net rate at which energy reaches that point due to conduction in the transverse (y-, z-) direction. However, in practice the fin is thin, and temperature changes in the transverse

t w

x

r

x (a)

(b)

x (c)

(d)

FIGURE 3.15 Fin configurations. (a) Straight fin of uniform cross section. (b) Straight fin of nonuniform cross section. (c) Annular fin. (d) Pin fin.

3.6

䊏

157

Heat Transfer from Extended Surfaces

dAs

qx

dqconv Ac(x) qx+dx

dx

x

z y

FIGURE 3.16 Energy balance for an extended surface.

x

direction within the fin are small compared with the temperature difference between the fin and the environment. Hence, we may assume that the temperature is uniform across the fin thickness, that is, it is only a function of x. We will consider steady-state conditions and also assume that the thermal conductivity is constant, that radiation from the surface is negligible, that heat generation effects are absent, and that the convection heat transfer coefficient h is uniform over the surface. Applying the conservation of energy requirement, Equation 1.12c, to the differential element of Figure 3.16, we obtain qx qxdx dqconv

(3.61)

qx kAc dT dx

(3.62)

From Fourier’s law we know that

where Ac is the cross-sectional area, which may vary with x. Since the conduction heat rate at x dx may be expressed as qxdx qx

dqx dx dx

(3.63)

it follows that

冢

冣

qxdx kAc dT k d Ac dT dx dx dx dx

(3.64)

The convection heat transfer rate may be expressed as dqconv hdAs(T T )

(3.65)

where dAs is the surface area of the differential element. Substituting the foregoing rate equations into the energy balance, Equation 3.61, we obtain

冢

冣

d A dT h dAs (T T ) 0 앝 dx c dx k dx

158

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

or

冢

冣

冢

冣

d 2T 1 dAc dT 1 h dAs (T T ) 0

Ac dx dx Ac k dx dx 2

(3.66)

This result provides a general form of the energy equation for an extended surface. Its solution for appropriate boundary conditions provides the temperature distribution, which may be used with Equation 3.62 to calculate the conduction rate at any x.

3.6.2

Fins of Uniform Cross-Sectional Area

To solve Equation 3.66 it is necessary to be more specific about the geometry. We begin with the simplest case of straight rectangular and pin fins of uniform cross section (Figure 3.17). Each fin is attached to a base surface of temperature T(0) Tb and extends into a fluid of temperature T앝. For the prescribed fins, Ac is a constant and As Px, where As is the surface area measured from the base to x and P is the fin perimeter. Accordingly, with dAc /dx 0 and dAs /dx P, Equation 3.66 reduces to d 2T hP (T T ) 0 앝 dx 2 kAc

(3.67)

To simplify the form of this equation, we transform the dependent variable by defining an excess temperature as (x) ⬅ T(x) T앝

(3.68)

where, since T앝 is a constant, d/dx dT/dx. Substituting Equation 3.68 into Equation 3.67, we then obtain d 2 m2 0 dx 2

(3.69)

T∞, h qconv T ∞, h Tb

qconv

t Ac

Tb

qf

qf

D

w x L P = 2w + 2t Ac = wt (a)

x Ac L P = πD Ac = π D2/4 (b)

FIGURE 3.17 Straight fins of uniform cross section. (a) Rectangular fin. (b) Pin fin.

3.6

䊏

159

Heat Transfer from Extended Surfaces

where m2 ⬅ hP kAc

(3.70)

Equation 3.69 is a linear, homogeneous, second-order differential equation with constant coefficients. Its general solution is of the form (x) C1emx C2emx

(3.71)

By substitution it may readily be verified that Equation 3.71 is indeed a solution to Equation 3.69. To evaluate the constants C1 and C2 of Equation 3.71, it is necessary to specify appropriate boundary conditions. One such condition may be specified in terms of the temperature at the base of the fin (x 0) (0) Tb T앝 ⬅ b

(3.72)

The second condition, specified at the fin tip (x L), may correspond to one of four different physical situations. The first condition, Case A, considers convection heat transfer from the fin tip. Applying an energy balance to a control surface about this tip (Figure 3.18), we obtain hAc[T(L) T앝] kAc dT dx or h(L) k

d dx

冏

xL

冏

(3.73)

xL

That is, the rate at which energy is transferred to the fluid by convection from the tip must equal the rate at which energy reaches the tip by conduction through the fin. Substituting Equation 3.71 into Equations 3.72 and 3.73, we obtain, respectively, b C1 C2

(3.74)

and h(C1emL C2emL) km(C2emL C1emL) Solving for C1 and C2, it may be shown, after some manipulation, that cosh m(L x) (h /mk) sin h m(L x) b cosh mL (h /mk) sin h mL

(3.75)

The form of this temperature distribution is shown schematically in Figure 3.18. Note that the magnitude of the temperature gradient decreases with increasing x. This trend is a consequence of the reduction in the conduction heat transfer qx(x) with increasing x due to continuous convection losses from the fin surface. We are particularly interested in the amount of heat transferred from the entire fin. From Figure 3.18, it is evident that the fin heat transfer rate qf may be evaluated in two

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Fluid, T∞

qconv Tb dT dx x=L

–kAc __

qb = qf

hAc[T(L) – T∞]

θb θ (x)

160

0

FIGURE 3.18 Conduction and convection in a fin of uniform cross section.

L

0

x

alternative ways, both of which involve use of the temperature distribution. The simpler procedure, and the one that we will use, involves applying Fourier’s law at the fin base. That is, qf qb kAc dT dx

冏

kAc

x0

d dx

冏

(3.76)

x0

Hence, knowing the temperature distribution, (x), qf may be evaluated, giving qf 兹hPkAcb

sinh mL (h /mk) cosh mL cosh mL (h/mk) sinh mL

(3.77)

However, conservation of energy dictates that the rate at which heat is transferred by convection from the fin must equal the rate at which it is conducted through the base of the fin. Accordingly, the alternative formulation for qf is

冕 h[T(x) T ] dA q 冕 h(x) dA

qf

Af

f

s

s

(3.78)

Af

where Af is the total, including the tip, fin surface area. Substitution of Equation 3.75 into Equation 3.78 would yield Equation 3.77. The second tip condition, Case B, corresponds to the assumption that the convective heat loss from the fin tip is negligible, in which case the tip may be treated as adiabatic and d dx

冏

0

xL

Substituting from Equation 3.71 and dividing by m, we then obtain C1emL C2emL 0

(3.79)

3.6

䊏

161

Heat Transfer from Extended Surfaces

Using this expression with Equation 3.74 to solve for C1 and C2 and substituting the results into Equation 3.71, we obtain cosh m(L x) b cosh mL

(3.80)

Using this temperature distribution with Equation 3.76, the fin heat transfer rate is then qf 兹hPkAc b tanh mL

(3.81)

In the same manner, we can obtain the fin temperature distribution and heat transfer rate for Case C, where the temperature is prescribed at the fin tip. That is, the second boundary condition is (L) L, and the resulting expressions are of the form (L /b) sinh mx sinh m(L x) b sinh mL qf 兹hPkAc b

(3.82)

cosh mL L /b sinh mL

(3.83)

The very long fin, Case D, is an interesting extension of these results. In particular, as L l 앝, L l 0 and it is easily verified that emx b

(3.84)

qf 兹hPkAc b

(3.85)

The foregoing results are summarized in Table 3.4. A table of hyperbolic functions is provided in Appendix B.1.

TABLE 3.4 Case A

Temperature distribution and heat loss for fins of uniform cross section Tip Condition (x ⴝ L) Convection heat transfer: h(L) kd/dx冨x⫽L

B

Adiabatic: d/dx冨x⫽L 0

C

Prescribed temperature: (L) L

D

Infinite fin (L l 앝): (L) 0

⬅ T T앝 b (0) Tb T앝

m2 ⬅ hP/kAc M ⬅ 兹h 苶P 苶kA 苶苶 c b

Temperature Distribution /b

Fin Heat Transfer Rate qƒ

cosh m(L x) (h/mk) sinh m(L x) cosh mL (h/mk) sinh mL (3.75) cosh m(L x) cosh mL (3.80) (L/b) sinh mx sinh m(L x) sinh mL (3.82) emx

(3.84)

M

sinh mL (h/mk) cosh mL cosh mL (h/mk) sinh mL (3.77) M tanh mL (3.81) M

(cosh mL L/b) sinh mL (3.83) M

(3.85)

162

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

EXAMPLE 3.9 A very long rod 5 mm in diameter has one end maintained at 100 C. The surface of the rod is exposed to ambient air at 25 C with a convection heat transfer coefficient of 100 W/m2 䡠 K. 1. Determine the temperature distributions along rods constructed from pure copper, 2024 aluminum alloy, and type AISI 316 stainless steel. What are the corresponding heat losses from the rods? 2. Estimate how long the rods must be for the assumption of infinite length to yield an accurate estimate of the heat loss.

SOLUTION Known: A long circular rod exposed to ambient air. Find: 1. Temperature distribution and heat loss when rod is fabricated from copper, an aluminum alloy, or stainless steel. 2. How long rods must be to assume infinite length. Schematic: Air

Tb = 100°C

T∞ = 25°C h = 100 W/m2•K

k, L→∞, D = 5 mm

Assumptions: 1. Steady-state conditions. 2. One-dimensional conduction along the rod. 3. Constant properties. 4. Negligible radiation exchange with surroundings. 5. Uniform heat transfer coefficient. 6. Infinitely long rod. Properties: Table A.1, copper [T (Tb T앝)/2 62.5 C ⬇ 335 K]: k 398 W/m 䡠 K. Table A.1, 2024 aluminum (335 K): k 180 W/m 䡠 K. Table A.1, stainless steel, AISI 316 (335 K): k 14 W/m 䡠 K. Analysis: 1. Subject to the assumption of an infinitely long fin, the temperature distributions are determined from Equation 3.84, which may be expressed as T T앝 (Tb T앝)emx

3.6

䊏

163

Heat Transfer from Extended Surfaces

where m (hP/kAc)1/2 (4h/kD)1/2. Substituting for h and D, as well as for the thermal conductivities of copper, the aluminum alloy, and the stainless steel, respectively, the values of m are 14.2, 21.2, and 75.6 m1. The temperature distributions may then be computed and plotted as follows: 100 316 SS

T (°C)

80

2024 Al Cu

60

40

T∞

20

0

50

100

150

200

250

300

x (mm)

From these distributions, it is evident that there is little additional heat transfer associated with extending the length of the rod much beyond 50, 200, and 300 mm, respectively, for the stainless steel, the aluminum alloy, and the copper. From Equation 3.85, the heat loss is qf 兹hPkAc b Hence for copper,

冤

qf 100 W/m2 䡠 K 0.005 m

冥

398 W/m 䡠 K (0.005 m)2 4

1/2

(100 25) C

8.3 W

䉰

Similarly, for the aluminum alloy and stainless steel, respectively, the heat rates are qf 5.6 W and 1.6 W. 2. Since there is no heat loss from the tip of an infinitely long rod, an estimate of the validity of this approximation may be made by comparing Equations 3.81 and 3.85. To a satisfactory approximation, the expressions provide equivalent results if tanh mL 0.99 or mL 2.65. Hence a rod may be assumed to be infinitely long if L L앝 ⬅ 2.65 m 2.65

冢 冣 kAc hP

1/2

For copper, L앝 2.65

冤

冥

398 W/m 䡠 K (/4)(0.005 m)2 100 W/m2 䡠 K (0.005 m)

1/2

0.19 m

䉰

Results for the aluminum alloy and stainless steel are L앝 0.13 m and L앝 0.04 m, respectively.

164

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Comments: 1. The foregoing results suggest that the fin heat transfer rate may accurately be predicted from the infinite fin approximation if mL 2.65. However, if the infinite fin approximation is to accurately predict the temperature distribution T(x), a larger value of mL would be required. This value may be inferred from Equation 3.84 and the requirement that the tip temperature be very close to the fluid temperature. Hence, if we require that (L)/b exp(mL) 0.01, it follows that mL 4.6, in which case L앝 ⬇ 0.33, 0.23, and 0.07 m for the copper, aluminum alloy, and stainless steel, respectively. These results are consistent with the distributions plotted in part 1. 2. This example is solved in the Advanced section of IHT.

3.6.3

Fin Performance

Recall that fins are used to increase the heat transfer from a surface by increasing the effective surface area. However, the fin itself represents a conduction resistance to heat transfer from the original surface. For this reason, there is no assurance that the heat transfer rate will be increased through the use of fins. An assessment of this matter may be made by evaluating the fin effectiveness f. It is defined as the ratio of the fin heat transfer rate to the heat transfer rate that would exist without the fin. Therefore f

qf hAc,bb

(3.86)

where Ac,b is the fin cross-sectional area at the base. In any rational design the value of f should be as large as possible, and in general, the use of fins may rarely be justified unless 2. f ⬃ Subject to any one of the four tip conditions that have been considered, the effectiveness for a fin of uniform cross section may be obtained by dividing the appropriate expression for qf in Table 3.4 by hAc,bb. Although the installation of fins will alter the surface convection coefficient, this effect is commonly neglected. Hence, assuming the convection coefficient of the finned surface to be equivalent to that of the unfinned base, it follows that, for the infinite fin approximation (Case D), the result is

冢 冣

f kP hAc

1/2

(3.87)

Several important trends may be inferred from this result. Obviously, fin effectiveness is enhanced by the choice of a material of high thermal conductivity. Aluminum alloys and copper come to mind. However, although copper is superior from the standpoint of thermal conductivity, aluminum alloys are the more common choice because of additional benefits related to lower cost and weight. Fin effectiveness is also enhanced by increasing the ratio of the perimeter to the cross-sectional area. For this reason, the use of thin, but closely spaced fins, is preferred, with the proviso that the fin gap not be reduced to a value for which flow between the fins is severely impeded, thereby reducing the convection coefficient. Equation 3.87 also suggests that the use of fins can be better justified under conditions for which the convection coefficient h is small. Hence from Table 1.1 it is evident that the need for fins is stronger when the fluid is a gas rather than a liquid and when the surface heat transfer is by free convection. If fins are to be used on a surface separating a gas and a liquid, they are

3.6

䊏

165

Heat Transfer from Extended Surfaces

generally placed on the gas side, which is the side of lower convection coefficient. A common example is the tubing in an automobile radiator. Fins are applied to the outer tube surface, over which there is flow of ambient air (small h), and not to the inner surface, through which there is flow of water (large h). Note that, if f 2 is used as a criterion to justify the implementation of fins, Equation 3.87 yields the requirement that (kP/hAc) 4. Equation 3.87 provides an upper limit to f, which is reached as L approaches infinity. However, it is certainly not necessary to use very long fins to achieve near maximum heat transfer enhancement. As seen in Example 3.8, 99% of the maximum possible fin heat transfer rate is achieved for mL 2.65. Hence, it would make no sense to extend the fins beyond L 2.65/m. Fin performance may also be quantified in terms of a thermal resistance. Treating the difference between the base and fluid temperatures as the driving potential, a fin resistance may be defined as Rt,f qb

(3.88)

f

This result is extremely useful, particularly when representing a finned surface by a thermal circuit. Note that, according to the fin tip condition, an appropriate expression for qf may be obtained from Table 3.4. Dividing Equation 3.88 into the expression for the thermal resistance due to convection at the exposed base, Rt,b 1 hAc,b

(3.89)

and substituting from Equation 3.86, it follows that f

Rt,b Rt, f

(3.90)

Hence the fin effectiveness may be interpreted as a ratio of thermal resistances, and to increase f it is necessary to reduce the conduction/convection resistance of the fin. If the fin is to enhance heat transfer, its resistance must not exceed that of the exposed base. Another measure of fin thermal performance is provided by the fin efficiency f. The maximum driving potential for convection is the temperature difference between the base (x 0) and the fluid, b Tb – T앝. Hence the maximum rate at which a fin could dissipate energy is the rate that would exist if the entire fin surface were at the base temperature. However, since any fin is characterized by a finite conduction resistance, a temperature gradient must exist along the fin and the preceding condition is an idealization. A logical definition of fin efficiency is therefore qf f ⬅ q max

qf hAf b

(3.91)

where Af is the surface area of the fin. For a straight fin of uniform cross section and an adiabatic tip, Equations 3.81 and 3.91 yield hf M tanh mL tanh mL hPLb mL

(3.92)

Referring to Table B.1, this result tells us that f approaches its maximum and minimum values of 1 and 0, respectively, as L approaches 0 and 앝.

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

In lieu of the somewhat cumbersome expression for heat transfer from a straight rectangular fin with an active tip, Equation 3.77, it has been shown that approximate, yet accurate, predictions may be obtained by using the adiabatic tip result, Equation 3.81, with a corrected fin length of the form Lc L (t/2) for a rectangular fin and Lc L (D/4) for a pin fin [14]. The correction is based on assuming equivalence between heat transfer from the actual fin with tip convection and heat transfer from a longer, hypothetical fin with an adiabatic tip. Hence, with tip convection, the fin heat rate may be approximated as qf M tanh mLc

(3.93)

and the corresponding efficiency as hf

tanh mLc mLc

(3.94)

Errors associated with the approximation are negligible if (ht/k) or (hD/2k) 0.0625 [15]. If the width of a rectangular fin is much larger than its thickness, w t, the perimeter may be approximated as P 2w, and

冢 冣

mLc hP kAc

冢 冣

1/2

Lc 2h kt

1/2

Lc

Multiplying numerator and denominator by L1/2 c and introducing a corrected fin profile area, Ap Lc t, it follows that

冢 冣

mLc 2h kAp

1/2

L3/2 c

(3.95)

Hence, as shown in Figures 3.19 and 3.20, the efficiency of a rectangular fin with tip convection may be represented as a function of Lc3/2(h/kAp)1/2. 100

y ~ x2

80

y

x

Lc = L Ap = Lt /3

t/2 L

60

η f (%)

166

Lc = L + t/2 Ap = Lc t

40

L

t/2

y~x y

20

t/2

x Lc = L Ap = Lt /2

L 0

0

0.5

1.0

1.5

2.0

1/2 L3/2 c (h/kAp)

FIGURE 3.19 Efficiency of straight fins (rectangular, triangular, and parabolic profiles).

2.5

3.6

䊏

167

Heat Transfer from Extended Surfaces

100

80

η f (%)

60 1 = r2c /r1 40

2

20

L

r2c = r2 + t/2 t Lc = L + t/2 Ap = Lc t

3

5

r1 r2 0

0

0.5

1.0

1.5

2.0

2.5

1/2 L3/2 c (h/kAp)

FIGURE 3.20 Efficiency of annular fins of rectangular profile.

3.6.4

Fins of Nonuniform Cross-Sectional Area

Analysis of fin thermal behavior becomes more complex if the fin is of nonuniform cross section. For such cases the second term of Equation 3.66 must be retained, and the solutions are no longer in the form of simple exponential or hyperbolic functions. As a special case, consider the annular fin shown in the inset of Figure 3.20. Although the fin thickness is uniform (t is independent of r), the cross-sectional area, Ac 2rt, varies with r. Replacing x by r in Equation 3.66 and expressing the surface area as As 2(r 2 r 12), the general form of the fin equation reduces to d 2T 1 dT 2h (T T ) 0 앝 kt dr 2 r dr or, with m2 ⬅ 2h/kt and ⬅ T – T앝, d 2 1 d m2 0 dr 2 r dr The foregoing expression is a modified Bessel equation of order zero, and its general solution is of the form (r) C1I0(mr) C2K0(mr) where I0 and K0 are modified, zero-order Bessel functions of the first and second kinds, respectively. If the temperature at the base of the fin is prescribed, (r1) b, and an adiabatic tip is presumed, d/dr冨r 2 0, C1 and C2 may be evaluated to yield a temperature distribution of the form I (mr)K1(mr2) K0(mr)I1(mr2) 0 b I0(mr1)K1(mr2) K0(mr1)I1(mr2)

168

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

where I1(mr) d[I0(mr)]/d(mr) and K1(mr) –d[K0(mr)]/d(mr) are modified, first-order Bessel functions of the first and second kinds, respectively. The Bessel functions are tabulated in Appendix B. With the fin heat transfer rate expressed as qf kAc,b dT dr

冏

k(2r1t)

rr1

d dr

冏

rr1

it follows that qf 2kr1tbm

K1(mr1)I1(mr2) I1(mr1)K1(mr2) K0(mr1)I1(mr2) I0(mr1)K1(mr2)

from which the fin efficiency becomes f

qf h2(r22

r21)b

2r1 K1(mr1)I1(mr2) I1(mr1)K1(mr2) 2 2 K (mr )I (mr ) I (mr )K (mr ) m(r2 r1) 0 1 1 2 0 1 1 2

(3.96)

This result may be applied for an active (convecting) tip, if the tip radius r2 is replaced by a corrected radius of the form r2c r2 (t/2). Results are represented graphically in Figure 3.20. Knowledge of the thermal efficiency of a fin may be used to evaluate the fin resistance, where, from Equations 3.88 and 3.91, it follows that Rt, f

1 hAff

(3.97)

Expressions for the efficiency and surface area of several common fin geometries are summarized in Table 3.5. Although results for the fins of uniform thickness or diameter

TABLE 3.5

Efficiency of common fin shapes

Straight Fins Rectangular a Aƒ 2wLc Lc L (t/2) Ap tL

tanh mLc mLc

(3.94)

1 I1(2mL) mL I0(2mL)

(3.98)

2 [4(mL)2 1]1/2 1

(3.99)

f

t w L

Triangular a Aƒ 2w[L2 (t/2)2]1/2 Ap (t/2)L

f

t w L

Parabolica Aƒ w[C1L (L2/t)ln (t/L C1)] C1 [1 (t/L)2]1/2 Ap (t/3)L

y = (t/2)(1 – x/L)2

f

t w L x

3.6

TABLE 3.5

䊏

169

Heat Transfer from Extended Surfaces

Continued

Circular Fin Rectangular a 2 Aƒ 2 (r 2c r 12) r2c r2 (t/2) V (r 22 r 12)t

W-121

t

L r1

f C2

K1(mr1)I1(mr2c) I1(mr1)K1(mr2c) I0(mr1)K1(mr2c) K0(mr1)I1(mr2c) (2r1/m) C2 2 (r 2c r 21)

(3.96)

r2

Pin Fins Rectangular b Aƒ DLc Lc L (D/4) V (D2/4)L

tanh mLc mLc

(3.100)

2 I2(2mL) mL I1(2mL)

(3.101)

2 [4/9(mL)2 1]1/2 1

(3.102)

f

D

L

Triangular b D 2 [L (D/2)2]1/2 2 V (/12)D2L

f

Aƒ

D

L

Parabolic b Aƒ

L3 {C3C4 8D L ln [(2DC4/L) C3]}

y = (D/2)(1 – x/L)2

D

2D

f

L

C3 1 2(D/L)2 C4 [1 (D/L)2]1/2 V (/20)D2 L

x

m (2h/kt)1/2. m (4h/kD)1/2.

a b

were obtained by assuming an adiabatic tip, the effects of convection may be treated by using a corrected length (Equations 3.94 and 3.100) or radius (Equation 3.96). The triangular and parabolic fins are of nonuniform thickness that reduces to zero at the fin tip. Expressions for the profile area, Ap, or the volume, V, of a fin are also provided in Table 3.5. The volume of a straight fin is simply the product of its width and profile area, V wAp. Fin design is often motivated by a desire to minimize the fin material and/or related manufacturing costs required to achieve a prescribed cooling effectiveness. Hence, a straight triangular fin is attractive because, for equivalent heat transfer, it requires much less volume (fin material) than a rectangular profile. In this regard, heat dissipation per unit volume, (q/V)f,

170

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

is largest for a parabolic profile. However, since (q/V)f for the parabolic profile is only slightly larger than that for a triangular profile, its use can rarely be justified in view of its larger manufacturing costs. The annular fin of rectangular profile is commonly used to enhance heat transfer to or from circular tubes.

3.6.5

Overall Surface Efficiency

In contrast to the fin efficiency f, which characterizes the performance of a single fin, the overall surface efficiency o characterizes an array of fins and the base surface to which they are attached. Representative arrays are shown in Figure 3.21, where S designates the fin pitch. In each case the overall efficiency is defined as q q ho q t t max hAtb

(3.103)

where qt is the total heat rate from the surface area At associated with both the fins and the exposed portion of the base (often termed the prime surface). If there are N fins in the array, each of surface area Af , and the area of the prime surface is designated as Ab, the total surface area is At NAf Ab

(3.104)

The maximum possible heat rate would result if the entire fin surface, as well as the exposed base, were maintained at Tb. The total rate of heat transfer by convection from the fins and the prime (unfinned) surface may be expressed as qt Nf hAf b hAbb

(3.105)

where the convection coefficient h is assumed to be equivalent for the finned and prime surfaces and f is the efficiency of a single fin. Hence

冤

qt h[Nf Af (At NAf )]b hAt 1

NAf At

冥

(1 f ) b

(3.106)

r2 r1

t t S Tb

Tb

S

w T∞, h

L

(a)

FIGURE 3.21 Representative fin arrays. (a) Rectangular fins. (b) Annular fins.

(b)

3.6

䊏

171

Heat Transfer from Extended Surfaces

Substituting Equation (3.106) into (3.103), it follows that o 1

NAf At

(1 f)

(3.107)

From knowledge of o, Equation 3.103 may be used to calculate the total heat rate for a fin array. Recalling the definition of the fin thermal resistance, Equation 3.88, Equation 3.103 may be used to infer an expression for the thermal resistance of a fin array. That is, Rt,o qb t

1 hohAt

(3.108)

where Rt,o is an effective resistance that accounts for parallel heat flow paths by conduction/convection in the fins and by convection from the prime surface. Figure 3.22 illustrates the thermal circuits corresponding to the parallel paths and their representation in terms of an effective resistance. If fins are machined as an integral part of the wall from which they extend (Figure 3.22a), there is no contact resistance at their base. However, more commonly, fins are manufactured separately and are attached to the wall by a metallurgical or adhesive joint. Alternatively, the attachment may involve a press fit, for which the fins are forced into slots machined on the wall material. In such cases (Figure 3.22b), there is a thermal contact resistance Rt,c, which

(Nηf hAf)–1

qf Nqf

Tb

qb

T∞ qb

Tb

[h(At – NAf)]–1

qt

Tb

T∞, h

T∞

(ηo hAt)–1

(a)

R"t,c

(Nηf hA f)–1

R"t ,c /NAc,b

qf Tb

Nqf qb

Tb

T∞ qb [h(At – NAf)]–1

T∞, h

qt

Tb

( ηo(c)h At)–1 (b)

FIGURE 3.22 Fin array and thermal circuit. (a) Fins that are integral with the base. (b) Fins that are attached to the base.

T∞

172

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

may adversely influence overall thermal performance. An effective circuit resistance may again be obtained, where, with the contact resistance, Rt,o(c) qb t

1 ho(c)hAt

(3.109)

It is readily shown that the corresponding overall surface efficiency is ho(c) 1

NAf At

冢1 C 冣 hf

(3.110a)

1

where C1 1 f hAf (Rt,c /Ac,b)

(3.110b)

In manufacturing, care must be taken to render Rt,c Rt,f.

EXAMPLE 3.10 The engine cylinder of a motorcycle is constructed of 2024-T6 aluminum alloy and is of height H 0.15 m and outside diameter D 50 mm. Under typical operating conditions the outer surface of the cylinder is at a temperature of 500 K and is exposed to ambient air at 300 K, with a convection coefficient of 50 W/m2 䡠 K. Annular fins are integrally cast with the cylinder to increase heat transfer to the surroundings. Consider five such fins, which are of thickness t 6 mm, length L 20 mm, and equally spaced. What is the increase in heat transfer due to use of the fins?

SOLUTION Known: Operating conditions of a finned motorcycle cylinder. Find:

Increase in heat transfer associated with using fins.

Schematic: Engine cylinder cross section (2024 T6 Al alloy)

S H = 0.15 m

Tb = 500 K t = 6 mm

T∞ = 300 K h = 50 W/m2•K Air

r1 = 25 mm L = 20 mm r2 = 45 mm

Assumptions: 1. Steady-state conditions. 2. One-dimensional radial conduction in fins. 3. Constant properties.

3.6

䊏

173

Heat Transfer from Extended Surfaces

4. Negligible radiation exchange with surroundings. 5. Uniform convection coefficient over outer surface (with or without fins).

Properties: Table A.1, 2024-T6 aluminum (T 400 K): k 186 W/m 䡠 K. Analysis: With the fins in place, the heat transfer rate is given by Equation 3.106

冤

qt hAt 1

NAf At

冥

(1 f ) b

2 r 12) 2[(0.048 m)2 (0.025 m)2] 0.0105 m2 and, from Equawhere Af 2(r 2c tion 3.104, At NAƒ 2r1(H Nt) 0.0527 m2 2 (0.025 m) [0.15 m 0.03 m] 0.0716 m2. With r2c /r1 1.92, Lc 0.023 m, Ap 1.380 104 m2, we obtain 1/2 L3/2 0.15. Hence, from Figure 3.20, the fin efficiency is ƒ ⬇ 0.95. c (h/kAp) With the fins, the total heat transfer rate is then

冤

冥

2 qt 50 W/m2 䡠 K 0.0716 m2 1 0.0527 m2 (0.05) 200 K 690 W 0.0716 m Without the fins, the convection heat transfer rate would be

qwo h(2r1H)b 50 W/m2 䡠 K(2 0.025 m 0.15 m)200 K 236 W Hence

q qt qwo 454 W

䉰

Comments: 1. Although the fins significantly increase heat transfer from the cylinder, considerable improvement could still be obtained by increasing the number of fins. We assess this possibility by computing qt as a function of N, first by fixing the fin thickness at t 6 mm and increasing the number of fins by reducing the spacing between fins. Prescribing a fin clearance of 2 mm at each end of the array and a minimum fin gap of 4 mm, the maximum allowable number of fins is N H/S 0.15 m/(0.004 0.006) m 15. The parametric calculations yield the following variation of qt with N: 1600 1400

t = 6 mm qt (W)

1200 1000 800 600

5

7

9 11 Number of fins, N

13

15

The number of fins could also be increased by reducing the fin thickness. If the fin gap is fixed at (S t) 4 mm and manufacturing constraints dictate a minimum allowable fin thickness of 2 mm, up to N 25 fins may be accommodated. In this case the parametric calculations yield

Chapter 3

One-Dimensional, Steady-State Conduction

䊏

3000 2500 (S – t) = 4 mm 2000

qt (W)

174

1500 1000 500

5

10

15 Number of fins, N

25

20

The foregoing calculations are based on the assumption that h is not affected by a reduction in the fin gap. The assumption is reasonable as long as there is no interaction between boundary layers that develop on the opposing surfaces of adjoining fins. Note that, since NAf 2r1(H – Nt) for the prescribed conditions, qt increases nearly linearly with increasing N. 2. The Models/Extended Surfaces option in the Advanced section of IHT provides readyto-solve models for straight, pin, and circular fins, as well as for fin arrays. The models include the efficiency relations of Figures 3.19 and 3.20 and Table 3.5.

EXAMPLE 3.11 In Example 1.5, we saw that to generate an electrical power of P 9 W, the temperature of the PEM fuel cell had to be maintained at Tc ⬇ 56.4 C, which required removal of 11.25 W from the fuel cell and a cooling air velocity of V 9.4 m/s for T앝 25 C. To provide these convective conditions, the fuel cell is centered in a 50 mm 26 mm rectangular duct, with 10-mm gaps between the exterior of the 50 mm 50 mm 6 mm fuel cell and the top and bottom of the well-insulated duct wall. A small fan, powered by the fuel cell, is used to circulate the cooling air. Inspection of a particular fan vendor’s data sheets suggests that the ratio of the fan power consumption to the fan’s volumetric flow rate is ˙ 102 m3/s. Pf / ˙ f C 1000 W/(m3/s) for the range 104 f Duct

Duct

W

Without finned heat sink

W H

H

Lf

tc

tf

tb

Fuel cell Lc

Fuel cell

Wc a •

T∞, f Air

•

T∞, f Air

Lc Wc

With finned heat sink

3.6

䊏

175

Heat Transfer from Extended Surfaces

1. Determine the net electric power produced by the fuel cell–fan system, Pnet P Pf . 2. Consider the effect of attaching an aluminum (k 200 W/m 䡠 K) finned heat sink, of identical top and bottom sections, onto the fuel cell body. The contact joint has a thermal resistance of Rt,c 103 m2 䡠 K/W, and the base of the heat sink is of thickness tb 2 mm. Each of the N rectangular fins is of length Lf 8 mm and thickness tf 1 mm, and spans the entire length of the fuel cell, Lc 50 mm. With the heat sink in place, radiation losses are negligible and the convective heat transfer coefficient may be related to the size and geometry of a typical air channel by an expression of the form h 1.78 kair (Lf a)/(Lf 䡠 a), where a is the distance between fins. Draw an equivalent thermal circuit for part 2 and determine the total number of fins needed to reduce the fan power consumption to half of the value found in part 1.

SOLUTION Known: Dimensions of a fuel cell and finned heat sink, fuel cell operating temperature, rate of thermal energy generation, power production. Relationship between power consumed by a cooling fan and the fan airflow rate. Relationship between the convection coefficient and the air channel dimensions. Find: 1. The net power produced by the fuel cell–fan system when there is no heat sink. 2. The number of fins needed to reduce the fan power consumption found in part 1 by 50%. Schematic: Lc = 50 mm A

Fuel cell

T∞

Fan

H = 26 mm

tc = 6 mm

A

Finned heat sink

Finned heat sink Air

T∞ = 25°C, V Lf = 8 mm

a

tc = 6 mm

Fuel cell, Tc = 56.4°C

tb = 2 mm

tf = 1 mm

W = Wc = 50 mm Section A–A

H = 26 mm

176

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Assumptions: 1. Steady-state conditions. 2. Negligible heat transfer from the edges of the fuel cell, as well as from the front and back faces of the finned heat sink. 3. One-dimensional heat transfer through the heat sink. 4. Adiabatic fin tips. 5. Constant properties. 6. Negligible radiation when the heat sink is in place. – Properties: Table A.4. air (T 300 K): kair 0.0263 W/m 䡠 K, cp 1007 J/kg 䡠 K, 1.1614 kg/m3.

Analysis: 1. The volumetric flow rate of cooling air is ˙ f VAc, where Ac W (H – tc) is the crosssectional area of the flow region between the duct walls and the unfinned fuel cell. Therefore, ˙ f V[W(H tc)] 9.4 m/s [0.05 m (0.026 m 0.006 m)] 9.4 103 m3/s and Pnet P C ˙ f 9.0 W 1000 W/(m3/s) 9.4 103 m3/s 0.4 W

䉰

With this arrangement, the fan consumes more power than is generated by the fuel cell, and the system cannot produce net power. 2. To reduce the fan power consumption by 50%, the volumetric flow rate of air must be reduced to ˙ f 4.7 103 m3/s. The thermal circuit includes resistances for the contact joint, conduction through the base of the finned heat sink, and resistances for the exposed base of the finned side of the heat sink, as well as the fins. Rt,b Tc q

T∞ Rt,c

Rt,base Rt, f(N)

The thermal resistances for the contact joint and the base are /2LcWc (103 m2 䡠 K/W)/(2 0.05 m 0.05 m) 0.2 K/W Rt,c Rt,c and Rt,base tb /(2kLcWc) (0.002 m)/(2 200 W/m 䡠 K 0.05 m 0.05 m) 0.002 K/W

3.6

䊏

Heat Transfer from Extended Surfaces

177

where the factors of two account for the two sides of the heat sink assembly. For the portion of the base exposed to the cooling air, the thermal resistance is Rt,b 1/[h (2Wc Ntf )Lc ] 1/[h (2 0.05 m N 0.001 m) 0.05 m] which cannot be evaluated until the total number of fins on both sides, N, and h are determined. For a single fin, Rt, f b/qf , where, from Table 3.4 for a fin with an insulated fin tip, Rt, f (hPkAc)1/2/tanh(mLf). In our case, P 2(Lc tf) 2 (0.05 m 0.001 m) 0.102 m, Ac Lctf 0.05 m 0.001 m 0.00005 m2, and m 兹hP/kAc [h 0.102 m/(200 W/m 䡠 K 0.00005 m2)]1/2 Hence, Rt, f

(h 0.102 m 200 W/m 䡠 K 0.00005 m2)1/2 tanh(m 0.008 m)

and for N fins, Rt, f(N) Rt, f /N. As for Rt,b, Rt,f cannot be evaluated until h and N are determined. Also, h depends on a, the distance between fins, which in turn depends on N, according to a (2Wc Ntf)/N (2 0.05 m N 0.001 m)/N. Thus, specification of N will make it possible to calculate all resistances. From the thermal resistance network, the total thermal resistance is Rtot Rt,c Rt,base Requiv, where Requiv [Rt, b1 Rt, f(N)1]1. The equivalent fin resistance, Requiv, corresponding to the desired fuel cell temperature is found from the expression q

Tc T앝 Tc T앝 Rtot Rt,c Rt,base Requiv

in which case, Requiv

Tc T앝 (Rt,c Rt,base) q

(56.4 C 25 C)/11.25 W (0.2 0.002) K/W 2.59 K/W For N 22, the following values of the various parameters are obtained: a 0.0035 m, h 19.1 W/m2 䡠 K, m 13.9 m1, Rt,f(N) 2.94 K/W, Rt,b 13.5 K/W, Requiv 2.41 K/W, and Rtot 2.61 K/W, resulting in a fuel cell temperature of 54.4 C. Fuel cell temperatures associated with N 20 and N 24 fins are Tc 58.9 C and 50.7 C, respectively. The actual fuel cell temperature is closest to the desired value when N 22. Therefore, a total of 22 fins, 11 on top and 11 on the bottom, should be specified, resulting in Pnet P Pf 9.0 W 4.7 W 4.3 W

䉰

Comments: 1. The performance of the fuel cell–fan system is enhanced significantly by combining the finned heat sink with the fuel cell. Good thermal management can transform an impractical proposal into a viable concept. 2. The temperature of the cooling air increases as heat is transferred from the fuel cell. The temperature of the air leaving the finned heat sink may be calculated from an overall ˙ ). For part 1, T 25 C energy balance on the airflow, which yields To Ti q/(cp f o 3 3 3 10.28 W/(1.1614 kg/m 1007 J/kg 䡠 K 9.4 10 m /s) 25.9 C. For part 2, the

178

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

outlet air temperature is To 27.0 C. Hence, the operating temperature of the fuel cell will be slightly higher than predicted under the assumption that the cooling air temperature is constant at 25 C and will be closer to the desired value. 3. For the conditions in part 2, the convection heat transfer coefficient does not vary with the air velocity. The insensitivity of the value of h to the fluid velocity occurs frequently in cases where the flow is confined within passages of small cross-sectional area, as will be discussed in detail in Chapter 8. The fin’s influence on increasing or reducing the value of h relative to that of an unfinned surface should be taken into account in critical applications. 4. A more detailed analysis of the system would involve prediction of the pressure drop associated with the fan-induced flow of air through the gaps between the fins. 5. The adiabatic fin tip assumption is valid since the duct wall is well insulated.

3.7 The Bioheat Equation The topic of heat transfer within the human body is becoming increasingly important as new medical treatments are developed that involve extreme temperatures [16] and as we explore more adverse environments, such as the Arctic, underwater, or space. There are two main phenomena that make heat transfer in living tissues more complex than in conventional engineering materials: metabolic heat generation and the exchange of thermal energy between flowing blood and the surrounding tissue. Pennes [17] introduced a modification to the heat equation, now known as the Pennes or bioheat equation, to account for these effects. The bioheat equation is known to have limitations, but it continues to be a useful tool for understanding heat transfer in living tissues. In this section, we present a simplified version of the bioheat equation for the case of steady-state, one-dimensional heat transfer. Both the metabolic heat generation and exchange of thermal energy with the blood can be viewed as effects of thermal energy generation. Therefore, we can rewrite Equation 3.44 to account for these two heat sources as d2T q˙m q˙p (3.111) 0 k dx2 where q˙m and q˙p are the metabolic and perfusion heat source terms, respectively. The perfusion term accounts for energy exchange between the blood and the tissue and is an energy source or sink according to whether heat transfer is from or to the blood, respectively. The thermal conductivity has been assumed constant in writing Equation 3.111. Pennes proposed an expression for the perfusion term by assuming that within any small volume of tissue, the blood flowing in the small capillaries enters at an arterial temperature, Ta, and exits at the local tissue temperature, T. The rate at which heat is gained by the tissue is the rate at which heat is lost from the blood. If the perfusion rate is (m3/s of volumetric blood flow per m3 of tissue), the heat lost from the blood can be calculated from Equation 1.12e, or on a unit volume basis, q˙ p bcb(Ta T)

(3.112)

where b and cb are the blood density and specific heat, respectively. Note that b is the blood mass flow rate per unit volume of tissue.

3.7

䊏

179

The Bioheat Equation

Substituting Equation 3.112 into Equation 3.111, we find d 2T q˙m bcb(Ta T) 0 k dx 2

(3.113)

Drawing on our experience with extended surfaces, it is convenient to define an excess temperature of the form ⬅ T Ta q˙ m /bcb. Then, if we assume that Ta, q˙ m, , and the blood properties are all constant, Equation 3.113 can be rewritten as d 2 ˜ 2 m 0 (3.114) dx2 ˜ 2 bcb/k. This equation is identical in form to Equation 3.69. Depending on the where m form of the boundary conditions, it may therefore be possible to use the results of Table 3.4 to estimate the temperature distribution within the living tissue.

EXAMPLE 3.12 In Example 1.7, the temperature at the inner surface of the skin/fat layer was given as 35 C. In reality, this temperature depends on the existing heat transfer conditions, including phenomena occurring farther inside the body. Consider a region of muscle with a skin/fat layer over it. At a depth of Lm 30 mm into the muscle, the temperature can be assumed to be at the core body temperature of Tc 37 C. The muscle thermal conductivity . is km 0.5 W/m 䡠 K. The metabolic heat generation rate within the muscle is qm 700 W/m3. The perfusion rate is 0.0005 s1; the blood density and specific heat are b 1000 kg/m3 and cb 3600 J/kg 䡠 K, respectively, and the arterial blood temperature Ta is the same as the core body temperature. The thickness, emissivity, and thermal conductivity of the skin/fat layer are as given in Example 1.7; perfusion and metabolic heat generation within this layer can be neglected. We wish to predict the heat loss rate from the body and the temperature at the inner surface of the skin/fat layer for air and water environments of Example 1.7.

SOLUTION Known: Dimensions and thermal conductivities of a muscle layer and a skin/fat layer. Skin emissivity and surface area. Metabolic heat generation rate and perfusion rate within the muscle layer. Core body and arterial temperatures. Blood density and specific heat. Ambient conditions. Find:

Heat loss rate from body and temperature at inner surface of the skin/fat layer.

Schematic: Tc = 37°C

Muscle

Skin/Fat

q•m = 700 W/m3 q•p

Ti

ε = 0.95

ksf = 0.3 W/m•K

km = 0.5 W/m•K = 0.0005 sⴚ1

Lm = 30 mm x

Tsur = 297 K

Lsf = 3 mm

Air or water

T∞ = 297 K h = 2 W/m2•K (air) h = 200 W/m2•K (water)

180

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer through the muscle and skin/fat layers. 3. Metabolic heat generation rate, perfusion rate, arterial temperature, blood properties, and thermal conductivities are all uniform. 4. Radiation heat transfer coefficient is known from Example 1.7. 5. Solar irradiation is negligible. Analysis: We will combine an analysis of the muscle layer with a treatment of heat transfer through the skin/fat layer and into the environment. The rate of heat transfer through the skin/fat layer and into the environment can be expressed in terms of a total resistance, Rtot, as q

Ti T앝 Rtot

(1)

As in Example 3.1 and for exposure of the skin to the air, Rtot accounts for conduction through the skin/fat layer in series with heat transfer by convection and radiation, which act in parallel with each other. Thus, Rtot

冢

Lsf 1 1 ksf A 1/hA 1/hr A

冣

1

冢

L 1 sf 1 A ksf h hr

冣

Using the values from Example 1.7 for air, Rtot

冢

冣

1 0.003 m 1 0.076 K/W 1.8 m2 0.3 W/m 䡠 K (2 5.9) W/m2 䡠 K

For water, with hr 0 and h 200 W/m2 䡠 K, Rtot 0.0083 W/m2 䡠 K. Heat transfer in the muscle layer is governed by Equation 3.114. The boundary conditions are specified in terms of the temperatures, Tc and Ti, where Ti is, as yet, unknown. In terms of the excess temperature , the boundary conditions are then q˙ (0) Tc Ta mc c

and

b b

q˙ (Lm) Ti Ta mc i b b

Since we have two boundary conditions involving prescribed temperatures, the solution for is given by case C of Table 3.4, ˜ x sinh m ˜ (Lm x) ( / )sinh m i c ˜ Lm c sinh m The value of qf given in Table 3.4 would correspond to the heat transfer rate at x 0, but this is not of particular interest here. Rather, we seek the rate at which heat leaves the muscle and enters the skin/fat layer so that we can equate this quantity with the rate at which heat is transferred through the skin/fat layer and into the environment. Therefore, we calculate the heat transfer rate at x Lm as q

冏

km A dT dx xLm

冏

xLm

km A

d dx

冏

xLm

˜ c kmAm

˜ Lm 1 (i /c) cosh m ˜ Lm sinh m

(2)

3.7

䊏

181

The Bioheat Equation

Combining Equations 1 and 2 yields ˜ c km Am

˜ Lm 1 (i /c) cosh m T T앝 i ˜ Lm Rtot sinh m

This expression can be solved for Ti, recalling that Ti also appears in i.

Ti

冤 冢

冣

冥

q˙ ˜ Lm kmAm ˜ Rtot c Ta m cosh m ˜ Lm T앝 sinh m c b b

˜ Lm km Am ˜ Rtot cosh m ˜ Lm sinh m

where m˜ 兹bcb /km [0.0005 s1 1000 kg/m3 3600 J/kg 䡠 K/0.5 W/m 䡠 K]1/2 60 m1 sinh (m˜ Lm) sinh (60 m1 0.03 m) 2.94 and cosh (m˜ Lm) cosh (60 m1 0.03 m) 3.11 q˙ q˙ c Tc Ta mc mc b b

b b

1

0.0005 s

700 W/m3 1000 kg/m3 3600 J/kg 䡠 K

0.389 K The excess temperature can be expressed in kelvins or degrees Celsius, since it is a temperature difference. Thus, for air: {24 C 2.94 0.5 W/m 䡠 K 1.8 m2 60 m1 0.076 K/W[0.389 C (37 C 0.389 C) 3.11]} Ti 34.8 C 2.94 0.5 W/m 䡠 K 1.8 m2 60 m1 0.076 K/W 3.11

䉰

This result agrees well with the value of 35 C that was assumed for Example 1.7. Next we can find the heat loss rate: q

Ti T앝 34.8 C 24 C 142 W Rtot 0.076 K/W

䉰

Again this agrees well with the previous result. Repeating the calculation for water, we find Ti 28.2 C

䉰

q 514 W

䉰

Here the calculation of Example 1.7 was not accurate because it incorrectly assumed that the inside of the skin/fat layer would be at 35 C. Furthermore, the skin temperature in this case would be only 25.4 C based on this more complete calculation.

Comments: 1. In reality, our bodies adjust in many ways to the thermal environment. For example, if we are too cold, we will shiver, which increases our metabolic heat generation rate. If we are too warm, the perfusion rate near the skin surface will increase, locally raising the skin temperature to increase heat loss to the environment.

182

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

2. Measuring the true thermal conductivity of living tissue is very challenging, first because of the necessity of making invasive measurements in a living being, and second because it is difficult to experimentally separate the effects of heat conduction and perfusion. It is easier to measure an effective thermal conductivity that would account for the combined contributions of conduction and perfusion. However, this effective conductivity value necessarily depends on the perfusion rate, which in turn varies with the thermal environment and physical condition of the specimen. 3. The calculations can be repeated for a range of values of the perfusion rate, and the dependence of the heat loss rate on the perfusion rate is illustrated below. The effect is stronger for the case of the water environment, because the muscle temperature is lower and therefore the effect of perfusion by the warm arterial blood is more pronounced. 700 600 500

q(W)

400

Water environment

300

Air environment 200 100 0 0

0.0002

0.0004

0.0006

0.0008

0.001

(sⴚ1)

3.8 Thermoelectric Power Generation As noted in Section 1.6, approximately 60% of the energy consumed globally is wasted in the form of low-grade heat. As such, an opportunity exists to harvest this energy stream and convert some of it to useful power. One approach involves thermoelectric power generation, which operates on a fundamental principle termed the Seebeck effect that states when a temperature gradient is established within a material, a corresponding voltage gradient is induced. The Seebeck coefficient S is a material property representing the proportionality between voltage and temperature gradients and, accordingly, has units of volts/K. For a constant property material experiencing one-dimensional conduction, as illustrated in Figure 3.23a, (E1 E2) S(T1 T2)

(3.115)

Electrically conducting materials can exhibit either positive or negative values of the Seebeck coefficient, depending on how they scatter electrons. The Seebeck coefficient is very small in metals, but can be relatively large in some semiconducting materials. If the material of Figure 3.23a is installed in an electric circuit, the voltage difference induced by the Seebeck effect can drive an electric current I, and electric power can be

3.8

䊏

183

Thermoelectric Power Generation

T1, E1

q1

Thin metallic conductor

T1, E1

L

qP,1

n-type semiconductor, Sn

Thin metallic conductor

I

p-type semiconductor, Sp

I

x T2, E2 qP,2

+L T2, E2 I

q2/2

q2/2

I

Re,load (a)

(b)

FIGURE 3.23 Thermoelectric phenomena. (a) The Seebeck effect. (b) A simplified thermoelectric circuit consisting of one pair (N 1) of semiconducting pellets.

generated from waste heat that induces a temperature difference across the material. A simplified thermoelectric circuit, consisting of two pellets of semiconducting material, is shown in Figure 3.23b. By blending minute amounts of a secondary element into the pellet material, the direction of the current induced by the Seebeck effect can be manipulated. The resulting p- and n-type semiconductors, which are characterized by positive and negative Seebeck coefficients, respectively, can be arranged as shown in the figure. Heat is supplied to the top and lost from the bottom of the assembly, and thin metallic conductors connect the semiconductors to an external load represented by the electrical resistance, Re,load. Ultimately, the amount of electric power that is produced is governed by the heat transfer rates to and from the pair of semiconducting pellets shown in Figure 3.23b. In addition to inducing an electric current I, thermoelectric effects also induce the generation or absorption of heat at the interface between two dissimilar materials. This heat source or heat sink phenomenon is known as the Peltier effect, and the amount of heat absorbed qP is related to the Seebeck coefficients of the adjoining materials by an equation of the form qP I(Sp Sn)T ISp-nT

(3.116)

where the individual Seebeck coefficients in the preceding expression, Sp and Sn, correspond to the p- and n-type semiconductors, and the differential Seebeck coefficient is Sp-n ⬅ Sp – Sn. Temperature is expressed in kelvins in Equation 3.116. The heat absorption is positive (generation is negative) when the electric current flows from the n-type to the p-type semiconductor. Hence, in Figure 3.23b, Peltier heat absorption occurs at the warm interface between the semiconducting pellets and the upper, thin metallic conductor, while Peltier heat generation occurs at the cool interface between the pellets and the lower conductor. When T1 T2, the heat transfer rates to and from the device, q1 and q2, respectively, may be found by solving the appropriate form of the energy equation. For steady-state, one-dimensional conduction within the assembly of Figure 3.23b the analysis proceeds as follows. Assuming the thin metallic connectors are of relatively high thermal and electrical conductivity, Ohmic dissipation occurs exclusively within the semiconducting pellets, each of which has a cross-sectional area Ac,s. The thermal resistances of the metallic conductors are assumed to be negligible, as is heat transfer within any gas trapped between the semiconducting pellets. Recognizing that the electrical resistance of each of the two pellets may be

184

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

expressed as Re,s e,s(2L)/Ac,s where e,s is the electrical resistivity of the semiconducting material, Equation 3.43 may be used to find the uniform volumetric generation rate within each pellet q˙

I 2e,s

(3.117)

A2c,s

Assuming negligible contact resistances and identical, as well as constant, thermophysical properties in each of the two pellets (with the exception being Sp Sn), Equation C.7 may be used to write expressions for the heat conduction out of and into the semiconducting material q(x L) 2Ac,s

冤

q(x L) 2Ac,s

冥

I 2e,s L ks (T1 T2) 2 2L Ac,s

冤

冥

I 2e,sL ks (T1 T2) 2 2L Ac,s

(3.118a)

(3.118b)

The factor of 2 outside the brackets accounts for heat transfer in both pellets and, as evident, q(x L) q(x –L). Because of the Peltier effect, q1 and q2 are not equal to the heat transfer rates into and out of the pellets as expressed in Equations 3.118a,b. Incorporating Equation 3.116 in an energy balance for a control surface about the interface between the thin metallic conductor and the semiconductor material at x –L yields q1 q(x L) qP,1 q(x L) ISp-nT1

(3.119)

q2 q(x L) ISn-pT2 q(x L) ISp-nT2

(3.120)

Similarly at x L,

Combining Equations 3.118b and 3.119 yields Ac,sks I 2e,sL q1 (T1 T2) ISp-nT1 2 L Ac,s Similarly, combining Equations 3.118a and 3.120 gives Ac,sks I 2e,sL q2 (T1 T2) ISp-nT2 2 L Ac,s

(3.121)

(3.122)

From an overall energy balance on the thermoelectric device, the electric power produced by the Seebeck effect is P q1 q2 (3.123) Substituting Equations 3.121 and 3.122 into this expression yields P ISp-n(T1 T2) 4

I 2e,sL ISp-n(T1 T2) I 2 Re,tot Ac,s

(3.124)

where Re,tot 2Re,s. The voltage difference induced by the Seebeck effect is relatively small for a single pair of semiconducting pellets. To amplify the voltage difference, thermoelectric modules are fabricated, as shown schematically in Figure 3.24a where N 1 pairs of semiconducting pellets are wired in series. Thin layers of a dielectric material, usually a ceramic, sandwich the module to provide structural rigidity and electrical insulation from the surroundings. Assuming the

3.8

䊏

185

Thermoelectric Power Generation

Thin ceramic insulators q″

Thin metallic conductors

1

n

p n

p n

p

n

p n

p n

p

2L

I n=1

n=2

n=3

q″

n=N1 n=N

2

I

Re,load (a)

T∞,1 Rt,conv,1 qconv,1

ISp-n,effT1 I2Re,eff

T1 Thermoelectric module

Rt,cond,mod

ISp-n,eff(T1 T2) 2I2Re,eff I2Re,load

T2 qconv,2

ISp-n,effT2 I2Re,eff

Rt,conv,2

T∞,2 (b)

FIGURE 3.24 Thermoelectric module. (a) Cross-section of a module consisting of N semiconductor pairs. (b) Equivalent thermal circuit for a convectively heated and cooled module.

thermal resistances of the thin ceramic layers are negligible, q1, q2, and the total module electric power, PN, can be written by modifying Equations 3.121, 3.122, 3.124 as q1

1 (T T2) ISp-n,eff T1 I 2 Re,eff Rt,cond,mod 1

(3.125)

q2

1 (T T2) ISp-n,eff T2 I 2 Re,eff Rt,cond,mod 1

(3.126)

PN q1 q2 ISp-n,eff (T1 T2) 2 I 2 Re,eff

(3.127)

where Sp-n,eff NSp-n, and Re,eff NRe,s are the effective Seebeck coefficient and the total internal electrical resistance of the module while Rt,cond,mod L/NAsks is the conduction resistance associated with the module’s p-n semiconductor matrix. An equivalent thermal circuit for a convectively heated and cooled thermoelectric module is shown in Figure 3.24b. If heating or cooling were to be applied by radiation or conduction, the resistance network outside of the thermoelectric module portion of the circuit would be modified accordingly.

186

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Returning to the single thermoelectric circuit of Figure 3.23b, the efficiency is defined as TE ⬅ P/q1. From Equations 3.121 and 3.124, it can be seen that efficiency depends on the electrical current in a complex manner. However, the efficiency can be maximized by adjusting the current through changes in the load resistance. The resulting maximum efficiency is given as [18]

冢

TE 1

冣

T2 兹1 ZT 1 T1 兹1 ZT T /T 2 1

(3.128)

where T (T1 T2)/2, S ⬅ Sp Sn , and 2 Z S (3.129) e,s ks – – Since the efficiency increases with increasing ZT , ZT may be seen as a dimensionless – figure of merit associated with thermoelectric generation [19]. As ZT l 앝, TE l (1 T2 /T1) (1 Tc /Th) ⬅ C where C is the Carnot efficiency. As discussed in Section 1.3.2, the Carnot efficiency and, in turn, the thermoelectric efficiency cannot be determined until the appropriate hot and cold temperatures are calculated from a heat transfer analysis. – Because ZT is defined in terms of interrelated electrical and thermal conductivities, extensive research is being conducted to tailor the properties of the semiconducting pellets, primarily by manipulating the nanostructure of the material so as to independently control phonon and electron motion and, in turn, the thermal and electrical conductivities of the mater– ial. Currently, ZT values of approximately unity at room temperature are readily achieved. Finally, we note that thermoelectric modules can be operated in reverse; supplying electric power to the module allows one to control the heat transfer rates to or from the outer ceramic surfaces. Such thermoelectric chillers or thermoelectric heaters are used in a wide variety of applications. A comprehensive discussion of one-dimensional, steady-state heat transfer modeling associated with thermoelectric heating and cooling modules is available [20].

EXAMPLE 3.13 An array of M 48 thermoelectric modules is installed on the exhaust of a sports car. Each module has an effective Seebeck coefficient of Sp-n,eff 0.1435 V/K, and an internal electrical resistance of Re,eff 4 . In addition, each module is of width and length W 54 mm and contains N 100 pairs of semiconducting pellets. Each pellet has an overall length of 2L 5 mm and cross-sectional area Ac,s 1.2 105 m2 and is characterized by a thermal conductivity of ks 1.2 W/m 䡠 K. The hot side of each module is exposed to exhaust gases at T앝,1 550 C with h1 40 W/m2 䡠 K, while the opposite side of each module is cooled by pressurized water at T앝,2 105 C with h2 500 W/m2 䡠 K. If the modules are wired in series, and the load resistance is Re,load 400 , what is the electric power harvested from the hot exhaust gases? Pressurized water T∞,2 105°C h2 500 W/m2 • K 2L 5 mm

Exhaust gas T∞,1 550°C h1 40 W/m2 • K

W M 48 thermoelectric modules

W 54 mm N 100 pellet pairs

3.8

䊏

187

Thermoelectric Power Generation

SOLUTION Known: Thermoelectric module properties and dimensions, number of semiconductor pairs in each module, and number of modules in the array. Temperature of exhaust gas and pressurized water, as well as convection coefficients at the hot and cold module surfaces. Modules are wired in series, and the electrical resistance of the load is known. Find:

Power produced by the module array.

Schematic: 2L = 5 mm

Pressurized water T∞,2 = 105°C

h2 = 500

W/m2 • K

W = 54 mm I

h1 = 40 W/m2 • K

Exhaust gas

Re,load = 400 Ω

T∞,1 = 550°C I Pressurized water T∞,2 = 105°C

M = 48 Thermoelectric modules N = 100 semiconductor pairs per module

Assumptions: 1. Steady-state conditions. 2. One-dimensional heat transfer. 3. Constant properties. 4. Negligible electrical and thermal contact resistances. 5. Negligible radiation exchange and negligible heat transfer within the gas inside the modules. 6. Negligible conduction resistance posed by the metallic contacts and ceramic insulators of the modules. Analysis: We begin by analyzing a single module. The conduction resistance of each module’s semiconductor array is Rt,cond,mod

L 2.5 103 m 1.736 K/W NAc,s ks 100 1.2 105 m2 1.2 W/m 䡠 K

From Equation 3.125, q1

1 Rt,cond,mod

(T1 T2) ISp-n,eff T1 I 2 Re,eff

(T1 T2) 1.736 K/W

I 0.1435 V/K T1 I 2 4

(1)

while from Equation 3.126, q2

1 Rt,cond,mod

(T1 T2) ISp-n,eff T2 I 2 Re,eff

I 0.1435 V/K T2 I 2 4

(T1 T2) 1.736 K/W (2)

188

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

At the hot surface, Newton’s law of cooling may be written as q1 h1W 2(T앝,1 T1) 40 W/m2 䡠 K (0.054 m)2 [(550 273) K T1]

(3)

whereas at the cool surface, q2 h2W 2(T2 T앝,2) 500 W/m2 䡠 K (0.054 m)2 [T2 (105 273) K]

(4)

Four equations have been written that include five unknowns, q1, q2, T1, T2, and I. An additional equation is obtained from the electrical circuit. With the modules wired in series, the total electric power produced by all M 48 modules is equal to the electric power dissipated in the load resistance. Equation 3.127 yields Ptot MPN M[ISp-n,eff(T1 T2) 2I 2Re,eff] 48[I 0.1435 V/K (T1 T2) 2I 2 4 ] (5)

Since the electric power produced by the thermoelectric module is dissipated in the electrical load, it follows that Ptot I 2Rload I 2 400 Equations 1 through 6 may be solved simultaneously, yielding Ptot 46.9 W.

(6) 䉰

Comments: 1. Equations 1 through 5 can be readily written by inspecting the equivalent thermal circuit of Figure 3.24b. 2. The module surface temperatures are T1 173 C and T2 134 C, respectively. If these surface temperatures were specified in the problem statement, the electric power could be obtained directly from Equations 5 and 6. In any practical design of a thermoelectric generator, however, a heat transfer analysis must be conducted to determine the power generated. 3. Power generation is very sensitive to the convection heat transfer resistances. For h1 h2 l 앝, Ptot 5900 W. To reduce the thermal resistance between the module and fluid streams, finned heat sinks are often used to increase the temperature difference across the modules and, in turn, increase their power output. Good thermal management and design are crucial to maximizing the power generation. 4. Harvesting the thermal energy contained in the exhaust with thermoelectrics can eliminate the need for an alternator, resulting in an increase in the net power produced by the engine, a reduction in the automobile’s weight, and an increase in gas mileage of up to 10%. 5. Thermoelectric modules, operating in the heating mode, can be embedded in car seats and powered by thermoelectric exhaust harvesters, reducing energy costs associated with heating the entire passenger cabin. The seat modules can also be operated in the cooling mode, potentially eliminating the need for vapor compression air conditioning. Common refrigerants, such as R134a, are harmful greenhouse gases, and are emitted into the atmosphere by leakage through seals and connections, and by catastrophic leaks due to collisions. Replacing automobile vapor compression air conditioners with personalized thermoelectric seat coolers can eliminate the equivalent of 45 million metric tons of CO2 released into the atmosphere every year in the United States alone.

3.9

䊏

189

Micro- and Nanoscale Conduction

3.9 Micro- and Nanoscale Conduction We conclude the discussion of one-dimensional, steady-state conduction by considering situations for which the physical dimensions are on the order of, or smaller than, the mean free path of the energy carriers, leading to potentially important nano- or microscale effects.

Conduction Through Thin Gas Layers

3.9.1

Figure 3.25 shows instantaneous trajectories of gas molecules between two isothermal, solid surfaces separated by a distance L. As discussed in Section 1.2.1, even in the absence of bulk fluid motion individual molecules continually impinge on the two solid boundaries that are held at uniform surface temperatures Ts,1 and Ts,2, respectively. The molecules also collide with each other, exchanging energy within the gaseous medium. When the thickness of the gas layer is large, L L1 (Figure 3.25a), a particular gas molecule will collide more frequently with other gas molecules than with either of the solid boundaries. Alternatively, for a very thin gas layer, L L2 L1 (Figure 3.25b), the probability of a molecule striking either of the solid boundaries is high relative to the likelihood of it colliding with another molecule. The energy content of a gas molecule is associated with its translational, rotational, and vibrational kinetic energies. It is this molecular-scale kinetic energy that ultimately defines the temperature of the gas, and collisions between individual molecules determine the value of the thermal conductivity, as discussed in Section 2.2.1. However, the manner in which a gas molecule is reflected or scattered from the solid walls also affects its level of kinetic energy and, in turn, its temperature. Hence, wall–molecule collisions can become important in determining the heat rate, qx, as L/mfp becomes small. The collision with and subsequent scattering of an individual gas molecule from a solid wall can be described by a thermal accommodation coefficient, ␣t, ␣t

Ti Tsc Ti Ts

(3.130)

where Ti is the effective molecule temperature just prior to striking the solid surface, Tsc is the temperature of the molecule immediately after it is scattered or reflected by the surface, and Ts is the surface temperature. When the temperature of the scattered molecule is identical to the wall temperature, ␣t 1. Alternatively, if Tsc Ti, the molecule’s kinetic energy and temperature are unaffected by a collision with the wall and ␣t 0.

Ts,1

Ts,1

x

x L1 (a)

qx

qx

Ts,2

Ts,2

x

x L2 (b)

FIGURE 3.25 Molecule trajectories in (a) a relatively thick gas layer and (b) a relatively thin gas layer. Molecules collide with each other, and with the two solid walls.

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For one-dimensional conduction within an ideal gas contained between two surfaces held at temperatures Ts,1 and Ts,2 Ts,1, the heat rate through the gas layer may be expressed as [21] q

Ts,1 Ts,2 (Rt,mm Rt,ms)

(3.131)

where, at the molecular level, the thermal resistances are associated with molecule–molecule and molecule-surface collisions Rt,mm L kA

and

Rt,ms

mfp 2 ␣t ␣t kA

冤

冥冤9␥␥ 15冥

(3.132a,b)

In the preceding expression, ␥ ⬅ cp /cv is the specific heat ratio of the ideal gas. The two solids are assumed to be the same material with equal values of ␣t, and the temperature difference is assumed to be small relative to the cold wall, (Ts,1 – Ts,2)/Ts,2 1. Equations 3.132a,b may be combined to yield Rt,ms mfp 2 ␣t ␣t Rt,mm L

冤

冥冤9␥␥ 15冥

from which it is evident that Rt,ms may be neglected if L/mfp is large and ␣t 0. In this case, Equation 3.131 reduces to Equation 3.6. However, Rt,ms can be significant if L/mfp is small. From Equation 2.11 the mean free path increases as the gas pressure is decreased. Hence, Rt,ms increases with decreasing gas pressure, and the heat rate can be pressure dependent when L/mfp is small. Values of ␣t for specific gas and surface combinations range from 0.87 to 0.97 for air–aluminum and air–steel, but can be less than 0.02 when helium interacts with clean metallic surfaces [21]. Equations 3.131, 3.132a,b may be applied to situations for which L/mfp 0.1. For air at atmospheric pressure, this corresponds to L 10 nm.

3.9.2

Conduction Through Thin Solid Films

One-dimensional conduction across or along thin solid films was discussed in Section 2.2.1 in terms of the thermal conductivities kx and ky. The heat transfer rate across a thin solid film may be approximated by combining Equation 2.9a with Equation 3.5, yielding qx

k[1 mfp /(3L)]A kx A (Ts,1 Ts,2) (Ts,1 Ts,2) L L

(3.133)

When L/mfp is large, Equation (3.133) reduces to Equation 3.4. Many alternative expressions for kx are available and are discussed in the literature [21].

3.10 Summary Despite its inherent mathematical simplicity, one-dimensional, steady-state heat transfer occurs in numerous engineering applications. Although one-dimensional, steady-state conditions may not apply exactly, the assumptions may often be made to obtain results of reasonable accuracy. You should therefore be thoroughly familiar with the means by which such

3.10

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Summary

191

problems are treated. In particular, you should be comfortable with the use of equivalent thermal circuits and with the expressions for the conduction resistances that pertain to each of the three common geometries. You should also be familiar with how the heat equation and Fourier’s law may be used to obtain temperature distributions and the corresponding fluxes. The implications of an internally distributed source of energy should also be clearly understood. In addition, you should appreciate the important role that extended surfaces can play in the design of thermal systems and should have the facility to effect design and performance calculations for such surfaces. Finally, you should understand how the preceding concepts can be applied to analyze heat transfer in the human body, thermoelectric power generation, and micro- and nanoscale conduction. You may test your understanding of this chapter’s key concepts by addressing the following questions. • Under what conditions may it be said that the heat flux is a constant, independent of the direction of heat flow? For each of these conditions, use physical considerations to convince yourself that the heat flux would not be independent of direction if the condition were not satisfied. • For one-dimensional, steady-state conduction in a cylindrical or spherical shell without heat generation, is the radial heat flux independent of radius? Is the radial heat rate independent of radius? • For one-dimensional, steady-state conduction without heat generation, what is the shape of the temperature distribution in a plane wall? In a cylindrical shell? In a spherical shell? • What is the thermal resistance? How is it defined? What are its units? • For conduction across a plane wall, can you write the expression for the thermal resistance from memory? Similarly, can you write expressions for the thermal resistance associated with conduction across cylindrical and spherical shells? From memory, can you express the thermal resistances associated with convection from a surface and net radiation exchange between the surface and large surroundings? • What is the physical basis for existence of a critical insulation radius? How do the thermal conductivity and the convection coefficient affect its value? • How is the conduction resistance of a solid affected by its thermal conductivity? How is the convection resistance at a surface affected by the convection coefficient? How is the radiation resistance affected by the surface emissivity? • If heat is transferred from a surface by convection and radiation, how are the corresponding thermal resistances represented in a circuit? • Consider steady-state conduction through a plane wall separating fluids of different temperatures, T앝,i and T앝,o, adjoining the inner and outer surfaces, respectively. If the convection coefficient at the outer surface is five times larger than that at the inner surface, ho 5hi, what can you say about relative proximity of the corresponding surface temperatures, Ts,o and Ts,i, to their adjoining fluid temperatures? • Can a thermal conduction resistance be applied to a solid cylinder or sphere? • What is a contact resistance? How is it defined? What are its units for an interface of prescribed area? What are they for a unit area? • How is the contact resistance affected by the roughness of adjoining surfaces? • If the air in the contact region between two surfaces is replaced by helium, how is the thermal contact resistance affected? How is it affected if the region is evacuated? • What is the overall heat transfer coefficient? How is it defined, and how is it related to the total thermal resistance? What are its units? • In a solid circular cylinder experiencing uniform volumetric heating and convection heat transfer from its surface, how does the heat flux vary with radius? How does the heat rate vary with radius?

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• In a solid circular sphere experiencing uniform volumetric heating and convection heat transfer from its surface, how does the heat flux vary with radius? How does the heat rate vary with radius? • Is it possible to achieve steady-state conditions in a solid cylinder or sphere that is experiencing heat generation and whose surface is perfectly insulated? Explain. • Can a material experiencing heat generation be represented by a thermal resistance and included in a circuit analysis? If so, why? If not, why not? • What is the physical mechanism associated with cooking in a microwave oven? How do conditions differ from a conventional (convection or radiant) oven? • If radiation is incident on the surface of a semitransparent medium and is absorbed as it propagates through the medium, will the corresponding volumetric rate of heat generation q˙ be distributed uniformly in the medium? If not, how will q˙ vary with distance from the surface? • In what way is a plane wall that is of thickness 2L and experiences uniform volumetric heating and equivalent convection conditions at both surfaces similar to a plane wall that is of thickness L and experiences the same volumetric heating and convection conditions at one surface but whose opposite surface is well insulated? • What purpose is served by attaching fins to a surface? • In the derivation of the general form of the energy equation for an extended surface, why is the assumption of one-dimensional conduction an approximation? Under what conditions is it a good approximation? • Consider a straight fin of uniform cross section (Figure 3.15a). For an x-location in the fin, sketch the temperature distribution in the transverse (y-) direction, placing the origin of the coordinate at the midplane of the fin (t/2 y t/2). What is the form of a surface energy balance applied at the location (x, t/2)? • What is the fin effectiveness? What is its range of possible values? Under what conditions are fins most effective? • What is the fin efficiency? What is its range of possible values? Under what conditions will the efficiency be large? • What is the fin resistance? What are its units? • How are the effectiveness, efficiency, and thermal resistance of a fin affected if its thermal conductivity is increased? If the convection coefficient is increased? If the length of the fin is increased? If the thickness (or diameter) of the fin is increased? • Heat is transferred from hot water flowing through a tube to air flowing over the tube. To enhance the rate of heat transfer, should fins be installed on the tube interior or exterior surface? • A fin may be manufactured as an integral part of a surface by using a casting or extrusion process, or it may be separately brazed or adhered to the surface. From thermal considerations, which option is preferred? • Describe the physical origins of the two heat source terms in the bioheat equation. Under what conditions is the perfusion term a heat sink? • How do heat sinks increase the electric power generated by a thermoelectric device? • Under what conditions do thermal resistances associated with molecule–wall interactions become important?

䊏

Problems

193

References 1. Fried, E., “Thermal Conduction Contribution to Heat Transfer at Contacts,” in R. P. Tye, Ed., Thermal Conductivity, Vol. 2, Academic Press, London, 1969. 2. Eid, J. C., and V. W. Antonetti, “Small Scale Thermal Contact Resistance of Aluminum Against Silicon,” in C. L. Tien, V. P. Carey, and J. K. Ferrel, Eds., Heat Transfer—1986, Vol. 2, Hemisphere, New York, 1986, pp. 659–664. 3. Snaith, B., P. W. O’Callaghan, and S. D. Probert, Appl. Energy, 16, 175, 1984. 4. Yovanovich, M. M., “Theory and Application of Constriction and Spreading Resistance Concepts for Microelectronic Thermal Management,” Presented at the International Symposium on Cooling Technology for Electronic Equipment, Honolulu, 1987. 5. Peterson, G. P., and L. S. Fletcher, “Thermal Contact Resistance of Silicon Chip Bonding Materials,” Proceedings of the International Symposium on Cooling Technology for Electronic Equipment, Honolulu, 1987, pp. 438–448. 6. Yovanovich, M. M., and M. Tuarze, AIAA J. Spacecraft Rockets, 6, 1013, 1969. 7. Madhusudana, C. V., and L. S. Fletcher, AIAA J., 24, 510, 1986. 8. Yovanovich, M. M., “Recent Developments in Thermal Contact, Gap and Joint Conductance Theories and Experiment,” in C. L. Tien, V. P. Carey, and J. K. Ferrel, Eds., Heat Transfer—1986, Vol. 1, Hemisphere, New York, 1986, pp. 35–45.

9. Maxwell, J. C., A Treatise on Electricity and Magnetism, 3rd ed., Oxford University Press, Oxford, 1892. 10. Hamilton, R. L., and O. K. Crosser, I&EC Fund. 1, 187, 1962. 11. Jeffrey, D. J., Proc. Roy. Soc. A, 335, 355, 1973. 12. Hashin Z., and S. Shtrikman, J. Appl. Phys., 33, 3125, 1962. 13. Aichlmayr, H. T., and F. A. Kulacki, “The Effective Thermal Conductivity of Saturated Porous Media,” in J. P. Hartnett, A. Bar-Cohen, and Y. I Cho, Eds., Advances in Heat Transfer, Vol. 39, Academic Press, London, 2006. 14. Harper, D. R., and W. B. Brown, “Mathematical Equations for Heat Conduction in the Fins of Air Cooled Engines,” NACA Report No. 158, 1922. 15. Schneider, P. J., Conduction Heat Transfer, AddisonWesley, Reading, MA, 1957. 16. Diller, K. R., and T. P. Ryan, J. Heat Transfer, 120, 810, 1998. 17. Pennes, H. H., J. Applied Physiology, 85, 5, 1998. 18. Goldsmid, H. J., “Conversion Efficiency and Figure-ofMerit,” in D. M. Rowe, Ed., CRC Handbook of Thermoelectrics, Chap. 3, CRC Press, Boca Raton, 1995. 19. Majumdar, A., Science, 303, 777, 2004. 20. Hodes, M., IEEE Trans. Com. Pack. Tech., 28, 218, 2005. 21. Zhang, Z. M., Nano/Microscale Heat Transfer, McGrawHill, New York, 2007.

Problems Plane and Composite Walls 3.1 Consider the plane wall of Figure 3.1, separating hot and cold fluids at temperatures T앝,1 and T앝,2, respectively. Using surface energy balances as boundary conditions at x 0 and x L (see Equation 2.34), obtain the temperature distribution within the wall and the heat flux in terms of T앝,1, T앝,2, h1, h2, k, and L. 3.2 A new building to be located in a cold climate is being designed with a basement that has an L 200-mm-thick wall. Inner and outer basement wall temperatures are Ti 20 C and To 0 C, respectively. The architect can specify the wall material to be either aerated concrete block with kac 0.15 W/m 䡠 K, or stone mix concrete. To reduce the conduction heat flux through the stone mix wall to a level equivalent to that of the aerated concrete wall, what thickness of extruded polystyrene sheet must be applied onto the inner surface of the stone mix con-

crete wall? Floor dimensions of the basement are 20 m 30 m, and the expected rental rate is $50/m2/ month. What is the yearly cost, in terms of lost rental income, if the stone mix concrete wall with polystyrene insulation is specified? 3.3 The rear window of an automobile is defogged by passing warm air over its inner surface. (a) If the warm air is at T앝,i 40 C and the corresponding convection coefficient is hi 30 W/m2 䡠 K, what are the inner and outer surface temperatures of 4-mm-thick window glass, if the outside ambient air temperature is T앝,o 10 C and the associated convection coefficient is ho 65 W/m2 䡠 K? (b) In practice T앝,o and ho vary according to weather conditions and car speed. For values of ho 2, 65, and 100 W/m2 䡠 K, compute and plot the inner and outer surface temperatures as a function of T앝,o for –30 T앝,o 0 C.

194

Chapter 3

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One-Dimensional, Steady-State Conduction

3.4 The rear window of an automobile is defogged by attaching a thin, transparent, film-type heating element to its inner surface. By electrically heating this element, a uniform heat flux may be established at the inner surface.

(a) Show the thermal circuit representing the steady-state heat transfer situation. Be sure to label all elements, nodes, and heat rates. Leave in symbolic form. (b) Assume the following conditions: T앝 20 C, h 50 W/m2 䡠 K, and T1 30 C. Calculate the heat flux q0 that is required to maintain the bonded surface at T0 60 C.

(a) For 4-mm-thick window glass, determine the electrical power required per unit window area to maintain an inner surface temperature of 15 C when the interior air temperature and convection coefficient are T앝,i 25 C and hi 10 W/m2 䡠 K, while the exterior (ambient) air temperature and convection coefficient are T앝,o 10 C and ho 65 W/m2 䡠 K.

(c) Compute and plot the required heat flux as a function of the film thickness for 0 Lƒ 1 mm. (d) If the film is not transparent and all of the radiant heat flux is absorbed at its upper surface, determine the heat flux required to achieve bonding. Plot your results as a function of Lƒ for 0 Lƒ 1 mm.

(b) In practice T앝,o and ho vary according to weather conditions and car speed. For values of ho 2, 20, 65, and 100 W/m2 䡠 K, determine and plot the electrical power requirement as a function of T앝,o for 30 T앝,o 0 C. From your results, what can you conclude about the need for heater operation at low values of ho? How is this conclusion affected by the value of T앝,o? If h V n, where V is the vehicle speed and n is a positive exponent, how does the vehicle speed affect the need for heater operation?

3.7 The walls of a refrigerator are typically constructed by sandwiching a layer of insulation between sheet metal panels. Consider a wall made from fiberglass insulation of thermal conductivity ki 0.046 W/m 䡠 K and thickness Li 50 mm and steel panels, each of thermal conductivity kp 60 W/m 䡠 K and thickness Lp 3 mm. If the wall separates refrigerated air at T앝, i 4 C from ambient air at T앝,o 25 C, what is the heat gain per unit surface area? Coefficients associated with natural convection at the inner and outer surfaces may be approximated as hi ho 5 W/m2 䡠 K.

3.5 A dormitory at a large university, built 50 years ago, has exterior walls constructed of Ls 25-mm-thick sheathing with a thermal conductivity of ks 0.1 W/m 䡠 K. To reduce heat losses in the winter, the university decides to encapsulate the entire dormitory by applying an Li 25-mm-thick layer of extruded insulation characterized by ki 0.029 W/m 䡠 K to the exterior of the original sheathing. The extruded insulation is, in turn, covered with an Lg 5-mm-thick architectural glass with kg 1.4 W/m 䡠 K. Determine the heat flux through the original and retrofitted walls when the interior and exterior air temperatures are T앝,i 22 C and T앝,o 20 C, respectively. The inner and outer convection heat transfer coefficients are hi 5 W/m2 䡠 K and ho 25 W/m2 䡠 K, respectively. 3.6 In a manufacturing process, a transparent film is being bonded to a substrate as shown in the sketch. To cure the bond at a temperature T0, a radiant source is used to provide a heat flux q0 (W/m2), all of which is absorbed at the bonded surface. The back of the substrate is maintained at T1 while the free surface of the film is exposed to air at T앝 and a convection heat transfer coefficient h. Air

q0"

T∞, h Lf

Film

Ls

Substrate

Bond, T0

T1

Lf = 0.25 mm kf = 0.025 W/m•K Ls = 1.0 mm ks = 0.05 W/m•K

3.8 A t 10-mm-thick horizontal layer of water has a top surface temperature of Tc 4 C and a bottom surface temperature of Th 2 C. Determine the location of the solid–liquid interface at steady state. 3.9 A technique for measuring convection heat transfer coefficients involves bonding one surface of a thin metallic foil to an insulating material and exposing the other surface to the fluid flow conditions of interest. T∞, h Foil ( P"elec, Ts)

L

Foam Insulation (k)

Tb

By passing an electric current through the foil, heat is dissipated uniformly within the foil and the corresponding flux, Pelec, may be inferred from related voltage and current measurements. If the insulation thickness L and thermal conductivity k are known and the fluid, foil, and insulation temperatures (T앝, Ts, Tb) are measured, the convection coefficient may be determined. Consider conditions for which T앝 Tb 25 C, Pelec 2000 W/m2, L 10 mm, and k 0.040 W/m 䡠 K.

䊏

195

Problems

(a) With water flow over the surface, the foil temperature measurement yields Ts 27 C. Determine the convection coefficient. What error would be incurred by assuming all of the dissipated power to be transferred to the water by convection? (b) If, instead, air flows over the surface and the temperature measurement yields Ts 125 C, what is the convection coefficient? The foil has an emissivity of 0.15 and is exposed to large surroundings at 25 C. What error would be incurred by assuming all of the dissipated power to be transferred to the air by convection? (c) Typically, heat flux gages are operated at a fixed temperature (Ts), in which case the power dissipation provides a direct measure of the convection coefficient. For Ts 27 C, plot Pelec as a function of ho for 10 ho 1000 W/m2 䡠 K. What effect does ho have on the error associated with neglecting conduction through the insulation? 3.10 The wind chill, which is experienced on a cold, windy day, is related to increased heat transfer from exposed human skin to the surrounding atmosphere. Consider a layer of fatty tissue that is 3 mm thick and whose interior surface is maintained at a temperature of 36 C. On a calm day the convection heat transfer coefficient at the outer surface is 25 W/m2 䡠 K, but with 30 km/h winds it reaches 65 W/m2 䡠 K. In both cases the ambient air temperature is 15 C. (a) What is the ratio of the heat loss per unit area from the skin for the calm day to that for the windy day? (b) What will be the skin outer surface temperature for the calm day? For the windy day? (c) What temperature would the air have to assume on the calm day to produce the same heat loss occurring with the air temperature at 15 C on the windy day? 3.11 Determine the thermal conductivity of the carbon nanotube of Example 3.4 when the heating island temperature is measured to be Th 332.6 K, without evaluating the thermal resistances of the supports. The conditions are the same as in the example. 3.12 A thermopane window consists of two pieces of glass 7 mm thick that enclose an air space 7 mm thick. The window separates room air at 20 C from outside ambient air at 10 C. The convection coefficient associated with the inner (room-side) surface is 10 W/m2 䡠 K. (a) If the convection coefficient associated with the outer (ambient) air is ho 80 W/m2 䡠 K, what is the heat loss through a window that is 0.8 m long by 0.5 m wide? Neglect radiation, and assume the air enclosed between the panes to be stagnant. (b) Compute and plot the effect of ho on the heat loss for 10 ho 100 W/m2 䡠 K. Repeat this calculation for a

triple-pane construction in which a third pane and a second air space of equivalent thickness are added. 3.13 A house has a composite wall of wood, fiberglass insulation, and plaster board, as indicated in the sketch. On a cold winter day, the convection heat transfer coefficients are ho 60 W/m2 䡠 K and hi 30 W/m2 䡠 K. The total wall surface area is 350 m2. Glass fiber blanket (28 kg/m3), kb Plaster board, kp

Plywood siding, ks

Inside

Outside

hi, T∞, i = 20°C

ho, T∞, o = –15°C

10 mm

100 mm

Lp

20 mm

Ls

Lb

(a) Determine a symbolic expression for the total thermal resistance of the wall, including inside and outside convection effects for the prescribed conditions. (b) Determine the total heat loss through the wall. (c) If the wind were blowing violently, raising ho to 300 W/m2 䡠 K, determine the percentage increase in the heat loss. (d) What is the controlling resistance that determines the amount of heat flow through the wall? 3.14 Consider the composite wall of Problem 3.13 under conditions for which the inside air is still characterized by T앝,i 20 C and hi 30 W/m2 䡠 K. However, use the more realistic conditions for which the outside air is characterized by a diurnal (time) varying temperature of the form T앝,o(K) 273 5 sin

冢224 t冣

T앝,o(K) 273 11 sin

冢224 t冣

0 t 12 h 12 t 24 h

with ho 60 W/m2 䡠 K. Assuming quasi-steady conditions for which changes in energy storage within the wall may be neglected, estimate the daily heat loss through the wall if its total surface area is 200 m2. 3.15 Consider a composite wall that includes an 8-mm-thick hardwood siding, 40-mm by 130-mm hardwood studs on 0.65-m centers with glass fiber insulation (paper

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One-Dimensional, Steady-State Conduction

faced, 28 kg/m3), and a 12-mm layer of gypsum (vermiculite) wall board. Wood siding Stud 130 mm

x

Insulation Wall board

3.19 The wall of a drying oven is constructed by sandwiching an insulation material of thermal conductivity k 0.05 W/m 䡠 K between thin metal sheets. The oven air is at T앝,i 300 C, and the corresponding convection coefficient is hi 30 W/m2 䡠 K. The inner wall surface absorbs a radiant flux of qrad 100 W/m2 from hotter objects within the oven. The room air is at T앝,o 25 C, and the overall coefficient for convection and radiation from the outer surface is ho 10 W/m2 䡠 K.

40 mm

What is the thermal resistance associated with a wall that is 2.5 m high by 6.5 m wide (having 10 studs, each 2.5 m high)? Assume surfaces normal to the x-direction are isothermal. 3.16 Work Problem 3.15 assuming surfaces parallel to the x-direction are adiabatic. 3.17 Consider the oven of Problem 1.54. The walls of the oven consist of L 30-mm-thick layers of insulation characterized by kins 0.03 W/m 䡠 K that are sandwiched between two thin layers of sheet metal. The exterior surface of the oven is exposed to air at 23 C with hext 2 W/m2 䡠 K. The interior oven air temperature is 180 C. Neglecting radiation heat transfer, determine the steady-state heat flux through the oven walls when the convection mode is disabled and the free convection coefficient at the inner oven surface is hfr 3 W/m2 䡠 K. Determine the heat flux through the oven walls when the convection mode is activated, in which case the forced convection coefficient at the inner oven surface is hfo 27 W/m2 䡠 K. Does operation of the oven in its convection mode result in significantly increased heat losses from the oven to the kitchen? Would your conclusion change if radiation were included in your analysis? 3.18 The composite wall of an oven consists of three materials, two of which are of known thermal conductivity, kA 20 W/m 䡠 K and kC 50 W/m 䡠 K, and known thickness, LA 0.30 m and LC 0.15 m. The third material, B, which is sandwiched between materials A and C, is of known thickness, LB 0.15 m, but unknown thermal conductivity kB. Ts, i

kA

kB

kC

LA

LB

LC

Ts,o

Air

T∞, h

Under steady-state operating conditions, measurements reveal an outer surface temperature of Ts,o 20 C, an inner surface temperature of Ts,i 600 C, and an oven air temperature of T앝 800 C. The inside convection coefficient h is known to be 25 W/m2 䡠 K. What is the value of kB?

Absorbed radiation, q"rad

Insulation, k

To

Oven air

Room air

T∞,i, hi

T∞,o, ho

L

(a) Draw the thermal circuit for the wall and label all temperatures, heat rates, and thermal resistances. (b) What insulation thickness L is required to maintain the outer wall surface at a safe-to-touch temperature of To 40 C? 3.20 The t 4-mm-thick glass windows of an automobile have a surface area of A 2.6 m2. The outside temperature is T앝,o 32 C while the passenger compartment is maintained at T앝,i 22 C. The convection heat transfer coefficient on the exterior window surface is ho 90 W/m2 䡠 K. Determine the heat gain through the windows when the interior convection heat transfer coefficient is hi 15 W/m2 䡠 K. By controlling the airflow in the passenger compartment the interior heat transfer coefficient can be reduced to hi 5 W/m2 䡠 K without sacrificing passenger comfort. Determine the heat gain through the window for the reduced inside heat transfer coefficient. 3.21 The thermal characteristics of a small, dormitory refrigerator are determined by performing two separate experiments, each with the door closed and the refrigerator placed in ambient air at T앝 25 C. In one case, an electric heater is suspended in the refrigerator cavity, while the refrigerator is unplugged. With the heater dissipating 20 W, a steady-state temperature of 90 C is recorded within the cavity. With the heater removed and the refrigerator now in operation, the second experiment involves maintaining a steady-state cavity temperature of 5 C for a fixed time interval and recording the electrical energy required to operate the refrigerator. In such an experiment for which steady operation is maintained over a 12-hour period, the input electrical energy is 125,000 J. Determine the refrigerator’s coefficient of performance (COP). 3.22 In the design of buildings, energy conservation requirements dictate that the exterior surface area, As, be minimized. This requirement implies that, for a desired floor

䊏

197

Problems

space, there may be optimum values associated with the number of floors and horizontal dimensions of the building. Consider a design for which the total floor space, Af , and the vertical distance between floors, Hf , are prescribed.

aerogel (k 0.006 W/m 䡠 K). The temperatures of the surroundings and the ambient are Tsur 300 K and T앝 298 K, respectively. The outer surface is characterized by a convective heat transfer coefficient of h 12 W/m2 䡠 K.

(a) If the building has a square cross section of width W on a side, obtain an expression for the value of W that would minimize heat loss to the surroundings. Heat loss may be assumed to occur from the four vertical side walls and from a flat roof. Express your result in terms of Af and Hf.

(b) Calculate the outer surface temperature of the canister for the four cases (high and low thermal conductivity; high and low surface emissivity).

(b) If Af 32,768 m2 and Hf 4 m, for what values of W and Nf (the number of floors) is the heat loss minimized? If the average overall heat transfer coefficient is U 1 W/m2 䡠 K and the difference between the inside and ambient air temperatures is 25 C, what is the corresponding heat loss? What is the percentage reduction in heat loss compared with a building for Nf 2?

3.24 A firefighter’s protective clothing, referred to as a turnout coat, is typically constructed as an ensemble of three layers separated by air gaps, as shown schematically.

3.23 When raised to very high temperatures, many conventional liquid fuels dissociate into hydrogen and other components. Thus the advantage of a solid oxide fuel cell is that such a device can internally reform readily available liquid fuels into hydrogen that can then be used to produce electrical power in a manner similar to Example 1.5. Consider a portable solid oxide fuel cell, operating at a temperature of Tfc 800 C. The fuel cell is housed within a cylindrical canister of diameter D 75 mm and length L 120 mm. The outer surface of the canister is insulated with a low-thermal-conductivity material. For a particular application, it is desired that the thermal signature of the canister be small, to avoid its detection by infrared sensors. The degree to which the canister can be detected with an infrared sensor may be estimated by equating the radiation heat flux emitted from the exterior surface of the canister (Equation 1.5; Es sT 4s ) to the heat flux emitted from an equivalent black surface, (Eb T b4). If the equivalent black surface temperature Tb is near the surroundings temperature, the thermal signature of the canister is too small to be detected—the canister is indistinguishable from the surroundings. (a) Determine the required thickness of insulation to be applied to the cylindrical wall of the canister to ensure that the canister does not become highly visible to an infrared sensor (i.e., Tb Tsur 5 K). Consider cases where (i) the outer surface is covered with a very thin layer of dirt (s 0.90) and (ii) the outer surface is comprised of a very thin polished aluminum sheet (s 0.08). Calculate the required thicknesses for two types of insulating material, calcium silicate (k 0.09 W/m 䡠 K) and

(c) Calculate the heat loss from the cylindrical walls of the canister for the four cases.

Moisture barrier (mb)

Shell (s)

1 mm

Fire-side

ks, Ls

kmb Lmb

Air gap

Thermal liner (tl)

1 mm

k tl L tl

Firefighter

Air gap

Representative dimensions and thermal conductivities for the layers are as follows. Layer Shell (s) Moisture barrier (mb) Thermal liner (tl)

Thickness (mm) 0.8 0.55 3.5

k (W/m 䡠 K) 0.047 0.012 0.038

The air gaps between the layers are 1 mm thick, and heat is transferred by conduction and radiation exchange through the stagnant air. The linearized radiation coefficient for a gap may be approximated 3 as, hrad (T1 T2)(T 12 T 22) 艐 4T avg , where Tavg represents the average temperature of the surfaces comprising the gap, and the radiation flux across the gap may be expressed as qrad hrad (T1 T2). (a) Represent the turnout coat by a thermal circuit, labeling all the thermal resistances. Calculate and tabulate the thermal resistances per unit area (m2 䡠 K/W) for each of the layers, as well as for the conduction and radiation processes in the gaps. Assume that a value of Tavg 470 K may be used to approximate the radiation resistance of both gaps. Comment on the relative magnitudes of the resistances. (b) For a pre-flash-over fire environment in which firefighters often work, the typical radiant heat flux on the fire-side of the turnout coat is 0.25 W/cm2.

198

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One-Dimensional, Steady-State Conduction

What is the outer surface temperature of the turnout coat if the inner surface temperature is 66 C, a condition that would result in burn injury? 3.25 A particular thermal system involves three objects of fixed shape with conduction resistances of R1 1 K/W, R2 2 K/W and R3 4 K/W, respectively. An objective is to minimize the total thermal resistance Rtot associated with a combination of R1, R2, and R3. The chief engineer is willing to invest limited funds to specify an alternative material for just one of the three objects; the alternative material will have a thermal conductivity that is twice its nominal value. Which object (1, 2, or 3) should be fabricated of the higher thermal conductivity material to most significantly decrease Rtot? Hint: Consider two cases, one for which the three thermal resistances are arranged in series, and the second for which the three resistances are arranged in parallel.

Contact Resistance 3.26 A composite wall separates combustion gases at 2600 C from a liquid coolant at 100 C, with gas- and liquid-side convection coefficients of 50 and 1000 W/m2 䡠 K. The wall is composed of a 10-mm-thick layer of beryllium oxide on the gas side and a 20-mm-thick slab of stainless steel (AISI 304) on the liquid side. The contact resistance between the oxide and the steel is 0.05 m2 䡠 K/W. What is the heat loss per unit surface area of the composite? Sketch the temperature distribution from the gas to the liquid. 3.27 Approximately 106 discrete electrical components can be placed on a single integrated circuit (chip), with electrical heat dissipation as high as 30,000 W/m2. The chip, which is very thin, is exposed to a dielectric liquid at its outer surface, with ho 1000 W/m2 䡠 K and T앝,o 20 C, and is joined to a circuit board at its inner surface. The thermal contact resistance between the chip and the board is 104 m2 䡠 K/W, and the board thickness and thermal conductivity are Lb 5 mm and kb 1 W/m 䡠 K, respectively. The other surface of the board is exposed to ambient air for which hi 40 W/m2 䡠 K and T앝,i 20 C.

(a) Sketch the equivalent thermal circuit corresponding to steady-state conditions. In variable form, label appropriate resistances, temperatures, and heat fluxes. (b) Under steady-state conditions for which the chip heat dissipation is qc 30,000 W/m2, what is the chip temperature? (c) The maximum allowable heat flux, qc,m, is determined by the constraint that the chip temperature must not exceed 85 C. Determine qc,m for the foregoing conditions. If air is used in lieu of the dielectric liquid, the convection coefficient is reduced by approximately an order of magnitude. What is the value of qc,m for ho 100 W/m2 䡠 K? With air cooling, can significant improvements be realized by using an aluminum oxide circuit board and/or by using a conductive paste at the chip/board interface for which Rt, c 105 m2 䡠 K/W? 3.28 Two stainless steel plates 10 mm thick are subjected to a contact pressure of 1 bar under vacuum conditions for which there is an overall temperature drop of 100 C across the plates. What is the heat flux through the plates? What is the temperature drop across the contact plane? 3.29 Consider a plane composite wall that is composed of two materials of thermal conductivities kA 0.1 W/m 䡠 K and kB 0.04 W/m 䡠 K and thicknesses LA 10 mm and LB 20 mm. The contact resistance at the interface between the two materials is known to be 0.30 m2 䡠 K/W. Material A adjoins a fluid at 200 C for which h 10 W/m2 䡠 K, and material B adjoins a fluid at 40 C for which h 20 W/m2 䡠 K. (a) What is the rate of heat transfer through a wall that is 2 m high by 2.5 m wide? (b) Sketch the temperature distribution. 3.30 The performance of gas turbine engines may be improved by increasing the tolerance of the turbine blades to hot gases emerging from the combustor. One approach to achieving high operating temperatures involves application of a thermal barrier coating (TBC) to the exterior surface of a blade, while passing cooling air through the blade. Typically, the blade is made from a high-temperature superalloy, such as Inconel (k ⬇ 25 W/m 䡠 K), while a ceramic, such as zirconia (k ⬇ 1.3 W/m 䡠 K), is used as a TBC.

Coolant

T∞,o, ho

Superalloy Chip q"c, Tc Thermal contact resistance, R"t,c Board, kb

Lb

Cooling air

T∞,i, hi Hot gases

T∞,o, ho

Air

Bonding agent

T∞,i, hi

TBC

䊏

199

Problems

Consider conditions for which hot gases at T앝,o 1700 K and cooling air at T앝,i 400 K provide outer and inner surface convection coefficients of ho 1000 W/m2 䡠 K and hi 500 W/m2 䡠 K, respectively. If a 0.5-mm-thick zirconia TBC is attached to a 5-mmthick Inconel blade wall by means of a metallic bonding agent, which provides an interfacial thermal resistance of Rt,c 104 m2 䡠 K/W, can the Inconel be maintained at a temperature that is below its maximum allowable value of 1250 K? Radiation effects may be neglected, and the turbine blade may be approximated as a plane wall. Plot the temperature distribution with and without the TBC. Are there any limits to the thickness of the TBC? 3.31 A commercial grade cubical freezer, 3 m on a side, has a composite wall consisting of an exterior sheet of 6.35-mm-thick plain carbon steel, an intermediate layer of 100-mm-thick cork insulation, and an inner sheet of 6.35-mm-thick aluminum alloy (2024). Adhesive interfaces between the insulation and the metallic strips are each characterized by a thermal contact resistance of Rt,c 2.5 104 m2 䡠 K/W. What is the steady-state cooling load that must be maintained by the refrigerator under conditions for which the outer and inner surface temperatures are 22 C and 6 C, respectively? 3.32 Physicists have determined the theoretical value of the thermal conductivity of a carbon nanotube to be kcn,T 5000 W/m 䡠 K. (a) Assuming the actual thermal conductivity of the carbon nanotube is the same as its theoretical value, find the thermal contact resistance, Rt,c, that exists between the carbon nanotube and the top surfaces of the heated and sensing islands in Example 3.4 .

Tsur

Transistor case Ts,c, Pelec

Base plate, (k,ε ) Interface, Ac

W Enclosure

Air

T∞, h L

(a) If the air-filled aluminum-to-aluminum interface is characterized by an area of Ac 2 104 m2 and a roughness of 10 m, what is the maximum allowable power dissipation if the surface temperature of the case, Ts,c, is not to exceed 85 C? (b) The convection coefficient may be increased by subjecting the plate surface to a forced flow of air. Explore the effect of increasing the coefficient over the range 4 h 200 W/m2 䡠 K.

Porous Media 3.34 Ring-porous woods, such as oak, are characterized by grains. The dark grains consist of very low-density material that forms early in the springtime. The surrounding lighter-colored wood is composed of highdensity material that forms slowly throughout most of the growing season. Wood grain (low-density)

(b) Using the value of the thermal contact resistance calculated in part (a), plot the fraction of the total resistance between the heated and sensing islands that is due to the thermal contact resistances for island separation distances of 5 m s 20 m. 3.33 Consider a power transistor encapsulated in an aluminum case that is attached at its base to a square aluminum plate of thermal conductivity k 240 W/m 䡠 K, thickness L 6 mm, and width W 20 mm. The case is joined to the plate by screws that maintain a contact pressure of 1 bar, and the back surface of the plate transfers heat by natural convection and radiation to ambient air and large surroundings at T앝 Tsur 25 C. The surface has an emissivity of 0.9, and the convection coefficient is h 4 W/m2 䡠 K. The case is completely enclosed such that heat transfer may be assumed to occur exclusively through the base plate.

High-density material

Assuming the low-density material is highly porous and the oak is dry, determine the fraction of the oak crosssection that appears as being grained. Hint: Assume the thermal conductivity parallel to the grains is the same as the radial conductivity of Table A.3. 3.35 A batt of glass fiber insulation is of density 28 kg/m3. Determine the maximum and minimum possible values of the effective thermal conductivity of the insulation at T 300 K, and compare with the value reported in Table A.3.

200

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

3.36 Air usually constitutes up to half of the volume of commercial ice creams and takes the form of small spherical bubbles interspersed within a matrix of frozen matter. The thermal conductivity of ice cream that contains no air is kna 1.1 W/m 䡠 K at T –20 C. Determine the thermal conductivity of commercial ice cream characterized by 0.20, also at T –20 C. 3.37 Determine the density, specific heat, and thermal conductivity of a lightweight aggregate concrete that is composed of 65% stone mix concrete and 35% air by volume. Evaluate properties at T 300 K. 3.38 A one-dimensional plane wall of thickness L is constructed of a solid material with a linear, nonuniform porosity distribution described by (x) max(x/L). Plot the steady-state temperature distribution, T(x), for ks 10 W/m 䡠 K, kf 0.1 W/m 䡠 K, L 1 m, max 0.25, T(x 0) 30 C and qx 100 W/m2 using the expression for the minimum effective thermal conductivity of a porous medium, the expression for the maximum effective thermal conductivity of a porous medium, Maxwell’s expression, and for the case where keff(x) ks.

Alternative Conduction Analysis 3.39 The diagram shows a conical section fabricated from pure aluminum. It is of circular cross section having diameter D ax1/2, where a 0.5 m1/2. The small end is located at x1 25 mm and the large end at x2 125 mm. The end temperatures are T1 600 K and T2 400 K, while the lateral surface is well insulated. T2 T1

x1 x

x2

(a) Derive an expression for the temperature distribution T(x) in symbolic form, assuming one-dimensional conditions. Sketch the temperature distribution.

0

x1 T1 x2

T2

The sides are well insulated, while the top surface of the cone at x1 is maintained at T1 and the bottom surface at x2 is maintained at T2. (a) Obtain an expression for the temperature distribution T(x). (b) What is the rate of heat transfer across the cone if it is constructed of pure aluminum with x1 0.075 m, T1 100 C, x2 0.225 m, and T2 20 C? 3.41 From Figure 2.5 it is evident that, over a wide temperature range, the temperature dependence of the thermal conductivity of many solids may be approximated by a linear expression of the form k ko aT, where ko is a positive constant and a is a coefficient that may be positive or negative. Obtain an expression for the heat flux across a plane wall whose inner and outer surfaces are maintained at T0 and T1, respectively. Sketch the forms of the temperature distribution corresponding to a 0, a 0, and a 0. 3.42 Consider a tube wall of inner and outer radii ri and ro, whose temperatures are maintained at Ti and To, respectively. The thermal conductivity of the cylinder is temperature dependent and may be represented by an expression of the form k ko(1 aT), where ko and a are constants. Obtain an expression for the heat transfer per unit length of the tube. What is the thermal resistance of the tube wall? 3.43 Measurements show that steady-state conduction through a plane wall without heat generation produced a convex temperature distribution such that the midpoint temperature was To higher than expected for a linear temperature distribution.

T1 T ( x)

TL/2 ∆To

(b) Calculate the heat rate qx. 3.40 A truncated solid cone is of circular cross section, and its diameter is related to the axial coordinate by an expression of the form D ax3/2, where a 1.0 m1/2.

T2

L

䊏

201

Problems

Assuming that the thermal conductivity has a linear dependence on temperature, k ko(1 ␣T), where ␣ is a constant, develop a relationship to evaluate ␣ in terms of To, T1, and T2. 3.44 A device used to measure the surface temperature of an object to within a spatial resolution of approximately 50 nm is shown in the schematic. It consists of an extremely sharp-tipped stylus and an extremely small cantilever that is scanned across the surface. The probe tip is of circular cross section and is fabricated of polycrystalline silicon dioxide. The ambient temperature is measured at the pivoted end of the cantilever as T⬁ 25 C, and the device is equipped with a sensor to measure the temperature at the upper end of the sharp tip, Tsen. The thermal resistance between the sensing probe and the pivoted end is Rt 5 106 K/W. (a) Determine the thermal resistance between the surface temperature and the sensing temperature. (b) If the sensing temperature is Tsen 28.5 C, determine the surface temperature. Hint: Although nanoscale heat transfer effects may be important, assume that the conduction occurring in the air adjacent to the probe tip can be described by Fourier’s law and the thermal conductivity found in Table A.4. Tsen

T∞ = 25°C

Cantilever

(T앝 25 C) that maintains a convection coefficient of h 25 W/m2 䡠 K and to large surroundings for which Tsur T앝 25 C. The surface emissivity of calcium silicate is approximately 0.8. Compute and plot the temperature distribution in the insulation as a function of the dimensionless radial coordinate, (r r1)/(r2 r1), where r1 0.06 m and r2 is a variable (0.06 r2 0.20 m). Compute and plot the heat loss as a function of the insulation thickness for 0 (r2 r1) 0.14 m. 3.46 Consider the water heater described in Problem 1.48. We now wish to determine the energy needed to compensate for heat losses incurred while the water is stored at the prescribed temperature of 55 C. The cylindrical storage tank (with flat ends) has a capacity of 100 gal, and foamed urethane is used to insulate the side and end walls from ambient air at an annual average temperature of 20 C. The resistance to heat transfer is dominated by conduction in the insulation and by free convection in the air, for which h ⬇ 2 W/m2 䡠 K. If electric resistance heating is used to compensate for the losses and the cost of electric power is $0.18/kWh, specify tank and insulation dimensions for which the annual cost associated with the heat losses is less than $50. 3.47 To maximize production and minimize pumping costs, crude oil is heated to reduce its viscosity during transportation from a production field.

Stylus

Tsen

Surface

d = 100 nm Air

Tsurf

L = 50 nm

Cylindrical Wall 3.45 A steam pipe of 0.12-m outside diameter is insulated with a layer of calcium silicate. (a) If the insulation is 20 mm thick and its inner and outer surfaces are maintained at Ts,1 800 K and Ts,2 490 K, respectively, what is the heat loss per unit length (q) of the pipe? (b) We wish to explore the effect of insulation thickness on the heat loss q and outer surface temperature Ts,2, with the inner surface temperature fixed at Ts,1 800 K. The outer surface is exposed to an airflow

(a) Consider a pipe-in-pipe configuration consisting of concentric steel tubes with an intervening insulating material. The inner tube is used to transport warm crude oil through cold ocean water. The inner steel pipe (ks 35 W/m 䡠 K) has an inside diameter of Di,1 150 mm and wall thickness ti 10 mm while the outer steel pipe has an inside diameter of Di,2 250 mm and wall thickness to ti. Determine the maximum allowable crude oil temperature to ensure the polyurethane foam insulation (kp 0.075 W/m 䡠 K) between the two pipes does not exceed its maximum service temperature of Tp,max 70 C. The ocean water is at T앝,o –5 C and provides an external convection heat transfer coefficient of ho 500 W/m2 䡠 K. The convection coefficient associated with the flowing crude oil is hi 450 W/m2 䡠 K. (b) It is proposed to enhance the performance of the pipe-in-pipe device by replacing a thin (ta 5 mm) section of polyurethane located at the outside of the inner pipe with an aerogel insulation material (ka 0.012 W/m 䡠 K). Determine the maximum allowable crude oil temperature to ensure maximum polyurethane temperatures are below Tp,max 70 C.

202

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

3.48 A thin electrical heater is wrapped around the outer surface of a long cylindrical tube whose inner surface is maintained at a temperature of 5 C. The tube wall has inner and outer radii of 25 and 75 mm, respectively, and a thermal conductivity of 10 W/m 䡠 K. The thermal contact resistance between the heater and the outer surface of the tube (per unit length of the tube) is Rt,c 0.01 m 䡠 K/W. The outer surface of the heater is exposed to a fluid with T앝 10 C and a convection coefficient of h 100 W/m2 䡠 K. Determine the heater power per unit length of tube required to maintain the heater at To 25 C. 3.49 In Problem 3.48, the electrical power required to maintain the heater at To 25 C depends on the thermal conductivity of the wall material k, the thermal contact resistance Rt,c and the convection coefficient h. Compute and plot the separate effect of changes in k (1 k 200 W/m 䡠 K), Rt,c (0 Rt,c 0.1 m 䡠 K/W), and h (10 h 1000 W/m2 䡠 K) on the total heater power requirement, as well as the rate of heat transfer to the inner surface of the tube and to the fluid. 3.50 A stainless steel (AISI 304) tube used to transport a chilled pharmaceutical has an inner diameter of 36 mm and a wall thickness of 2 mm. The pharmaceutical and ambient air are at temperatures of 6 C and 23 C, respectively, while the corresponding inner and outer convection coefficients are 400 W/m2 䡠 K and 6 W/m2 䡠 K, respectively. (a) What is the heat gain per unit tube length? (b) What is the heat gain per unit length if a 10-mmthick layer of calcium silicate insulation (kins 0.050 W/m 䡠 K) is applied to the tube? 3.51 Superheated steam at 575 C is routed from a boiler to the turbine of an electric power plant through steel tubes (k 35 W/m 䡠 K) of 300-mm inner diameter and 30-mm wall thickness. To reduce heat loss to the surroundings and to maintain a safe-to-touch outer surface temperature, a layer of calcium silicate insulation (k 0.10 W/m 䡠 K) is applied to the tubes, while degradation of the insulation is reduced by wrapping it in a thin sheet of aluminum having an emissivity of 0.20. The air and wall temperatures of the power plant are 27 C. (a) Assuming that the inner surface temperature of a steel tube corresponds to that of the steam and the convection coefficient outside the aluminum sheet is 6 W/m2 䡠 K, what is the minimum insulation thickness needed to ensure that the temperature of the aluminum does not exceed 50 C? What is the corresponding heat loss per meter of tube length?

(b) Explore the effect of the insulation thickness on the temperature of the aluminum and the heat loss per unit tube length. 3.52 A thin electrical heater is inserted between a long circular rod and a concentric tube with inner and outer radii of 20 and 40 mm. The rod (A) has a thermal conductivity of kA 0.15 W/m 䡠 K, while the tube (B) has a thermal conductivity of kB 1.5 W/m 䡠 K and its outer surface is subjected to convection with a fluid of temperature T앝 15 C and heat transfer coefficient 50 W/m2 䡠 K. The thermal contact resistance between the cylinder surfaces and the heater is negligible. (a) Determine the electrical power per unit length of the cylinders (W/m) that is required to maintain the outer surface of cylinder B at 5 C. (b) What is the temperature at the center of cylinder A? 3.53 A wire of diameter D 2 mm and uniform temperature T has an electrical resistance of 0.01 /m and a current flow of 20 A. (a) What is the rate at which heat is dissipated per unit length of wire? What is the heat dissipation per unit volume within the wire? (b) If the wire is not insulated and is in ambient air and large surroundings for which T앝 Tsur 20 C, what is the temperature T of the wire? The wire has an emissivity of 0.3, and the coefficient associated with heat transfer by natural convection may be approximated by an expression of the form, h C[(T T앝)/D]1/4, where C 1.25 W/m7/4 䡠 K5/4. (c) If the wire is coated with plastic insulation of 2-mm thickness and a thermal conductivity of 0.25 W/m 䡠 K, what are the inner and outer surface temperatures of the insulation? The insulation has an emissivity of 0.9, and the convection coefficient is given by the expression of part (b). Explore the effect of the insulation thickness on the surface temperatures. 3.54 A 2-mm-diameter electrical wire is insulated by a 2-mm-thick rubberized sheath (k 0.13 W/m 䡠 K), and the wire/sheath interface is characterized by a thermal 3 104 m2 䡠 K/W. The concontact resistance of Rt,c vection heat transfer coefficient at the outer surface of the sheath is 10 W/m2 䡠 K, and the temperature of the ambient air is 20 C. If the temperature of the insulation may not exceed 50 C, what is the maximum allowable electrical power that may be dissipated per unit length of the conductor? What is the critical radius of the insulation?

䊏

203

Problems

3.55 Electric current flows through a long rod generating thermal energy at a uniform volumetric rate of q˙ 2 106 W/m3. The rod is concentric with a hollow ceramic cylinder, creating an enclosure that is filled with air. To = 25°C Tr

heater for which interfacial contact resistances are negligible.

Resistance heater q"h, Th

r3 Ceramic, k = 1.75 W/m•K Di = 40 mm Do = 120 mm

r2 r1

Enclosure, air space •

Rod, q, Dr = 20 mm

Internal flow

T∞,i, hi

The thermal resistance per unit length due to radiation between the enclosure surfaces is Rrad 0.30 m 䡠 K/W, and the coefficient associated with free convection in the enclosure is h 20 W/m2 䡠 K. (a) Construct a thermal circuit that can be used to calculate the surface temperature of the rod, Tr . Label all temperatures, heat rates, and thermal resistances, and evaluate each thermal resistance. (b) Calculate the surface temperature of the rod for the prescribed conditions. 3.56 The evaporator section of a refrigeration unit consists of thin-walled, 10-mm-diameter tubes through which refrigerant passes at a temperature of 18 C. Air is cooled as it flows over the tubes, maintaining a surface convection coefficient of 100 W/m2 䡠 K, and is subsequently routed to the refrigerator compartment. (a) For the foregoing conditions and an air temperature of 3 C, what is the rate at which heat is extracted from the air per unit tube length? (b) If the refrigerator’s defrost unit malfunctions, frost will slowly accumulate on the outer tube surface. Assess the effect of frost formation on the cooling capacity of a tube for frost layer thicknesses in the range 0 ␦ 4 mm. Frost may be assumed to have a thermal conductivity of 0.4 W/m 䡠 K. (c) The refrigerator is disconnected after the defrost unit malfunctions and a 2-mm-thick layer of frost has formed. If the tubes are in ambient air for which T앝 20 C and natural convection maintains a convection coefficient of 2 W/m2 䡠 K, how long will it take for the frost to melt? The frost may be assumed to have a mass density of 700 kg/m3 and a latent heat of fusion of 334 kJ/kg. 3.57 A composite cylindrical wall is composed of two materials of thermal conductivity kA and kB, which are separated by a very thin, electric resistance

B A Ambient air

T∞,o, ho

Liquid pumped through the tube is at a temperature T앝,i and provides a convection coefficient hi at the inner surface of the composite. The outer surface is exposed to ambient air, which is at T앝,o and provides a convection coefficient of ho. Under steady-state conditions, a uniform heat flux of qh is dissipated by the heater. (a) Sketch the equivalent thermal circuit of the system and express all resistances in terms of relevant variables. (b) Obtain an expression that may be used to determine the heater temperature, Th. (c) Obtain an expression for the ratio of heat flows to the outer and inner fluids, qo /qi. How might the variables of the problem be adjusted to minimize this ratio? 3.58 An electrical current of 700 A flows through a stainless steel cable having a diameter of 5 mm and an electrical resistance of 6 104 /m (i.e., per meter of cable length). The cable is in an environment having a temperature of 30 C, and the total coefficient associated with convection and radiation between the cable and the environment is approximately 25 W/m2 䡠 K. (a) If the cable is bare, what is its surface temperature? (b) If a very thin coating of electrical insulation is applied to the cable, with a contact resistance of 0.02 m2 䡠 K/W, what are the insulation and cable surface temperatures? (c) There is some concern about the ability of the insulation to withstand elevated temperatures. What thickness of this insulation (k 0.5 W/m 䡠 K) will yield the lowest value of the maximum insulation temperature? What is the value of the maximum temperature when this thickness is used?

204

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

3.59 A 0.20-m-diameter, thin-walled steel pipe is used to transport saturated steam at a pressure of 20 bars in a room for which the air temperature is 25 C and the convection heat transfer coefficient at the outer surface of the pipe is 20 W/m2 䡠 K. (a) What is the heat loss per unit length from the bare pipe (no insulation)? Estimate the heat loss per unit length if a 50-mm-thick layer of insulation (magnesia, 85%) is added. The steel and magnesia may each be assumed to have an emissivity of 0.8, and the steam-side convection resistance may be neglected. (b) The costs associated with generating the steam and installing the insulation are known to be $4/109 J and $100/m of pipe length, respectively. If the steam line is to operate 7500 h/yr, how many years are needed to pay back the initial investment in insulation? 3.60 An uninsulated, thin-walled pipe of 100-mm diameter is used to transport water to equipment that operates outdoors and uses the water as a coolant. During particularly harsh winter conditions, the pipe wall achieves a temperature of –15 C and a cylindrical layer of ice forms on the inner surface of the wall. If the mean water temperature is 3 C and a convection coefficient of 2000 W/m2 䡠 K is maintained at the inner surface of the ice, which is at 0 C, what is the thickness of the ice layer? 3.61 Steam flowing through a long, thin-walled pipe maintains the pipe wall at a uniform temperature of 500 K. The pipe is covered with an insulation blanket comprised of two different materials, A and B. The interface between the two materials may be assumed to have an infinite contact resistance, and the entire outer surface is exposed to air for which T앝 300 K and h 25 W/m2 䡠 K. r1 = 50 mm A

Ts,2(A)

kA = 2 W/m K •

kB = 0.25 W/m•K

Ts,2(B) Ts,1 = 500 K

r2 = 100 mm

B

T∞, h

(a) Sketch the thermal circuit of the system. Label (using the preceding symbols) all pertinent nodes and resistances. (b) For the prescribed conditions, what is the total heat loss from the pipe? What are the outer surface temperatures Ts,2(A) and Ts,2(B)?

3.62 A bakelite coating is to be used with a 10-mm-diameter conducting rod, whose surface is maintained at 200 C by passage of an electrical current. The rod is in a fluid at 25 C, and the convection coefficient is 140 W/m2 䡠 K. What is the critical radius associated with the coating? What is the heat transfer rate per unit length for the bare rod and for the rod with a coating of bakelite that corresponds to the critical radius? How much bakelite should be added to reduce the heat transfer associated with the bare rod by 25%?

Spherical Wall 3.63 A storage tank consists of a cylindrical section that has a length and inner diameter of L 2 m and Di 1 m, respectively, and two hemispherical end sections. The tank is constructed from 20-mm-thick glass (Pyrex) and is exposed to ambient air for which the temperature is 300 K and the convection coefficient is 10 W/m2 䡠 K. The tank is used to store heated oil, which maintains the inner surface at a temperature of 400 K. Determine the electrical power that must be supplied to a heater submerged in the oil if the prescribed conditions are to be maintained. Radiation effects may be neglected, and the Pyrex may be assumed to have a thermal conductivity of 1.4 W/m 䡠 K. 3.64 Consider the liquid oxygen storage system and the laboratory environmental conditions of Problem 1.49. To reduce oxygen loss due to vaporization, an insulating layer should be applied to the outer surface of the container. Consider using a laminated aluminum foil/glass mat insulation, for which the thermal conductivity and surface emissivity are k 0.00016 W/m 䡠 K and 0.20, respectively. (a) If the container is covered with a 10-mm-thick layer of insulation, what is the percentage reduction in oxygen loss relative to the uncovered container? (b) Compute and plot the oxygen evaporation rate (kg/s) as a function of the insulation thickness t for 0 t 50 mm. 3.65 A spherical Pyrex glass shell has inside and outside diameters of D1 0.1 m and D2 0.2 m, respectively. The inner surface is at Ts,1 100 C while the outer surface is at Ts,2 45 C. (a) Determine the temperature at the midpoint of the shell thickness, T(rm 0.075 m). (b) For the same surface temperatures and dimensions as in part (a), show how the midpoint temperature would change if the shell material were aluminum. 3.66 In Example 3.6, an expression was derived for the critical insulation radius of an insulated, cylindrical tube.

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205

Problems

Derive the expression that would be appropriate for an insulated sphere. 3.67 A hollow aluminum sphere, with an electrical heater in the center, is used in tests to determine the thermal conductivity of insulating materials. The inner and outer radii of the sphere are 0.15 and 0.18 m, respectively, and testing is done under steady-state conditions with the inner surface of the aluminum maintained at 250 C. In a particular test, a spherical shell of insulation is cast on the outer surface of the sphere to a thickness of 0.12 m. The system is in a room for which the air temperature is 20 C and the convection coefficient at the outer surface of the insulation is 30 W/m2 䡠 K. If 80 W are dissipated by the heater under steady-state conditions, what is the thermal conductivity of the insulation? 3.68 A spherical tank for storing liquid oxygen on the space shuttle is to be made from stainless steel of 0.80-m outer diameter and 5-mm wall thickness. The boiling point and latent heat of vaporization of liquid oxygen are 90 K and 213 kJ/kg, respectively. The tank is to be installed in a large compartment whose temperature is to be maintained at 240 K. Design a thermal insulation system that will maintain oxygen losses due to boiling below 1 kg/day. 3.69 A spherical, cryosurgical probe may be imbedded in diseased tissue for the purpose of freezing, and thereby destroying, the tissue. Consider a probe of 3-mm diameter whose surface is maintained at 30 C when imbedded in tissue that is at 37 C. A spherical layer of frozen tissue forms around the probe, with a temperature of 0 C existing at the phase front (interface) between the frozen and normal tissue. If the thermal conductivity of frozen tissue is approximately 1.5 W/m 䡠 K and heat transfer at the phase front may be characterized by an effective convection coefficient of 50 W/m2 䡠 K, what is the thickness of the layer of frozen tissue (assuming negligible perfusion)? 3.70 A spherical vessel used as a reactor for producing pharmaceuticals has a 10-mm-thick stainless steel wall (k 17 W/m 䡠 K) and an inner diameter of l m. The exterior surface of the vessel is exposed to ambient air (T앝 25 C) for which a convection coefficient of 6 W/m2 䡠 K may be assumed. (a) During steady-state operation, an inner surface temperature of 50 C is maintained by energy generated within the reactor. What is the heat loss from the vessel? (b) If a 20-mm-thick layer of fiberglass insulation (k 0.040 W/m 䡠 K) is applied to the exterior of the vessel and the rate of thermal energy generation is unchanged, what is the inner surface temperature of the vessel?

3.71 The wall of a spherical tank of 1-m diameter contains an exothermic chemical reaction and is at 200 C when the ambient air temperature is 25 C. What thickness of urethane foam is required to reduce the exterior temperature to 40 C, assuming the convection coefficient is 20 W/m2 䡠 K for both situations? What is the percentage reduction in heat rate achieved by using the insulation? 3.72 A composite spherical shell of inner radius r1 0.25 m is constructed from lead of outer radius r2 0.30 m and AISI 302 stainless steel of outer radius r3 0.31 m. The cavity is filled with radioactive wastes that generate heat at a rate of q˙ 5 105 W/m3. It is proposed to submerge the container in oceanic waters that are at a temperature of T앝 10 C and provide a uniform convection coefficient of h 500 W/m2 䡠 K at the outer surface of the container. Are there any problems associated with this proposal? 3.73 The energy transferred from the anterior chamber of the eye through the cornea varies considerably depending on whether a contact lens is worn. Treat the eye as a spherical system and assume the system to be at steady state. The convection coefficient ho is unchanged with and without the contact lens in place. The cornea and the lens cover one-third of the spherical surface area.

r1

r2 r3

Anterior chamber

T∞,i, hi

Cornea

k1

T∞,o, ho k2 Contact lens

Values of the parameters representing this situation are as follows: r1 10.2 mm r3 16.5 mm T앝,i 37 C k1 0.35 W/m 䡠 K hi 12 W/m2 䡠 K

r2 12.7 mm T앝,o 21 C k2 0.80 W/m 䡠 K ho 6 W/m2 䡠 K

(a) Construct the thermal circuits, labeling all potentials and flows for the systems excluding the contact lens and including the contact lens. Write resistance elements in terms of appropriate parameters. (b) Determine the heat loss from the anterior chamber with and without the contact lens in place. (c) Discuss the implication of your results.

206

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One-Dimensional, Steady-State Conduction

3.74 The outer surface of a hollow sphere of radius r2 is subjected to a uniform heat flux q2. The inner surface at r1 is held at a constant temperature Ts,1. (a) Develop an expression for the temperature distribution T(r) in the sphere wall in terms of q2, Ts,1, r1, r2, and the thermal conductivity of the wall material k. (b) If the inner and outer tube radii are r1 50 mm and r2 100 mm, what heat flux q2 is required to maintain the outer surface at Ts,2 50 C, while the inner surface is at Ts,1 20 C? The thermal conductivity of the wall material is k 10 W/m 䡠 K. 3.75 A spherical shell of inner and outer radii ri and ro, respectively, is filled with a heat-generating material that provides for a uniform volumetric generation rate (W/m3) of q˙. The outer surface of the shell is exposed to a fluid having a temperature T앝 and a convection coefficient h. Obtain an expression for the steady-state temperature distribution T(r) in the shell, expressing your result in terms of ri, ro, q˙, h, T앝, and the thermal conductivity k of the shell material. 3.76 A spherical tank of 3-m diameter contains a liquifiedpetroleum gas at 60 C. Insulation with a thermal conductivity of 0.06 W/m 䡠 K and thickness 250 mm is applied to the tank to reduce the heat gain. (a) Determine the radial position in the insulation layer at which the temperature is 0 C when the ambient air temperature is 20 C and the convection coefficient on the outer surface is 6 W/m2 䡠 K. (b) If the insulation is pervious to moisture from the atmospheric air, what conclusions can you reach about the formation of ice in the insulation? What effect will ice formation have on heat gain to the LP gas? How could this situation be avoided? 3.77 A transistor, which may be approximated as a hemispherical heat source of radius ro 0.1 mm, is embedded in a large silicon substrate (k 125 W/m 䡠 K) and dissipates heat at a rate q. All boundaries of the silicon are maintained at an ambient temperature of T앝 27 C, except for the top surface, which is well insulated.

ro

Silicon substrate

q T∞

Obtain a general expression for the substrate temperature distribution and evaluate the surface temperature of the heat source for q 4 W.

3.78 One modality for destroying malignant tissue involves imbedding a small spherical heat source of radius ro within the tissue and maintaining local temperatures above a critical value Tc for an extended period. Tissue that is well removed from the source may be assumed to remain at normal body temperature (Tb 37 C). Obtain a general expression for the radial temperature distribution in the tissue under steady-state conditions for which heat is dissipated at a rate q. If ro 0.5 mm, what heat rate must be supplied to maintain a tissue temperature of T Tc 42 C in the domain 0.5 r 5 mm? The tissue thermal conductivity is approximately 0.5 W/m 䡠 K. Assume negligible perfusion.

Conduction with Thermal Energy Generation 3.79 The air inside a chamber at T앝,i 50 C is heated convectively with hi 20 W/m2 䡠 K by a 200-mm-thick wall having a thermal conductivity of 4 W/m 䡠 K and a uniform heat generation of 1000 W/m3. To prevent any heat generated within the wall from being lost to the outside of the chamber at T앝,o 25 C with ho 5 W/m2 䡠 K, a very thin electrical strip heater is placed on the outer wall to provide a uniform heat flux, qo. Wall, k, q•

Strip heater, q"o Outside chamber

Inside chamber

T∞, o, ho

T∞, i, hi x

L

(a) Sketch the temperature distribution in the wall on T x coordinates for the condition where no heat generated within the wall is lost to the outside of the chamber. (b) What are the temperatures at the wall boundaries, T(0) and T(L), for the conditions of part (a)? (c) Determine the value of qo that must be supplied by the strip heater so that all heat generated within the wall is transferred to the inside of the chamber. (d) If the heat generation in the wall were switched off while the heat flux to the strip heater remained constant, what would be the steady-state temperature, T(0), of the outer wall surface? 3.80 Consider cylindrical and spherical shells with inner and outer surfaces at r1 and r2 maintained at uniform temperatures Ts,1 and Ts,2, respectively. If there is uniform heat generation within the shells, obtain expressions for the steady-state, one-dimensional radial distributions of the temperature, heat flux, and heat rate. Contrast your results with those summarized in Appendix C.

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207

Problems

3.81 A plane wall of thickness 0.1 m and thermal conductivity 25 W/m 䡠 K having uniform volumetric heat generation of 0.3 MW/m3 is insulated on one side, while the other side is exposed to a fluid at 92 C. The convection heat transfer coefficient between the wall and the fluid is 500 W/m2 䡠 K. Determine the maximum temperature in the wall. 3.82 Large, cylindrical bales of hay used to feed livestock in the winter months are D 2 m in diameter and are stored end-to-end in long rows. Microbial energy generation occurs in the hay and can be excessive if the farmer bales the hay in a too-wet condition. Assuming the thermal conductivity of baled hay to be k 0.04 W/m 䡠 K, determine the maximum steady-state . hay temperature for dry hay (q 1W/m3), moist hay . . 3 (q 10 W/m ), and wet hay (q 100 W/m3). Ambient conditions are T앝 0 C and h 25 W/m2 䡠 K. 3.83 Consider the cylindrical bales of hay in Problem 3.82. It is proposed to utilize the microbial energy generation associated with wet hay to heat water. Consider a 30-mm diameter, thin-walled tube inserted lengthwise through the middle of a cylindrical bale. The tube carries water at T앝,i 20 C with hi 200 W/m2 䡠 K. (a) Determine the steady-state heat transfer to the water per unit length of tube. (b) Plot the radial temperature distribution in the hay, T(r). (c) Plot the heat transfer to the water per unit length of tube for bale diameters of 0.2 m D 2 m. 3.84 Consider one-dimensional conduction in a plane composite wall. The outer surfaces are exposed to a fluid at 25 C and a convection heat transfer coefficient of 1000 W/m2 䡠 K. The middle wall B experiences uniform . heat generation qB, while there is no generation in walls A and C. The temperatures at the interfaces are T1 261 C and T2 211 C. T1

T2

T∞, h

T∞, h A

B

(b) Plot the temperature distribution, showing its important features. (c) Consider conditions corresponding to a loss of coolant at the exposed surface of material A (h 0). Determine T1 and T2 and plot the temperature distribution throughout the system. 3.85 Consider a plane composite wall that is composed of three materials (materials A, B, and C are arranged left to right) of thermal conductivities kA 0.24 W/m 䡠 K, kB 0.13 W/m 䡠 K, and kC 0.50 W/m 䡠 K. The thicknesses of the three sections of the wall are LA 20 mm, L B 13 mm, and LC 20 mm. A contact resistance of Rt,c 102 m2 䡠 K/W exists at the interface between materials A and B, as well as at the interface between materials B and C. The left face of the composite wall is insulated, while the right face is exposed to convective conditions characterized by h 10 W/m2 䡠 K, T앝 20 C. For Case 1, thermal energy is generated within . material A at the rate qA 5000 W/m3. For Case 2, thermal energy is generated within material C at the . rate qC 5000 W/m3. (a) Determine the maximum temperature within the composite wall under steady-state conditions for Case 1. (b) Sketch the steady-state temperature distribution on T x coordinates for Case 1. (c) Sketch the steady-state temperature distribution for Case 2 on the same T x coordinates used for Case 1. 3.86 An air heater may be fabricated by coiling Nichrome wire and passing air in cross flow over the wire. Consider a heater fabricated from wire of diameter D 1 mm, electrical resistivity e 106 䡠 m, thermal conductivity k 25 W/m 䡠 K, and emissivity 0.20. The heater is designed to deliver air at a temperature of T앝 50 C under flow conditions that provide a convection coefficient of h 250 W/m2 䡠 K for the wire. The temperature of the housing that encloses the wire and through which the air flows is Tsur 50 C. Wire (D, L, ρe, k, ε , Tmax)

Housing, Tsur Air

C

q• B ∆E

LA kA = 25 W/m•K kC = 50 W/m•K

2LB

LC

LA = 30 mm LB = 30 mm LC = 20 mm

(a) Assuming negligible contact resistance at the inter. faces, determine the volumetric heat generation qB and the thermal conductivity kB.

T∞ , h

I

If the maximum allowable temperature of the wire is Tmax 1200 C, what is the maximum allowable electric current I? If the maximum available voltage is E 110 V, what is the corresponding length L of wire that may be used in the heater and the power rating of the heater? Hint: In your solution, assume

208

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

negligible temperature variations within the wire, but after obtaining the desired results, assess the validity of this assumption. 3.87 Consider the composite wall of Example 3.7. In the Comments section, temperature distributions in the wall were determined assuming negligible contact resistance between materials A and B. Compute and plot the temperature distributions if the thermal contact resistance is Rt, c 104 m2 䡠 K/W.

3.90 A nuclear fuel element of thickness 2L is covered with a steel cladding of thickness b. Heat generated within . the nuclear fuel at a rate q is removed by a fluid at T앝, which adjoins one surface and is characterized by a convection coefficient h. The other surface is well insulated, and the fuel and steel have thermal conductivities of kƒ and ks, respectively. Nuclear fuel Steel

3.88 Consider uniform thermal energy generation inside a one-dimensional plane wall of thickness L with one surface held at Ts,1 and the other surface insulated.

Insulation

b

(a) Find an expression for the conduction heat flux to the cold surface and the temperature of the hot surface Ts,2, . expressing your results in terms of k, q, L, and Ts,1.

Case 1

Case 2

To

•

q, k

–L

0

+L

x

•

–L

To

•

q, k

A

B

0

+L

x

(a) Sketch the temperature distribution for Case 1 on T ⫺ x coordinates. Describe the key features of this distribution. Identify the location of the maximum temperature in the wall and calculate this temperature. (b) Sketch the temperature distribution for Case 2 on the same T ⫺ x coordinates. Describe the key features of this distribution. (c) What is the temperature difference between the two walls at x 0 for Case 2? (d) What is the location of the maximum temperature in the composite wall of Case 2? Calculate this temperature.

L

b

(a) Obtain an equation for the temperature distribution T(x) in the nuclear fuel. Express your results in . terms of q, kƒ, L, b, ks, h, and T앝. (b) Sketch the temperature distribution T(x) for the entire system. 3.91 Consider the clad fuel element of Problem 3.90. (a) Using appropriate relations from Tables C.1 and C.2, obtain an expression for the temperature distribution T(x) in the fuel element. For kf 60 W/m 䡠 K, L 15 mm, b 3 mm, ks 15 W/m 䡠 K, h 10,000 W/m2 䡠 K, and T앝 200 C, what are the largest and smallest temperatures in the fuel element if heat is generated uniformly at a volumetric rate of q˙ 2 107 W/m3? What are the corresponding locations?

Thin dielectric strip, R"t

q, k

T∞, h L

x

(b) Compare the heat flux found in part (a) with the heat flux associated with a plane wall without energy generation whose surface temperatures are Ts,1 and Ts,2. 3.89 A plane wall of thickness 2L and thermal conductivity k . experiences a uniform volumetric generation rate q. As shown in the sketch for Case 1, the surface at x L is perfectly insulated, while the other surface is maintained at a uniform, constant temperature To. For Case 2, a very thin dielectric strip is inserted at the midpoint of the wall (x 0) in order to electrically isolate the two sections, A and B. The thermal resistance of the strip is R t 0.0005 m2 䡠 K/W. The parameters associated with the wall are k 50 W/m 䡠 K, L . 20 mm, q 5 106 W/m3, and To 50⬚C.

Steel

(b) If the insulation is removed and equivalent convection conditions are maintained at each surface, what is the corresponding form of the temperature distribution in the fuel element? For the conditions of part (a), what are the largest and smallest temperatures in the fuel? What are the corresponding locations? (c) For the conditions of parts (a) and (b), plot the temperature distributions in the fuel element.

3.92 In Problem 3.79 the strip heater acts to guard against heat losses from the wall to the outside, and the required heat flux qo depends on chamber operating . conditions such as q and T앝,i. As a first step in designing a controller for the guard heater, compute . . and plot qo and T(0) as a function of q for 200 q 3 2000 W/m and T앝,i 30, 50, and 70 C. 3.93 The exposed surface (x 0) of a plane wall of thermal conductivity k is subjected to microwave radiation that causes volumetric heating to vary as x . . q(x) qo 1 L

冢

冣

䊏

Problems

. where qo (W/m3) is a constant. The boundary at x L is perfectly insulated, while the exposed surface is maintained at a constant temperature To. Determine the tem. perature distribution T(x) in terms of x, L, k, qo, and To. 3.94 A quartz window of thickness L serves as a viewing port in a furnace used for annealing steel. The inner surface (x 0) of the window is irradiated with a uniform heat flux qo due to emission from hot gases in the furnace. A fraction, , of this radiation may be assumed to be absorbed at the inner surface, while the remaining radiation is partially absorbed as it passes through the quartz. The volumetric heat generation due to this absorption may be described by an expression of the form . q(x) (1 )qo␣e␣x where ␣ is the absorption coefficient of the quartz. Convection heat transfer occurs from the outer surface (x L) of the window to ambient air at T앝 and is characterized by the convection coefficient h. Convection and radiation emission from the inner surface may be neglected, along with radiation emission from the outer surface. Determine the temperature distribution in the quartz, expressing your result in terms of the foregoing parameters.

209 (a) It is proposed that, under steady-state conditions, . the system operates with a generation rate of q 8 3 7 10 W/m and cooling system characteristics of T앝 95 C and h 7000 W/m2 䡠 K. Is this proposal satisfactory? . (b) Explore the effect of variations in q and h by plotting temperature distributions T(r) for a range of parameter values. Suggest an envelope of acceptable operating conditions. 3.98 A nuclear reactor fuel element consists of a solid cylindrical pin of radius r1 and thermal conductivity kf. The fuel pin is in good contact with a cladding material of outer radius r2 and thermal conductivity kc. Consider steady-state conditions for which uniform heat genera. tion occurs within the fuel at a volumetric rate q and the outer surface of the cladding is exposed to a coolant that is characterized by a temperature T앝 and a convection coefficient h. (a) Obtain equations for the temperature distributions Tf (r) and Tc(r) in the fuel and cladding, respectively. Express your results exclusively in terms of the foregoing variables.

3.95 For the conditions described in Problem 1.44, determine the temperature distribution, T(r), in the container, . expressing your result in terms of qo, ro, T앝, h, and the thermal conductivity k of the radioactive wastes.

(b) Consider a uranium oxide fuel pin for which kƒ 2 W/m 䡠 K and r1 6 mm and cladding for which . kc 25 W/m 䡠 K and r2 9 mm. If q 2 108 3 2 W/m , h 2000 W/m 䡠 K, and T앝 300 K, what is the maximum temperature in the fuel element?

3.96 A cylindrical shell of inner and outer radii, ri and ro, respectively, is filled with a heat-generating material that provides a uniform volumetric generation rate . (W/m3) of q. The inner surface is insulated, while the outer surface of the shell is exposed to a fluid at T앝 and a convection coefficient h.

(c) Compute and plot the temperature distribution, T(r), for values of h 2000, 5000, and 10,000 W/m2 䡠 K. If the operator wishes to maintain the centerline temperature of the fuel element below 1000 K, can she do so by adjusting the coolant flow and hence the value of h?

(a) Obtain an expression for the steady-state temperature distribution T(r) in the shell, expressing your . result in terms of ri, ro, q, h, T앝, and the thermal conductivity k of the shell material.

3.99 Consider the configuration of Example 3.8, where uniform volumetric heating within a stainless steel tube is induced by an electric current and heat is transferred by convection to air flowing through the tube. The tube wall has inner and outer radii of r1 25 mm and r2 35 mm, a thermal conductivity of k 15 W/m 䡠 K, an electrical resistivity of e 0.7 106 䡠 m, and a maximum allowable operating temperature of 1400 K.

(b) Determine an expression for the heat rate, q(ro), at . the outer radius of the shell in terms of q and shell dimensions. 3.97 The cross section of a long cylindrical fuel element in a nuclear reactor is shown. Energy generation occurs uniformly in the thorium fuel rod, which is of diameter D 25 mm and is wrapped in a thin aluminum cladding. Coolant

T∞, h

Thorium fuel rod

D

Thin aluminum cladding

(a) Assuming the outer tube surface to be perfectly insulated and the airflow to be characterized by a temperature and convection coefficient of T앝,1 400 K and h1 100 W/m2 䡠 K, determine the maximum allowable electric current I. (b) Compute and plot the radial temperature distribution in the tube wall for the electric current of part (a) and three values of h1 (100, 500, and 1000 W/m2 䡠 K). For each value of h1, determine the rate of heat transfer to the air per unit length of tube.

210

Chapter 3

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One-Dimensional, Steady-State Conduction

(c) In practice, even the best of insulating materials would be unable to maintain adiabatic conditions at the outer tube surface. Consider use of a refractory insulating material of thermal conductivity k 1.0 W/m 䡠 K and neglect radiation exchange at its outer surface. For h1 100 W/m2 䡠 K and the maximum allowable current determined in part (a), compute and plot the temperature distribution in the composite wall for two values of the insulation thickness (␦ 25 and 50 mm). The outer surface of the insulation is exposed to room air for which T앝, 2 300 K and h2 25 W/m2 䡠 K. For each insulation thickness, determine the rate of heat transfer per unit tube length to the inner airflow and the ambient air. 3.100 A high-temperature, gas-cooled nuclear reactor consists of a composite cylindrical wall for which a thorium fuel element (k ⬇ 57 W/m 䡠 K) is encased in graphite (k ⬇ 3 W/m 䡠 K) and gaseous helium flows through an annular coolant channel. Consider conditions for which the helium temperature is T앝 600 K and the convection coefficient at the outer surface of the graphite is h 2000 W/m2 䡠 K.

r1 = 8 mm r2 = 11 mm r3 = 14 mm

Coolant channel with helium flow (T∞, h) Graphite Thorium, q•

T1 T2 T3

(a) If thermal energy is uniformly generated in the fuel . element at a rate q 108 W/m3, what are the temperatures T1 and T2 at the inner and outer surfaces, respectively, of the fuel element? (b) Compute and plot the temperature distribution in . the composite wall for selected values of q. What . is the maximum allowable value of q? 3.101 A long cylindrical rod of diameter 200 mm with thermal conductivity of 0.5 W/m 䡠 K experiences uniform volumetric heat generation of 24,000 W/m3. The rod is encapsulated by a circular sleeve having an outer diameter of 400 mm and a thermal conductivity of 4 W/m 䡠 K. The outer surface of the sleeve is exposed to cross flow of air at 27 C with a convection coefficient of 25 W/m2 䡠 K. (a) Find the temperature at the interface between the rod and sleeve and on the outer surface. (b) What is the temperature at the center of the rod?

3.102 A radioactive material of thermal conductivity k is cast as a solid sphere of radius ro and placed in a liquid bath for which the temperature T앝 and convection coefficient h are known. Heat is uniformly generated within . the solid at a volumetric rate of q. Obtain the steadystate radial temperature distribution in the solid, . expressing your result in terms of ro, q, k, h, and T앝. 3.103 Radioactive wastes are packed in a thin-walled spherical container. The wastes generate thermal energy nonuni. . formly according to the relation q qo[1 (r/ro)2] where . . q is the local rate of energy generation per unit volume, q is a constant, and ro is the radius of the container. Steadystate conditions are maintained by submerging the container in a liquid that is at T앝 and provides a uniform convection coefficient h. ro

Coolant T∞, h

q• = q• o [1 – (r/ro)2]

Determine the temperature distribution, T(r), in the con. tainer. Express your result in terms of qo, ro, T앝, h, and the thermal conductivity k of the radioactive wastes. 3.104 Radioactive wastes (krw 20 W/m 䡠 K) are stored in a spherical, stainless steel (kss 15 W/m 䡠 K) container of inner and outer radii equal to ri 0.5 m and ro 0.6 m. Heat is generated volumetrically within the wastes at a . uniform rate of q 105 W/m3, and the outer surface of the container is exposed to a water flow for which h 1000 W/m2 䡠 K and T앝 25 C.

Water T∞, h

ri

Radioactive wastes, krw, q• Stainless steel,

ro

Ts, o

kss

Ts, i

(a) Evaluate the steady-state outer surface temperature, Ts,o. (b) Evaluate the steady-state inner surface temperature, Ts,i. (c) Obtain an expression for the temperature distribution, T(r), in the radioactive wastes. Express your . result in terms of ri, Ts,i, krw, and q. Evaluate the temperature at r 0.

䊏

211

Problems

(d) A proposed extension of the foregoing design involves storing waste materials having the same thermal conductivity but twice the heat generation . (q 2 105 W/m3) in a stainless steel container of equivalent inner radius (ri 0.5 m). Safety considerations dictate that the maximum system temperature not exceed 475 C and that the container wall thickness be no less than t 0.04 m and preferably at or close to the original design (t 0.1 m). Assess the effect of varying the outside convection coefficient to a maximum achievable value of h 5000 W/m2 䡠 K (by increasing the water velocity) and the container wall thickness. Is the proposed extension feasible? If so, recommend suitable operating and design conditions for h and t, respectively. 3.105 Unique characteristics of biologically active materials such as fruits, vegetables, and other products require special care in handling. Following harvest and separation from producing plants, glucose is catabolized to produce carbon dioxide, water vapor, and heat, with attendant internal energy generation. Consider a carton of apples, each of 80-mm diameter, which is ventilated with air at 5 C and a velocity of 0.5 m/s. The corresponding value of the heat transfer coefficient is 7.5 W/m2 䡠 K. Within each apple thermal energy is uniformly generated at a total rate of 4000 J/kg 䡠 day. The density and thermal conductivity of the apple are 840 kg/m3 and 0.5 W/m 䡠 K, respectively. Apple, 80 mm diameter

3.106 Consider the plane wall, long cylinder, and sphere shown schematically, each with the same characteristic length a, thermal conductivity k, and uniform volu. metric energy generation rate q. Plane wall

Long cylinder

•

Sphere

•

q, k

•

q, k

q, k

r=a a

x

a

x

(a) On the same graph, plot the steady-state dimen. sionless temperature, [T(x or r) T(a)]/[(qa2)/2k], versus the dimensionless characteristic length, x/a or r/a, for each shape. (b) Which shape has the smallest temperature difference between the center and the surface? Explain this behavior by comparing the ratio of the volumeto-surface area. (c) Which shape would be preferred for use as a nuclear fuel element? Explain why.

Extended Surfaces 3.107 The radiation heat gage shown in the diagram is made from constantan metal foil, which is coated black and is in the form of a circular disk of radius R and thickness t. The gage is located in an evacuated enclosure. The incident radiation flux absorbed by the foil, qi, diffuses toward the outer circumference and into the larger copper ring, which acts as a heat sink at the constant temperature T(R). Two copper lead wires are attached to the center of the foil and to the ring to complete a thermocouple circuit that allows for measurement of the temperature difference between the foil center and the foil edge, T T(0) T(R).

Air

T∞ = 5°C

(a) Determine the temperatures.

apple

center

and

q"i

surface

(b) For the stacked arrangement of apples within the crate, the convection coefficient depends on the velocity as h C1V 0.425, where C1 10.1 W/m2 䡠 K 䡠 (m/s)0.425. Compute and plot the center and surface temperatures as a function of the air velocity for 0.1 V 1 m/s.

Evacuated enclosure

R Foil

T(0)

T(R)

Copper ring

Copper wires

212

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Obtain the differential equation that determines T(r), the temperature distribution in the foil, under steady-state conditions. Solve this equation to obtain an expression relating T to qi. You may neglect radiation exchange between the foil and its surroundings.

nanowire that may be grown for conditions characterized by h 105 W/m2 䡠 K and T앝 8000 K. Assume properties of the nanowire are the same as for bulk silicon carbide. Gas absorption

3.108 Copper tubing is joined to the absorber of a flat-plate solar collector as shown. Cover plate

Solid deposition Evacuated space

q"rad

Nanowire

h, T∞

Absorber plate

Liquid catalyst Water

t

Tw Insulation

L Initial time

The aluminum alloy (2024-T6) absorber plate is 6 mm thick and well insulated on its bottom. The top surface of the plate is separated from a transparent cover plate by an evacuated space. The tubes are spaced a distance L of 0.20 m from each other, and water is circulated through the tubes to remove the collected energy. The water may be assumed to be at a uniform temperature of Tw 60 C. Under steady-state operating conditions for which the net radiation heat flux to the surface is qrad 800 W/m2, what is the maximum temperature on the plate and the heat transfer rate per unit length of tube? Note that qrad represents the net effect of solar radiation absorption by the absorber plate and radiation exchange between the absorber and cover plates. You may assume the temperature of the absorber plate directly above a tube to be equal to that of the water. 3.109 One method that is used to grow nanowires (nanotubes with solid cores) is to initially deposit a small droplet of a liquid catalyst onto a flat surface. The surface and catalyst are heated and simultaneously exposed to a higher-temperature, low-pressure gas that contains a mixture of chemical species from which the nanowire is to be formed. The catalytic liquid slowly absorbs the species from the gas through its top surface and converts these to a solid material that is deposited onto the underlying liquid-solid interface, resulting in construction of the nanowire. The liquid catalyst remains suspended at the tip of the nanowire. Consider the growth of a 15-nm-diameter silicon carbide nanowire onto a silicon carbide surface. The surface is maintained at a temperature of Ts 2400 K, and the particular liquid catalyst that is used must be maintained in the range 2400 K Tc 3000 K to perform its function. Determine the maximum length of a

Intermediate time

Maximum length

3.110 Consider the manufacture of photovoltaic silicon, as described in Problem 1.42. The thin sheet of silicon is pulled from the pool of molten material very slowly and is subjected to an ambient temperature of T앝 527 C within the growth chamber. A convection coefficient of h 7.5 W/m2 䡠 K is associated with the exposed surfaces of the silicon sheet when it is inside the growth chamber. Calculate the maximum allowable velocity of the silicon sheet Vsi. The latent heat of fusion for silicon is hsf 1.8 106 J/kg. It can be assumed that the thermal energy released due to solidification is removed by conduction along the sheet. 3.111 Copper tubing is joined to a solar collector plate of thickness t, and the working fluid maintains the temperature of the plate above the tubes at To. There is a uniform net radiation heat flux qrad to the top surface of the plate, while the bottom surface is well insulated. The top surface is also exposed to a fluid at T앝 that provides for a uniform convection coefficient h. Air

T∞, h q"rad To

To Absorber plate

t Working fluid

Working fluid

x 2L

(a) Derive the differential equation that governs the temperature distribution T(x) in the plate.

䊏

213

Problems

(b) Obtain a solution to the differential equation for appropriate boundary conditions. 3.112 A thin flat plate of length L, thickness t, and width W L is thermally joined to two large heat sinks that are maintained at a temperature To. The bottom of the plate is well insulated, while the net heat flux to the top surface of the plate is known to have a uniform value of qo. L

distance between the two legs of the sting, L L1 L2, to ensure that the sting temperature does not influence the junction temperature and, in turn, invalidate the gas temperature measurement. Consider two different types of thermocouple junctions consisting of (i) copper and constantan wires and (ii) chromel and alumel wires. Evaluate the thermal conductivity of copper and constantan at T 300 K. Use kCh 19 W/m 䡠 K and kAl 29 W/m 䡠 K for the thermal conductivities of the chromel and alumel wires, respectively.

x Heat sink

q"o

Thermocouple junction

Heat sink

To

To t

(a) Derive the differential equation that determines the steady-state temperature distribution T(x) in the plate. (b) Solve the foregoing equation for the temperature distribution, and obtain an expression for the rate of heat transfer from the plate to the heat sinks. 3.113 Consider the flat plate of Problem 3.112, but with the heat sinks at different temperatures, T(0) To and T(L) TL, and with the bottom surface no longer insulated. Convection heat transfer is now allowed to occur between this surface and a fluid at T앝, with a convection coefficient h. (a) Derive the differential equation that determines the steady-state temperature distribution T(x) in the plate. (b) Solve the foregoing equation for the temperature distribution, and obtain an expression for the rate of heat transfer from the plate to the heat sinks. (c) For qo 20,000 W/m2, To 100 C, TL 35 C, T앝 25 C, k 25 W/m 䡠 K, h 50 W/m2 䡠 K, L 100 mm, t 5 mm, and a plate width of W 30 mm, plot the temperature distribution and determine the sink heat rates, qx(0) and qx(L). On the same graph, plot three additional temperature distributions corresponding to changes in the following parameters, with the remaining parameters unchanged: (i) qo 30,000 W/m2, (ii) h 200 W/m2 䡠 K, and (iii) the value of qo for which qx(0) 0 when h 200 W/m2 䡠 K. 3.114 The temperature of a flowing gas is to be measured with a thermocouple junction and wire stretched between two legs of a sting, a wind tunnel test fixture. The junction is formed by butt-welding two wires of different material, as shown in the schematic. For wires of diameter D 125 m and a convection coefficient of h 700 W/m2 䡠 K, determine the minimum separation

L1 Gas h, T∞

L

Sting

L2 D

3.115 A bonding operation utilizes a laser to provide a constant heat flux, qo, across the top surface of a thin adhesivebacked, plastic film to be affixed to a metal strip as shown in the sketch. The metal strip has a thickness d 1.25 mm, and its width is large relative to that of the film. The thermophysical properties of the strip are 7850 kg/m3, cp 435 J/kg 䡠 K, and k 60 W/m 䡠 K. The thermal resistance of the plastic film of width w1 40 mm is negligible. The upper and lower surfaces of the strip (including the plastic film) experience convection with air at 25 C and a convection coefficient of 10 W/m2 䡠 K. The strip and film are very long in the direction normal to the page. Assume the edges of the metal strip are at the air temperature (T앝). Laser source, q"o Plastic film

T∞, h Metal strip

d

w1 w2 x T∞, h

(a) Derive an expression for the temperature distribution in the portion of the steel strip with the plastic film (w1/2 x w1/2). (b) If the heat flux provided by the laser is 10,000 W/m2, determine the temperature of the plastic film at the center (x 0) and its edges (x w1/2). (c) Plot the temperature distribution for the entire strip and point out its special features.

214

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

3.116 A thin metallic wire of thermal conductivity k, diameter D, and length 2L is annealed by passing an electrical current through the wire to induce a uniform volu. metric heat generation q. The ambient air around the wire is at a temperature T앝, while the ends of the wire at x L are also maintained at T앝. Heat transfer from the wire to the air is characterized by the convection coefficient h. Obtain expressions for the following:

experiences uniform volumetric energy generation at a . rate of q 10 106 W/m3. Air at Ta 80 C provides a convection coefficient of ha 35 W/m2 䡠 K on one side of the membrane, while hydrogen at Th 80 C, hh 235 W/m2 䡠 K flows on the opposite side of the membrane. The flow channels are 2L 3 mm wide. The membrane is clamped between bipolar plates, each of which is at a temperature Tbp 80 C. Membrane

(a) The steady-state temperature distribution T(x) along the wire, (b) The maximum wire temperature.

t

(c) The average wire temperature.

2L

3.117 A motor draws electric power Pelec from a supply line and delivers mechanical power Pmech to a pump through a rotating copper shaft of thermal conductivity ks, length L, and diameter D. The motor is mounted on a square pad of width W, thickness t, and thermal conductivity kp. The surface of the housing exposed to ambient air at T앝 is of area Ah, and the corresponding convection coefficient is hh. Opposite ends of the shaft are at temperatures of Th and T앝, and heat transfer from the shaft to the ambient air is characterized by the convection coefficient hs. The base of the pad is at T앝. T∞, hh

T∞, hs

Pelec

x

Bipolar plate, Tbp

(a) Derive the differential equation that governs the temperature distribution T(x) in the membrane. (b) Obtain a solution to the differential equation, assuming the membrane is at the bipolar plate temperature at x 0 and x 2L.

T∞

(c) Plot the temperature distribution T(x) from x 0 to x L for carbon nanotube loadings of 0% and 10% by volume. Comment on the ability of the carbon nanotubes to keep the membrane below its softening temperature of 85 C.

Pump

D Th

L

t

Pad, kp

W

Ta , ha

Th , hh

Motor housing, Th, Ah

Electric motor

Air

Hydrogen

Shaft, ks, Pmech

T∞

(a) Expressing your result in terms of Pelec, Pmech, ks, L, D, W, t, kp, Ah, hh, and hs, obtain an expression for (Th T앝).

3.119 Consider a rod of diameter D, thermal conductivity k, and length 2L that is perfectly insulated over one portion of its length, L x 0, and experiences convection with a fluid (T앝, h) over the other portion, 0 x L. One end is maintained at T1, while the other is separated from a heat sink at T3 by an interfa. cial thermal contact resistance Rt,c

(b) What is the value of Th if Pelec 25 kW, Pmech 15 kW, ks 400 W/m 䡠 K, L 0.5 m, D 0.05 m, W 0.7 m, t 0.05 m, kp 0.5 W/m 䡠 K, Ah 2 m2, hh 10 W/m2 䡠 K, hs 300 W/m2 䡠 K, and T1 T앝 25 C? 3.118 Consider the fuel cell stack of Problem 1.58. The t 0.42-mm-thick membranes have a nominal thermal conductivity of k 0.79 W/m 䡠 K that can be increased to keff,x 15.1 W/m 䡠 K by loading 10%, by volume, carbon nanotubes into the catalyst layers. The membrane

Insulation

R"t,c = 4 × 10–4 m2•K/W T2

–L

Rod 0 D = 5 mm L = 50 mm k = 100 W/m•K

T3

x +L T∞ = 20°C h = 500 W/m2•K

215

Problems

䊏

(a) Sketch the temperature distribution on T x coordinates and identify its key features. Assume that T1 T3 T앝.

Duct wall

Ambient air

Water

T∞,o, ho

T∞,i, hi

(b) Derive an expression for the midpoint temperature T2 in terms of the thermal and geometric parameters of the system. (c) For T1 200 C, T3 100 C, and the conditions provided in the schematic, calculate T2 and plot the temperature distribution. Describe key features of the distribution and compare it to your sketch of part (a). 3.120 A carbon nanotube is suspended across a trench of width s 5 m that separates two islands, each at T앝 300 K. A focused laser beam irradiates the nanotube at a distance from the left island, delivering q 10 W of energy to the nanotube. The nanotube temperature is measured at the midpoint of the trench using a point probe. The measured nanotube temperature is T1 324.5 K for 1 1.5 m and T2 326.4 K for 2 3.5 m.

Temperature measurement Laser irradiation s/2

ξ

T∞ Tsur 300 K

s 5 µm

Rt,c,L

Carbon nanotube

Rt,c,R

Determine the two contact resistances, Rt,c,L and Rt,c,R at the left and right ends of the nanotube, respectively. The experiment is performed in a vacuum with Tsur 300 K. The nanotube thermal conductivity and diameter are kcn 3100 W/m 䡠 K and D 14 nm, respectively. 3.121 A probe of overall length L 200 mm and diameter D 12.5 mm is inserted through a duct wall such that a portion of its length, referred to as the immersion length Li, is in contact with the water stream whose temperature, T앝, i, is to be determined. The convection coefficients over the immersion and ambient-exposed lengths are hi 1100 W/m2 䡠 K and ho 10 W/m2 䡠 K, respectively. The probe has a thermal conductivity of 177 W/m 䡠 K and is in poor thermal contact with the duct wall.

Sensor, Ttip Leads

D

Lo

Li L

(a) Derive an expression for evaluating the measurement error, Terr Ttip T앝,i, which is the difference between the tip temperature, Ttip, and the water temperature, T앝,i. Hint: Define a coordinate system with the origin at the duct wall and treat the probe as two fins extending inward and outward from the duct, but having the same base temperature. Use Case A results from Table 3.4. (b) With the water and ambient air temperatures at 80 and 20 C, respectively, calculate the measurement error, Terr, as a function of immersion length for the conditions Li /L 0.225, 0.425, and 0.625. (c) Compute and plot the effects of probe thermal conductivity and water velocity (hi) on the measurement error. 3.122 A rod of diameter D 25 mm and thermal conductivity k 60 W/m 䡠 K protrudes normally from a furnace wall that is at Tw 200 C and is covered by insulation of thickness Lins 200 mm. The rod is welded to the furnace wall and is used as a hanger for supporting instrumentation cables. To avoid damaging the cables, the temperature of the rod at its exposed surface, To, must be maintained below a specified operating limit of Tmax 100 C. The ambient air temperature is T앝 25 C, and the convection coefficient is h 15 W/m2 䡠 K. Air

T∞, h

Tw

D

To

Hot furnace wall

Insulation

Lins

Lo

(a) Derive an expression for the exposed surface temperature To as a function of the prescribed thermal and

216

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

geometrical parameters. The rod has an exposed length Lo, and its tip is well insulated.

(a) Calculate the steady-state temperature To of the rod at the midpoint of the heated portion in the coil.

(b) Will a rod with Lo 200 mm meet the specified operating limit? If not, what design parameters would you change? Consider another material, increasing the thickness of the insulation, and increasing the rod length. Also, consider how you might attach the base of the rod to the furnace wall as a means to reduce To.

(b) Calculate the temperature of the rod Tb at the edge of the heated portion.

3.123 A metal rod of length 2L, diameter D, and thermal conductivity k is inserted into a perfectly insulating wall, exposing one-half of its length to an airstream that is of temperature T앝 and provides a convection coefficient h at the surface of the rod. An electromagnetic field induces volumetric energy generation at . a uniform rate q within the embedded portion of the rod.

3.125 From Problem 1.71, consider the wire leads connecting the transistor to the circuit board. The leads are of thermal conductivity k, thickness t, width w, and length L. One end of a lead is maintained at a temperature Tc corresponding to the transistor case, while the other end assumes the temperature Tb of the circuit board. During steady-state operation, current flow through the leads provides for uniform volumetric heating in the amount . q, while there is convection cooling to air that is at T앝 and maintains a convection coefficient h.

Air T∞ = 20°C h = 100 W/m2•K

Tb

To

Transistor case(Tc) Wire lead(k)

T∞, h

Rod, D, k

x

•

q L

Circuit board(Tb)

L L = 50 mm D = 5 mm k = 25 W/m•K • q = 1 × 106 W/m3

x

(a) Derive an expression for the steady-state temperature Tb at the base of the exposed half of the rod. The exposed region may be approximated as a very long fin. (b) Derive an expression for the steady-state temperature To at the end of the embedded half of the rod. (c) Using numerical values provided in the schematic, plot the temperature distribution in the rod and describe key features of the distribution. Does the rod behave as a very long fin? 3.124 A very long rod of 5-mm diameter and uniform thermal conductivity k 25 W/m 䡠 K is subjected to a heat treatment process. The center, 30-mm-long portion of the rod within the induction heating coil experiences uniform volumetric heat generation of 7.5 106 W/m3. Induction heating coil

To

t w

Gap

(a) Derive an equation from which the temperature distribution in a wire lead may be determined. List all pertinent assumptions. (b) Determine the temperature distribution in a wire lead, expressing your results in terms of the prescribed variables. 3.126 Turbine blades mounted to a rotating disc in a gas turbine engine are exposed to a gas stream that is at T앝 1200 C and maintains a convection coefficient of h 250 W/m2 䡠 K over the blade. Blade tip

L

Gas stream

T∞, h

Tb

x •

Region experiencing q

30 mm

Very long rod, 5-mm dia.

The unheated portions of the rod, which protrude from the heating coil on either side, experience convection with the ambient air at T앝 20 C and h 10 W/m2 䡠 K. Assume that there is no convection from the surface of the rod within the coil.

Tb Rotating disk Air coolant

䊏

217

Problems

The blades, which are fabricated from Inconel, k ⬇ 20 W/m 䡠 K, have a length of L 50 mm. The blade profile has a uniform cross-sectional area of Ac 6 104 m2 and a perimeter of P 110 mm. A proposed blade-cooling scheme, which involves routing air through the supporting disc, is able to maintain the base of each blade at a temperature of Tb 300 C. (a) If the maximum allowable blade temperature is 1050 C and the blade tip may be assumed to be adiabatic, is the proposed cooling scheme satisfactory? (b) For the proposed cooling scheme, what is the rate at which heat is transferred from each blade to the coolant? 3.127 In a test to determine the friction coefficient associated with a disk brake, one disk and its shaft are rotated at a constant angular velocity , while an equivalent disk/shaft assembly is stationary. Each disk has an outer radius of r2 180 mm, a shaft radius of r1 20 mm, a thickness of t 12 mm, and a thermal conductivity of k 15 W/m 䡠 K. A known force F is applied to the system, and the corresponding torque required to maintain rotation is measured. The disk contact pressure may be assumed to be uniform (i.e., independent of location on the interface), and the disks may be assumed to be well insulated from the surroundings. t r2

ω

T1

r1

F

T∞, h

Tb

Ts (x) 2t

t y

To (x)

x x

In this problem we seek to determine conditions for which the transverse (y-direction) temperature difference within the extended surface is negligible compared to the temperature difference between the surface and the environment, such that the one-dimensional analysis of Section 3.6.1 is valid. (a) Assume that the transverse temperature distribution is parabolic and of the form

冢冣

T(y) To(x) y t Ts(x) To(x)

2

where Ts(x) is the surface temperature and To(x) is the centerline temperature at any x-location. Using Fourier’s law, write an expression for the conduction heat flux at the surface, qy (t), in terms of Ts and To. (b) Write an expression for the convection heat flux at the surface for the x-location. Equating the two expressions for the heat flux by conduction and convection, identify the parameter that determines the ratio (To Ts)/(Ts T앝). (c) From the foregoing analysis, develop a criterion for establishing the validity of the onedimensional assumption used to model an extended surface.

τ Disk interface, friction coefficient, µ

(a) Obtain an expression that may be used to evaluate from known quantities. (b) For the region r1 r r2, determine the radial temperature distribution T(r) in the disk, where T(r1) T1 is presumed to be known. (c) Consider test conditions for which F 200 N, 40 rad/s, 8 N 䡠 m, and T1 80 C. Evaluate the friction coefficient and the maximum disk temperature. 3.128 Consider an extended surface of rectangular cross section with heat flow in the longitudinal direction.

Simple Fins 3.129 A long, circular aluminum rod is attached at one end to a heated wall and transfers heat by convection to a cold fluid. (a) If the diameter of the rod is tripled, by how much would the rate of heat removal change? (b) If a copper rod of the same diameter is used in place of the aluminum, by how much would the rate of heat removal change? 3.130 A brass rod 100 mm long and 5 mm in diameter extends horizontally from a casting at 200 C. The rod is in an air environment with T앝 20 C and h 30 W/m2 䡠 K. What is the temperature of the rod 25, 50, and 100 mm from the casting?

218

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

3.131 The extent to which the tip condition affects the thermal performance of a fin depends on the fin geometry and thermal conductivity, as well as the convection coefficient. Consider an alloyed aluminum (k 180 W/m 䡠 K) rectangular fin of length L 10 mm, thickness t 1 mm, and width w t. The base temperature of the fin is Tb l00 C, and the fin is exposed to a fluid of temperature T앝 25 C. (a) Assuming a uniform convection coefficient of h 100 W/m2 䡠 K over the entire fin surface, determine the fin heat transfer rate per unit width qf , efficiency f , effectiveness f , thermal resistance per unit width Rt, f , and the tip temperature T(L) for Cases A and B of Table 3.4. Contrast your results with those based on an infinite fin approximation. (b) Explore the effect of variations in the convection coefficient on the heat rate for 10 h 1000 W/m2 䡠 K. Also consider the effect of such variations for a stainless steel fin (k 15 W/m 䡠 K). 3.132 A pin fin of uniform, cross-sectional area is fabricated of an aluminum alloy (k 160 W/m 䡠 K). The fin diameter is D 4 mm, and the fin is exposed to convective conditions characterized by h 220 W/m2 䡠 K. It is reported that the fin efficiency is f 0.65. Determine the fin length L and the fin effectiveness f. Account for tip convection. 3.133 The extent to which the tip condition affects the thermal performance of a fin depends on the fin geometry and thermal conductivity, as well as the convection coefficient. Consider an alloyed aluminum (k 180 W/m 䡠 K) rectangular fin whose base temperature is Tb 100 C. The fin is exposed to a fluid of temperature T앝 25 C, and a uniform convection coefficient of h 100 W/m2 䡠 K may be assumed for the fin surface. (a) For a fin of length L 10 mm, thickness t 1 mm, and width w t, determine the fin heat transfer rate per unit width qf , efficiency f, effectiveness f, thermal resistance per unit width Rt,f, and tip temperature T(L) for Cases A and B of Table 3.4. Contrast your results with those based on an infinite fin approximation. (b) Explore the effect of variations in L on the heat rate for 3 L 50 mm. Also consider the effect of such variations for a stainless steel fin (k 15 W/m 䡠 K). 3.134 A straight fin fabricated from 2024 aluminum alloy (k 185 W/m 䡠 K) has a base thickness of t 3 mm and a length of L 15 mm. Its base temperature is Tb 100 C, and it is exposed to a fluid for which T앝 20 C and h 50 W/m2 䡠 K. For the foregoing conditions and a fin of unit width, compare the fin heat

rate, efficiency, and volume for rectangular, triangular, and parabolic profiles. 3.135 Triangular and parabolic straight fins are subjected to the same thermal conditions as the rectangular straight fin of Problem 3.134. (a) Determine the length of a triangular fin of unit width and base thickness t 3 mm that will provide the same fin heat rate as the straight rectangular fin. Determine the ratio of the mass of the triangular straight fin to that of the rectangular straight fin. (b) Repeat part (a) for a parabolic straight fin. 3.136 Two long copper rods of diameter D 10 mm are soldered together end to end, with solder having a melting point of 650 C. The rods are in air at 25 C with a convection coefficient of 10 W/m2 䡠 K. What is the minimum power input needed to effect the soldering? 3.137 Circular copper rods of diameter D 1 mm and length L 25 mm are used to enhance heat transfer from a surface that is maintained at Ts,1 100 C. One end of the rod is attached to this surface (at x 0), while the other end (x 25 mm) is joined to a second surface, which is maintained at Ts,2 0 C. Air flowing between the surfaces (and over the rods) is also at a temperature of T앝 0 C, and a convection coefficient of h 100 W/m2 䡠 K is maintained. (a) What is the rate of heat transfer by convection from a single copper rod to the air? (b) What is the total rate of heat transfer from a 1 m 1 m section of the surface at 100 C, if a bundle of the rods is installed on 4-mm centers? 3.138 During the initial stages of the growth of the nanowire of Problem 3.109, a slight perturbation of the liquid catalyst droplet can cause it to be suspended on the top of the nanowire in an off-center position. The resulting nonuniform deposition of solid at the solid-liquid interface can be manipulated to form engineered shapes such as a nanospring, that is characterized by a spring radius r, spring pitch s, overall chord length Lc (length running along the spring), and end-to-end length L, as shown in the sketch. Consider a silicon carbide nanospring of diameter D 15 nm, r 30 nm, s 25 nm, and Lc 425 nm. From experiments, it is known that the average spring pitch s– varies with average tem– – perature T by the relation ds–/dT 0.1 nm/K. Using this information, a student suggests that a nanoactuator can be constructed by connecting one end of the nanospring to a small heater and raising the temperature of that end of the nano spring above its initial value. Calculate the actuation distance L for conditions where h 106 W/m2 䡠 K, T앝 Ti 25 C, with a base

䊏

temperature of Tb 50 C. If the base temperature can be controlled to within 1 C, calculate the accuracy to which the actuation distance can be controlled. Hint: Assume the spring radius does not change when the spring is heated. The overall spring length may be approximated by the formula, L

Lc s 2 兹 r2 (s2)2 L

x

Tb

• D

219

Problems

s

T∞, h

3.139 Consider two long, slender rods of the same diameter but different materials. One end of each rod is attached to a base surface maintained at 100 C, while the surfaces of the rods are exposed to ambient air at 20 C. By traversing the length of each rod with a thermocouple, it was observed that the temperatures of the rods were equal at the positions xA 0.15 m and xB 0.075 m, where x is measured from the base surface. If the thermal conductivity of rod A is known to be kA 70 W/m 䡠 K, determine the value of kB for rod B. 3.140 A 40-mm-long, 2-mm-diameter pin fin is fabricated of an aluminum alloy (k 140 W/m 䡠 K). (a) Determine the fin heat transfer rate for Tb 50 C, T앝 25 C, h 1000 W/m2 䡠 K, and an adiabatic tip condition. (b) An engineer suggests that by holding the fin tip at a low temperature, the fin heat transfer rate can be increased. For T(x L) 0 C, determine the new fin heat transfer rate. Other conditions are as in part (a). (c) Plot the temperature distribution, T(x), over the range 0 x L for the adiabatic tip case and the prescribed tip temperature case. Also show the ambient temperature in your graph. Discuss relevant features of the temperature distribution. (d) Plot the fin heat transfer rate over the range 0 h 1000 W/m2 䡠 K for the adiabatic tip case and the prescribed tip temperature case. For the prescribed tip temperature case, what would the

calculated fin heat transfer rate be if Equation 3.78 were used to determine qf rather than Equation 3.76? 3.141 An experimental arrangement for measuring the thermal conductivity of solid materials involves the use of two long rods that are equivalent in every respect, except that one is fabricated from a standard material of known thermal conductivity kA while the other is fabricated from the material whose thermal conductivity kB is desired. Both rods are attached at one end to a heat source of fixed temperature Tb, are exposed to a fluid of temperature T앝, and are instrumented with thermocouples to measure the temperature at a fixed distance x1 from the heat source. If the standard material is aluminum, with kA 200 W/m 䡠 K, and measurements reveal values of TA 75 C and TB 60 C at x1 for Tb 100 C and T앝 25 C, what is the thermal conductivity kB of the test material?

Fin Systems and Arrays 3.142 Finned passages are frequently formed between parallel plates to enhance convection heat transfer in compact heat exchanger cores. An important application is in electronic equipment cooling, where one or more air-cooled stacks are placed between heat-dissipating electrical components. Consider a single stack of rectangular fins of length L and thickness t, with convection conditions corresponding to h and T앝. 200 mm 100 mm

14 mm

To

x

L

Air T∞, h

TL

(a) Obtain expressions for the fin heat transfer rates, qf,o and qf,L, in terms of the base temperatures, To and TL. (b) In a specific application, a stack that is 200 mm wide and 100 mm deep contains 50 fins, each of length L 12 mm. The entire stack is made from aluminum, which is everywhere 1.0 mm thick. If temperature limitations associated with electrical components joined to opposite plates dictate maximum allowable plate temperatures of To 400 K

220

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

and TL 350 K, what are the corresponding maximum power dissipations if h 150 W/m2 䡠 K and T앝 300 K?

(a) Consider limitations for which the array has N 11 fins, in which case values of the fin thickness t 0.182 mm and pitch S 1.982 mm are obtained from the requirements that W (N 1)S t and S t 1.8 mm. If the maximum allowable chip temperature is Tc 85 C, what is the corresponding value of the chip power qc? An adiabatic fin tip condition may be assumed, and airflow along the outer surfaces of the heat sink may be assumed to provide a convection coefficient equivalent to that associated with airflow through the channels.

3.143 The fin array of Problem 3.142 is commonly found in compact heat exchangers, whose function is to provide a large surface area per unit volume in transferring heat from one fluid to another. Consider conditions for which the second fluid maintains equivalent temperatures at the parallel plates, To TL, thereby establishing symmetry about the midplane of the fin array. The heat exchanger is 1 m long in the direction of the flow of air (first fluid) and 1 m wide in a direction normal to both the airflow and the fin surfaces. The length of the fin passages between adjoining parallel plates is L 8 mm, whereas the fin thermal conductivity and convection coefficient are k 200 W/m 䡠 K (aluminum) and h 150 W/m2 䡠 K, respectively. (a) If the fin thickness and pitch are t 1 mm and S 4 mm, respectively, what is the value of the thermal resistance Rt,o for a one-half section of the fin array? (b) Subject to the constraints that the fin thickness and pitch may not be less than 0.5 and 3 mm, respectively, assess the effect of changes in t and S. 3.144 An isothermal silicon chip of width W 20 mm on a side is soldered to an aluminum heat sink (k 180 W/m 䡠 K) of equivalent width. The heat sink has a base thickness of Lb 3 mm and an array of rectangular fins, each of length Lf 15 mm. Airflow at T앝 20 C is maintained through channels formed by the fins and a cover plate, and for a convection coefficient of h 100 W/m2 䡠 K, a minimum fin spacing of 1.8 mm is dictated by limitations on the flow pressure drop. The solder joint has a thermal resistance of Rt, c 2 106 m2 䡠 K/W. Chip, Tc, qc

(b) With (S t) and h fixed at 1.8 mm and 100 W/m2 䡠 K, respectively, explore the effect of increasing the fin thickness by reducing the number of fins. With N 11 and S t fixed at 1.8 mm, but relaxation of the constraint on the pressure drop, explore the effect of increasing the airflow, and hence the convection coefficient. 3.145 As seen in Problem 3.109, silicon carbide nanowires of diameter D 15 nm can be grown onto a solid silicon carbide surface by carefully depositing droplets of catalyst liquid onto a flat silicon carbide substrate. Silicon carbide nanowires grow upward from the deposited drops, and if the drops are deposited in a pattern, an array of nanowire fins can be grown, forming a silicon carbide nano-heat sink. Consider finned and unfinned electronics packages in which an extremely small, 10 m 10 m electronics device is sandwiched between two d 100-nm-thick silicon carbide sheets. In both cases, the coolant is a dielectric liquid at 20 C. A heat transfer coefficient of h 1 105 W/m2 䡠 K exists on the top and bottom of the unfinned package and on all surfaces of the exposed silicon carbide fins, which are each L 300 nm long. Each nano-heat sink includes a 200 200 array of nanofins. Determine the maximum allowable heat rate that can be generated by the electronic device so that its temperature is maintained at Tt 85 C for the unfinned and finned packages.

Solder, R"t ,c

W

T∞, h

Cover plate

D

T∞, h

Heat sink, k

Tt

d

L

Lb t

W = 10 µm

Lf

S

T∞, h T∞, h

Air

T∞, h

Unfinned

Nano-finned

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221

Problems

3.146 As more and more components are placed on a single integrated circuit (chip), the amount of heat that is dissipated continues to increase. However, this increase is limited by the maximum allowable chip operating temperature, which is approximately 75 C. To maximize heat dissipation, it is proposed that a 4 4 array of copper pin fins be metallurgically joined to the outer surface of a square chip that is 12.7 mm on a side.

Top view

W

Pin fins, Dp

D 200 mm Tsur,t

ht, T∞,t

Lb 10 mm

Lf 25 mm w 80 mm hb, T∞,b t 5 mm

Tsur,b

Sideview

3.148 In Problem 3.146, the prescribed value of ho 1000 W/m2 䡠 K is large and characteristic of liquid cooling. In T∞,o, ho practice it would be far more preferable to use air coolLp Chip, ing, for which a reasonable upper limit to the convecqc, Tc Chip tion coefficient would be ho 250 W/m2 䡠 K. Assess the Lb effect of changes in the pin fin geometry on the chip heat rate if the remaining conditions of Problem 3.146, Contact Air W = 12.7 mm including a maximum allowable chip temperature of resistance, T∞,i, hi 75 C, remain in effect. Parametric variations that may R"t, c /Ac be considered include the total number of pins N in the Board, kb square array, the pin diameter Dp, and the pin length Lp. However, the product N1/2Dp should not exceed 9 mm (a) Sketch the equivalent thermal circuit for the pin– to ensure adequate airflow passage through the array. chip–board assembly, assuming one-dimensional, Recommend a design that enhances chip cooling. steady-state conditions and negligible contact resistance between the pins and the chip. In vari- 3.149 Water is heated by submerging 50-mm-diameter, thinable form, label appropriate resistances, temperawalled copper tubes in a tank and passing hot combustures, and heat rates. tion gases (Tg 750 K) through the tubes. To enhance heat transfer to the water, four straight fins of uniform (b) For the conditions prescribed in Problem 3.27, cross section, which form a cross, are inserted in each what is the maximum rate at which heat can be tube. The fins are 5 mm thick and are also made of dissipated in the chip when the pins are in place? copper (k 400 W/m 䡠 K). That is, what is the value of qc for Tc 75 C? The pin diameter and length are Dp 1.5 mm and D = 50 mm Lp 15 mm. 3.147 A homeowner’s wood stove is equipped with a top burner for cooking. The D 200-mm-diameter burner is fabricated of cast iron (k 65 W/m 䡠 K). The bottom (combustion) side of the burner has 8 straight fins of uniform cross section, arranged as shown in the sketch. A very thin ceramic coating ( 0.95) is applied to all surfaces of the burner. The top of the burner is exposed to room conditions (Tsur,t T앝,t 20 C, ht 40 W/m2 䡠 K), while the bottom of the burner is exposed to combustion conditions (Tsur,b T앝.b 450 C, hb 50 W/m2 䡠 K). Compare the top surface temperature of the finned burner to that which would exist for a burner without fins. Hint: Use the same expression for radiation heat transfer to the bottom of the finned burner as for the burner with no fins.

Ts = 350 K

Water

Fins (t = 5 mm)

Gases

Tg = 750 K

hg = 30 W/m2•K Tube wall

If the tube surface temperature is Ts 350 K and the gas-side convection coefficient is hg 30 W/m2 䡠 K, what is the rate of heat transfer to the water per meter of pipe length? 3.150 As a means of enhancing heat transfer from highperformance logic chips, it is common to attach a

222

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

heat sink to the chip surface in order to increase the surface area available for convection heat transfer. Because of the ease with which it may be manufactured (by taking orthogonal sawcuts in a block of material), an attractive option is to use a heat sink consisting of an array of square fins of width w on a side. The spacing between adjoining fins would be determined by the width of the sawblade, with the sum of this spacing and the fin width designated as the fin pitch S. The method by which the heat sink is joined to the chip would determine the interfacial contact resistance, Rt,c. Wc

(S w) 0.25 mm, and/or increasing Lƒ (subject to manufacturing constraints that Lƒ 10 mm). Assess the effect of such changes. 3.151 Because of the large number of devices in today’s PC chips, finned heat sinks are often used to maintain the chip at an acceptable operating temperature. Two fin designs are to be evaluated, both of which have base (unfinned) area dimensions of 53 mm 57 mm. The fins are of square cross section and fabricated from an extruded aluminum alloy with a thermal conductivity of 175 W/m 䡠 K. Cooling air may be supplied at 25 C, and the maximum allowable chip temperature is 75 C. Other features of the design and operating conditions are tabulated.

Heat sink Top View

T∞, h

Fin Dimensions Cross Section Design w ⴛ w (mm) A B

w Square fins

33 11

Length L (mm) 30 7

Convection Number of Coefficient Fins in Array (W/m2 䡠 K) 69 14 17

125 375

57 mm

Lf

L = 30 mm

S Heat sink Interface,

Lb

53 mm

R"t,c Chip,

3 mm × 3 mm Tb = 75°C cross section

qc, Tc

Consider a square chip of width Wc 16 mm and conditions for which cooling is provided by a dielectric liquid with T앝 25 C and h 1500 W/m2 䡠 K. The heat sink is fabricated from copper (k 400 W/m 䡠 K), and its characteristic dimensions are w 0.25 mm, S 0.50 mm, Lƒ 6 mm, and Lb 3 mm. The prescribed values of w and S represent minima imposed by manufacturing constraints and the need to maintain adequate flow in the passages between fins. (a) If a metallurgical joint provides a contact resistance of Rt,c 5 106 m2 䡠 K/W and the maximum allowable chip temperature is 85 C, what is the maximum allowable chip power dissipation qc? Assume all of the heat to be transferred through the heat sink. (b) It may be possible to increase the heat dissipation by increasing w, subject to the constraint that

54 pins, 9 × 6 array (Design A)

Determine which fin arrangement is superior. In your analysis, calculate the heat rate, efficiency, and effectiveness of a single fin, as well as the total heat rate and overall efficiency of the array. Since real estate inside the computer enclosure is important, compare the total heat rate per unit volume for the two designs. 3.152 Consider design B of Problem 3.151. Over time, dust can collect in the fine grooves that separate the fins. Consider the buildup of a dust layer of thickness Ld, as shown in the sketch. Calculate and plot the total heat rate for design B for dust layers in the range 0 Ld 5 mm. The thermal conductivity of the dust can be taken as kd = 0.032 W/m 䡠 K. Include the effects of convection from the fin tip.

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223

Problems

while air at T앝,o 25 C flows through the annular region formed by the larger concentric tube.

L = 7 mm Ld

r1 Dust

r2 Air

r3 1 mm 1 mm cross section

Tb = 75°C

•

Spider with 12 ribs

Tube

r2

r1 t

Insulating sleeve

Water

T∞,i, hi

3.153 A long rod of 20-mm diameter and a thermal conductivity of 1.5 W/m 䡠 K has a uniform internal volumetric thermal energy generation of 106 W/m3. The rod is covered with an electrically insulating sleeve of 2-mm thickness and thermal conductivity of 0.5 W/m 䡠 K. A spider with 12 ribs and dimensions as shown in the sketch has a thermal conductivity of 175 W/m 䡠 K, and is used to support the rod and to maintain concentricity with an 80mm-diameter tube. Air at T앝 25 C passes over the spider surface, and the convection coefficient is 20 W/m2 䡠 K. The outer surface of the tube is well insulated. We wish to increase volumetric heating within the rod, while not allowing its centerline temperature to exceed 100 C. Determine the impact of the following changes, which may be effected independently or concurrently: (i) increasing the air speed and hence the convection coefficient; (ii) changing the number and/or thickness of the ribs; and (iii) using an electrically nonconducting sleeve material of larger thermal conductivity (e.g., amorphous carbon or quartz). Recommend a realis. tic configuration that yields a significant increase in q. Rod, q

T∞,o, ho

r3

Air

T∞ = 25°C

t

(a) Sketch the equivalent thermal circuit of the heater and relate each thermal resistance to appropriate system parameters. (b) If hi 5000 W/m2 䡠 K and ho 200 W/m2 䡠 K, what is the heat rate per unit length? (c) Assess the effect of increasing the number of fins N and/or the fin thickness t on the heat rate, subject to the constraint that Nt 50 mm. 3.155 Determine the percentage increase in heat transfer associated with attaching aluminum fins of rectangular profile to a plane wall. The fins are 50 mm long, 0.5 mm thick, and are equally spaced at a distance of 4 mm (250 fins/m). The convection coefficient associated with the bare wall is 40 W/m2 䡠 K, while that resulting from attachment of the fins is 30 W/m2 䡠 K. 3.156 Heat is uniformly generated at the rate of 2 105 W/m3 in a wall of thermal conductivity 25 W/m 䡠 K and thickness 60 mm. The wall is exposed to convection on both sides, with different heat transfer coefficients and temperatures as shown. There are straight rectangular fins on the right-hand side of the wall, with dimensions as shown and thermal conductivity of 250 W/m 䡠 K. What is the maximum temperature that will occur in the wall?

r1 = 12 mm r2 = 17 mm r3 = 40 mm t = 4 mm L = r3 – r2 = 23 mm

3.154 An air heater consists of a steel tube (k 20 W/m 䡠 K), with inner and outer radii of r1 13 mm and r2 16 mm, respectively, and eight integrally machined longitudinal fins, each of thickness t 3 mm. The fins extend to a concentric tube, which is of radius r3 40 mm and insulated on its outer surface. Water at a temperature T앝,i 90 C flows through the inner tube,

Lf = 20 mm

k = 25 W/m•K q• = 2 105 W/m3 h1 = 50 W/m2•K T∞,1 = 30°C

t = 2 mm

δ = 2 mm 2L = 60 mm

kf = 250 W/m•K

h2 = 12 W/m2•K T∞,2 = 15°C

224

Chapter 3

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One-Dimensional, Steady-State Conduction

3.157 Aluminum fins of triangular profile are attached to a plane wall whose surface temperature is 250 C. The fin base thickness is 2 mm, and its length is 6 mm. The system is in ambient air at a temperature of 20 C, and the surface convection coefficient is 40 W/m2 䡠 K. (a) What are the fin efficiency and effectiveness? (b) What is the heat dissipated per unit width by a single fin? 3.158 An annular aluminum fin of rectangular profile is attached to a circular tube having an outside diameter of 25 mm and a surface temperature of 250 C. The fin is 1 mm thick and 10 mm long, and the temperature and the convection coefficient associated with the adjoining fluid are 25 C and 25 W/m2 䡠 K, respectively. (a) What is the heat loss per fin? (b) If 200 such fins are spaced at 5-mm increments along the tube length, what is the heat loss per meter of tube length? 3.159 Annular aluminum fins of rectangular profile are attached to a circular tube having an outside diameter of 50 mm and an outer surface temperature of 200 C. The fins are 4 mm thick and 15 mm long. The system is in ambient air at a temperature of 20 C, and the surface convection coefficient is 40 W/m2 䡠 K. (a) What are the fin efficiency and effectiveness? (b) If there are 125 such fins per meter of tube length, what is the rate of heat transfer per unit length of tube?

qi 105 W/m2. Assuming negligible contact resistance between the wall and the casing, determine the wall inner temperature Ti, the interface temperature T1, and the fin base temperature Tb. Determine these temperatures if the interface contact resistance is Rt, c 104 m2 䡠 K/W. 3.161 Consider the air-cooled combustion cylinder of Problem 3.160, but instead of imposing a uniform heat flux at the inner surface, consider conditions for which the time-averaged temperature of the combustion gases is Tg 1100 K and the corresponding convection coefficient is hg 150 W/m2 䡠 K. All other conditions, including the cylinder/casing contact resistance, remain the same. Determine the heat rate per unit length of cylinder (W/m), as well as the cylinder inner temperature Ti, the interface temperatures T1,i and T1,o, and the fin base temperature Tb. Subject to the constraint that the fin gap is fixed at ␦ 2 mm, assess the effect of increasing the fin thickness at the expense of reducing the number of fins. 3.162 Heat transfer from a transistor may be enhanced by inserting it in an aluminum sleeve (k 200 W/m 䡠 K) having 12 integrally machined longitudinal fins on its outer surface. The transistor radius and height are r1 2.5 mm and H 4 mm, respectively, while the fins are of length L r3 r2 8 mm and uniform thickness t 0.8 mm. The thickness of the sleeve base is r2 r1 1 mm, and the contact resistance of the sleeve-transistor interface is Rt,c 0.6 103 m2 䡠 K/W. Air at T앝 20 C flows over the fin surface, providing an approximately uniform convection coeffficient of h 30 W/m2 䡠 K.

3.160 It is proposed to air-cool the cylinders of a combustion chamber by joining an aluminum casing with annular fins (k 240 W/m 䡠 K) to the cylinder wall (k 50 W/m 䡠 K). Cylinder wall

Ti

t

Aluminum casing

T∞, h

T1 Tb

Transistor

R"t,c, T1

t = 2 mm q"i

H

δ = 2 mm

Sleeve with longitudinal fins

ri = 60 mm

T∞, h

r1 = 66 mm r2 = 70 mm ro = 95 mm

The air is at 320 K and the corresponding convection coefficient is 100 W/m2 䡠 K. Although heating at the inner surface is periodic, it is reasonable to assume steady-state conditions with a time-averaged heat flux of

r1 r2

r3

(a) When the transistor case temperature is 80 C, what is the rate of heat transfer from the sleeve? (b) Identify all of the measures that could be taken to improve design and/or operating conditions, such that heat dissipation may be increased while still maintaining a case temperature of 80 C. In words, assess the relative merits of each measure. Choose

䊏

225

Problems

what you believe to be the three most promising measures, and numerically assess the effect of corresponding changes in design and/or operating conditions on thermal performance. 3.163 Consider the conditions of Problem 3.149 but now allow for a tube wall thickness of 5 mm (inner and outer diameters of 50 and 60 mm), a fin-to-tube thermal contact resistance of 104 m2 䡠 K/W, and the fact that the water temperature, Tw 350 K, is known, not the tube surface temperature. The water-side convection coefficient is hw 2000 W/m2 䡠 K. Determine the rate of heat transfer per unit tube length (W/m) to the water. What would be the separate effect of each of the following design changes on the heat rate: (i) elimination of the contact resistance; (ii) increasing the number of fins from four to eight; and (iii) changing the tube wall and fin material from copper to AISI 304 stainless steel (k 20 W/m 䡠 K)? 3.164 A scheme for concurrently heating separate water and air streams involves passing them through and over an array of tubes, respectively, while the tube wall is heated electrically. To enhance gas-side heat transfer, annular fins of rectangular profile are attached to the outer tube surface. Attachment is facilitated with a dielectric adhesive that electrically isolates the fins from the current-carrying tube wall. Gas flow

(a) Assuming uniform volumetric heat generation within the tube wall, obtain expressions for the heat rate per unit tube length (W/m) at the inner (ri) and outer (ro) surfaces of the wall. Express your results in terms of the tube inner and outer surface temperatures, Ts,i and Ts,o, and other pertinent parameters. (b) Obtain expressions that could be used to determine Ts,i and Ts,o in terms of parameters associated with the water- and air-side conditions. (c) Consider conditions for which the water and air are at T앝,i T앝,o 300 K, with corresponding convection coefficients of hi 2000 W/m2 䡠 K and ho 100 W/m2 䡠 K. Heat is uniformly dissipated in a stainless steel tube (kw 15 W/m 䡠 K), having inner and outer radii of ri 25 mm and ro 30 mm, and aluminum fins (t ␦ 2 mm, rt 55 mm) are attached to the outer surface, with Rt,c 104 m2 䡠 K/W. Determine the heat rates and temperatures at the inner and outer surfaces as a func. tion of the rate of volumetric heating q. The upper . limit to q will be determined by the constraints that Ts,i not exceed the boiling point of water (100 C) and Ts,o not exceed the decomposition temperature of the adhesive (250 C).

The Bioheat Equation 3.165 Consider the conditions of Example 3.12, except that the person is now exercising (in the air environment), which increases the metabolic heat generation rate by a factor of 8, to 5600 W/m3. At what rate would the person have to perspire (in liters/s) to maintain the same skin temperature as in that example? 3.166 Consider the conditions of Example 3.12 for an air environment, except now the air and surroundings temperatures are both 15 C. Humans respond to cold by shivering, which increases the metabolic heat generation rate. What would the metabolic heat generation rate (per unit volume) have to be to maintain a comfortable skin temperature of 33 C under these conditions?

Liquid flow

Air

T∞,o, ho t

Ts,o

δ

Ts,i

rt T∞,i, hi

ri ro

I Tube, q•, k w Adhesive, R"t,c

3.167 Consider heat transfer in a forearm, which can be approximated as a cylinder of muscle of radius 50 mm (neglecting the presence of bones), with an outer layer of skin and fat of thickness 3 mm. There is metabolic heat generation and perfusion within the muscle. The metabolic heat generation rate, perfusion rate, arterial temperature, and properties of blood, muscle, and skin/fat layer are identical to those in Example 3.12.

226

Chapter 3

䊏

One-Dimensional, Steady-State Conduction with the flowing gases is h h1 h2 80 W/m2 䡠 K while the electrical resistance of the load is Re,load 4 .

The environment and surroundings are the same as for the air environment in Example 3.12. ri = 50 mm Skin/fat

Cover plate Heat sink 1, k

Re,load Air T∞,1, h1

Muscle δ sf = 3 mm

Thermoelectric module Lb

(a) Write the bioheat transfer equation in radial coordinates. Write the boundary conditions that express symmetry at the centerline of the forearm and specified temperature at the outer surface of the muscle. Solve the differential equation and apply the boundary conditions to find an expression for the temperature distribution. Note that the derivatives of the modified Bessel functions are given in Section 3.6.4. (b) Equate the heat flux at the outer surface of the muscle to the heat flux through the skin/fat layer and into the environment to determine the temperature at the outer surface of the muscle. (c) Find the maximum forearm temperature.

Thermoelectric Power Generation 3.168 For one of the M 48 modules of Example 3.13, determine a variety of different efficiency values concerning the conversion of waste heat to electrical energy. (a) Determine the thermodynamic efficiency, therm ⬅ PM1/q1. (b) Determine the figure of merit ZT for one module, and the thermoelectric efficiency, TE using Equation 3.128.

2L Lf

t

Air T∞,2, h2

Solder, Rt,c Heat sink 2, k

Cover plate S W

(a) Sketch the equivalent thermal circuit and determine the electric power generated by the module for the situation where the hot and cold gases provide convective heating and cooling directly to the module (no heat sinks). (b) Two heat sinks (k 180 W/m 䡠 K; see sketch), each with a base thickness of Lb 4 mm and fin length Lf 20 mm, are soldered to the upper and lower sides of the module. The fin spacing is 3 mm, while the solder joints each have a thermal 2.5 106 m2 䡠 K/W. Each resistance of Rt,c heat sink has N 11 fins, so that t 2.182 mm and S 5.182 mm, as determined from the requirements that W (N 1)S t and S t 3 mm. Sketch the equivalent thermal circuit and determine the electric power generated by the module. Compare the electric power generated to your answer for part (a). Assume adiabatic fin tips and convection coefficients that are the same as in part (a).

(c) Determine the Carnot efficiency, Carnot 1 – T2/T1. (d) Determine both the thermoelectric efficiency and the Carnot efficiency for the case where h1 h2 l 앝. (e) The energy conversion efficiency of thermoelectric devices is commonly reported by evaluating Equation 3.128, but with T앝,1 and T앝,2 used instead of T1 and T2, respectively. Determine the value of TE based on the inappropriate use of T앝,1 and T앝,2, and compare with your answers for parts (b) and (d). 3.169 One of the thermoelectric modules of Example 3.13 is installed between a hot gas at T앝,1 450 C and a cold gas at T앝,2 20 C. The convection coefficient associated

3.170 Thermoelectric modules have been used to generate electric power by tapping the heat generated by wood stoves. Consider the installation of the thermoelectric module of Example 3.13 on a vertical surface of a wood stove that has a surface temperature of 5 Ts 375 C. A thermal contact resistance of Rt,c 106 m2 䡠 K/W exists at the interface between the stove and the thermoelectric module, while the room air and walls are at T앝 Tsur 25 C. The exposed surface of the thermoelectric module has an emissivity of 0.90 and is subjected to a convection coefficient of h 15 W/m2 䡠 K. Sketch the equivalent thermal circuit and determine the electric power

䊏

227

Problems

generated by the module. The load electrical resistance is Re,load 3 . 3.171 The electric power generator for an orbiting satellite is composed of a long, cylindrical uranium heat source that is housed within an enclosure of square cross section. The only way for heat that is generated by the uranium to leave the enclosure is through four rows of the thermoelectric modules of Example 3.13. The thermoelectric modules generate electric power and also radiate heat into deep space characterized by Tsur 4 K. Consider the situation for which there are 20 modules in each row for a total of M 4 20 80 modules. The modules are wired in series with an electrical load of Re,load 250 , and have an emissivity of 0.93. Determine the electric power generated for E˙ g 1, 10, and 100 kW. Also determine the surface temperatures of the modules for the three thermal energy generation rates.

Tsur W 2L

•

Heat source, Eg Insulation Thermoelectric module,

Re, load

3.172 Rows of the thermoelectric modules of Example 3.13 are attached to the flat absorber plate of Problem 3.108. The rows of modules are separated by Lsep 0.5 m and the backs of the modules are cooled by water at a temperature of Tw 40 C, with h 45 W/m2 䡠 K. Cover plate Evacuated space Absorber plate

q″rad

W

Water Tw , h

Insulation

Lsep

Thermoelectric module

Determine the electric power produced by one row of thermoelectric modules connected in series electrically with a load resistance of 60 . Calculate the heat

transfer rate to the flowing water. Assume rows of 20 immediately adjacent modules, with the lengths of both the module rows and water tubing to be Lrow 20W where W 54 mm is the module dimension taken from Example 3.13. Neglect thermal contact resistances and the temperature drop across the tube wall, and assume that the high thermal conductivity tube wall creates a uniform temperature around the tube perimeter. Because of the thermal resistance provided by the thermoelectric modules, it is no longer appropriate to assume that the temperature of the absorber plate directly above a tube is equal to that of the water.

Micro- and Nanoscale Conduction 3.173 Determine the conduction heat transfer through an air layer held between two 10 mm 10 mm parallel aluminum plates. The plates are at temperatures Ts,1 305 K and Ts,2 295 K, respectively, and the air is at atmospheric pressure. Determine the conduction heat rate for plate spacings of L 1 mm, L 1 m, and L 10 nm. Assume a thermal accommodation coefficient of ␣t 0.92. 3.174 Determine the parallel plate separation distance L, above which the thermal resistance associated with molecule-surface collisions Rt,ms is less than 1% of the resistance associated with molecule–molecule collisions, Rt,mm for (i) air between steel plates with ␣t 0.92 and (ii) helium between clean aluminum plates with ␣t 0.02. The gases are at atmospheric pressure, and the temperature is T 300 K. 3.175 Determine the conduction heat flux through various plane layers that are subjected to boundary temperatures of Ts,1 301 K and Ts,2 299 K at atmospheric pressure. Hint: Do not account for micro- or nanoscale effects within the solid, and assume the thermal accommodation coefficient for an aluminum–air interface is ␣t 0.92. (a) Case A: The plane layer is aluminum. Determine the heat flux qx for Ltot 600 m and Ltot 600 nm. (b) Case B: Conduction occurs through an air layer. Determine the heat flux qx for Ltot 600 m and Ltot 600 nm. (c) Case C: The composite wall is composed of air held between two aluminum sheets. Determine the heat flux qx for Ltot 600 m (with aluminum sheet thicknesses of ␦ 40 m) and Ltot 600 nm (with aluminum sheet thicknesses of ␦ 40 nm). (d) Case D: The composite wall is composed of 7 air layers interspersed between 8 aluminum sheets.

228

Chapter 3

䊏

One-Dimensional, Steady-State Conduction

Determine the heat flux qx for Ltot 600 m (with aluminum sheet and air layer thicknesses of ␦ 40 m) and Ltot 600 nm (with aluminum sheet and air layer thicknesses of ␦ 40 nm).

Ts,1

Ts,1 Aluminum

Air

Ts,2 x Case A

Ts,2 x

Ltot

Case B

Ltot

Ts,1

Ts,1 Air

δ

Aluminum

δ Ts,2

Ts,2 x Case C

Ltot

Air x

Aluminum Case D

Ltot

3.176 The Knudsen number, Kn mfp/L, is a dimensionless parameter used to describe potential micro- or nanoscale effects. Derive an expression for the ratio of the thermal resistance due to molecule–surface collisions to the thermal resistance associated with molecule–molecule collisions, Rt,ms/Rt,mm, in terms of the Knudsen number, the thermal accommodation coefficient ␣t , and the ratio of specific heats ␥, for an ideal gas. Plot the critical Knudsen number, Kncrit, that is associated with Rt,ms /Rt,mm 0.01 versus ␣t, for ␥ 1.4 and 1.67 (corresponding to air and helium, respectively). 3.177 A nanolaminated material is fabricated with an atomic layer deposition process, resulting in a series of

stacked, alternating layers of tungsten and aluminum oxide, each layer being ␦ 0.5 nm thick. Each tungsten–aluminum oxide interface is associated with a thermal resistance of Rt,i 3.85 109 m2 䡠 K/W. The theoretical values of the thermal conductivities of the thin aluminum oxide and tungsten layers are kA 1.65 W/m 䡠 K and kT 6.10 W/m 䡠 K, respectively. The properties are evaluated at T 300 K. (a) Determine the effective thermal conductivity of the nanolaminated material. Compare the value of the effective thermal conductivity to the bulk thermal conductivities of aluminum oxide and tungsten, given in Tables A.1 and A.2. (b) Determine the effective thermal conductivity of the nanolaminated material assuming that the thermal conductivities of the tungsten and aluminum oxide layers are equal to their bulk values. 3.178 Gold is commonly used in semiconductor packaging to form interconnections (also known as interconnects) that carry electrical signals between different devices in the package. In addition to being a good electrical conductor, gold interconnects are also effective at protecting the heat-generating devices to which they are attached by conducting thermal energy away from the devices to surrounding, cooler regions. Consider a thin film of gold that has a cross section of 60 nm 250 nm. (a) For an applied temperature difference of 20 C, determine the energy conducted along a 1-mlong, thin-film interconnect. Evaluate properties at 300 K. (b) Plot the lengthwise (in the 1-m direction) and spanwise (in the thinnest direction) thermal conductivities of the gold film as a function of the film thickness L for 30 L 140 nm.

C H A P T E R

Two-Dimensional, Steady-State Conduction

4

230

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

T

o this point, we have restricted our attention to conduction problems in which the temperature gradient is significant for only one coordinate direction. However, in many cases such problems are grossly oversimplified if a one-dimensional treatment is used, and it is necessary to account for multidimensional effects. In this chapter, we consider several techniques for treating two-dimensional systems under steady-state conditions. We begin our consideration of two-dimensional, steady-state conduction by briefly reviewing alternative approaches to determining temperatures and heat rates (Section 4.1). The approaches range from exact solutions, which may be obtained for idealized conditions, to approximate methods of varying complexity and accuracy. In Section 4.2 we consider some of the mathematical issues associated with obtaining an exact solution. In Section 4.3, we present compilations of existing exact solutions for a variety of simple geometries. Our objective in Sections 4.4 and 4.5 is to show how, with the aid of a computer, numerical ( finite-difference or finite-element) methods may be used to accurately predict temperatures and heat rates within the medium and at its boundaries.

4.1 Alternative Approaches Consider a long, prismatic solid in which there is two-dimensional heat conduction (Figure 4.1). With two surfaces insulated and the other surfaces maintained at different temperatures, T1 ⬎ T2, heat transfer by conduction occurs from surface 1 to 2. According to Fourier’s law, Equation 2.3 or 2.4, the local heat flux in the solid is a vector that is everywhere perpendicular to lines of constant temperature (isotherms). The directions of the heat flux vector are represented by the heat flow lines of Figure 4.1, and the vector itself is the resultant of heat flux components in the x- and y-directions. These components are determined by Equation 2.6. Since the heat flow lines are, by definition, in the direction of heat flow, no heat can be conducted across a heat flow line, and they are therefore sometimes referred to as adiabats. Conversely, adiabatic surfaces (or symmetry lines) are heat flow lines. Recall that, in any conduction analysis, there exist two major objectives. The first objective is to determine the temperature distribution in the medium, which, for the present problem, necessitates determining T(x, y). This objective is achieved by solving the appropriate form of the heat equation. For two-dimensional, steady-state conditions with no generation and constant thermal conductivity, this form is, from Equation 2.22, ⭸2T ⭸2T ⫹ ⫽0 ⭸x2 ⭸y2

(4.1)

y q"y

T1

q" = iq"x + jq"y

T2 < T1

q"x Isotherms

Heat flow lines

Isotherm x

FIGURE 4.1 Two-dimensional conduction.

4.2

䊏

The Method of Separation of Variables

231

If Equation 4.1 can be solved for T(x, y), it is then a simple matter to satisfy the second major objective, which is to determine the heat flux components q⬙x and q⬙y by applying the rate equations (2.6). Methods for solving Equation 4.1 include the use of analytical, graphical, and numerical (finite-difference, finite-element, or boundary-element) approaches. The analytical method involves effecting an exact mathematical solution to Equation 4.1. The problem is more difficult than those considered in Chapter 3, since it now involves a partial, rather than an ordinary, differential equation. Although several techniques are available for solving such equations, the solutions typically involve complicated mathematical series and functions and may be obtained for only a restricted set of simple geometries and boundary conditions [1–5]. Nevertheless, the solutions are of value, since the dependent variable T is determined as a continuous function of the independent variables (x, y). Hence the solution could be used to compute the temperature at any point of interest in the medium. To illustrate the nature and importance of analytical techniques, an exact solution to Equation 4.1 is obtained in Section 4.2 by the method of separation of variables. Conduction shape factors and dimensionless conduction heat rates (Section 4.3) are compilations of existing solutions for geometries that are commonly encountered in engineering practice. In contrast to the analytical methods, which provide exact results at any point, graphical and numerical methods can provide only approximate results at discrete points. Although superseded by computer solutions based on numerical procedures, the graphical, or flux-plotting, method may be used to obtain a quick estimate of the temperature distribution. Its use is restricted to two-dimensional problems involving adiabatic and isothermal boundaries. The method is based on the fact that isotherms must be perpendicular to heat flow lines, as noted in Figure 4.1. Unlike the analytical or graphical approaches, numerical methods (Sections 4.4 and 4.5) may be used to obtain accurate results for complex, two- or three-dimensional geometries involving a wide variety of boundary conditions.

4.2 The Method of Separation of Variables To appreciate how the method of separation of variables may be used to solve twodimensional conduction problems, we consider the system of Figure 4.2. Three sides of a thin rectangular plate or a long rectangular rod are maintained at a constant temperature T1, while the fourth side is maintained at a constant temperature T2 ⫽ T1. Assuming negligible heat transfer from the surfaces of the plate or the ends of the rod, temperature gradients normal to the x–y plane may be neglected (⭸2T/⭸z2 ⬇ 0) and conduction heat transfer is primarily in the x- and y-directions. We are interested in the temperature distribution T(x, y), but to simplify the solution we introduce the transformation ⬅

T ⫺ T1 T2 ⫺ T1

(4.2)

Substituting Equation 4.2 into Equation 4.1, the transformed differential equation is then ⭸2 ⭸2 ⫹ ⫽0 ⭸x2 ⭸y2

The graphical method is described, and its use is demonstrated, in Section 4S.1.

(4.3)

232

Chapter 4

y

䊏

Two-Dimensional, Steady-State Conduction

T2, θ = 1

W

T1, θ = 0

T1, θ = 0

T(x, y)

0 0

L T1, θ = 0

x

FIGURE 4.2 Two-dimensional conduction in a thin rectangular plate or a long rectangular rod.

Since the equation is second order in both x and y, two boundary conditions are needed for each of the coordinates. They are (0, y) ⫽ 0 (L, y) ⫽ 0

and and

(x, 0) ⫽ 0 (x, W) ⫽ 1

Note that, through the transformation of Equation 4.2, three of the four boundary conditions are now homogeneous and the value of is restricted to the range from 0 to 1. We now apply the separation of variables technique by assuming that the desired solution can be expressed as the product of two functions, one of which depends only on x while the other depends only on y. That is, we assume the existence of a solution of the form (x, y) ⫽ X(x) 䡠 Y(y)

(4.4)

Substituting into Equation 4.3 and dividing by XY, we obtain 2 2 ⫺ 1 d X2 ⫽ 1 d Y2 (4.5) X dx Y dy and it is evident that the differential equation is, in fact, separable. That is, the left-hand side of the equation depends only on x and the right-hand side depends only on y. Hence the equality can apply in general (for any x or y) only if both sides are equal to the same constant. Identifying this, as yet unknown, separation constant as 2, we then have

d 2X ⫹ 2X ⫽ 0 (4.6) dx 2 d 2Y ⫺ 2Y ⫽ 0 (4.7) dy 2 and the partial differential equation has been reduced to two ordinary differential equations. Note that the designation of 2 as a positive constant was not arbitrary. If a negative value were selected or a value of 2 ⫽ 0 was chosen, it is readily shown (Problem 4.1) that it would be impossible to obtain a solution that satisfies the prescribed boundary conditions. The general solutions to Equations 4.6 and 4.7 are, respectively, X ⫽ C1 cos x ⫹ C2 sin x Y ⫽ C3e⫺y ⫹ C4e⫹y in which case the general form of the two-dimensional solution is ⫽ (C1 cos x ⫹ C2 sin x)(C3e⫺y ⫹ C4ey)

(4.8)

4.2

䊏

233

The Method of Separation of Variables

Applying the condition that (0, y) ⫽ 0, it is evident that C1 ⫽ 0. In addition from the requirement that (x, 0) ⫽ 0, we obtain C2 sin x(C3 ⫹ C4) ⫽ 0 which may only be satisfied if C3 ⫽ ⫺C4. Although the requirement could also be satisfied by having C2 ⫽ 0, this would result in (x, y) ⫽ 0, which does not satisfy the boundary condition (x, W) ⫽ 1. If we now invoke the requirement that (L, y) ⫽ 0, we obtain C2C4 sin L(ey ⫺ e⫺y) ⫽ 0 The only way in which this condition may be satisfied (and still have a nonzero solution) is by requiring that assume discrete values for which sin L ⫽ 0. These values must then be of the form ⫽ n L

n ⫽ 1, 2, 3, . . .

(4.9)

where the integer n ⫽ 0 is precluded, since it implies (x, y) ⫽ 0. The desired solution may now be expressed as ⫽ C2C4 sin nx (eny/L ⫺ e⫺ny/L) L

(4.10)

Combining constants and acknowledging that the new constant may depend on n, we obtain ny (x, y) ⫽ Cn sin nx sinh L L where we have also used the fact that (eny/L ⫺ e⫺ny/L) ⫽ 2 sinh (ny/L). In this form we have really obtained an infinite number of solutions that satisfy the differential equation and boundary conditions. However, since the problem is linear, a more general solution may be obtained from a superposition of the form (x, y) ⫽

⬁

sinh 兺 C sin nx L n

n⫽1

ny L

(4.11)

To determine Cn we now apply the remaining boundary condition, which is of the form (x, W) ⫽ 1 ⫽

⬁

兺C

n⫽1

n

sin nx sinh nW L L

(4.12)

Although Equation 4.12 would seem to be an extremely complicated relation for evaluating Cn, a standard method is available. It involves writing an infinite series expansion in terms of orthogonal functions. An infinite set of functions g1(x), g2(x), … , gn(x), … is said to be orthogonal in the domain a ⱕ x ⱕ b if

冕 g (x)g (x) dx ⫽ 0 b

m

a

n

m⫽n

(4.13)

234

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

Many functions exhibit orthogonality, including the trigonometric functions sin (nx/L) and cos (nx/L) for 0 ⱕ x ⱕ L. Their utility in the present problem rests with the fact that any function f(x) may be expressed in terms of an infinite series of orthogonal functions ⬁

兺 A g (x)

f (x) ⫽

(4.14)

n n

n⫽1

The form of the coefficients An in this series may be determined by multiplying each side of the equation by gm(x) and integrating between the limits a and b.

冕 f(x)g (x) dx ⫽ 冕 g (x) 兺 A g (x) dx b

앝

b

m

m

a

n n

a

(4.15)

n⫽1

However, from Equation 4.13 it is evident that all but one of the terms on the right-hand side of Equation 4.15 must be zero, leaving us with

冕 f(x)g (x) dx ⫽ A 冕 g (x) dx b

b

m

2 m

m

a

a

Hence, solving for Am, and recognizing that this holds for any An by switching m to n:

冕 f (x)g (x) dx A ⫽ 冕 g (x) dx b

n

a

n

(4.16)

b

2 n

a

The properties of orthogonal functions may be used to solve Equation 4.12 for Cn by formulating an infinite series for the appropriate form of f(x). From Equation 4.14 it is evident that we should choose f(x) ⫽ 1 and the orthogonal function gn(x) ⫽ sin (nx/L). Substituting into Equation 4.16 we obtain

冕 sin nxL dx 2 (⫺1) ⫹ 1 ⫽ A ⫽ n nx 冕 sin L dx L

n⫹1

0 L

n

2

0

Hence from Equation 4.14, we have 1⫽

n⫹1 2 (⫺1) ⫹ 1 sin nx n L n⫽1 ⬁

兺

(4.17)

which is simply the expansion of unity in a Fourier series. Comparing Equations 4.12 and 4.17 we obtain Cn ⫽

2[(⫺1)n⫹1 ⫹ 1] n sinh (nW/L)

n ⫽ 1, 2, 3, . . .

(4.18)

Substituting Equation 4.18 into Equation 4.11, we then obtain for the final solution 2 (x, y) ⫽

(⫺1)n⫹1 ⫹ 1 sinh (ny/L) sin nx n L sinh (nW/L) n⫽1 ⬁

兺

(4.19)

4.3

䊏

The Conduction Shape Factor and the Dimensionless Conduction Heat Rate

235

y W

θ =1

0.75 0.5 0.25

θ =0

0 0

θ =0

θ = 0.1

θ =0

L

x

FIGURE 4.3 Isotherms and heat flow lines for two-dimensional conduction in a rectangular plate.

Equation 4.19 is a convergent series, from which the value of may be computed for any x and y. Representative results are shown in the form of isotherms for a schematic of the rectangular plate (Figure 4.3). The temperature T corresponding to a value of may be obtained from Equation 4.2, and components of the heat flux may be determined by using Equation 4.19 with Equation 2.6. The heat flux components determine the heat flow lines, which are shown in the figure. We note that the temperature distribution is symmetric about x ⫽ L/2, with ⭸T/⭸x ⫽ 0 at that location. Hence, from Equation 2.6, we know the symmetry plane at x ⫽ L/2 is adiabatic and therefore is a heat flow line. However, note that the discontinuities prescribed at the upper corners of the plate are physically untenable. In reality, large temperature gradients could be maintained in proximity to the corners, but discontinuities could not exist. Exact solutions have been obtained for a variety of other geometries and boundary conditions, including cylindrical and spherical systems. Such solutions are presented in specialized books on conduction heat transfer [1–5].

4.3 The Conduction Shape Factor and the Dimensionless Conduction Heat Rate In general, finding analytical solutions to the two- or three-dimensional heat equation is time-consuming and, in many cases, not possible. Therefore, a different approach is often taken. For example, in many instances, two- or three-dimensional conduction problems may be rapidly solved by utilizing existing solutions to the heat diffusion equation. These solutions are reported in terms of a shape factor S or a steady-state dimensionless conduction heat rate, q*ss. The shape factor is defined such that q ⫽ Sk⌬T1⫺2

(4.20)

where ⌬T1⫺2 is the temperature difference between boundaries, as shown in, for example, Figure 4.2. It also follows that a two-dimensional conduction resistance may be expressed as Rt,cond(2D) ⫽ 1 Sk

(4.21)

236

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

Shape factors have been obtained analytically for numerous two- and three-dimensional systems, and results are summarized in Table 4.1 for some common configurations. Results are also available for other configurations [6–9]. In cases 1 through 8 and case 11, twodimensional conduction is presumed to occur between the boundaries that are maintained at uniform temperatures, with ⌬T1⫺2 ⫽ T1 ⫺ T2. In case 9, three-dimensional conduction exists in the corner region, while in case 10 conduction occurs between an isothermal disk (T1) and a semi-infinite medium of uniform temperature (T2) at locations well removed from the disk. Shape factors may also be defined for one-dimensional geometries, and from the results of Table 3.3, it follows that for plane, cylindrical, and spherical walls, respectively, the shape factors are A/L, 2L/ln(r2/r1), and 4r1r2/(r2 ⫺ r1). Cases 12 through 15 are associated with conduction from objects held at an isothermal temperature (T1) that are embedded within an infinite medium of uniform temperature (T2)

Shape factors for two-dimensional geometries may also be estimated with the graphical method that is described in Section 4S.1.

TABLE 4.1 Conduction shape factors and dimensionless conduction heat rates for selected systems. (a) Shape factors [q ⴝ Sk(T1 ⴚ T2)] System

Schematic

Restrictions

Shape Factor

z ⬎ D/2

2D 1 ⫺ D/4z

LⰇD

2L cosh⫺1 (2z/D)

LⰇD z ⬎ 3D/2

2L ln (4z/D)

LⰇD

2L ln (4L/D)

T2

Case 1 Isothermal sphere buried in a semiinfinite medium

z T1

D T2

Case 2 Horizontal isothermal cylinder of length L buried in a semi-infinite medium

z L T1

Case 3 Vertical cylinder in a semi-infinite medium

D T2 L T1 D

Case 4 Conduction between two cylinders of length L in infinite medium

T1

D1

D2 T2

w

L Ⰷ D1, D2 LⰇw

2L

冢

cosh⫺1

4w2 ⫺ D21 ⫺ D22 2D1D2

冣

4.3

TABLE 4.1

䊏

The Conduction Shape Factor and the Dimensionless Conduction Heat Rate

237

Continued

System Case 5 Horizontal circular cylinder of length L midway between parallel planes of equal length and infinite width

Schematic

Restrictions

T2

∞

∞

z

z Ⰷ D/2 LⰇz

D

z T1

∞

Shape Factor

2L ln (8z/D)

∞

T2

Case 6 Circular cylinder of length L centered in a square solid of equal length

T2 D w

w⬎D LⰇw

2L ln (1.08 w/D)

T1

Case 7 Eccentric circular cylinder of length L in a cylinder of equal length

T1

d D

T2

z

Case 8 Conduction through the edge of adjoining walls

2L D⬎d LⰇD

冢D

cosh⫺1

2

⫹ d2 ⫺ 4z2 2Dd

冣

T2

L D

D ⬎ 5L

0.54D

L Ⰶ length and width of wall

0.15L

None

2D

W w ⬍ 1.4

2L 0.785 ln (W/w)

W w ⬎ 1.4

2L 0.930 ln (W/w) ⫺ 0.050

T1 L

Case 9 Conduction through corner of three walls with a temperature difference ⌬T1⫺2 across the walls Case 10 Disk of diameter D and temperature T1 on a semi-infinite medium of thermal conductivity k and temperature T2

L

L L

D

T1

k T2

Case 11 Square channel of length L

L

T1 T2 w W

L ⰇW

238

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

TABLE 4.1 Continued (b) Dimensionless conduction heat rates [q ⴝ q*ss kAs(T1 ⴚ T2)/Lc; Lc ⬅ (As/4)1/2] System

Schematic

Case 12 Isothermal sphere of diameter D and temperature T1 in an infinite medium of temperature T2

T1

q*ss

D2

1

D 2 2

2兹2 ⫽ 0.900

2wL

0.932

D T2

Case 13 Infinitely thin, isothermal disk of diameter D and temperature T1 in an infinite medium of temperature T2

T1 D T2

Case 14 Infinitely thin rectangle of length L, width w, and temperature T1 in an infinite medium of temperature T2

Case 15 Cuboid shape of height d with a square footprint of width D and temperature T1 in an infinite medium of temperature T2

Active Area, As

L w

T1 T2

D

2D2 ⫹ 4Dd T1

d

T2

d/D

q*ss

0.1 1.0 2.0 10

0.943 0.956 0.961 1.111

at locations removed from the object. For these infinite medium cases, useful results may be obtained by defining a characteristic length Lc ⬅ (As /4)1/2

(4.22)

where As is the surface area of the object. Conduction heat transfer rates from the object to the infinite medium may then be reported in terms of a dimensionless conduction heat rate [10] q* ss ⬅ qLc /kAs(T1 ⫺ T2)

(4.23)

From Table 4.1, it is evident that the values of q*ss, which have been obtained analytically and numerically, are similar for a wide range of geometrical configurations. As a consequence of this similarity, values of q*ss may be estimated for configurations that are similar to those for which q*ss is known. For example, dimensionless conduction heat rates from cuboid shapes (case 15) over the range 0.1 ⱕ d/D ⱕ 10 may be closely approximated by interpolating the values of q*ss reported in Table 4.1. Additional procedures that may be exploited to estimate values of q*ss for other geometries are explained in [10]. Note that results for q*ss in Table 4.1b may be converted to expressions for S listed in Table 4.1a. For example, the shape factor of case 10 may be derived from the dimensionless conduction heat rate of case 13 (recognizing that the infinite medium can be viewed as two adjacent semi-infinite media).

4.3

䊏

The Conduction Shape Factor and the Dimensionless Conduction Heat Rate

239

The shape factors and dimensionless conduction heat rates reported in Table 4.1 are associated with objects that are held at uniform temperatures. For uniform heat flux conditions, the object’s temperature is no longer uniform but varies spatially with the coolest temperatures located near the periphery of the heated object. Hence, the temperature difference that is used to define S or q*ss is replaced by a temperature difference involving the spatially averaged surface temperature of the object (T1 ⫺ T2) or by the difference between the maximum surface temperature of the heated object and the far field temperature of the surrounding medium, (T1,max ⫺ T2). For the uniformly heated geometry of case 10 (a disk of diameter D in contact with a semi-infinite medium of thermal conductivity k and temperature T2), the values of S are 32D/16 and D/2 for temperature differences based on the average and maximum disk temperatures, respectively.

EXAMPLE 4.1 A metallic electrical wire of diameter d ⫽ 5 mm is to be coated with insulation of thermal conductivity k ⫽ 0.35 W/m 䡠 K. It is expected that, for the typical installation, the coated wire will be exposed to conditions for which the total coefficient associated with convection and radiation is h ⫽ 15 W/m2 䡠 K. To minimize the temperature rise of the wire due to ohmic heating, the insulation thickness is specified so that the critical insulation radius is achieved (see Example 3.5). During the wire coating process, however, the insulation thickness sometimes varies around the periphery of the wire, resulting in eccentricity of the wire relative to the coating. Determine the change in the thermal resistance of the insulation due to an eccentricity that is 50% of the critical insulation thickness.

SOLUTION Known: Wire diameter, convective conditions, and insulation thermal conductivity. Find: Thermal resistance of the wire coating associated with peripheral variations in the coating thickness. Schematic: d = 5 mm

tcr /2 D

tcr z

Insulation, k (a) Concentric wire

Assumptions: 1. Steady-state conditions. 2. Two-dimensional conduction.

(b) Eccentric wire

T∞, h

240

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

3. Constant properties. 4. Both the exterior and interior surfaces of the coating are at uniform temperatures.

Analysis: From Example 3.5, the critical insulation radius is 䡠 K ⫽ 0.023 m ⫽ 23 mm rcr ⫽ k ⫽ 0.35 W/m h 15 W/m2 䡠 K Therefore, the critical insulation thickness is tcr ⫽ rcr ⫺ d/2 ⫽ 0.023 m ⫺ 0.005 m ⫽ 0.021 m ⫽ 21 mm 2 The thermal resistance of the coating associated with the concentric wire may be evaluated using Equation 3.33 and is R⬘t,cond ⫽

ln[rcr/(d/2)] ln[0.023 m/(0.005 m/2)] ⫽ ⫽ 1.0 m 䡠 K/W 2k 2(0.35 W/m 䡠 K)

For the eccentric wire, the thermal resistance of the insulation may be evaluated using case 7 of Table 4.1, where the eccentricity is z ⫽ 0.5 ⫻ tcr ⫽ 0.5 ⫻ 0.021 m ⫽ 0.010 m cosh⫺1 R⬘t,cond(2D) ⫽ 1 ⫽ Sk cosh⫺1 ⫽

d ⫺ 4z 冢D ⫹2Dd 冣 2

2

2

2k

冢

(2 ⫻ 0.023 m)2 ⫹ (0.005 m)2 ⫺ 4(0.010 m)2 2 ⫻ (2 ⫻ 0.023 m) ⫻ 0.005 m

冣

2 ⫻ 0.35 W/m 䡠 K

⫽ 0.91 m 䡠 K/W Therefore, the reduction in the thermal resistance of the insulation is 0.10 m 䡠 K/W, or 10%. 䉰

Comments: 1. Reduction in the local insulation thickness leads to a smaller local thermal resistance of the insulation. Conversely, locations associated with thicker coatings have increased local thermal resistances. These effects offset each other, but not exactly; the maximum resistance is associated with the concentric wire case. For this application, eccentricity of the wire relative to the coating provides enhanced thermal performance relative to the concentric wire case. 2. The interior surface of the coating will be at nearly uniform temperature if the thermal conductivity of the wire is high relative to that of the insulation. Such is the case for metallic wire. However, the exterior surface temperature of the coating will not be perfectly uniform due to the variation in the local insulation thickness.

4.4

䊏

241

Finite-Difference Equations

4.4 Finite-Difference Equations As discussed in Sections 4.1 and 4.2, analytical methods may be used, in certain cases, to effect exact mathematical solutions to steady, two-dimensional conduction problems. These solutions have been generated for an assortment of simple geometries and boundary conditions and are well documented in the literature [1–5]. However, more often than not, two-dimensional problems involve geometries and/or boundary conditions that preclude such solutions. In these cases, the best alternative is often one that uses a numerical technique such as the finite-difference, finite-element, or boundary-element method. Another strength of numerical methods is that they can be readily extended to three-dimensional problems. Because of its ease of application, the finite-difference method is well suited for an introductory treatment of numerical techniques.

4.4.1

The Nodal Network

In contrast to an analytical solution, which allows for temperature determination at any point of interest in a medium, a numerical solution enables determination of the temperature at only discrete points. The first step in any numerical analysis must therefore be to select these points. Referring to Figure 4.4, this may be done by subdividing the medium of interest into a number of small regions and assigning to each a reference point that is at its center. ∆x

∆y

m, n + 1

y, n m, n

m + 1, n

x, m

m – 1, n

m, n – 1 (a)

∂T ___ ∂x

Tm,n – Tm –1, n = ______________ m–1/2,n

∂T ___ ∂x

m–1

T(x)

m

∆x Tm+1,n – Tm,n = ______________

m+1/2,n

∆x m – 12_

m + 12_

∆x

∆x

x (b)

FIGURE 4.4 Two-dimensional conduction. (a) Nodal network. (b) Finite-difference approximation.

m+1

242

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

The reference point is frequently termed a nodal point (or simply a node), and the aggregate of points is termed a nodal network, grid, or mesh. The nodal points are designated by a numbering scheme that, for a two-dimensional system, may take the form shown in Figure 4.4a. The x and y locations are designated by the m and n indices, respectively. Each node represents a certain region, and its temperature is a measure of the average temperature of the region. For example, the temperature of the node (m, n) of Figure 4.4a may be viewed as the average temperature of the surrounding shaded area. The selection of nodal points is rarely arbitrary, depending often on matters such as geometric convenience and the desired accuracy. The numerical accuracy of the calculations depends strongly on the number of designated nodal points. If this number is large (a fine mesh), accurate solutions can be obtained.

4.4.2

Finite-Difference Form of the Heat Equation

Determination of the temperature distribution numerically dictates that an appropriate conservation equation be written for each of the nodal points of unknown temperature. The resulting set of equations may then be solved simultaneously for the temperature at each node. For any interior node of a two-dimensional system with no generation and uniform thermal conductivity, the exact form of the energy conservation requirement is given by the heat equation, Equation 4.1. However, if the system is characterized in terms of a nodal network, it is necessary to work with an approximate, or finite-difference, form of this equation. A finite-difference equation that is suitable for the interior nodes of a two-dimensional system may be inferred directly from Equation 4.1. Consider the second derivative, ⭸2T/⭸x2. From Figure 4.4b, the value of this derivative at the (m, n) nodal point may be approximated as ⭸2T ⭸x2

冏

⭸T/⭸x兩 m⫹1/2,n ⫺ ⭸T/⭸x兩 m⫺1/2,n ⌬x

艐

m,n

(4.24)

The temperature gradients may in turn be expressed as a function of the nodal temperatures. That is, ⭸T ⭸x ⭸T ⭸x

冏 冏

艐

Tm⫹1,n ⫺ Tm,n ⌬x

(4.25)

艐

Tm,n ⫺ Tm⫺1,n ⌬x

(4.26)

m⫹1/2,n

m⫺1/2,n

Substituting Equations 4.25 and 4.26 into 4.24, we obtain ⭸2T ⭸x2

冏

艐

m,n

Tm⫹1,n ⫹ Tm⫺1,n ⫺ 2Tm,n (⌬x)2

(4.27)

Proceeding in a similar fashion, it is readily shown that ⭸2T ⭸y2

冏

艐

m,n

艐

⭸T/⭸y 兩m,n⫹1/2 ⫺ ⭸T/⭸y 兩m,n⫺1/2 ⌬y Tm,n⫹1 ⫹ Tm,n⫺1 ⫺ 2Tm,n (⌬y)2

(4.28)

4.4

䊏

243

Finite-Difference Equations

Using a network for which ⌬x ⫽ ⌬y and substituting Equations 4.27 and 4.28 into Equation 4.1, we obtain Tm,n⫹1 ⫹ Tm,n⫺1 ⫹ Tm⫹1,n ⫹ Tm⫺1,n ⫺ 4Tm,n ⫽ 0

(4.29)

Hence for the (m, n) node, the heat equation, which is an exact differential equation, is reduced to an approximate algebraic equation. This approximate, finite-difference form of the heat equation may be applied to any interior node that is equidistant from its four neighboring nodes. It requires simply that the temperature of an interior node be equal to the average of the temperatures of the four neighboring nodes.

4.4.3

The Energy Balance Method

In many cases, it is desirable to develop the finite-difference equations by an alternative method called the energy balance method. As will become evident, this approach enables one to analyze many different phenomena such as problems involving multiple materials, embedded heat sources, or exposed surfaces that do not align with an axis of the coordinate system. In the energy balance method, the finite-difference equation for a node is obtained by applying conservation of energy to a control volume about the nodal region. Since the actual direction of heat flow (into or out of the node) is often unknown, it is convenient to formulate the energy balance by assuming that all the heat flow is into the node. Such a condition is, of course, impossible, but if the rate equations are expressed in a manner consistent with this assumption, the correct form of the finite-difference equation is obtained. For steady-state conditions with generation, the appropriate form of Equation 1.12c is then E˙ in ⫹ E˙ g ⫽ 0

(4.30)

Consider applying Equation 4.30 to a control volume about the interior node (m, n) of Figure 4.5. For two-dimensional conditions, energy exchange is influenced by conduction between (m, n) and its four adjoining nodes, as well as by generation. Hence Equation 4.30 reduces to 4

兺q

(i) l (m,n)

⫹ q˙(⌬x 䡠 ⌬y 䡠 1) ⫽ 0

i⫽1

∆x m, n + 1 ∆y ∆y m – 1, n

m, n

m + 1, n

m, n – 1 ∆x

FIGURE 4.5 Conduction to an interior node from its adjoining nodes.

244

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

where i refers to the neighboring nodes, q(i) l (m, n) is the conduction rate between nodes, and unit depth is assumed. To evaluate the conduction rate terms, we assume that conduction transfer occurs exclusively through lanes that are oriented in either the x- or y-direction. Simplified forms of Fourier’s law may therefore be used. For example, the rate at which energy is transferred by conduction from node (m – 1, n) to (m, n) may be expressed as q(m⫺1,n) l (m,n) ⫽ k(⌬y 䡠 1)

Tm⫺1,n ⫺ Tm,n ⌬x

(4.31)

The quantity (⌬y 䡠 1) is the heat transfer area, and the term (Tm–1,n ⫺ Tm,n)/⌬x is the finitedifference approximation to the temperature gradient at the boundary between the two nodes. The remaining conduction rates may be expressed as q(m⫹1,n) l (m,n) ⫽ k(⌬y 䡠 1)

Tm⫹1,n ⫺ Tm,n ⌬x

(4.32)

q(m,n⫹1) l (m,n) ⫽ k(⌬x 䡠 1)

Tm,n⫹1 ⫺ Tm,n ⌬y

(4.33)

q(m,n⫺1) l (m,n) ⫽ k(⌬x 䡠 1)

Tm,n⫺1 ⫺ Tm,n ⌬y

(4.34)

Note that, in evaluating each conduction rate, we have subtracted the temperature of the (m, n) node from the temperature of its adjoining node. This convention is necessitated by the assumption of heat flow into (m, n) and is consistent with the direction of the arrows shown in Figure 4.5. Substituting Equations 4.31 through 4.34 into the energy balance and remembering that ⌬x ⫽ ⌬y, it follows that the finite-difference equation for an interior node with generation is Tm,n⫹1 ⫹ Tm,n⫺1 ⫹ Tm⫹1,n ⫹ Tm⫺1,n ⫹

q˙ (⌬x)2 ⫺ 4Tm,n ⫽ 0 k

(4.35)

If there is no internally distributed source of energy (q˙ ⫽ 0), this expression reduces to Equation 4.29. It is important to note that a finite-difference equation is needed for each nodal point at which the temperature is unknown. However, it is not always possible to classify all such points as interior and hence to use Equation 4.29 or 4.35. For example, the temperature may be unknown at an insulated surface or at a surface that is exposed to convective conditions. For points on such surfaces, the finite-difference equation must be obtained by applying the energy balance method. To further illustrate this method, consider the node corresponding to the internal corner of Figure 4.6. This node represents the three-quarter shaded section and exchanges energy by convection with an adjoining fluid at T앝. Conduction to the nodal region (m, n) occurs along four different lanes from neighboring nodes in the solid. The conduction heat rates qcond may be expressed as q(m⫺1,n)l(m,n) ⫽ k(⌬y 䡠 1)

Tm⫺1,n ⫺ Tm,n ⌬x

(4.36)

4.4

䊏

245

Finite-Difference Equations

m, n + 1 qcond

∆y qcond

m – 1, n qcond

qconv

qcond m, n – 1

m + 1, n

T∞, h

FIGURE 4.6 Formulation of the finite-difference equation for an internal corner of a solid with surface convection.

∆x

q(m,n⫹1)l(m,n) ⫽ k(⌬x 䡠 1)

Tm,n⫹1 ⫺ Tm,n ⌬y

(4.37)

冢⌬y2 䡠 1冣

Tm⫹1,n ⫺ Tm,n ⌬x

(4.38)

Tm,n⫺1 ⫺ Tm,n q(m,n⫺1)l(m,n) ⫽ k ⌬x 䡠 1 2 ⌬y

(4.39)

q(m⫹1,n)l(m,n) ⫽ k

冢 冣

Note that the areas for conduction from nodal regions (m ⫺ 1, n) and (m, n ⫹ 1) are proportional to ⌬y and ⌬x, respectively, whereas conduction from (m ⫹ 1, n) and (m, n – 1) occurs along lanes of width ⌬y/2 and ⌬x/2, respectively. Conditions in the nodal region (m, n) are also influenced by convective exchange with the fluid, and this exchange may be viewed as occurring along half-lanes in the x- and ydirections. The total convection rate qconv may be expressed as

冢

冣

冢

冣

⌬y q(⬁)l(m,n) ⫽ h ⌬x 䡠 1 (T⬁ ⫺ Tm,n) ⫹ h 䡠 1 (T⬁ ⫺ Tm,n) 2 2

(4.40)

Implicit in this expression is the assumption that the exposed surfaces of the corner are at a uniform temperature corresponding to the nodal temperature Tm,n. This assumption is consistent with the concept that the entire nodal region is characterized by a single temperature, which represents an average of the actual temperature distribution in the region. In the absence of transient, three-dimensional, and generation effects, conservation of energy, Equation 4.30, requires that the sum of Equations 4.36 through 4.40 be zero. Summing these equations and rearranging, we therefore obtain

冢

冣

Tm⫺1,n ⫹ Tm,n⫹1 ⫹ 1 (Tm⫹1,n ⫹ Tm,n⫺1) ⫹ h⌬x T⬁ ⫺ 3 ⫹ h⌬x Tm,n ⫽ 0 2 k k

(4.41)

where again the mesh is such that ⌬x ⫽ ⌬y. Nodal energy balance equations pertinent to several common geometries for situations where there is no internal energy generation are presented in Table 4.2.

246

Chapter 4

TABLE 4.2

䊏

Two-Dimensional, Steady-State Conduction

Summary of nodal finite-difference equations Finite-Difference Equation for ⌬x ⴝ ⌬y

Configuration m, n + 1 ∆y m, n m + 1, n

m – 1, n

(4.29)

Tm,n⫹1 ⫹ Tm,n⫺1 ⫹ Tm⫹1,n ⫹ Tm⫺1,n ⫺ 4Tm,n ⫽ 0 Case 1.

m, n – 1

∆x ∆x

Interior node

m, n + 1

2(Tm⫺1,n ⫹ Tm,n⫹1) ⫹ (Tm⫹1,n ⫹ Tm,n⫺1) m – 1, n

m + 1, n

m, n

⫹2

T∞, h

∆y

m, n – 1

Case 2.

冢

冣

h ⌬x h ⌬x T ⫺2 3⫹ Tm,n ⫽ 0 k 앝 k

(4.41)

Node at an internal corner with convection

m, n + 1 ∆y m, n

(2Tm⫺1,n ⫹ Tm,n⫹1 ⫹ Tm,n⫺1) ⫹

T∞, h

m – 1, n

Case 3.

m, n – 1

冢

冣

2h ⌬x h ⌬x T앝 ⫺ 2 ⫹ 2 Tm,n ⫽ 0 k k

(4.42)a

Node at a plane surface with convection

∆x T∞ , h

m – 1, n

(Tm,n⫺1 ⫹ Tm⫺1,n) ⫹ 2

m, n ∆y m, n – 1

Case 4.

∆x

冢

冣

h ⌬x h ⌬x ⫹ 1 Tm,n ⫽ 0 T ⫺2 k ⬁ k

(4.43)

Node at an external corner with convection

m, n + 1 ∆y m, n

(2Tm⫺1,n ⫹ Tm,n⫹1 ⫹ Tm,n⫺1) ⫹

q"

m – 1, n

m, n – 1

Case 5.

2q⬙ ⌬x ⫺4Tm,n ⫽ 0 k

(4.44)b

Node at a plane surface with uniform heat flux

∆x a,b

To obtain the finite-difference equation for an adiabatic surface (or surface of symmetry), simply set h or q⬙ equal to zero.

EXAMPLE 4.2 Using the energy balance method, derive the finite-difference equation for the (m, n) nodal point located on a plane, insulated surface of a medium with uniform heat generation.

4.4

䊏

247

Finite-Difference Equations

SOLUTION Known: Network of nodal points adjoining an insulated surface. Find: Finite-difference equation for the surface nodal point. Schematic: m, n + 1 q4

Insulated surface

•

k, q

y, n

m – 1, n

m, n q3

q1

x, m

q2

∆y = ∆ x ∆y Unit depth (normal to paper)

m, n – 1

∆x ___ 2

Assumptions: 1. Steady-state conditions. 2. Two-dimensional conduction. 3. Constant properties. 4. Uniform internal heat generation. Analysis: Applying the energy conservation requirement, Equation 4.30, to the control surface about the region (⌬x/2 䡠 ⌬y 䡠 1) associated with the (m, n) node, it follows that, with volumetric heat generation at a rate q˙ ,

冢

冣

q1 ⫹ q2 ⫹ q3 ⫹ q4 ⫹ q˙ ⌬x 䡠 ⌬y 䡠 1 ⫽ 0 2 where Tm⫺1,n ⫺ Tm,n ⌬x T m,n⫺1 ⫺ Tm,n q2 ⫽ k ⌬x 䡠 1 2 ⌬y

q1 ⫽ k(⌬y 䡠 1)

冢

冣

q3 ⫽ 0

冢

冣

Tm,n⫹1 ⫺ Tm,n q4 ⫽ k ⌬x 䡠 1 2 ⌬y Substituting into the energy balance and dividing by k/2, it follows that 2Tm⫺1,n ⫹ Tm,n⫺1 ⫹ Tm,n⫹1 ⫺ 4Tm,n ⫹

q˙(⌬x 䡠 ⌬y) ⫽0 k

䉰

Comments: 1. The same result could be obtained by using the symmetry condition, Tm⫹1,n ⫽ Tm⫺1,n, with the finite-difference equation (Equation 4.35) for an interior nodal point.

248

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

If q˙ ⫽ 0, the desired result could also be obtained by setting h ⫽ 0 in Equation 4.42 (Table 4.2). 2. As an application of the foregoing finite-difference equation, consider the following two-dimensional system within which thermal energy is uniformly generated at an unknown rate q˙. The thermal conductivity of the solid is known, as are convection conditions at one of the surfaces. In addition, temperatures have been measured at locations corresponding to the nodal points of a finite-difference mesh. Ta

•

k, q

Tb

Tc

∆y

Tc ⫽ 230.9⬚C

Td ⫽ 220.1⬚C

Te ⫽ 222.4⬚C

T⬁ ⫽ 200.0⬚C

⌬x ⫽ 10 mm

∆x

Td

Tb ⫽ 227.6⬚C

h ⫽ 50 WⲐm2 䡠 K

y x

Ta ⫽ 235.9⬚C

k ⫽ 1 WⲐm 䡠 K ⌬y ⫽ 10 mm

Te T∞, h

The generation rate can be determined by applying the finite-difference equation to node c. q˙ (⌬x 䡠 ⌬y) ⫽0 k q˙ (0.01 m)2 (2 ⫻ 227.6 ⫹ 222.4 ⫹ 235.9 ⫺ 4 ⫻ 230.9)⬚C ⫹ ⫽0 1 W/m 䡠 K 2Tb ⫹ Te ⫹ Ta ⫺ 4Tc ⫹

q˙ ⫽ 1.01 ⫻ 105 W/m3 From the prescribed thermal conditions and knowledge of q˙, we can also determine whether the conservation of energy requirement is satisfied for node e. Applying an energy balance to a control volume about this node, it follows that q1 ⫹ q2 ⫹ q3 ⫹ q4 ⫹ q˙ (⌬x/2 䡠 ⌬y/2 䡠 1) ⫽ 0 k(⌬x/2 䡠 1)

Tc ⫺ Te T ⫺ Te ⫹ 0 ⫹ h(⌬x/2 䡠 1)(T⬁ ⫺ Te) ⫹ k(⌬y/2 䡠 1) d ⌬y ⌬x ⫹ q˙(⌬x/2 䡠 ⌬y/2 䡠 1) ⫽ 0 ∆x •

k, q

Tc q1 ∆y

q2 q4 Td

Te q3 T∞, h

4.4

䊏

249

Finite-Difference Equations

If the energy balance is satisfied, the left-hand side of this equation will be identically equal to zero. Substituting values, we obtain (230.9 ⫺ 222.4)⬚C 0.010 m ⫹ 0 ⫹ 50 W/m2 䡠 K(0.005 m2) (200 ⫺ 222.4)⬚C (220.1 ⫺ 222.4)⬚C ⫹ 1 W/m 䡠 K(0.005 m2) ⫹ 1.01 ⫻ 105 W/m3(0.005)2 m3 ⫽ 0(?) 0.010 m 4.250 W ⫹ 0 ⫺ 5.600 W ⫺ 1.150 W ⫹ 2.525 W ⫽ 0(?) 0.025 W 艐 0 1 W/m 䡠 K(0.005 m2)

The inability to precisely satisfy the energy balance is attributable to temperature measurement errors, the approximations employed in developing the finite-difference equations, and the use of a relatively coarse mesh.

It is useful to note that heat rates between adjoining nodes may also be formulated in terms of the corresponding thermal resistances. Referring, for example, to Figure 4.6, the rate of heat transfer by conduction from node (m ⫺ 1, n) to (m, n) may be expressed as q(m⫺1,n) l (m,n) ⫽

Tm⫺1,n ⫺ Tm,n Tm⫺1,n ⫺ Tm,n ⫽ Rt,cond ⌬x/k (⌬y 䡠 1)

yielding a result that is equivalent to that of Equation 4.36. Similarly, the rate of heat transfer by convection to (m, n) may be expressed as q(⬁) l (m,n) ⫽

T⬁ ⫺ Tm,n T⬁ ⫺ Tm,n ⫽ Rt,conv {h[(⌬x/2) 䡠 1 ⫹ (⌬y/2) 䡠 1]}⫺1

which is equivalent to Equation 4.40. As an example of the utility of resistance concepts, consider an interface that separates two dissimilar materials and is characterized by a thermal contact resistance R⬙t,c (Figure 4.7). The rate of heat transfer from node (m, n) to (m, n ⫺ 1) may be expressed as q(m,n) l (m,n⫺1) ⫽

Tm,n ⫺ Tm,n⫺1 Rtot

(4.45)

where, for a unit depth, Rtot ⫽

R⬙t,c ⌬y/2 ⌬y/2 ⫹ ⫹ kA(⌬x 䡠 1) ⌬x 䡠 1 kB(⌬x 䡠 1)

(4.46)

∆x

∆y

(m, n)

Material A

kA R"t,c

∆y

(m, n – 1)

Material B

kB

FIGURE 4.7 Conduction between adjoining, dissimilar materials with an interface contact resistance.

250

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

4.5 Solving the Finite-Difference Equations Once the nodal network has been established and an appropriate finite-difference equation has been written for each node, the temperature distribution may be determined. The problem reduces to one of solving a system of linear, algebraic equations. In this section, we formulate the system of linear, algebraic equations as a matrix equation and briefly discuss its solution by the matrix inversion method. We also present some considerations for verifying the accuracy of the solution.

4.5.1

Formulation as a Matrix Equation

Consider a system of N finite-difference equations corresponding to N unknown temperatures. Identifying the nodes by a single integer subscript, rather than by the double subscript (m, n), the procedure for performing a matrix inversion begins by expressing the equations as

...

...

...

...

...

...

a11T1 ⫹ a12T2 ⫹ a13T3 ⫹ … ⫹ a1NTN ⫽ C1 a21T1 ⫹ a22T2 ⫹ a23T3 ⫹ … ⫹ a2NTN ⫽ C2 aN1T1 ⫹ aN2T2 ⫹ aN3T3 ⫹ … ⫹ aNNTN ⫽ CN

(4.47)

where the quantities a11, a12, . . . , C1, . . . are known coefficients and constants involving quantities such as ⌬x, k, h, and T앝. Using matrix notation, these equations may be expressed as [A][T] ⫽ [C]

(4.48)

where

aNN

T⬅

TN

,

C⬅

C1 C2 ...

,

T1 T2 ...

aN1 aN2 …

a1N a2N ...

a12 … a22 … ...

...

A⬅

a11 a21

CN

The coefficient matrix [A] is square (N ⫻ N), and its elements are designated by a double subscript notation, for which the first and second subscripts refer to rows and columns, respectively. The matrices [T] and [C] have a single column and are known as column vectors. Typically, they are termed the solution and right-hand side vectors, respectively. If the matrix multiplication implied by the left-hand side of Equation 4.48 is performed, Equations 4.47 are obtained. Numerous mathematical methods are available for solving systems of linear, algebraic equations [11, 12], and many computational software programs have the built-in capability to solve Equation 4.48 for the solution vector [T]. For small matrices, the solution can be found using a programmable calculator or by hand. One method suitable for hand or computer calculation is the Gauss–Seidel method, which is presented in Appendix D.

4.5

䊏

4.5.2

251

Solving the Finite-Difference Equations

Verifying the Accuracy of the Solution

It is good practice to verify that a numerical solution has been correctly formulated by performing an energy balance on a control surface surrounding all nodal regions whose temperatures have been evaluated. The temperatures should be substituted into the energy balance equation, and if the balance is not satisfied to a high degree of precision, the finitedifference equations should be checked for errors. Even when the finite-difference equations have been properly formulated and solved, the results may still represent a coarse approximation to the actual temperature field. This behavior is a consequence of the finite spacings (⌬x, ⌬y) between nodes and of finite-difference approximations, such as k(⌬y 䡠 1)(Tm⫺1,n ⫺ Tm,n)/⌬x, to Fourier’s law of conduction, ⫺k(dy 䡠 1)⭸T/⭸x. The finite-difference approximations become more accurate as the nodal network is refined (⌬x and ⌬y are reduced). Hence, if accurate results are desired, grid studies should be performed, whereby results obtained for a fine grid are compared with those obtained for a coarse grid. One could, for example, reduce ⌬x and ⌬y by a factor of 2, thereby increasing the number of nodes and finite-difference equations by a factor of 4. If the agreement is unsatisfactory, further grid refinements could be made until the computed temperatures no longer depend significantly on the choice of ⌬x and ⌬y. Such grid-independent results would provide an accurate solution to the physical problem. Another option for validating a numerical solution involves comparing results with those obtained from an exact solution. For example, a finite-difference solution of the physical problem described in Figure 4.2 could be compared with the exact solution given by Equation 4.19. However, this option is limited by the fact that we seldom seek numerical solutions to problems for which there exist exact solutions. Nevertheless, if we seek a numerical solution to a complex problem for which there is no exact solution, it is often useful to test our finitedifference procedures by applying them to a simpler version of the problem.

EXAMPLE 4.3 A major objective in advancing gas turbine engine technologies is to increase the temperature limit associated with operation of the gas turbine blades. This limit determines the permissible turbine gas inlet temperature, which, in turn, strongly influences overall system performance. In addition to fabricating turbine blades from special, high-temperature, high-strength superalloys, it is common to use internal cooling by machining flow channels within the blades and routing air through the channels. We wish to assess the effect of such a scheme by approximating the blade as a rectangular solid in which rectangular channels are machined. The blade, which has a thermal conductivity of k ⫽ 25 W/m 䡠 K, is 6 mm thick, and each channel has a 2 mm ⫻ 6 mm rectangular cross section, with a 4-mm spacing between adjoining channels. Combustion gases

T∞,o, ho

Air channel T∞,i, hi

2 mm 6 mm 4 mm

Combustion gases

6 mm

Turbine blade, k

T∞,o, ho

252

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

Under operating conditions for which ho ⫽ 1000 W/m2 䡠 K, T앝,o ⫽ 1700 K, hi ⫽ 200 W/m2 䡠 K, and T앝,i ⫽ 400 K, determine the temperature field in the turbine blade and the rate of heat transfer per unit length to the channel. At what location is the temperature a maximum?

SOLUTION Known: Dimensions and operating conditions for a gas turbine blade with embedded channels. Find: Temperature field in the blade, including a location of maximum temperature. Rate of heat transfer per unit length to the channel. Schematic: T∞,o, ho 1

2

3

7

8

9

Symmetry adiabat 13

14

15

19

20

21

4

5

6

10

11

12

16

17

18

x ∆y = 1 mm

∆x = 1 mm

Symmetry adiabat

T∞,i, hi

Symmetry adiabat

y

Assumptions: 1. Steady-state, two-dimensional conduction. 2. Constant properties. Analysis: Adopting a grid space of ⌬x ⫽ ⌬y ⫽ 1 mm and identifying the three lines of symmetry, the foregoing nodal network is constructed. The corresponding finite-difference equations may be obtained by applying the energy balance method to nodes 1, 6, 18, 19, and 21 and by using the results of Table 4.2 for the remaining nodes. Heat transfer to node 1 occurs by conduction from nodes 2 and 7, as well as by convection from the outer fluid. Since there is no heat transfer from the region beyond the symmetry adiabat, application of an energy balance to the one-quarter section associated with node 1 yields a finite-difference equation of the form Node 1:

冢

T2 ⫹ T7 ⫺ 2 ⫹

冣

ho⌬x h ⌬x T1 ⫽ ⫺ o T⬁,o k k

A similar result may be obtained for nodal region 6, which is characterized by equivalent surface conditions (2 conduction, 1 convection, 1 adiabatic). Nodes 2 to 5 correspond to case 3 of Table 4.2, and choosing node 3 as an example, it follows that Node 3:

T2 ⫹ T4 ⫹ 2T9 ⫺2

冢h k⌬x ⫹ 2冣T ⫽ ⫺ 2hk⌬x T o

o

3

⬁,o

4.5

䊏

253

Solving the Finite-Difference Equations

Nodes 7, 12, 13, and 20 correspond to case 5 of Table 4.2, with q⬙ ⫽ 0, and choosing node 12 as an example, it follows that Node 12:

T6 ⫹ 2T11 ⫹ T18 ⫺ 4T12 ⫽ 0

Nodes 8 to 11 and 14 are interior nodes (case 1), in which case the finite-difference equation for node 8 is Node 8:

T2 ⫹ T7 ⫹ T9 ⫹ T14 ⫺ 4T8 ⫽ 0

Node 15 is an internal corner (case 2) for which Node 15:

冢

2T9 ⫹ 2T14 ⫹ T16 ⫹ T21 ⫺ 2 3 ⫹

冣

hi ⌬x h ⌬x T15 ⫽ ⫺ 2 i T⬁,i k k

while nodes 16 and 17 are situated on a plane surface with convection (case 3): Node 16:

冢h k⌬x ⫹ 2冣T

2T10 ⫹ T15 ⫹ T17 ⫺ 2

i

16

⫽⫺

2hi ⌬x T⬁,i k

In each case, heat transfer to nodal regions 18 and 21 is characterized by conduction from two adjoining nodes and convection from the internal flow, with no heat transfer occurring from an adjoining adiabat. Performing an energy balance for nodal region 18, it follows that Node 18:

冢

T12 ⫹ T17 ⫺ 2 ⫹

冣

hi ⌬x h ⌬x T18 ⫽ ⫺ i T⬁, i k k

The last special case corresponds to nodal region 19, which has two adiabatic surfaces and experiences heat transfer by conduction across the other two surfaces. Node 19:

T13 ⫹ T20 ⫺ 2T19 ⫽ 0

The equations for nodes 1 through 21 may be solved simultaneously using IHT, another commercial code, or a handheld calculator. The following results are obtained: T1

T2

T3

T4

T5

T6

1526.0 K

1525.3 K

1523.6 K

1521.9 K

1520.8 K

1520.5 K

T7

T8

T9

T10

T11

T12

1519.7 K

1518.8 K

1516.5 K

1514.5 K

1513.3 K

1512.9 K

T13

T14

T15

T16

T17

T18

1515.1 K

1513.7 K

1509.2 K

1506.4 K

1505.0 K

1504.5 K

T19

T20

T21

1513.4 K

1511.7 K

1506.0 K

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

The temperature field may also be represented in the form of isotherms, and four such lines of constant temperature are shown schematically. Also shown are heat flux lines that have been carefully drawn so that they are everywhere perpendicular to the isotherms and coincident with the symmetry adiabat. The surfaces that are exposed to the combustion gases and air are not isothermal, and therefore the heat flow lines are not perpendicular to these boundaries.

1521.7

1517.4

Symmetry adiabat

1513.1 1508.9

As expected, the maximum temperature exists at the location farthest removed from the coolant, which corresponds to node 1. Temperatures along the surface of the turbine blade exposed to the combustion gases are of particular interest. The finite-difference predictions are plotted below (with straight lines connecting the nodal temperatures). 1528

1526 T (K)

254

1524

1522

1520

0

1

2

3

4

5

x (mm)

The rate of heat transfer per unit length of channel may be calculated in two ways. Based on heat transfer from the blade to the air, it is q⬘ ⫽ 4hi[(⌬y/2)(T21 ⫺ T앝,i) ⫹ (⌬y/2 ⫹ ⌬x/2)(T15 ⫺ T앝,i) ⫹ (⌬x)(T16 ⫺ T⬁,i) ⫹ ⌬x(T17 ⫺ T⬁,i) ⫹ (⌬x/2)(T18 ⫺ T⬁,i)] Alternatively, based on heat transfer from the combustion gases to the blade, it is q⬘ ⫽ 4ho[(⌬x/2)(T⬁,o ⫺ T1) ⫹ (⌬x)(T⬁,o ⫺ T2) ⫹ (⌬x)(T⬁,o ⫺ T3) ⫹ (⌬x)(T⬁,o ⫺ T4) ⫹ (⌬x)(T⬁,o ⫺ T5) ⫹ (⌬x/2)(T⬁,o ⫺ T6)]

4.5

䊏

255

Solving the Finite-Difference Equations

where the factor of 4 originates from the symmetry conditions. In both cases, we obtain q⬘ ⫽ 3540.6 W/m

䉰

Comments: 1. In matrix notation, following Equation 4.48, the equations for nodes 1 through 21 are of the form [A][T] ⫽ [C], where ⎡ −a ⎢ 1 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 1 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 [A] = ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢⎣ 0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

−b 1 0 1 −b 1 0 1 −b

1

0

0 0 1

0 0 0

0 0 0

2 0 0

0 2 0

0 0 2

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0 0 −4 2 1 −4

0 0 0 1

0 0 0 0

2 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 1

0 0 0

0 0 0

1 0 0

0 1 0

0 0 1

0 0 0

0 0 0

0 0 0

0 2

0 0

0 0

0 0

1 0

0 1

0 0

0 0 1

0 0 0

0 0 0

1 0 0

0 0 0 1

0 0 0 0

1 −b 1 0 1 −a 0 0 0 0 0 0

0 0 0

1 0 0

0 1 0

0 0 1

0 0 0

0 0 0

1 −4 1 0 0 1 −4 1 0 0 1 −4

0 0

0 0

0 0

0 0

1 0

0 1

0 0

0 0

0 0

2 −4 0 0 0 −4

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

1 0 0

0 2 0

0 0 2

0 0 0

0 0 0

1 −4 1 0 0 2 −c 1 0 0 1 −d

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

2 0

0 1

0 0

0 0

0 0

1 0

−d 1

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

1 0

0 2

0 0

0 0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1 0 0 0 0 −e 0 −2 1 0 1 −4 0

0

1

0⎤ 0⎥ ⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0 ⎥⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 1⎥ ⎥ 0 ⎥ 0⎥ ⎥ 0 ⎥ 0⎥ ⎥ 1⎥ −e ⎥⎦

⎡ −f ⎤ ⎢ −2f ⎥ ⎥ ⎢ ⎢−2f ⎥ ⎥ ⎢ ⎢ −2f ⎥ ⎢ −2f ⎥ ⎥ ⎢ ⎢ −f ⎥ ⎢0 ⎥ ⎥ ⎢ ⎢0 ⎥ ⎢0 ⎥ ⎥ ⎢ ⎢0 ⎥ [C] = ⎢0 ⎥ ⎥ ⎢ ⎢0 ⎥ ⎥ ⎢ ⎢0 ⎥ ⎢0 ⎥ ⎢ ⎥ ⎢ −2g ⎥ ⎢ −2g ⎥ ⎢ ⎥ ⎢ −2g ⎥ ⎢ −g ⎥ ⎢ ⎥ ⎢0 ⎥ ⎢0 ⎥ ⎢ ⎥ ⎢⎣ −g ⎥⎦

With ho⌬x/k ⫽ 0.04 and hi⌬x/k ⫽ 0.008, the following coefficients in the equations can be calculated: a ⫽ 2.04, b ⫽ 4.08, c ⫽ 6.016, d ⫽ 4.016, e ⫽ 2.008, f ⫽ 68, and g ⫽ 3.2. By framing the equations as a matrix equation, standard tools for solving matrix equations may be used. 2. To ensure that no errors have been made in formulating and solving the finite-difference equations, the calculated temperatures should be used to verify that conservation of energy is satisfied for a control surface surrounding all nodal regions. This check has already been performed, since it was shown that the heat transfer rate from the combustion gases to the blade is equal to that from the blade to the air. 3. The accuracy of the finite-difference solution may be improved by refining the grid. If, for example, we halve the grid spacing (⌬x ⫽ ⌬y ⫽ 0.5 mm), thereby increasing the number of unknown nodal temperatures to 65, we obtain the following results for selected temperatures and the heat rate: T1 ⫽ 1525.9 K, T6 ⫽ 1520.5 K, T18 ⫽ 1504.5 K, T19 ⫽ 1513.5 K, q⬘ ⫽ 3539.9 W/m

T15 ⫽ 1509.2 K, T21 ⫽ 1505.7 K,

Agreement between the two sets of results is excellent. Of course, use of the finer mesh increases setup and computation time, and in many cases the results obtained from a coarse grid are satisfactory. Selection of the appropriate grid is a judgment that the engineer must make.

256

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

4. In the gas turbine industry, there is great interest in adopting measures that reduce blade temperatures. Such measures could include use of a different alloy of larger thermal conductivity and/or increasing coolant flow through the channel, thereby increasing hi. Using the finite-difference solution with ⌬x ⫽ ⌬y ⫽ 1 mm, the following results are obtained for parametric variations of k and hi: k (W/m 䡠 K)

hi (W/m2 䡠 K)

T1 (K)

qⴕ (W/m)

25 50 25 50

200 200 1000 1000

1526.0 1523.4 1154.5 1138.9

3540.6 3563.3 11,095.5 11,320.7

Why do increases in k and hi reduce temperature in the blade? Why is the effect of the change in hi more significant than that of k? 5. Note that, because the exterior surface of the blade is at an extremely high temperature, radiation losses to its surroundings may be significant. In the finite-difference analysis, such effects could be considered by linearizing the radiation rate equation (see Equations 1.8 and 1.9) and treating radiation in the same manner as convection. However, because the radiation coefficient hr depends on the surface temperature, an iterative finite-difference solution would be necessary to ensure that the resulting surface temperatures correspond to the temperatures at which hr is evaluated at each nodal point. 6. See Example 4.3 in IHT. This problem can also be solved using Tools, FiniteDifference Equations in the Advanced section of IHT. 7. A second software package accompanying this text, Finite-Element Heat Transfer (FEHT), may also be used to solve one- and two-dimensional forms of the heat equation. This example is provided as a solved model in FEHT and may be accessed through Examples on the Toolbar.

4.6 Summary The primary objective of this chapter was to develop an appreciation for the nature of a twodimensional conduction problem and the methods that are available for its solution. When confronted with a two-dimensional problem, one should first determine whether an exact solution is known. This may be done by examining some of the excellent references in which exact solutions to the heat equation are obtained [1–5]. One may also want to determine whether the shape factor or dimensionless conduction heat rate is known for the system of interest [6–10]. However, often, conditions are such that the use of a shape factor, dimensionless conduction heat rate, or an exact solution is not possible, and it is necessary to use a finite-difference or finite-element solution. You should therefore appreciate the inherent nature of the discretization process and know how to formulate and solve the finite-difference

䊏

257

Problems

equations for the discrete points of a nodal network. You may test your understanding of related concepts by addressing the following questions. • What is an isotherm? What is a heat flow line? How are the two lines related geometrically? • What is an adiabat? How is it related to a line of symmetry? How is it intersected by an isotherm? • What parameters characterize the effect of geometry on the relationship between the heat rate and the overall temperature difference for steady conduction in a two-dimensional system? How are these parameters related to the conduction resistance? • What is represented by the temperature of a nodal point, and how does the accuracy of a nodal temperature depend on prescription of the nodal network?

References 1. Schneider, P. J., Conduction Heat Transfer, AddisonWesley, Reading, MA, 1955. 2. Carslaw, H. S., and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, London, 1959. 3. Özisik, M. N., Heat Conduction, Wiley Interscience, New York, 1980. 4. Kakac, S., and Y. Yener, Heat Conduction, Hemisphere Publishing, New York, 1985. 5. Poulikakos, D., Conduction Heat Transfer, PrenticeHall, Englewood Cliffs, NJ, 1994. 6. Sunderland, J. E., and K. R. Johnson, Trans. ASHRAE, 10, 237–241, 1964. 7. Kutateladze, S. S., Fundamentals of Heat Transfer, Academic Press, New York, 1963.

8. General Electric Co. (Corporate Research and Development), Heat Transfer Data Book, Section 502, General Electric Company, Schenectady, NY, 1973. 9. Hahne, E., and U. Grigull, Int. J. Heat Mass Transfer, 18, 751–767, 1975. 10. Yovanovich, M. M., in W. M. Rohsenow, J. P. Hartnett, and Y. I. Cho, Eds., Handbook of Heat Transfer, McGraw-Hill, New York, 1998, pp. 3.1–3.73. 11. Gerald, C. F., and P. O. Wheatley, Applied Numerical Analysis, Pearson Education, Upper Saddle River, NJ, 1998. 12. Hoffman, J. D., Numerical Methods for Engineers and Scientists, McGraw-Hill, New York, 1992.

Problems Exact Solutions 4.1 In the method of separation of variables (Section 4.2) for two-dimensional, steady-state conduction, the separation constant 2 in Equations 4.6 and 4.7 must be a positive constant. Show that a negative or zero value of 2 will result in solutions that cannot satisfy the prescribed boundary conditions. 4.2 A two-dimensional rectangular plate is subjected to prescribed boundary conditions. Using the results of the exact solution for the heat equation presented in Section 4.2, calculate the temperature at the midpoint (1, 0.5) by considering the first five nonzero terms of the infinite series that must be evaluated. Assess the error resulting from using only the first three terms of the infinite series. Plot the temperature distributions T(x, 0.5) and T(1.0, y).

y (m) T2 = 150°C 1

T1 = 50°C 0 0

T1 = 50°C

2

T1 = 50°C x (m)

4.3 Consider the two-dimensional rectangular plate of Problem 4.2 having a thermal conductivity of 50 W/m 䡠 K. Beginning with the exact solution for the temperature distribution, derive an expression for the heat transfer rate per unit thickness from the plate along the lower surface (0 ⱕ x ⱕ 2, y ⫽ 0). Evaluate the heat rate considering the first five nonzero terms of the infinite series.

258

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

4.4 A two-dimensional rectangular plate is subjected to the boundary conditions shown. Derive an expression for the steady-state temperature distribution T(x, y). y T = Ax

b T=0

T=0

4.8 Consider Problem 4.5 for the case where the plate is of square cross section, W ⫽ L.

x

a

T=0

4.5 A two-dimensional rectangular plate is subjected to prescribed temperature boundary conditions on three sides and a uniform heat flux into the plate at the top surface. Using the general approach of Section 4.2, derive an expression for the temperature distribution in the plate. q"s

y

T1

0

L

0

(a) Derive an expression for the shape factor, Smax, associated with the maximum top surface temperature, such that q ⫽ Smax k (T2,max ⫺ T1) where T2,max is the maximum temperature along y ⫽ W. (b) Derive an expression for the shape factor, Savg, associated with the average top surface tempera– – ture, q ⫽ Savg k(T 2 ⫺ T1) where T2 is the average temperature along y ⫽ W. (c) Evaluate the shape factors that can be used to determine the maximum and average temperatures along y ⫽ W. Evaluate the maximum and average temperatures for T1 ⫽ 0°C, L ⫽ W ⫽ 10 mm, k ⫽ 20 W/m 䡠 K, and q⬙s ⫽ 1000 W/m2.

W

T1

An experiment for the configuration shown yields a heat transfer rate per unit length of q⬘conv ⫽ 110 W/m for surface temperatures of T1 ⫽ 53°C and T2 ⫽ 15°C, respectively. For inner and outer cylinders of diameters d ⫽ 20 mm and D ⫽ 60 mm, and an eccentricity factor of z ⫽ 10 mm, determine the value of keff. The actual thermal conductivity of the fluid is k ⫽ 0.255 W/m 䡠 K.

x

T1

Shape Factors and Dimensionless Conduction Heat Rates 4.6 Using the thermal resistance relations developed in Chapter 3, determine shape factor expressions for the following geometries: (a) Plane wall, cylindrical shell, and spherical shell. (b) Isothermal sphere of diameter D buried in an infinite medium. 4.7 Free convection heat transfer is sometimes quantified by writing Equation 4.20 as qconv ⫽ Skeff ⌬T1⫺2, where keff is an effective thermal conductivity. The ratio keff /k is greater than unity because of fluid motion driven by buoyancy forces, as represented by the dashed streamlines.

4.9 Radioactive wastes are temporarily stored in a spherical container, the center of which is buried a distance of 10 m below the earth’s surface. The outside diameter of the container is 2 m, and 500 W of heat are released as a result of radioactive decay. If the soil surface temperature is 20°C, what is the outside surface temperature of the container under steady-state conditions? On a sketch of the soil–container system drawn to scale, show representative isotherms and heat flow lines in the soil. 4.10 Based on the dimensionless conduction heat rates for cases 12–15 in Table 4.1b, find shape factors for the following objects having temperature T1, located at the surface of a semi-infinite medium having temperature T2. The surface of the semi-infinite medium is adiabatic. (a) A buried hemisphere, flush with the surface. (b) A disk on the surface. Compare your result to Table 4.1a, case 10. (c) A square on the surface. (d) A buried cube, flush with the surface.

T1

D

D

z D

g

T1

T2

d

(a)

T2

T1

T2 (b) and (c)

D T1

T2 (d)

4.11 Determine the heat transfer rate between two particles of diameter D ⫽ 100 m and temperatures T1 ⫽ 300.1 K

259

Problems

䊏

and T2 ⫽ 299.9 K, respectively. The particles are in contact and are surrounded by air. Air

D

4.15 A small water droplet of diameter D ⫽ 100 m and temperature Tmp ⫽ 0°C falls on a nonwetting metal surface that is at temperature Ts ⫽ –15°C. Determine how long it will take for the droplet to freeze completely. The latent heat of fusion is hsf ⫽ 334 kJ/kg. Air

Water droplet D, Tmp

T1

T2

4.12 A two-dimensional object is subjected to isothermal conditions at its left and right surfaces, as shown in the schematic. Both diagonal surfaces are adiabatic and the depth of the object is L ⫽ 100 mm.

y θ = π/2

x

T1 T2

a

Nonwetting metal, Ts

4.16 A tube of diameter 50 mm having a surface temperature of 85°C is embedded in the center plane of a concrete slab 0.1 m thick with upper and lower surfaces at 20°C. Using the appropriate tabulated relation for this configuration, find the shape factor. Determine the heat transfer rate per unit length of the tube. 4.17 Pressurized steam at 450 K flows through a long, thinwalled pipe of 0.5-m diameter. The pipe is enclosed in a concrete casing that is of square cross section and 1.5 m on a side. The axis of the pipe is centered in the casing, and the outer surfaces of the casing are maintained at 300 K. What is the heat loss per unit length of pipe?

b

(a) Determine the two-dimensional shape factor for the object for a ⫽ 10 mm, b ⫽ 12 mm. (b) Determine the two-dimensional shape factor for the object for a ⫽ 10 mm, b ⫽ 15 mm. (c) Use the alternative conduction analysis of Section 3.2 to estimate the shape factor for parts (a) and (b). Compare the values of the approximate shape factors of the alternative conduction analysis to the two-dimensional shape factors of parts (a) and (b). (d) For T1 ⫽ 100°C and T2 ⫽ 60°C, determine the heat transfer rate per unit depth for k ⫽ 15 W/m 䡠 K for parts (a) and (b). 4.13 An electrical heater 100 mm long and 5 mm in diameter is inserted into a hole drilled normal to the surface of a large block of material having a thermal conductivity of 5 W/m 䡠 K. Estimate the temperature reached by the heater when dissipating 50 W with the surface of the block at a temperature of 25°C. 4.14 Two parallel pipelines spaced 0.5 m apart are buried in soil having a thermal conductivity of 0.5 W/m 䡠 K. The pipes have outer diameters of 100 and 75 mm with surface temperatures of 175°C and 5°C, respectively. Estimate the heat transfer rate per unit length between the two pipelines.

4.18 The temperature distribution in laser-irradiated materials is determined by the power, size, and shape of the laser beam, along with the properties of the material being irradiated. The beam shape is typically Gaussian, and the local beam irradiation flux (often referred to as the laser fluence) is q⬙(x, y) ⫽ q⬙(x ⫽ y ⫽ 0)exp(⫺xⲐrb)2 exp(⫺yⲐrb)2 The x- and y-coordinates determine the location of interest on the surface of the irradiated material. Consider the case where the center of the beam is located at x ⫽ y ⫽ r ⫽ 0. The beam is characterized by a radius rb, defined as the radial location where the local fluence is q⬙(rb) ⫽ q⬙(r ⫽ 0)/e 艐 0.368q⬙(r ⫽ 0). A shape factor for Gaussian heating is S ⫽ 21/2rb, where S is defined in terms of T1,max ⫺ T2 [Nissin, Y. I., A. Lietoila, R. G. Gold, and J. F. Gibbons, J. Appl. Phys., 51, 274, 1980]. Calculate the maximum steadystate surface temperature associated with irradiation of a material of thermal conductivity k ⫽ 27 W/m 䡠 K and absorptivity ␣ ⫽ 0.45 by a Gaussian beam with rb ⫽ 0.1 mm and power P ⫽ 1 W. Compare your result with the maximum temperature that would occur if the irradiation was from a circular beam of the same diameter and power, but characterized by a uniform fluence (a flat beam). Also calculate the average temperature of the irradiated surface for the uniform fluence case. The temperature far from the irradiated spot is T2 ⫽ 25°C.

260

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

4.19 Hot water at 85°C flows through a thin-walled copper tube of 30-mm diameter. The tube is enclosed by an eccentric cylindrical shell that is maintained at 35°C and has a diameter of 120 mm. The eccentricity, defined as the separation between the centers of the tube and shell, is 20 mm. The space between the tube and shell is filled with an insulating material having a thermal conductivity of 0.05 W/m 䡠 K. Calculate the heat loss per unit length of the tube, and compare the result with the heat loss for a concentric arrangement. 4.20 A furnace of cubical shape, with external dimensions of 0.35 m, is constructed from a refractory brick (fireclay). If the wall thickness is 50 mm, the inner surface temperature is 600°C, and the outer surface temperature is 75°C, calculate the heat loss from the furnace. 4.21 Laser beams are used to thermally process materials in a wide range of applications. Often, the beam is scanned along the surface of the material in a desired pattern. Consider the laser heating process of Problem 4.18, except now the laser beam scans the material at a scanning velocity of U. A dimensionless maximum surface temperature can be well correlated by an expression of the form [Nissin, Y. I., A. Lietoila, R. G. Gold, and J. F. Gibbons, J. Appl. Phys., 51, 274, 1980] T1,max,U⫽0 ⫺ T2 ⫽ 1 ⫹ 0.301Pe ⫺ 0.0108Pe2 T1,max,U⫽0 ⫺ T2 for the range 0 ⬍ Pe ⬍ 10, where Pe is a dimensionless velocity known as the Peclet number. For this problem, Pe ⫽ Urb /兹2␣ where ␣ is the thermal diffusivity of the material. The maximum material temperature does not occur directly below the laser beam, but at a lag distance ␦ behind the center of the moving beam. The dimensionless lag distance can be correlated to Pe by [Sheng, I. C., and Y. Chen, J. Thermal Stresses, 14, 129, 1991] ␦U 1.55 ␣ ⫽ 0.944Pe (a) For the laser beam size and shape and material of Problem 4.18, determine the laser power required to achieve T1,max ⫽ 200°C for U ⫽ 2 m/s. The density and specific heat of the material are ⫽ 2000 kg/m3 and c ⫽ 800 J/kg 䡠 K, respectively.

is evacuated, eliminating conduction and convection across the gap. Small cylindrical pillars, each L ⫽ 0.2 mm long and D ⫽ 0.15 mm in diameter, are inserted between the glass sheets to ensure that the glass does not break due to stresses imposed by the pressure difference across each glass sheet. A con⬙ ⫽ 1.5 ⫻ 10⫺6 m2 䡠 K/W exists tact resistance of Rt,c between the pillar and the sheet. For nominal glass temperatures of T1 ⫽ 20°C and T2 ⫽ ⫺10°C, determine the conduction heat transfer through an individual stainless steel pillar. 4.23 A pipeline, used for the transport of crude oil, is buried in the earth such that its centerline is a distance of 1.5 m below the surface. The pipe has an outer diameter of 0.5 m and is insulated with a layer of cellular glass 100 mm thick. What is the heat loss per unit length of pipe when heated oil at 120°C flows through the pipe and the surface of the earth is at a temperature of 0°C? 4.24 A long power transmission cable is buried at a depth (ground-to-cable-centerline distance) of 2 m. The cable is encased in a thin-walled pipe of 0.1-m diameter, and, to render the cable superconducting (with essentially zero power dissipation), the space between the cable and pipe is filled with liquid nitrogen at 77 K. If the pipe is covered with a superinsulator (ki ⫽ 0.005 W/m 䡠 K) of 0.05-m thickness and the surface of the earth (kg ⫽ 1.2 W/m 䡠 K) is at 300 K, what is the cooling load (W/m) that must be maintained by a cryogenic refrigerator per unit pipe length? 4.25 A small device is used to measure the surface temperature of an object. A thermocouple bead of diameter D ⫽ 120 m is positioned a distance z ⫽ 100 m from the surface of interest. The two thermocouple wires, each of diameter d ⫽ 25 m and length L ⫽ 300 m, are held by a large manipulator that is at a temperature of Tm ⫽ 23°C. Manipulator, Tm d L Air

(b) Determine the lag distance ␦ associated with U ⫽ 2 m/s. (c) Plot the required laser power to achieve Tmax,1 ⫽ 200⬚C for 0 ⱕ U ⱕ 2 m/s.

Shape Factors with Thermal Circuits 4.22 A double-glazed window consists of two sheets of glass separated by an L ⫽ 0.2-mm-thick gap. The gap

z

D, Ttc

Thermocouple bead

Ts

If the thermocouple registers a temperature of Ttc ⫽ 29°C, what is the surface temperature? The thermal

䊏

261

Problems

conductivities of the chromel and alumel thermocouple wires are kCh ⫽ 19 W/m 䡠 K and kAl ⫽ 29 W/m 䡠 K, respectively. You may neglect radiation and convection effects. 4.26 A cubical glass melting furnace has exterior dimensions of width W ⫽ 5 m on a side and is constructed from refractory brick of thickness L ⫽ 0.35 m and thermal conductivity k ⫽ 1.4 W/m 䡠 K. The sides and top of the furnace are exposed to ambient air at 25°C, with free convection characterized by an average coefficient of h ⫽ 5 W/m2 䡠 K. The bottom of the furnace rests on a framed platform for which much of the surface is exposed to the ambient air, and a convection coefficient of h ⫽ 5 W/m2 䡠 K may be assumed as a first approximation. Under operating conditions for which combustion gases maintain the inner surfaces of the furnace at 1100°C, what is the heat loss from the furnace? 4.27 A hot fluid passes through circular channels of a cast iron platen (A) of thickness LA ⫽ 30 mm which is in poor contact with the cover plates (B) of thickness LB ⫽ 7.5 mm. The channels are of diameter D ⫽ 15 mm with a centerline spacing of Lo ⫽ 60 mm. The thermal conductivities of the materials are kA ⫽ 20 W/m 䡠 K and kB ⫽ 75 W/m 䡠 K, while the contact resistance between the two materials is R⬙t,c ⫽ 2.0 ⫻ 10⫺4 m2 䡠 K/W. The hot fluid is at Ti ⫽ 150°C, and the convection coefficient is 1000 W/m2 䡠 K. The cover plate is exposed to ambient air at T앝 ⫽ 25°C with a convection coefficient of 200 W/m2 䡠 K. The shape factor between one channel and the platen top and bottom surfaces is 4.25. Air

T∞, h

Ts

Cover plate, B

LB

4.28 An aluminum heat sink (k ⫽ 240 W/m 䡠 K), used to cool an array of electronic chips, consists of a square channel of inner width w ⫽ 25 mm, through which liquid flow may be assumed to maintain a uniform surface temperature of T1 ⫽ 20⬚C. The outer width and length of the channel are W ⫽ 40 mm and L ⫽ 160 mm, respectively. Chip, Tc

Rt,c Heat sink

Coolant

w

L

W

If N ⫽ 120 chips attached to the outer surfaces of the heat sink maintain an approximately uniform surface temperature of T2 ⫽ 50⬚C and all of the heat dissipated by the chips is assumed to be transferred to the coolant, what is the heat dissipation per chip? If the contact resistance between each chip and the heat sink is Rt,c ⫽ 0.2 K/W, what is the chip temperature? 4.29 Hot water is transported from a cogeneration power station to commercial and industrial users through steel pipes of diameter D ⫽ l50 mm, with each pipe centered in concrete (k ⫽ 1.4 W/m ⭈ K) of square cross section (w ⫽ 300 mm). The outer surfaces of the concrete are exposed to ambient air for which T앝 ⫽ 0⬚C and h ⫽ 25 W/m2 䡠 K.

R"t,c

Concrete, k

D

Contact resistance

T2

T1

To Fluid

Air

Ti, hi

T∞, h

LA Platen, A

Lo

Cover plate, B

LB

T1

R"t,c Water

Air

w D

Ts

L

•

Ti, m

T∞, h

(b) Determine the outer surface temperature of the cover plate, Ts.

(a) If the inlet temperature of water flowing through the pipe is Ti ⫽ 90⬚C, what is the heat loss per unit length of pipe in proximity to the inlet? The temperature of the pipe T1 may be assumed to be that of the inlet water.

(c) Comment on the effects that changing the centerline spacing will have on q⬘i and Ts. How would insulating the lower surface affect q⬘i and Ts?

(b) If the difference between the inlet and outlet temperatures of water flowing through a 100-m-long pipe is not to exceed 5⬚C, estimate the minimum

(a) Determine the heat rate from a single channel per unit length of the platen normal to the page, q⬘i.

262

Chapter 4

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Two-Dimensional, Steady-State Conduction

. allowable flow rate m . A value of c ⫽ 4207 J/kg 䡠 K may be used for the specific heat of the water.

material exceeds the fluid temperature, attachment of a fin depresses the junction temperature Tj below the original temperature of the base, and heat flow from the base material to the fin is two-dimensional.

4.30 A long constantan wire of 1-mm diameter is butt welded to the surface of a large copper block, forming a thermocouple junction. The wire behaves as a fin, permitting heat to flow from the surface, thereby depressing the sensing junction temperature Tj below that of the block To.

T∞, h

Tj

D

Air

Aluminum or stainless steel base

Thermocouple wire, D

T∞, h

Tb

Aluminum pin fin

Tj Copper block, To

(a) If the wire is in air at 25°C with a convection coefficient of 10 W/m2 䡠 K, estimate the measurement error (Tj ⫺ To) for the thermocouple when the block is at 125°C. (b) For convection coefficients of 5, 10, and 25 W/m2 䡠 K, plot the measurement error as a function of the thermal conductivity of the block material over the range 15 to 400 W/m 䡠 K. Under what circumstances is it advantageous to use smaller diameter wire? 4.31 A hole of diameter D ⫽ 0.25 m is drilled through the center of a solid block of square cross section with w ⫽ 1 m on a side. The hole is drilled along the length, l ⫽ 2 m, of the block, which has a thermal conductivity of k ⫽ 150 W/m 䡠 K. The four outer surfaces are exposed to ambient air, with T앝,2 ⫽ 25°C and h2 ⫽ 4 W/m2 䡠 K, while hot oil flowing through the hole is characterized by T앝,1 ⫽ 300°C and h1 ⫽ 50 W/m2 䡠 K. Determine the corresponding heat rate and surface temperatures.

D = 0.25 m

h1, T∞,1

Consider conditions for which a long aluminum pin fin of diameter D ⫽ 5 mm is attached to a base material whose temperature far from the junction is maintained at Tb ⫽ 100°C. Fin convection conditions correspond to h ⫽ 50 W/m2 䡠 K and T앝 ⫽ 25°C. (a) What are the fin heat rate and junction temperature when the base material is (i) aluminum (k ⫽ 240 W/m 䡠 K) and (ii) stainless steel (k ⫽ 15 W/m 䡠 K)? (b) Repeat the foregoing calculations if a thermal contact resistance of R⬙t, j ⫽ 3 ⫻ 10⫺5 m2 䡠 K/W is associated with the method of joining the pin fin to the base material. (c) Considering the thermal contact resistance, plot the heat rate as a function of the convection coefficient over the range 10 ⱕ h ⱕ 100 W/m2 䡠 K for each of the two materials. 4.33 An igloo is built in the shape of a hemisphere, with an inner radius of 1.8 m and walls of compacted snow that are 0.5 m thick. On the inside of the igloo, the surface heat transfer coefficient is 6 W/m2 䡠 K; on the outside, under normal wind conditions, it is 15 W/m2 䡠 K. The thermal conductivity of compacted snow is 0.15 W/m 䡠 K. The temperature of the ice cap on which the igloo sits is ⫺20°C and has the same thermal conductivity as the compacted snow.

h2, T∞,2 Arctic wind, T∞

w=1m

4.32 In Chapter 3 we assumed that, whenever fins are attached to a base material, the base temperature is unchanged. What in fact happens is that, if the temperature of the base

Igloo

Tair

Ice cap, Tic

䊏

263

Problems

(a) Assuming that the occupants’ body heat provides a continuous source of 320 W within the igloo, calculate the inside air temperature when the outside air temperature is T앝 ⫽ ⫺40°C. Be sure to consider heat losses through the floor of the igloo. (b) Using the thermal circuit of part (a), perform a parameter sensitivity analysis to determine which variables have a significant effect on the inside air temperature. For instance, for very high wind conditions, the outside convection coefficient could double or even triple. Does it make sense to construct the igloo with walls half or twice as thick? 4.34 Consider the thin integrated circuit (chip) of Problem 3.150. Instead of attaching the heat sink to the chip surface, an engineer suggests that sufficient cooling might be achieved by mounting the top of the chip onto a large copper (k ⫽ 400 W/m 䡠 K) surface that is located nearby. The metallurgical joint between the chip and the substrate provides a contact resistance of R⬙t,c ⫽ 5 ⫻ 10⫺6 m2 䡠 K/W, and the maximum allowable chip temperature is 85°C. If the large substrate temperature is T2 ⫽ 25°C at locations far from the chip, what is the maximum allowable chip power dissipation qc? 4.35 An electronic device, in the form of a disk 20 mm in diameter, dissipates 100 W when mounted flush on a large aluminum alloy (2024) block whose temperature is maintained at 27°C. The mounting arrangement is such that a contact resistance of R⬙t,c ⫽ 5 ⫻ 10⫺5 m2 䡠 K/W exists at the interface between the device and the block. Air

T∞, h Electronic device, Td, P

Pin fins (30), D = 1.5 mm L = 15 mm

Copper, 5-mm thickness Device

Epoxy,

R"t ,c

Epoxy, Aluminum block, Tb

R"t ,c

(a) Calculate the temperature the device will reach, assuming that all the power generated by the device must be transferred by conduction to the block. (b) To operate the device at a higher power level, a circuit designer proposes to attach a finned heat sink to the top of the device. The pin fins and base material are fabricated from copper (k ⫽ 400 W/m 䡠 K) and are exposed to an airstream at 27°C for which the convection coefficient is 1000 W/m2 䡠 K. For the device temperature computed in part (a), what is the permissible operating power?

4.36 The elemental unit of an air heater consists of a long circular rod of diameter D, which is encapsulated by a finned sleeve and in which thermal energy is generated by ohmic heating. The N fins of thickness t and length L are integrally fabricated with the square sleeve of width w. Under steady-state operating conditions, the rate of thermal energy generation corresponds to the rate of heat transfer to airflow over the sleeve. Fins, N Sleeve, ks Airflow

T∞, h

t D Ts w

Heater • (q, kh)

L

(a) Under conditions for which a uniform surface temperature Ts is maintained around the circumference of the heater and the temperature T앝 and convection coefficient h of the airflow are known, obtain an expression for the rate of heat transfer per unit length to the air. Evaluate the heat rate for Ts ⫽ 300⬚C, D ⫽ 20 mm, an aluminum sleeve (ks ⫽ 240 W/m 䡠 K), w ⫽ 40 mm, N ⫽ 16, t ⫽ 4 mm, L ⫽ 20 mm, T앝 ⫽ 50⬚C, and h ⫽ 500 W/m2 䡠 K. (b) For the foregoing heat rate and a copper heater of thermal conductivity kh ⫽ 400 W/m 䡠 K, what is the required volumetric heat generation within the heater and its corresponding centerline temperature? (c) With all other quantities unchanged, explore the effect of variations in the fin parameters (N, L, t) on the heat rate, subject to the constraint that the fin thickness and the spacing between fins cannot be less than 2 mm. 4.37 For a small heat source attached to a large substrate, the spreading resistance associated with multidimensional conduction in the substrate may be approximated by the expression [Yovanovich, M. M., and V. W. Antonetti, in Adv. Thermal Modeling Elec. Comp. and Systems, Vol. 1, A. Bar-Cohen and A. D. Kraus, Eds., Hemisphere, NY, 79–128, 1988] Rt(sp) ⫽

1 ⫺ 1.410 Ar ⫹ 0.344 A3r ⫹ 0.043 A5r ⫹ 0.034 A7r 4ksub A1/2 s, h

where Ar ⫽ As,h /As,sub is the ratio of the heat source area to the substrate area. Consider application of the expression to an in-line array of square chips of width Lh ⫽ 5 mm on a side and pitch Sh ⫽ 10 mm. The interface

264

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

between the chips and a large substrate of thermal conductivity ksub ⫽ 80 W/m 䡠 K is characterized by a thermal contact resistance of R⬙t,c ⫽ 0.5 ⫻ 10⫺4 m2 䡠 K/W. Top view Substrate, ksub

the sketch, the boundary condition changes from specified heat flux q⬙s (into the domain) to convection, at the location of the node (m, n). Write the steadystate, two-dimensional finite difference equation at this node.

Chip, Th Side view

Air

q"s

T∞, h Sh

h, T∞

Lh

Lh

m, n Sh

Substrate

∆y

R"t,c

∆x

If a convection heat transfer coefficient of h ⫽ 100 W/m2 䡠 K is associated with airflow (T앝 ⫽ 15⬚C) over the chips and substrate, what is the maximum allowable chip power dissipation if the chip temperature is not to exceed Th ⫽ 85⬚C?

4.42 Determine expressions for q(m⫺1,n) → (m,n), q(m⫹1,n) → (m,n), q(m,n⫹1) → (m,n) and q(m,n⫺1) → (m,n) for conduction associated with a control volume that spans two different materials. There is no contact resistance at the interface between the materials. The control volumes are L units long into the page. Write the finite difference equation under steadystate conditions for node (m, n).

Finite-Difference Equations: Derivations ∆x

4.38 Consider nodal configuration 2 of Table 4.2. Derive the finite-difference equations under steady-state conditions for the following situations. (a) The horizontal boundary of the internal corner is perfectly insulated and the vertical boundary is subjected to the convection process (T앝, h). (b) Both boundaries of the internal corner are perfectly insulated. How does this result compare with Equation 4.41?

∆y

•

(m 1, n)

•

(m, n 1)

•

(m, n)

•

(m, n 1)

Material A kA

•

(m 1, n) Material B kB

4.39 Consider nodal configuration 3 of Table 4.2. Derive the finite-difference equations under steady-state conditions for the following situations. (a) The boundary is insulated. Explain how Equation 4.42 can be modified to agree with your result. (b) The boundary is subjected to a constant heat flux. 4.40 Consider nodal configuration 4 of Table 4.2. Derive the finite-difference equations under steady-state conditions for the following situations. (a) The upper boundary of the external corner is perfectly insulated and the side boundary is subjected to the convection process (T앝, h). (b) Both boundaries of the external corner are perfectly insulated. How does this result compare with Equation 4.43? 4.41 One of the strengths of numerical methods is their ability to handle complex boundary conditions. In

4.43 Consider heat transfer in a one-dimensional (radial) cylindrical coordinate system under steady-state conditions with volumetric heat generation. (a) Derive the finite-difference equation for any interior node m. (b) Derive the finite-difference equation for the node n located at the external boundary subjected to the convection process (T앝, h). 4.44 In a two-dimensional cylindrical configuration, the radial (⌬r) and angular (⌬) spacings of the nodes are uniform. The boundary at r ⫽ ri is of uniform temperature Ti. The boundaries in the radial direction are adiabatic (insulated) and exposed to surface convection (T앝 , h), as illustrated. Derive the finite-difference equations for (i) node 2, (ii) node 3, and (iii) node 1.

䊏

5

4

T∞, h

265

Problems

6

2

1

(b) Node (m, n) at the tip of a cutting tool with the upper surface exposed to a constant heat flux q⬙o, and the diagonal surface exposed to a convection cooling process with the fluid at T앝 and a heat transfer coefficient h. Assume ⌬x ⫽ ⌬y.

3

∆r

q"o

∆φ

∆φ Uniform temperature surface, Ti

m + 1, n

m, n 45°

ri

∆y

∆x m + 1, n – 1

4.45 Upper and lower surfaces of a bus bar are convectively cooled by air at T앝, with hu ⫽ hl. The sides are cooled by maintaining contact with heat sinks at To, through a thermal contact resistance of R⬙t,c. The bar is of thermal conductivity k, and its width is twice its thickness L. T∞, hu 1

2

To

•

2 4

3

∆y

∆ x = ∆y

∆y 8

9

10

11

12

13

14

15

To

Consider steady-state conditions for which heat is uni. formly generated at a volumetric rate q due to passage of an electric current. Using the energy balance method, derive finite-difference equations for nodes 1 and 13.

kB

Derive the finite-difference equation, assuming no internal generation. 4.48 Consider the two-dimensional grid (⌬x ⫽ ⌬y) representing steady-state conditions with no internal volumetric generation for a system with thermal conductivity k. One of the boundaries is maintained at a constant temperature Ts while the others are adiabatic.

4.46 Derive the nodal finite-difference equations for the following configurations. (a) Node (m, n) on a diagonal boundary subjected to convection with a fluid at T앝 and a heat transfer coefficient h. Assume ⌬x ⫽ ⌬y.

∆y

y

12

11

10

9

8

13

4

5

6

7

14

3

15

2

∆x

m + 1, n + 1 ∆y

T∞, h m, n

m + 1, n m, n – 1 ∆x

kA

L

R"t,c

T∞, hl

Material A

Material B

4

7

m – 1, n – 1

0

1

5

6

R"t,c

4.47 Consider the nodal point 0 located on the boundary between materials of thermal conductivity kA and kB.

q, k

3

∆x

T∞, h

x

16

Insulation

Isothermal boundary, Ts

1 Insulation

Derive an expression for the heat rate per unit length normal to the page crossing the isothermal boundary (Ts). 4.49 Consider a one-dimensional fin of uniform crosssectional area, insulated at its tip, x ⫽ L. (See Table 3.4,

266

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

case B). The temperature at the base of the fin Tb and of the adjoining fluid T앝, as well as the heat transfer coefficient h and the thermal conductivity k, are known. (a) Derive the finite-difference equation for any interior node m. (b) Derive the finite-difference equation for a node n located at the insulated tip.

Finite-Difference Equations: Analysis 4.50 Consider the network for a two-dimensional system without internal volumetric generation having nodal temperatures shown below. If the grid spacing is 125 mm and the thermal conductivity of the material is 50 W/m 䡠 K, calculate the heat rate per unit length normal to the page from the isothermal surface (Ts).

1

2

3

4

5

6 7

Node

Ti (°C)

1 2 3 4 5 6 7

120.55 120.64 121.29 123.89 134.57 150.49 147.14

Ts = 100°C

4.51 An ancient myth describes how a wooden ship was destroyed by soldiers who reflected sunlight from their polished bronze shields onto its hull, setting the ship ablaze. To test the validity of the myth, a group of college students are given mirrors and they reflect sunlight onto a 100 mm ⫻ 100 mm area of a t ⫽ 10-mm-thick plywood mockup characterized by k ⫽ 0.8 W/m 䡠 K. The bottom of the mockup is in water at Tw ⫽ 20°C, while the air temperature is T앝 ⫽ 25°C. The surroundings are at Tsur ⫽ 23°C. The wood has an emissivity of ⫽ 0.90; both the front and back surfaces of the plywood are characterized by h ⫽ 5 W/m2 䡠 K. The absorbed irradiation from the N students’ mirrors is GS,N ⫽ 70,000 W/m2 on the front surface of the mockup. Tsur 23°C

T∞ 25°C h 5 W/m2·K

L2 800 mm

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

Irradiation location A

H = 300 mm

Irradiation location B L1 500 mm

Tw 20°C

(a) A debate ensues concerning where the beam should be focused, location A or location B. Using a finite

difference method with ⌬x ⫽ ⌬y ⫽ 100 mm and treating the wood as a two-dimensional extended surface (Figure 3.17a), enlighten the students as to whether location A or location B will be more effective in igniting the wood by determining the maximum local steady-state temperature. (b) Some students wonder whether the same technique can be used to melt a stainless steel hull. Repeat part (a) considering a stainless steel mockup of the same dimensions with k ⫽ 15 W/m 䡠 K and ⫽ 0.2. The value of the absorbed irradiation is the same as in part (a). 4.52 Consider the square channel shown in the sketch operating under steady-state conditions. The inner surface of the channel is at a uniform temperature of 600 K, while the outer surface is exposed to convection with a fluid at 300 K and a convection coefficient of 50 W/m2 䡠 K. From a symmetrical element of the channel, a twodimensional grid has been constructed and the nodes labeled. The temperatures for nodes 1, 3, 6, 8, and 9 are identified. T∞ = 300 K h = 50 W/m2• K

1

2

3

5

6

7

4

∆ x = ∆y = 0.01 m 8

T = 600 K

9

y x

k = 1 W/m•K T1 = 430 K T3 = 394 K

T8 = T9 = 600 K T6 = 492 K

(a) Beginning with properly defined control volumes, derive the finite-difference equations for nodes 2, 4, and 7 and determine the temperatures T2, T4, and T7 (K). (b) Calculate the heat loss per unit length from the channel. 4.53 A long conducting rod of rectangular cross section (20 mm ⫻ 30 mm) and thermal conductivity k ⫽ 20 W/m 䡠 K experiences uniform heat generation at a . rate q ⫽ 5 ⫻ 107 W/m3, while its surfaces are maintained at 300 K. (a) Using a finite-difference method with a grid spacing of 5 mm, determine the temperature distribution in the rod. (b) With the boundary conditions unchanged, what heat generation rate will cause the midpoint temperature to reach 600 K?

䊏

267

Problems

4.54 A flue passing hot exhaust gases has a square cross section, 300 mm to a side. The walls are constructed of refractory brick 150 mm thick with a thermal conductivity of 0.85 W/m 䡠 K. Calculate the heat loss from the flue per unit length when the interior and exterior surfaces are maintained at 350 and 25°C, respectively. Use a grid spacing of 75 mm. 4.55 Steady-state temperatures (K) at three nodal points of a long rectangular rod are as shown. The rod experiences a uniform volumetric generation rate of 5 ⫻ 107 W/m3 and has a thermal conductivity of 20 W/m 䡠 K. Two of its sides are maintained at a constant temperature of 300 K, while the others are insulated. 5 mm 1

2

398.0 5 mm

348.5

3

374.6

Uniform temperature, 300 K

(a) Determine the temperatures at nodes 1, 2, and 3. (b) Calculate the heat transfer rate per unit length (W/m) from the rod using the nodal temperatures. Compare this result with the heat rate calculated from knowledge of the volumetric generation rate and the rod dimensions. 4.56 Functionally graded materials are intentionally fabricated to establish a spatial distribution of properties in the final product. Consider an L ⫻ L two-dimensional object with L ⫽ 20 mm. The thermal conductivity distribution of the functionally graded material is k(x) ⫽ 20 W/m 䡠 K ⫹ (7070 W/m5/2 䡠 K) x3/2. Two sets of boundary conditions, denoted as cases 1 and 2, are applied.

Case 1 — — — 2 — — —

Surface

Boundary Condition

1 2 3 4 1 2 3 4

T ⫽ 100°C T ⫽ 50°C Adiabatic Adiabatic Adiabatic Adiabatic T ⫽ 50°C T ⫽ 100°C

(a) Determine the spatially averaged value of the thermal conductivity k. Use this value to estimate the heat rate per unit length for cases 1 and 2. (b) Using a grid spacing of 2 mm, determine the heat rate per unit depth for case 1. Compare your result to the estimated value calculated in part (a). (c) Using a grid spacing of 2 mm, determine the heat rate per unit depth for case 2. Compare your result to the estimated value calculated in part (a). 4.57 Steady-state temperatures at selected nodal points of the symmetrical section of a flow channel are known to be T2 ⫽ 95.47⬚C, T3 ⫽ 117.3⬚C, T5 ⫽ 79.79⬚C, T6 ⫽ 77.29⬚C, T8 ⫽ 87.28⬚C, and T10 ⫽ 77.65⬚C. The wall experiences uniform volumetric heat generation of . q ⫽ 106 W/m3 and has a thermal conductivity of k ⫽ 10 W/m 䡠 K. The inner and outer surfaces of the channel experience convection with fluid temperatures of T앝,i ⫽ 50⬚C and T앝,o ⫽ 25⬚C and convection coefficients of hi ⫽500 W/m2 䡠 K and ho ⫽ 250 W/m2 䡠 K. y 1

T∞,i, hi

2

Surface B

Insulation 4

3

5

•

k, q 6

Symmetry plane

∆x = ∆y = 25 mm 7

8

9

10

x

Surface A

T∞,o, ho

Surface 3

Surface 2 Surface 1 y

k(x) x Surface 4

(a) Determine the temperatures at nodes 1, 4, 7, and 9. (b) Calculate the heat rate per unit length (W/m) from the outer surface A to the adjacent fluid. (c) Calculate the heat rate per unit length from the inner fluid to surface B. (d) Verify that your results are consistent with an overall energy balance on the channel section. 4.58 Consider an aluminum heat sink (k ⫽ 240 W/m 䡠 K), such as that shown schematically in Problem 4.28. The

268

Chapter 4

Two-Dimensional, Steady-State Conduction

䊏

inner and outer widths of the square channel are w ⫽ 20 mm and W ⫽ 40 mm, respectively, and an outer surface temperature of Ts ⫽ 50⬚C is maintained by the array of electronic chips. In this case, it is not the inner surface temperature that is known, but conditions (T앝, h) associated with coolant flow through the channel, and we wish to determine the rate of heat transfer to the coolant per unit length of channel. For this purpose, consider a symmetrical section of the channel and a two-dimensional grid with ⌬x ⫽ ⌬y ⫽ 5 mm. (a) For T앝 ⫽ 20⬚C and h ⫽ 5000 W/m2 䡠 K, determine the unknown temperatures, T1, . . ., T7, and the rate of heat transfer per unit length of channel, q⬘. (b) Assess the effect of variations in h on the unknown temperatures and the heat rate. Heat sink, k T4

Ts

T∞ , h

T1

T5

T2

T6

T3

T7

Ts

Calculate the heat transfer per unit depth into the page, q⬘, using ⌬x ⫽ ⌬y ⫽ ⌬r ⫽ 10 mm and ⌬ ⫽ /8. The base of the rectangular subdomain is held at Th ⫽ 20°C, while the vertical surface of the cylindrical subdomain and the surface at outer radius ro are at Tc ⫽ 0°C. The remaining surfaces are adiabatic, and the thermal conductivity is k ⫽ 10 W/m 䡠 K. 4.60 Consider the two-dimensional tube of a noncircular cross section formed by rectangular and semicylindrical subdomains patched at the common dashed control surfaces in a manner similar to that described in Problem 4.59. Note that, along the dashed control surfaces, temperatures in the two subdomains are identical and local conduction heat fluxes to the semicylindrical subdomain are identical to local conduction heat fluxes from the rectangular subdomain. The bottom of the domain is held at Ts ⫽ 100°C by condensing steam, while the flowing fluid is characterized by the temperature and convection coefficient shown in the sketch. The remaining surfaces are insulated, and the thermal conductivity is k ⫽ 15 W/m 䡠 K.

k = 15 W/m⋅K

Coolant, T∞, h

4.59 Conduction within relatively complex geometries can sometimes be evaluated using the finite-difference methods of this text that are applied to subdomains and patched together. Consider the two-dimensional domain formed by rectangular and cylindrical subdomains patched at the common, dashed control surface. Note that, along the dashed control surface, temperatures in the two subdomains are identical and local conduction heat fluxes to the cylindrical subdomain are identical to local conduction heat fluxes from the rectangular subdomain.

Tc = 0°C

Adiabatic surfaces

T∞,i = 20°C hi = 240 W/m2·K

r Di = 40 mm

y

t = 10 mm

x L = Do = 80 mm

Find the heat transfer rate per unit length of tube, q⬘, using ⌬x ⫽ ⌬y ⫽ ⌬r ⫽ 10 mm and ⌬ ⫽ /8. Hint: Take advantage of the symmetry of the problem by considering only half of the entire domain. 4.61 The steady-state temperatures (°C) associated with selected nodal points of a two-dimensional system having a thermal conductivity of 1.5 W/m 䡠 K are shown on the accompanying grid. Insulated boundary

ro = 50 mm

T2

129.4 0.1 m

y

ri = 30 mm

H = 30 mm W = 20 mm x Th = 20°C

Ts = 100°C

0.1 m 172.9

137.0 103.5

T1

132.8

Isothermal boundary

T0 = 200°C

45.8

T3

67.0

T∞ = 30°C h = 50 W/m2•K

䊏

269

Problems

(a) Determine the temperatures at nodes 1, 2, and 3. (b) Calculate the heat transfer rate per unit thickness normal to the page from the system to the fluid. 4.62 A steady-state, finite-difference analysis has been performed on a cylindrical fin with a diameter of 12 mm and a thermal conductivity of 15 W/m 䡠 K. The convection process is characterized by a fluid temperature of 25°C and a heat transfer coefficient of 25 W/m2 䡠 K. T∞, h

T0

T1

T2

T3

D

T0 = 100.0°C T1 = 93.4°C T2 = 89.5°C

∆x

x

(a) The temperatures for the first three nodes, separated by a spatial increment of x ⫽ 10 mm, are given in the sketch. Determine the fin heat rate. (b) Determine the temperature at node 3, T3. 4.63 Consider the two-dimensional domain shown. All surfaces are insulated except for the isothermal surfaces at x ⫽ 0 and L.

T2

2H/3

H

T1 0.8 L

y

(a) Determine the temperatures at nodes 1, 2, 3, and 4. Estimate the midpoint temperature. (b) Reducing the mesh size by a factor of 2, determine the corresponding nodal temperatures. Compare your results with those from the coarser grid. (c) From the results for the finer grid, plot the 75, 150, and 250°C isotherms. 4.65 Consider a long bar of square cross section (0.8 m to the side) and of thermal conductivity 2 W/m 䡠 K. Three of these sides are maintained at a uniform temperature of 300°C. The fourth side is exposed to a fluid at 100°C for which the convection heat transfer coefficient is 10 W/m2 䡠 K. (a) Using an appropriate numerical technique with a grid spacing of 0.2 m, determine the midpoint temperature and heat transfer rate between the bar and the fluid per unit length of the bar. (b) Reducing the grid spacing by a factor of 2, determine the midpoint temperature and heat transfer rate. Plot the corresponding temperature distribution across the surface exposed to the fluid. Also, plot the 200 and 250°C isotherms. 4.66 Consider a two-dimensional, straight triangular fin of length L ⫽ 50 mm and base thickness t ⫽ 20 mm. The thermal conductivity of the fin is k ⫽ 25 W/m 䡠 K. The base temperature is Tb ⫽ 50°C, and the fin is exposed to convection conditions characterized by h ⫽ 50 W/m2 䡠 K, T앝 ⫽ 20°C. Using a finite difference mesh with ⌬x ⫽ 10 mm and ⌬y ⫽ 2 mm, and taking advantage of symmetry, determine the fin efficiency, f. Compare your value of the fin efficiency with that reported in Figure 3.19.

x L = 50 mm

L = 5H/3 6 5 4 3 2 1

(a) Use a one-dimensional analysis to estimate the shape factor S. (b) Estimate the shape factor using a finite difference analysis with ⌬x ⫽ ⌬y ⫽ 0.05L. Compare your answer with that of part (a), and explain the difference between the two solutions.

t = 20 mm

15 14 13 12

18 17 16

100°C

T∞ = 20°C h = 50 W/m2·K

x

2

Air duct

T2 = 30°C

T1 = 80°C

200°C 3

4 1.5L

x 300°C

21

4.67 A common arrangement for heating a large surface area is to move warm air through rectangular ducts below the surface. The ducts are square and located midway between the top and bottom surfaces that are exposed to room air and insulated, respectively.

y

50°C

20 19

y

4.64 Consider two-dimensional, steady-state conduction in a square cross section with prescribed surface temperatures.

1

11 10 9 8 7

L

L L

Concrete

270

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

For the condition when the floor and duct temperatures are 30 and 80°C, respectively, and the thermal conductivity of concrete is 1.4 W/m 䡠 K, calculate the heat rate from each duct, per unit length of duct. Use a grid spacing with ⌬x ⫽ 2 ⌬y, where ⌬y ⫽ 0.125L and L ⫽150 mm. 4.68 Consider the gas turbine cooling scheme of Example 4.3. In Problem 3.23, advantages associated with applying a thermal barrier coating (TBC) to the exterior surface of a turbine blade are described. If a 0.5-mm-thick zirconia coating (k ⫽ 1.3 W/m 䡠 K, R⬙t,c ⫽ 10⫺4 m2 䡠 K/W) is applied to the outer surface of the air-cooled blade, determine the temperature field in the blade for the operating conditions of Example 4.3. 4.69 A long, solid cylinder of diameter D ⫽ 25 mm is formed of an insulating core that is covered with a very thin (t ⫽ 50 m), highly polished metal sheathing of thermal conductivity k ⫽ 25 W/m 䡠 K. Electric current flows through the stainless steel from one end of the cylinder to the other, inducing uniform volumetric heating within the . sheathing of q ⫽ 5 ⫻ 106 W/m3. As will become evident in Chapter 6, values of the convection coefficient between the surface and air for this situation are spatially nonuniform, and for the airstream conditions of the experiment, the convection heat transfer coefficient varies with the angle as h() ⫽ 26 ⫹ 0.637 ⫺ 8.922 for 0 ⱕ ⬍ /2 and h() ⫽ 5 for /2 ⱕ ⱕ .

device to nonintrusively determine the surface temperature distribution. Predict the temperature distribution of the painted surface, accounting for radiation heat transfer with large surroundings at Tsur ⫽ 25°C. 4.71 Consider using the experimental methodology of Problem 4.70 to determine the convection coefficient distribution about an airfoil of complex shape.

Tsur = 25°C

3 Air T∞ = 25°C

Location Metal sheathing q• = 5 106 W/m3 1 k = 25 W/m • K

(b) Accounting for -direction conduction in the stainless steel, determine temperatures in the stainless steel at increments of ⌬ ⫽ /20 for 0 ⱕ ⱕ . Compare the temperature distribution with that of part (a). Hint: The temperature distribution is symmetrical about the horizontal centerline of the cylinder. 4.70 Consider Problem 4.69. An engineer desires to measure the surface temperature of the thin sheathing by painting it black ( ⫽ 0.98) and using an infrared measurement

9 10 11

Insulation

1 29

28 27

26 25 24 23 22

12

13

14

15

16 21 20 19 18 17

Accounting for conduction in the metal sheathing and radiation losses to the large surroundings, determine the convection heat transfer coefficients at the locations shown. The surface locations at which the temperatures are measured are spaced 2 mm apart. The thickness of the metal sheathing is t ⫽ 20 m, the volumetric gener. ation rate is q ⫽ 20 ⫻ 106 W/m3, the sheathing’s thermal conductivity is k ⫽ 25 W/m 䡠 K, and the emissivity of the painted surface is ⫽ 0.98. Compare your results to cases where (i) both conduction along the sheathing and radiation are neglected, and (ii) when only radiation is neglected.

θ

(a) Neglecting conduction in the -direction within the stainless steel, plot the temperature distribution T() for 0 ⱕ ⱕ for T앝 ⫽ 25°C.

7 8

Metal sheathing

D = 25 mm

Insulation

2

30

t = 50 µm Air T∞ = 25°C

5 6

4

Temperature Temperature Temperature (°C) Location (°C) Location (°C) 27.77

11

34.29

21

31.13

2

27.67

12

36.78

22

30.64

3

27.71

13

39.29

23

30.60

4

27.83

14

41.51

24

30.77

5

28.06

15

42.68

25

31.16

6

28.47

16

42.84

26

31.52

7

28.98

17

41.29

27

31.85

8

29.67

18

37.89

28

31.51

9

30.66

19

34.51

29

29.91

10

32.18

20

32.36

30

28.42

4.72 A thin metallic foil of thickness 0.25 mm with a pattern of extremely small holes serves as an acceleration grid to control the electrical potential of an ion beam. Such a grid is used in a chemical vapor deposition (CVD) process for the fabrication of semiconductors. The top surface of the grid is exposed to a uniform heat flux

䊏

271

Problems

caused by absorption of the ion beam, q⬙s ⫽ 600 W/m2. The edges of the foil are thermally coupled to watercooled sinks maintained at 300 K. The upper and lower surfaces of the foil experience radiation exchange with the vacuum enclosure walls maintained at 300 K. The effective thermal conductivity of the foil material is 40 W/m 䡠 K, and its emissivity is 0.45.

Vacuum enclosure, Tsur Ion beam, q"s

Grid hole pattern Grid

x

L = 115 mm Water-cooled electrode sink, Tsink

Assuming one-dimensional conduction and using a finite-difference method representing the grid by 10 nodes in the x-direction, estimate the temperature distribution for the grid. Hint: For each node requiring an energy balance, use the linearized form of the radiation rate equation, Equation 1.8, with the radiation coefficient hr, Equation 1.9, evaluated for each node. 4.73 A long bar of rectangular cross section, 0.4 m ⫻ 0.6 m on a side and having a thermal conductivity of 1.5 W/m 䡠 K, is subjected to the boundary conditions shown.

w

w/4 w/2 w/2 T2

(a) Using a finite-difference method with a mesh size of ⌬x ⫽ ⌬y ⫽ 40 mm, calculate the unknown nodal temperatures and the heat transfer rate per width of groove spacing (w) and per unit length normal to the page. (b) With a mesh size of ⌬x ⫽ ⌬y ⫽ 10 mm, repeat the foregoing calculations, determining the temperature field and the heat rate. Also, consider conditions for which the bottom surface is not at a uniform temperature T2 but is exposed to a fluid at T앝 ⫽ 20°C. With ⌬x ⫽ ⌬y ⫽ 10 mm, determine the temperature field and heat rate for values of h ⫽ 5, 200, and 1000 W/m2 䡠 K, as well as for h → 앝. 4.75 Refer to the two-dimensional rectangular plate of Problem 4.2. Using an appropriate numerical method with ⌬x ⫽ ⌬y ⫽ 0.25 m, determine the temperature at the midpoint (1, 0.5). 4.76 The shape factor for conduction through the edge of adjoining walls for which D ⬎ L/5, where D and L are the wall depth and thickness, respectively, is shown in Table 4.1. The two-dimensional symmetrical element of the edge, which is represented by inset (a), is bounded by the diagonal symmetry adiabat and a section of the wall thickness over which the temperature distribution is assumed to be linear between T1 and T2.

Uniform temperature, T = 200°C

T∞, h Insulated

T1

w

y T2

T2 T2 T2 T2

Linear temperature distribution

Symmetry adiabat

T2 Uniform temperature, T = 200°C

Two of the sides are maintained at a uniform temperature of 200°C. One of the sides is adiabatic, and the remaining side is subjected to a convection process with T앝 ⫽ 30°C and h ⫽ 50 W/m2 䡠 K. Using an appropriate numerical technique with a grid spacing of 0.1 m, determine the temperature distribution in the bar and the heat transfer rate between the bar and the fluid per unit length of the bar. 4.74 The top surface of a plate, including its grooves, is maintained at a uniform temperature of T1 ⫽ 200°C. The lower surface is at T2 ⫽ 20°C, the thermal conductivity is 15 W/m 䡠 K, and the groove spacing is 0.16 m.

∆y ∆x

T1

x

(a)

T1 T2

y T2 a

b

L

L a

b

x

n•L (b)

(a) Using the nodal network of inset (a) with L ⫽ 40 mm, determine the temperature distribution in the element for T1 ⫽ 100°C and T2 ⫽ 0°C. Evaluate the heat rate

272

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

per unit depth (D ⫽ 1 m) if k ⫽ 1 W/m 䡠 K. Determine the corresponding shape factor for the edge, and compare your result with that from Table 4.1. (b) Choosing a value of n ⫽ 1 or n ⫽ 1.5, establish a nodal network for the trapezoid of inset (b) and determine the corresponding temperature field. Assess the validity of assuming linear temperature distributions across sections a–a and b–b. 4.77 The diagonal of a long triangular bar is well insulated, while sides of equivalent length are maintained at uniform temperatures Ta and Tb.

Ta = 100°C Insulation

Tb = 0°C

(a) Establish a nodal network consisting of five nodes along each of the sides. For one of the nodes on the diagonal surface, define a suitable control volume and derive the corresponding finite-difference equation. Using this form for the diagonal nodes and appropriate equations for the interior nodes, find the temperature distribution for the bar. On a scale drawing of the shape, show the 25, 50, and 75°C isotherms. (b) An alternate and simpler procedure to obtain the finite-difference equations for the diagonal nodes follows from recognizing that the insulated diagonal surface is a symmetry plane. Consider a square 5 ⫻ 5 nodal network, and represent its diagonal as a symmetry line. Recognize which nodes on either side of the diagonal have identical temperatures. Show that you can treat the diagonal nodes as “interior” nodes and write the finite-difference equations by inspection. 4.78 A straight fin of uniform cross section is fabricated from a material of thermal conductivity 50 W/m 䡠 K, thickness w ⫽ 6 mm, and length L ⫽ 48 mm, and it is very long in the direction normal to the page. The convection heat transfer coefficient is 500 W/m2 䡠 K with an ambient air temperature of T앝 ⫽ 30°C. The base of the fin is maintained at Tb ⫽ 100°C, while the fin tip is well insulated. T∞, h w

Tb T∞, h L

Insulated

(a) Using a finite-difference method with a space increment of 4 mm, estimate the temperature distribution within the fin. Is the assumption of onedimensional heat transfer reasonable for this fin? (b) Estimate the fin heat transfer rate per unit length normal to the page. Compare your result with the one-dimensional, analytical solution, Equation 3.81. (c) Using the finite-difference mesh of part (a), compute and plot the fin temperature distribution for values of h ⫽ 10, 100, 500, and 1000 W/m2 䡠 K. Determine and plot the fin heat transfer rate as a function of h. 4.79 A rod of 10-mm diameter and 250-mm length has one end maintained at 100°C. The surface of the rod experiences free convection with the ambient air at 25°C and a convection coefficient that depends on the difference between the temperature of the surface and the ambient air. Specifically, the coefficient is prescribed by a correlation of the form, hfc ⫽ 2.89[0.6 ⫹ 0.624 (T ⫺ T앝)1/6]2, where the units are hfc (W/m2 䡠 K) and T (K). The surface of the rod has an emissivity ⫽ 0.2 and experiences radiation exchange with the surroundings at Tsur ⫽ 25°C. The fin tip also experiences free convection and radiation exchange. Tsur = 25°C Quiescent air,

T∞ = 25°C Stainless steel rod

Tb = 100°C

k = 14 W/m•K, ε = 0.2

D= 10 mm

L = 250 mm x

Assuming one-dimensional conduction and using a finite-difference method representing the fin by five nodes, estimate the temperature distribution for the fin. Determine also the fin heat rate and the relative contributions of free convection and radiation exchange. Hint: For each node requiring an energy balance, use the linearized form of the radiation rate equation, Equation 1.8, with the radiation coefficient hr, Equation 1.9, evaluated for each node. Similarly, for the convection rate equation associated with each node, the free convection coefficient hfc must be evaluated for each node. 4.80 A simplified representation for cooling in very large-scale integration (VLSI) of microelectronics is shown in the sketch. A silicon chip is mounted in a dielectric substrate, and one surface of the system is convectively cooled, while the remaining surfaces are well insulated from the surroundings. The problem is rendered two-dimensional

䊏

273

Problems

by assuming the system to be very long in the direction perpendicular to the paper. Under steady-state operation, electric power dissipation in the chip provides for uni. form volumetric heating at a rate of q . However, the heating rate is limited by restrictions on the maximum temperature that the chip is allowed to achieve.

(b) The grid spacing used in the foregoing finite-difference solution is coarse, resulting in poor precision for the temperature distribution and heat removal rate. Investigate the effect of grid spacing by considering spatial increments of 50 and 25 m. (c) Consistent with the requirement that a ⫹ b ⫽ 400 m, can the heat sink dimensions be altered in a manner that reduces the overall thermal resistance?

Coolant Chip kc = 50 W/m•K q• = 107 W/m3

T∞ = 20°C h = 500 W/m2•K H/4 L/3 Substrate, ks = 5 W/m•K

H= 12 mm

4.82 A plate (k ⫽ 10 W/m 䡠 K) is stiffened by a series of longitudinal ribs having a rectangular cross section with length L ⫽ 8 mm and width w ⫽ 4 mm. The base of the plate is maintained at a uniform temperature Tb ⫽ 45°C, while the rib surfaces are exposed to air at a temperature of T앝 ⫽ 25°C and a convection coefficient of h ⫽ 600 W/m2 䡠 K.

L = 27 mm

y

For the conditions shown on the sketch, will the maximum temperature in the chip exceed 85°C, the maximum allowable operating temperature set by industry standards? A grid spacing of 3 mm is suggested. 4.81 A heat sink for cooling computer chips is fabricated from copper (ks ⫽ 400 W/m 䡠 K), with machined microchannels passing a cooling fluid for which T ⫽ 25°C and h ⫽ 30,000 W/m2 䡠 K. The lower side of the sink experiences no heat removal, and a preliminary heat sink design calls for dimensions of a ⫽ b ⫽ ws ⫽ wf ⫽ 200 m. A symmetrical element of the heat path from the chip to the fluid is shown in the inset. y Tc Chips, Tc

a

ws

wf

Sink, ks Microchannel

b

T∞, h Insulation

x ws ___ 2

wf ___ 2

(a) Using the symmetrical element with a square nodal network of ⌬x ⫽ ⌬y ⫽ 100 m, determine the corresponding temperature field and the heat rate q⬘ to the coolant per unit channel length (W/m) for a maximum allowable chip temperature Tc, max ⫽ 75°C. Estimate the corresponding thermal resistance between the chip surface and the fluid, R⬘t,c⫺ƒ (m 䡠 K/W). What is the maximum allowable heat dissipation for a chip that measures 10 mm ⫻ 10 mm on a side?

T∞, h

Rib Plate

w

Tb

x L

T∞, h

(a) Using a finite-difference method with ⌬x ⫽ ⌬y ⫽ 2 mm and a total of 5 ⫻ 3 nodal points and regions, estimate the temperature distribution and the heat rate from the base. Compare these results with those obtained by assuming that heat transfer in the rib is one-dimensional, thereby approximating the behavior of a fin. (b) The grid spacing used in the foregoing finitedifference solution is coarse, resulting in poor precision for estimates of temperatures and the heat rate. Investigate the effect of grid refinement by reducing the nodal spacing to ⌬x ⫽ ⌬y ⫽ 1 mm (a 9 ⫻ 3 grid) considering symmetry of the center line. (c) Investigate the nature of two-dimensional conduction in the rib and determine a criterion for which the one-dimensional approximation is reasonable. Do so by extending your finite-difference analysis to determine the heat rate from the base as a function of the length of the rib for the range 1.5 ⱕ L/w ⱕ 10, keeping the length L constant. Compare your results with those determined by approximating the rib as a fin. 4.83 The bottom half of an I-beam providing support for a furnace roof extends into the heating zone. The web is well insulated, while the flange surfaces experience

274

Chapter 4

Two-Dimensional, Steady-State Conduction

䊏

(a) Using a grid spacing of 30 mm and the Gauss-Seidel iteration method, determine the nodal temperatures and the heat rate per unit length normal to the page into the bar from the air.

convection with hot gases at T앝 ⫽ 400°C and a convection coefficient of h ⫽ 150 W/m2 䡠 K. Consider the symmetrical element of the flange region (inset a), assuming that the temperature distribution across the web is uniform at Tw ⫽ 100°C. The beam thermal conductivity is 10 W/m 䡠 K, and its dimensions are wƒ ⫽ 80 mm, ww ⫽ 30 mm, and L ⫽ 30 mm.

(b) Determine the effect of grid spacing on the temperature field and heat rate. Specifically, consider a grid spacing of 15 mm. For this grid, explore the effect of changes in h on the temperature field and the isotherms.

Oven roof I-beam

Insulation Flange Gases T∞, h

Assume uniform

Web

y

Uniform ?

wo

w ___w

4.85 A long trapezoidal bar is subjected to uniform temperatures on two surfaces, while the remaining surfaces are well insulated. If the thermal conductivity of the material is 20 W/m 䡠 K, estimate the heat transfer rate per unit length of the bar using a finite-difference method. Use the Gauss–Seidel method of solution with a space increment of 10 mm.

T∞, h

2

Insulation

Tw T2 = 25°C

L

(b)

50 mm

x

wf ___ 2

20 mm

T∞, h

(a)

T1 = 100°C

(a) Calculate the heat transfer rate per unit length to the beam using a 5 ⫻ 4 nodal network. (b) Is it reasonable to assume that the temperature distribution across the web–flange interface is uniform? Consider the L-shaped domain of inset (b) and use a fine grid to obtain the temperature distribution across the web–flange interface. Make the distance wo ⱖ ww /2. 4.84 A long bar of rectangular cross section is 60 mm ⫻ 90 mm on a side and has a thermal conductivity of 1 W/m 䡠 K. One surface is exposed to a convection process with air at 100°C and a convection coefficient of 100 W/m2 䡠 K, while the remaining surfaces are maintained at 50°C.

4.86 Small-diameter electrical heating elements dissipating 50 W/m (length normal to the sketch) are used to heat a ceramic plate of thermal conductivity 2 W/m 䡠 K. The upper surface of the plate is exposed to ambient air at 30°C with a convection coefficient of 100 W/m2 䡠 K, while the lower surface is well insulated. Air

T∞, h Ceramic plate

Ts Ts = 50°C

Ts

Heating element

y

6 mm

x 2 mm

T∞, h

30 mm

24 mm

24 mm

(a) Using the Gauss–Seidel method with a grid spacing of ⌬x ⫽ 6 mm and ⌬y ⫽ 2 mm, obtain the temperature distribution within the plate. (b) Using the calculated nodal temperatures, sketch four isotherms to illustrate the temperature distribution in the plate. (c) Calculate the heat loss by convection from the plate to the fluid. Compare this value with the element dissipation rate.

䊏

275

Problems

(d) What advantage, if any, is there in not making ⌬x ⫽ ⌬y for this situation? (e) With ⌬x ⫽ ⌬y ⫽ 2 mm, calculate the temperature field within the plate and the rate of heat transfer from the plate. Under no circumstances may the temperature at any location in the plate exceed 400°C. Would this limit be exceeded if the airflow were terminated and heat transfer to the air were by natural convection with h ⫽ 10 W/m2 䡠 K?

Special Applications: Finite Element Analysis 4.87 A straight fin of uniform cross section is fabricated from a material of thermal conductivity k ⫽ 5 W/m 䡠 K, thickness w ⫽ 20 mm, and length L ⫽ 200 mm. The fin is very long in the direction normal to the page. The base of the fin is maintained at Tb ⫽ 200°C, and the tip condition allows for convection (case A of Table 3.4), with h ⫽ 500 W/m2 䡠 K and T앝 ⫽ 25°C. T∞ = 100°C h = 500 W/m2•K Tb = 200°C

k = 5 W/m•K

T∞, h

q'f

w = 20 mm x

L = 200 mm

T∞, h

(a) Assuming one-dimensional heat transfer in the fin, calculate the fin heat rate, q⬘f (W/m), and the tip temperature TL. Calculate the Biot number for the fin to determine whether the one-dimensional assumption is valid. (b) Using the finite-element method of FEHT, perform a two-dimensional analysis on the fin to determine the fin heat rate and tip temperature. Compare your results with those from the one-dimensional, analytical solution of part (a). Use the View/Temperature Contours option to display isotherms, and discuss key features of the corresponding temperature field and heat flow pattern. Hint: In drawing the outline of the fin, take advantage of symmetry. Use a fine mesh near the base and a coarser mesh near the tip. Why? (c) Validate your FEHT model by comparing predictions with the analytical solution for a fin with thermal conductivities of k ⫽ 50 W/m 䡠 K and 500 W/m 䡠 K. Is the one-dimensional heat transfer assumption valid for these conditions?

4.88 Consider the long rectangular bar of Problem 4.84 with the prescribed boundary conditions. (a) Using the finite-element method of FEHT, determine the temperature distribution. Use the View/ Temperature Contours command to represent the isotherms. Identify significant features of the distribution. (b) Using the View/Heat Flows command, calculate the heat rate per unit length (W/m) from the bar to the airstream. (c) Explore the effect on the heat rate of increasing the convection coefficient by factors of two and three. Explain why the change in the heat rate is not proportional to the change in the convection coefficient. 4.89 Consider the long rectangular rod of Problem 4.53, which experiences uniform heat generation while its surfaces are maintained at a fixed temperature. (a) Using the finite-element method of FEHT, determine the temperature distribution. Use the View/ Temperature Contours command to represent the isotherms. Identify significant features of the distribution. (b) With the boundary conditions unchanged, what heat generation rate will cause the midpoint temperature to reach 600 K? 4.90 Consider the symmetrical section of the flow channel of . Problem 4.57, with the prescribed values of q , k, T앝,i, T앝,o, hi, and ho. Use the finite-element method of FEHT to obtain the following results. (a) Determine the temperature distribution in the symmetrical section, and use the View/Temperature Contours command to represent the isotherms. Identify significant features of the temperature distribution, including the hottest and coolest regions and the region with the steepest gradients. Describe the heat flow field. (b) Using the View/Heat Flows command, calculate the heat rate per unit length (W/m) from the outer surface A to the adjacent fluid. (c) Calculate the heat rate per unit length from the inner fluid to surface B. (d) Verify that your results are consistent with an overall energy balance on the channel section. 4.91 The hot-film heat flux gage shown schematically may be used to determine the convection coefficient of an adjoining fluid stream by measuring the electric power dissipation per unit area, P⬙e (W/m2), and the average surface temperature, Ts,f , of the film. The power dissipated in the film is transferred directly to the fluid by convection, as well as by conduction into the substrate.

276

Chapter 4

䊏

Two-Dimensional, Steady-State Conduction

If substrate conduction is negligible, the gage measurements can be used to determine the convection coefficient without application of a correction factor. Your assignment is to perform a two-dimensional, steadystate conduction analysis to estimate the fraction of the power dissipation that is conducted into a 2-mm-thick quartz substrate of width W ⫽ 40 mm and thermal conductivity k ⫽ 1.4 W/m 䡠 K. The thin, hot-film gage has a width of w ⫽ 4 mm and operates at a uniform power dissipation of 5000 W/m2. Consider cases for which the fluid temperature is 25°C and the convection coefficient is 500, 1000, and 2000 W/m2 䡠 K. Ts, f

Hot-thin film, P"e = 5000 W/m2

Fluid

T∞, h

Quartz substrate k = 1.4 W/m•K

w = 4 mm

2 mm

(b) Determine the effect of grid spacing on the temperature field and heat loss per unit length to the air. Specifically, consider a grid spacing of 25 mm and plot appropriately spaced isotherms on a schematic of the system. Explore the effect of changes in the convection coefficients on the temperature field and heat loss. 4.93 Electronic devices dissipating electrical power can be cooled by conduction to a heat sink. The lower surface of the sink is cooled, and the spacing of the devices ws, the width of the device wd, and the thickness L and thermal conductivity k of the heat sink material each affect the thermal resistance between the device and the cooled surface. The function of the heat sink is to spread the heat dissipated in the device throughout the sink material. ws = 48 mm

P"e

Device, Td = 85°C

wd = 18 mm

T∞ , h

L = 24 mm

W = 40 mm w/2

W/2

Sink material,

Use the finite-element method of FEHT to analyze a symmetrical half-section of the gage and the quartz substrate. Assume that the lower and end surfaces of the substrate are perfectly insulated, while the upper surface experiences convection with the fluid. (a) Determine the temperature distribution and the conduction heat rate into the region below the hot film for the three values of h. Calculate the fractions of electric power dissipation represented by these rates. Hint: Use the View/Heat Flow command to find the heat rate across the boundary elements. (b) Use the View/Temperature Contours command to view the isotherms and heat flow patterns. Describe the heat flow paths, and comment on features of the gage design that influence the paths. What limitations on applicability of the gage have been revealed by your analysis? 4.92 Consider the system of Problem 4.54. The interior surface is exposed to hot gases at 350°C with a convection coefficient of 100 W/m2 䡠 K, while the exterior surface experiences convection with air at 25°C and a convection coefficient of 5 W/m2 䡠 K. (a) Using a grid spacing of 75 mm, calculate the temperature field within the system and determine the heat loss per unit length by convection from the outer surface of the flue to the air. Compare this result with the heat gained by convection from the hot gases to the air.

k = 300 W/m•K

Cooled surface, Ts = 25°C

(a) Beginning with the shaded symmetrical element, use a coarse (5 ⫻ 5) nodal network to estimate the thermal resistance per unit depth between the device and lower surface of the sink, R⬘t,d⫺s (m 䡠 K/W). How does this value compare with thermal resistances based on the assumption of one-dimensional conduction in rectangular domains of (i) width wd and length L and (ii) width ws and length L? (b) Using nodal networks with grid spacings three and five times smaller than that in part (a), determine the effect of grid size on the precision of the thermal resistance calculation. (c) Using the finer nodal network developed for part (b), determine the effect of device width on the thermal resistance. Specifically, keeping ws and L fixed, find the thermal resistance for values of wd /ws ⫽ 0.175, 0.275, 0.375, and 0.475. 4.94 Consider one-dimensional conduction in a plane

composite wall. The exposed surfaces of materials A and B are maintained at T1 ⫽ 600 K and T2 ⫽ 300 K, respectively. Material A, of thickness La ⫽ 20 mm, has a temperature-dependent thermal conductivity of ka ⫽ ko [1 ⫹ ␣(T ⫺ To)], where ko ⫽ 4.4 W/m 䡠 K, ␣ ⫽ 0.008 K⫺1, To ⫽ 300 K, and T is in kelvins. Material B is of thickness Lb ⫽ 5 mm and has a thermal conductivity of kb ⫽ 1 W/m 䡠 K.

䊏

277

Problems

that of IHT, or the finite-element method of FEHT to obtain the following results.

kb

ka = ka(T)

T1 = 600 K

T2 = 300 K

Fluid

T∞,o = 25°C ho = 250 W/m2•K Temperature uniformity of 5°C required

A

B L

x

La

Heating channel T∞,i = 200°C hi = 500 W/m2•K

L

La + Lb L

(a) Calculate the heat flux through the composite wall by assuming material A to have a uniform thermal conductivity evaluated at the average temperature of the section. (b) Using a space increment of 1 mm, obtain the finitedifference equations for the internal nodes and calculate the heat flux considering the temperaturedependent thermal conductivity for Material A. If the IHT software is employed, call-up functions from Tools/Finite-Difference Equations may be used to obtain the nodal equations. Compare your result with that obtained in part (a). (c) As an alternative to the finite-difference method of part (b), use the finite-element method of FEHT to calculate the heat flux, and compare the result with that from part (a). Hint: In the Specify/Material Properties box, properties may be entered as a function of temperature (T), the space coordinates (x, y), or time (t). See the Help section for more details. 4.95 A platen of thermal conductivity k ⫽ 15 W/m 䡠 K is heated by flow of a hot fluid through channels of width L ⫽ 20 mm, with T앝,i ⫽ 200⬚C and hi ⫽ 500 W/m2 䡠 K. The upper surface of the platen is used to heat a process fluid at T앝,o ⫽ 25⬚C with a convection coefficient of ho ⫽ 250 W/m2 䡠 K. The lower surface of the platen is insulated. To heat the process fluid uniformly, the temperature of the platen’s upper surface must be uniform to within 5⬚C. Use a finite-difference method, such as

Platen, k = 15 W/m•K

L /2 Insulation

W

(a) Determine the maximum allowable spacing W between the channel centerlines that will satisfy the specified temperature uniformity requirement. (b) What is the corresponding heat rate per unit length from a flow channel? 4.96 Consider the cooling arrangement for the very large-scale integration (VLSI) chip of Problem 4.93. Use the finiteelement method of FEHT to obtain the following results. (a) Determine the temperature distribution in the chipsubstrate system. Will the maximum temperature exceed 85°C? (b) Using the FEHT model developed for part (a), determine the volumetric heating rate that yields a maximum temperature of 85°C. (c) What effect would reducing the substrate thickness have on the maximum operating temperature? For a . volumetric generation rate of q ⫽ 107 W/m3, reduce the thickness of the substrate from 12 to 6 mm, keeping all other dimensions unchanged. What is the maximum system temperature for these conditions? What fraction of the chip power generation is removed by convection directly from the chip surface?

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4S.1 The Graphical Method The graphical method may be employed for two-dimensional problems involving adiabatic and isothermal boundaries. Although the approach has been superseded by computer solutions based on numerical procedures, it may be used to obtain a first estimate of the temperature distribution and to develop a physical appreciation for the nature of the temperature field and heat flow.

4S.1.1 Methodology of Constructing a Flux Plot The rationale for the graphical method comes from the fact that lines of constant temperature must be perpendicular to lines that indicate the direction of heat flow (see Figure 4.1). The objective of the graphical method is to systematically construct such a network of isotherms and heat flow lines. This network, commonly termed a flux plot, is used to infer the temperature distribution and the rate of heat flow through the system. Consider a square, two-dimensional channel whose inner and outer surfaces are maintained at T1 and T2, respectively. A cross section of the channel is shown in Figure 4S.1a. A procedure for constructing the flux plot, a portion of which is shown in Figure 4S.1b, is as follows. 1. The first step is to identify all relevant lines of symmetry. Such lines are determined by thermal, as well as geometrical, conditions. For the square channel of Figure 4S.1a, such lines include the designated vertical, horizontal, and diagonal lines. For this system it is therefore possible to consider only one-eighth of the configuration, as shown in Figure 4S.1b. 2. Lines of symmetry are adiabatic in the sense that there can be no heat transfer in a direction perpendicular to the lines. They are therefore heat flow lines and should be treated as such. Since there is no heat flow in a direction perpendicular to a heat flow line, it can be termed an adiabat. 3. After all known lines of constant temperature are identified, an attempt should be made to sketch lines of constant temperature within the system. Note that isotherms should always be perpendicular to adiabats. 4. Heat flow lines should then be drawn with an eye toward creating a network of curvilinear squares. This is done by having the heat flow lines and isotherms intersect at right angles and by requiring that all sides of each square be of approximately the same length. It is often impossible to satisfy this second requirement exactly, and it is more realistic to strive for equivalence between the sums of the opposite sides of each square, as shown in Figure 4S.1c. Assigning the x-coordinate to the direction of heat

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4S.1

䊏

The Graphical Method

y

b a

T1 T2 Adiabats

∆x

x

T2 qi c

Symmetry lines

∆y ∆Tj

d

(c)

(a)

T1 qi qi

∆Tj Isotherms (b)

FIGURE 4S.1 Two-dimensional conduction in a square channel of length l. (a) Symmetry planes. (b) Flux plot. (c) Typical curvilinear square.

flow and the y-coordinate to the direction normal to this flow, the requirement may be expressed as x ⬅ ab cd 艐 y ⬅ ac bd 2 2

(4S.1)

It is difficult to create a satisfactory network of curvilinear squares in the first attempt, and several iterations must often be made. This trial-and-error process involves adjusting the isotherms and adiabats until satisfactory curvilinear squares are obtained for most of the network.1 Once the flux plot has been obtained, it may be used to infer the temperature distribution in the medium. From a simple analysis, the heat transfer rate may then be obtained.

4S.1.2

Determination of the Heat Transfer Rate

The rate at which energy is conducted through a lane, which is the region between adjoining adiabats, is designated as qi. If the flux plot is properly constructed, the value of qi will be approximately the same for all lanes and the total heat transfer rate may be expressed as q艐

M

兺q

i

Mqi

(4S.2)

i1

where M is the number of lanes associated with the plot. From the curvilinear square of Figure 4S.1c and the application of Fourier’s law, qi may be expressed as qi 艐 kAi

1

Tj x

艐 k(y 䡠 l)

Tj x

(4S.3)

In certain regions, such as corners, it may be impossible to approach the curvilinear square requirements. However, such difficulties generally have a small effect on the overall accuracy of the results obtained from the flux plot.

4S.1

䊏

W-3

The Graphical Method

where Tj is the temperature difference between successive isotherms, Ai is the conduction heat transfer area for the lane, and l is the length of the channel normal to the page. However, since the temperature increment is approximately the same for all adjoining isotherms, the overall temperature difference between boundaries, T12, may be expressed as T12

N

兺 T 艐 N T j

j

(4S.4)

j1

where N is the total number of temperature increments. Combining Equations 4S.2 through 4S.4 and recognizing that x ⬇ y for curvilinear squares, we obtain q 艐 Ml k T12 N

(4S.5)

The manner in which a flux plot may be used to obtain the heat transfer rate for a twodimensional system is evident from Equation 4S.5. The ratio of the number of heat flow lanes to the number of temperature increments (the value of M/N) may be obtained from the plot. Recall that specification of N is based on step 3 of the foregoing procedure, and the value, which is an integer, may be made large or small depending on the desired accuracy. The value of M is then a consequence of following step 4. Note that M is not necessarily an integer, since a fractional lane may be needed to arrive at a satisfactory network of curvilinear squares. For the network of Figure 4S.1b, N 6 and M 5. Of course, as the network, or mesh, of curvilinear squares is made finer, N and M increase and the estimate of M/N becomes more accurate.

4S.1.3

The Conduction Shape Factor

Equation 4S.5 may be used to define the shape factor, S, of a two-dimensional system. That is, the heat transfer rate may be expressed as q SkT12

(4S.6)

S ⬅ Ml N

(4S.7)

where, for a flux plot,

From Equation 4S.6, it also follows that a two-dimensional conduction resistance may be expressed as Rt,cond(2D) 1 Sk

(4S.8)

Shape factors have been obtained for numerous two-dimensional systems, and results are summarized in Table 4.1 for some common configurations. In cases 1 through 9 and case 11, conduction is presumed to occur between boundaries that are maintained at uniform temperatures, with T12 ⬅ T1 T2. In case 10 conduction is between an isothermal surface (T1) and a semi-infinite medium of uniform temperature (T2) at locations well removed from the surface. Shape factors may also be defined for one-dimensional geometries, and from the

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4S.1

䊏

The Graphical Method

results of Table 3.3, it follows that for plane, cylindrical, and spherical walls, respectively, the shape factors are A/L, 2L/ln (r2/r1), and 4r1r2/(r2 r1). Results are available for many other configurations [1–4].

EXAMPLE 4S.1 A hole of diameter D 0.25 m is drilled through the center of a solid block of square cross section with w 1 m on a side. The hole is drilled along the length, l 2 m, of the block, which has a thermal conductivity of k 150 W/m 䡠 K. A warm fluid passing through the hole maintains an inner surface temperature of T1 75°C, while the outer surface of the block is kept at T2 25°C. 1. Using the flux plot method, determine the shape factor for the system. 2. What is the rate of heat transfer through the block?

SOLUTION Known: Dimensions and thermal conductivity of a block with a circular hole drilled along its length. Find: 1. Shape factor. 2. Heat transfer rate for prescribed surface temperatures. Schematic: k = 150 W/m•K

T2 = 25°C T1 = 75°C D1 = 0.25 m

w =1m

Symmetrical section

w =1m

Assumptions: 1. Steady-state conditions. 2. Two-dimensional conduction. 3. Constant properties. 4. Ends of block are well insulated. Analysis: 1. The flux plot may be simplified by identifying lines of symmetry and reducing the system to the one-eighth section shown in the schematic. Using a fairly coarse grid involving N 6 temperature increments, the flux plot was generated. The resulting network of curvilinear squares is as follows.

4S.2

䊏

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The Gauss-Seidel Method: Example of Usage

T1

Line of symmetry and adiabat N=1 2 3 4 5

6

M=1 T2 2

3 Line of symmetry and adiabat

With the number of heat flow lanes for the section corresponding to M 3, it follows from Equation 4S.7 that the shape factor for the entire block is S 8 Ml 8 3 2 m 8 m N 6

䉰

where the factor of 8 results from the number of symmetrical sections. The accuracy of this result may be determined by referring to Table 4.1, where, for the prescribed system, case 6, it follows that S

2L 2 2 m 8.59 m ln (1.08 w/D) ln (1.08 1 m/0.25 m)

Hence the result of the flux plot underpredicts the shape factor by approximately 7%. Note that, although the requirement l w is not satisfied for this problem, the shape factor result from Table 4.1 remains valid if there is negligible axial conduction in the block. This condition is satisfied if the ends are insulated. 2. Using S 8.59 m with Equation 4S.6, the heat rate is q Sk (T1 T2) q 8.59 m 150 W/m 䡠 K (75 25)C 64.4 kW

䉰

Comments: The accuracy of the flux plot may be improved by using a finer grid (increasing the value of N). How would the symmetry and heat flow lines change if the vertical sides were insulated? If one vertical and one horizontal side were insulated? If both vertical and one horizontal side were insulated?

4S.2 The Gauss-Seidel Method: Example of Usage The Gauss-Seidel method, described in Appendix D, is utilized in the following example.

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4S.1

䊏

The Graphical Method

EXAMPLE 4S.2 A large industrial furnace is supported on a long column of fireclay brick, which is 1 m 1 m on a side. During steady-state operation, installation is such that three surfaces of the column are maintained at 500 K while the remaining surface is exposed to an airstream for which T앝 300 K and h 10 W/m2 䡠 K. Using a grid of x y 0.25 m, determine the two-dimensional temperature distribution in the column and the heat rate to the airstream per unit length of column.

SOLUTION Known: Dimensions and surface conditions of a support column. Find: Temperature distribution and heat rate per unit length. Schematic: ∆x = 0.25 m

Ts = 500 K

∆y = 0.25 m 1

2

1

3

4

3

5

6

5

7

8

7

Fireclay brick

Ts = 500 K

Ts = 500 K

Air

T∞ = 300 K h = 10 W/m2•K

Assumptions: 1. Steady-state conditions. 2. Two-dimensional conduction. 3. Constant properties. 4. No internal heat generation. Properties: Table A.3, fireclay brick (T ⬇ 478 K): k 1 W/m 䡠 K. Analysis: The prescribed grid consists of 12 nodal points at which the temperature is unknown. However, the number of unknowns is reduced to 8 through symmetry, in which case the temperature of nodal points to the left of the symmetry line must equal the temperature of those to the right.

4S.2

䊏

The Gauss-Seidel Method: Example of Usage

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Nodes 1, 3, and 5 are interior points for which the finite-difference equations may be inferred from Equation 4.29. Hence Node 1:

T2 T3 1000 4T1 0

Node 3:

T1 T4 T5 500 4T3 0

Node 5:

T3 T6 T7 500 4T5 0

Equations for points 2, 4, and 6 may be obtained in a like manner or, since they lie on a symmetry adiabat, by using Equation 4.42 with h 0. Hence Node 2:

2T1 T4 500 4T2 0

Node 4:

T2 2T3 T6 4T4 0

Node 6:

T4 2T5 T8 4T6 0

From Equation 4.42 and the fact that h x/k 2.5, it also follows that Node 7:

2T5 T8 2000 9T7 0

Node 8:

2T6 2T7 1500 9T8 0

Having the required finite-difference equations, the temperature distribution will be determined by using the Gauss–Seidel iteration method. Referring to the arrangement of finite-difference equations, it is evident that the order is already characterized by diagonal dominance. This behavior is typical of finite-difference solutions to conduction problems. We therefore begin with step 2 and express the equations in explicit form T 1(k) 0.25T 2(k1) 0.25T (k1) 250 3 T 2(k) 0.50T 1(k) 0.25T 4(k1) 125 (k) (k1) T (k) 0.25T 5(k1) 125 3 0.25T 1 0.25T 4 (k1) T 4(k) 0.25T 2(k) 0.50T (k) 3 0.25T 6

T 5(k) 0.25T 3(k) 0.25T (k1) 0.25T 7(k1) 125 6 (k1) T 6(k) 0.25T 4(k) 0.50T (k) 5 0.25T 8

T 7(k) 0.2222T 5(k) 0.1111T (k1) 222.22 8 T 8(k) 0.2222T 6(k) 0.2222T (k) 7 166.67 Having the finite-difference equations in the required form, the iteration procedure may be implemented by using a table that has one column for the iteration (step) number and a column for each of the nodes labeled as Ti. The calculations proceed as follows: 1. For each node, the initial temperature estimate is entered on the row for k 0. Values are selected rationally to reduce the number of required iterations. 2. Using the N finite-difference equations and values of Ti from the first and second rows, the new values of Ti are calculated for the first iteration (k 1). These new values are entered on the second row. 3. This procedure is repeated to calculate T i(k) from the previous values of T i(k1) and the current values of T i(k), until the temperature difference between iterations meets the prescribed criterion, 0.2 K, at every nodal point.

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4S.1

䊏

The Graphical Method

k

T1

T2

T3

T4

T5

T6

T7

T8

0 1 2 3 4 5 6 7 8

480 477.5 480.8 484.6 487.0 488.1 488.7 489.0 489.1

470 471.3 475.7 480.6 482.9 484.0 484.5 484.8 485.0

440 451.9 462.5 467.6 469.7 470.8 471.4 471.7 471.9

430 441.3 453.1 457.4 459.6 460.7 461.3 461.6 461.8

400 428.0 432.6 434.3 435.5 436.1 436.5 436.7 436.8

390 411.8 413.9 415.9 417.2 417.9 418.3 418.5 418.6

370 356.2 355.8 356.2 356.6 356.7 356.9 356.9 356.9

350 337.3 337.7 338.3 338.6 338.8 338.9 339.0 339.0

The results given in row 8 are in excellent agreement with those that would be obtained by an exact solution of the matrix equation, although better agreement could be obtained by reducing the value of . However, given the approximate nature of the finite-difference equations, the results still represent approximations to the actual temperatures. The accuracy of the approximation may be improved by using a finer grid (increasing the number of nodes). The heat rate from the column to the airstream may be computed from the expression

冢Lq冣 2h 冤冢x2 冣 (T T ) x (T T ) 冢x2 冣 (T T )冥 앝

s

7

앝

8

앝

where the factor of 2 outside the brackets originates from the symmetry condition. Hence

冢Lq冣 2 10 W/m 䡠 K[0.125 m (200 K) 2

0.25 m (56.9 K) 0.125 m (39.0 K)] 882 W/m

䉰

Comments: 1. To ensure that no errors have been made in formulating the finite-difference equations or in effecting their solution, a check should be made to verify that the results satisfy conservation of energy for the nodal network. For steady-state conditions, the requirement dictates that the rate of energy inflow be balanced by the rate of outflow for a control surface surrounding all nodal regions whose temperatures have been evaluated. Ts q1(1) q1(2)

Ts

q3

q2

1

2

3

4

5

6

7

8

q5

q7(1)

q7(2) T∞, h

q8

䊏

W-9

References

For the one-half symmetrical section shown schematically above, it follows that conduction into the nodal regions must be balanced by convection from the regions. Hence (2) (1) (2) q(1) 1 q1 q2 q3 q5 q7 q7 q8

The cumulative conduction rate is then

冤

qcond (T T1) (T T1) x (Ts T2) k x s y s L y x 2 y y

(Ts T3) (T T5) y (Ts T7) y s x x 2 x

冥

192.1 W/m and the convection rate is

冤

冥

qconv h x(T7 T앝) x (T8 T앝) 191.0 W/m L 2 Agreement between the conduction and convection rates is excellent, confirming that mistakes have not been made in formulating and solving the finite-difference equations. Note that convection transfer from the entire bottom surface (882 W/m) is obtained by adding transfer from the edge node at 500 K (250 W/m) to that from the interior nodes (191.0 W/m) and multiplying by 2 from symmetry. 2. Although the computed temperatures satisfy the finite-difference equations, they do not provide us with the exact temperature field. Remember that the equations are approximations whose accuracy may be improved by reducing the grid size (increasing the number of nodal points). 3. See Example 4S.2 in the Advanced section of IHT. 4. A second software package accompanying this text, Finite-Element Heat Transfer (FEHT), may also be used to solve one- and two-dimensional forms of the heat equation. This example is provided as a solved model in FEHT and may be accessed through Examples on the Toolbar.

References 1. Sunderland, J. E., and K. R. Johnson, Trans. ASHRAE, 10, 237–241, 1964. 2. Kutateladze, S. S., Fundamentals of Heat Transfer, Academic Press, New York, 1963.

3. General Electric Co. (Corporate Research and Development), Heat Transfer Data Book, Section 502, General Electric Company, Schenectady, NY, 1973. 4. Hahne, E., and U. Grigull, Int. J. Heat Mass Transfer, 18, 751–767, 1975.

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4S.1

The Graphical Method

䊏

Problems (b) Using the flux plot method, estimate the shape factor and the heat transfer rate through the strut per unit length.

Flux Plotting 4S.1 A long furnace, constructed from refractory brick with a thermal conductivity of 1.2 W/m • K, has the cross section shown with inner and outer surface temperatures of 600 and 60°C, respectively. Determine the shape factor and the heat transfer rate per unit length using the flux plot method.

1m2m

(c) Sketch the 25, 50, and 75°C isotherms. (d) Consider the same geometry, but now with the 0.1-m-wide surfaces insulated, the 45° surface maintained at T1 100°C, and the 0.2-m-wide surfaces maintained at T2 0°C. Using the flux plot method, estimate the corresponding shape factor and the heat rate per unit length. Sketch the 25, 50, and 75°C isotherms. 4S.4 A hot liquid flows along a V-groove in a solid whose top and side surfaces are well insulated and whose bottom surface is in contact with a coolant.

1.5 m

W/4

2.5 m

T1 W/4

4S.2 A hot pipe is embedded eccentrically as shown in a material of thermal conductivity 0.5 W/m 䡠 K. Using the flux plot method, determine the shape factor and the heat transfer per unit length when the pipe and outer surface temperatures are 150 and 35°C, respectively.

20 mm 40 mm

4S.3 A supporting strut fabricated from a material with a thermal conductivity of 75 W/m 䡠 K has the cross section shown. The end faces are at different temperatures T1 100°C and T2 0°C, while the remaining sides are insulated.

W/2 T2 W

Accordingly, the V-groove surface is at a temperature T1, which exceeds that of the bottom surface, T2. Construct an appropriate flux plot and determine the shape factor of the system. 4S.5 A very long conduit of inner circular cross section and a thermal conductivity of 1 W/m 䡠 K passes a hot fluid, which maintains the inner surface at T1 50°C. The outer surfaces of square cross section are insulated or maintained at a uniform temperature of T2 20°C, depending on the application. Find the shape factor and the heat rate for each case. T2

T2 T2

T2

0.2 m

T1

40 mm

T1

0.1 m 0.2 m

P T2 45°

T1 0.1 m

(a) Estimate the temperature at the location P.

120 mm

4S.6 A long support column of trapezoidal cross section is well insulated on its sides, and temperatures of 100 and 0°C are maintained at its top and bottom surfaces, respectively. The column is fabricated from AISI 1010 steel,

䊏

W-11

Problems

and its widths at the top and bottom surfaces are 0.3 and 0.6 m, respectively. 0.3 m

0.3 m

4S.8 The two-dimensional, square shapes, 1 m to a side, are maintained at uniform temperatures, T1 100°C and T2 0°C, on portions of their boundaries and are well insulated elsewhere.

H

0.3 m

T1

T1

0.6 m

(a) Using the flux plot method, determine the heat transfer rate per unit length of the column. (b) If the trapezoidal column is replaced by a bar of rectangular cross section 0.3 m wide and the same material, what height H must the bar be to provide an equivalent thermal resistance? 4S.7 Hollow prismatic bars fabricated from plain carbon steel are 1 m long with top and bottom surfaces, as well as both ends, well insulated. For each bar, find the shape factor and the heat rate per unit length of the bar when T1 500 K and T2 300 K. 100 mm

100 mm

T2

T2

T2

35 mm 100 mm

T1

100 mm 35 mm

T2

35 mm

T1

35 mm

T2

T1

T2

T2 (a)

(b)

Use the flux plot method to estimate the heat rate per unit length normal to the page if the thermal conductivity is 50 W/m 䡠 K.

C H A P T E R

Transient Conduction

5

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280

Chapter 5

䊏

Transient Conduction

I

n our treatment of conduction we have gradually considered more complicated conditions. We began with the simple case of one-dimensional, steady-state conduction with no internal generation, and we subsequently considered more realistic situations involving multidimensional and generation effects. However, we have not yet considered situations for which conditions change with time. We now recognize that many heat transfer problems are time dependent. Such unsteady, or transient, problems typically arise when the boundary conditions of a system are changed. For example, if the surface temperature of a system is altered, the temperature at each point in the system will also begin to change. The changes will continue to occur until a steadystate temperature distribution is reached. Consider a hot metal billet that is removed from a furnace and exposed to a cool airstream. Energy is transferred by convection and radiation from its surface to the surroundings. Energy transfer by conduction also occurs from the interior of the metal to the surface, and the temperature at each point in the billet decreases until a steady-state condition is reached. The final properties of the metal will depend significantly on the time-temperature history that results from heat transfer. Controlling the heat transfer is one key to fabricating new materials with enhanced properties. Our objective in this chapter is to develop procedures for determining the time dependence of the temperature distribution within a solid during a transient process, as well as for determining heat transfer between the solid and its surroundings. The nature of the procedure depends on assumptions that may be made for the process. If, for example, temperature gradients within the solid may be neglected, a comparatively simple approach, termed the lumped capacitance method, may be used to determine the variation of temperature with time. The method is developed in Sections 5.1 through 5.3. Under conditions for which temperature gradients are not negligible, but heat transfer within the solid is one-dimensional, exact solutions to the heat equation may be used to compute the dependence of temperature on both location and time. Such solutions are considered for finite solids (plane walls, long cylinders and spheres) in Sections 5.4 through 5.6 and for semi-infinite solids in Section 5.7. Section 5.8 presents the transient thermal response of a variety of objects subject to a step change in either surface temperature or surface heat flux. In Section 5.9, the response of a semi-infinite solid to periodic heating conditions at its surface is explored. For more complex conditions, finite-difference or finite-element methods must be used to predict the time dependence of temperatures within the solid, as well as heat rates at its boundaries (Section 5.10).

5.1 The Lumped Capacitance Method A simple, yet common, transient conduction problem is one for which a solid experiences a sudden change in its thermal environment. Consider a hot metal forging that is initially at a uniform temperature Ti and is quenched by immersing it in a liquid of lower temperature T앝 Ti (Figure 5.1). If the quenching is said to begin at time t 0, the temperature of the solid will decrease for time t 0, until it eventually reaches T앝. This reduction is due to convection heat transfer at the solid–liquid interface. The essence of the lumped capacitance method is the assumption that the temperature of the solid is spatially uniform at any instant during the transient process. This assumption implies that temperature gradients within the solid are negligible.

5.1

䊏

281

The Lumped Capacitance Method

Ti

t 1

T∞, h x

L

FIGURE 5.3 Effect of Biot number on steady-state temperature distribution in a plane wall with surface convection.

284

Chapter 5

䊏

Transient Conduction

The quantity (hL/k) appearing in Equation 5.9 is a dimensionless parameter. It is termed the Biot number, and it plays a fundamental role in conduction problems that involve surface convection effects. According to Equation 5.9 and as illustrated in Figure 5.3, the Biot number provides a measure of the temperature drop in the solid relative to the temperature difference between the solid’s surface and the fluid. From Equation 5.9, it is also evident that the Biot number may be interpreted as a ratio of thermal resistances. In particular, if Bi 1, the resistance to conduction within the solid is much less than the resistance to convection across the fluid boundary layer. Hence, the assumption of a uniform temperature distribution within the solid is reasonable if the Biot number is small. Although we have discussed the Biot number in the context of steady-state conditions, we are reconsidering this parameter because of its significance to transient conduction problems. Consider the plane wall of Figure 5.4, which is initially at a uniform temperature Ti and experiences convection cooling when it is immersed in a fluid of T앝 Ti. The problem may be treated as one-dimensional in x, and we are interested in the temperature variation with position and time, T(x, t). This variation is a strong function of the Biot number, and three conditions are shown in Figure 5.4. Again, for Bi 1 the temperature gradients in the solid are small and the assumption of a uniform temperature distribution, T(x, t) ⬇ T(t) is reasonable. Virtually all the temperature difference is between the solid and the fluid, and the solid temperature remains nearly uniform as it decreases to T앝. For moderate to large values of the Biot number, however, the temperature gradients within the solid are significant. Hence T T(x, t). Note that for Bi 1, the temperature difference across the solid is much larger than that between the surface and the fluid. We conclude this section by emphasizing the importance of the lumped capacitance method. Its inherent simplicity renders it the preferred method for solving transient heating and cooling problems. Hence, when confronted with such a problem, the very first thing that one should do is calculate the Biot number. If the following condition is satisfied Bi

hLc 0.1 k

(5.10)

the error associated with using the lumped capacitance method is small. For convenience, it is customary to define the characteristic length of Equation 5.10 as the ratio of the solid’s

T(x, 0) = Ti

T(x, 0) = Ti

T∞, h t

T∞, h

T∞ –L

L x

T∞ –L

Bi > 1 T = T(x, t)

FIGURE 5.4 Transient temperature distributions for different Biot numbers in a plane wall symmetrically cooled by convection.

5.2

䊏

Validity of the Lumped Capacitance Method

285

volume to surface area Lc ⬅ V/As. Such a definition facilitates calculation of Lc for solids of complicated shape and reduces to the half-thickness L for a plane wall of thickness 2L (Figure 5.4), to ro /2 for a long cylinder, and to ro /3 for a sphere. However, if one wishes to implement the criterion in a conservative fashion, Lc should be associated with the length scale corresponding to the maximum spatial temperature difference. Accordingly, for a symmetrically heated (or cooled) plane wall of thickness 2L, Lc would remain equal to the half-thickness L. However, for a long cylinder or sphere, Lc would equal the actual radius ro, rather than ro /2 or ro /3. Finally, we note that, with Lc ⬅ V/As, the exponent of Equation 5.6 may be expressed as hL k t hL hAs t c ␣t2 ht c c 2 Vc cLc k k Lc Lc or hAs t Bi 䡠 Fo Vc

(5.11)

where Fo ⬅ ␣t2 Lc

(5.12)

is termed the Fourier number. It is a dimensionless time, which, with the Biot number, characterizes transient conduction problems. Substituting Equation 5.11 into 5.6, we obtain T T앝 exp(Bi 䡠 Fo) i Ti T앝

(5.13)

EXAMPLE 5.1 A thermocouple junction, which may be approximated as a sphere, is to be used for temperature measurement in a gas stream. The convection coefficient between the junction surface and the gas is h 400 W/m2 䡠 K, and the junction thermophysical properties are k 20 W/m 䡠 K, c 400 J/kg 䡠 K, and 8500 kg/m3. Determine the junction diameter needed for the thermocouple to have a time constant of 1 s. If the junction is at 25 C and is placed in a gas stream that is at 200 C, how long will it take for the junction to reach 199 C?

SOLUTION Known: Thermophysical properties of thermocouple junction used to measure temperature of a gas stream. Find: 1. Junction diameter needed for a time constant of 1 s. 2. Time required to reach 199 C in gas stream at 200 C.

286

Chapter 5

䊏

Transient Conduction

Schematic: Leads

T∞ = 200°C h = 400 W/m2•K

Gas stream

Thermocouple junction Ti = 25°C

k = 20 W/m•K c = 400 J/kg•K ρ = 8500 kg/m3

D

Assumptions: 1. Temperature of junction is uniform at any instant. 2. Radiation exchange with the surroundings is negligible. 3. Losses by conduction through the leads are negligible. 4. Constant properties. Analysis: 1. Because the junction diameter is unknown, it is not possible to begin the solution by determining whether the criterion for using the lumped capacitance method, Equation 5.10, is satisfied. However, a reasonable approach is to use the method to find the diameter and to then determine whether the criterion is satisfied. From Equation 5.7 and the fact that As D2 and V D3/6 for a sphere, it follows that t

1 D c 6 hD2 3

Rearranging and substituting numerical values, 2 6h 䡠 K 1 s 7.06 104 m D c t 6 400 W/m 3 8500 kg/m 400 J/kg 䡠 K

䉰

With Lc ro /3 it then follows from Equation 5.10 that Bi

h(ro /3) 400 W/m2 䡠 K 3.53 104 m 2.35 103 k 3 20 W/m 䡠 K

Accordingly, Equation 5.10 is satisfied (for Lc ro, as well as for Lc ro /3) and the lumped capacitance method may be used to an excellent approximation. 2. From Equation 5.5 the time required for the junction to reach T 199 C is (D3/6)c Ti T앝 Dc Ti T앝 ln ln T T앝 6h T T앝 h(D2) 3 4 8500 kg/m 7.06 10 m 400 J/kg 䡠 K 25 200 t ln 199 200 6 400 W/m2 䡠 K t 5.2 s 艐 5t t

䉰

Comments: Heat transfer due to radiation exchange between the junction and the surroundings and conduction through the leads would affect the time response of the junction and would, in fact, yield an equilibrium temperature that differs from T앝.

5.3

䊏

287

General Lumped Capacitance Analysis

5.3 General Lumped Capacitance Analysis Although transient conduction in a solid is commonly initiated by convection heat transfer to or from an adjoining fluid, other processes may induce transient thermal conditions within the solid. For example, a solid may be separated from large surroundings by a gas or vacuum. If the temperatures of the solid and surroundings differ, radiation exchange could cause the internal thermal energy, and hence the temperature, of the solid to change. Temperature changes could also be induced by applying a heat flux at a portion, or all, of the surface or by initiating thermal energy generation within the solid. Surface heating could, for example, be applied by attaching a film or sheet electrical heater to the surface, while thermal energy could be generated by passing an electrical current through the solid. Figure 5.5 depicts the general situation for which thermal conditions within a solid may be influenced simultaneously by convection, radiation, an applied surface heat flux, and internal energy generation. It is presumed that, initially (t 0), the temperature of the solid Ti differs from that of the fluid T앝, and the surroundings Tsur , and that both surface and volu. metric heating (qs and q) are initiated. The imposed heat flux qs and the convection–radiation heat transfer occur at mutually exclusive portions of the surface, As(h) and As(c,r), respectively, and convection–radiation transfer is presumed to be from the surface. Moreover, although convection and radiation have been prescribed for the same surface, the surfaces may, in fact, differ (As,c As,r). Applying conservation of energy at any instant t, it follows from Equation 1.12c that qs As,h E˙ g (qconv qrad )As(c,r) Vc dT dt

(5.14)

or, from Equations 1.3a and 1.7, qs As,h E˙ g [h(T T앝) (T 4 T 4sur)]As(c,r) Vc dT dt

(5.15)

Equation 5.15 is a nonlinear, first-order, nonhomogeneous, ordinary differential equation that cannot be integrated to obtain an exact solution.1 However, exact solutions may be obtained for simplified versions of the equation.

Surroundings

Tsur ρ, c, V, T (0) = Ti

q"rad

q"s

•

•

Eg, Est

T∞, h q"conv

As, h

1

As(c, r)

FIGURE 5.5 Control surface for general lumped capacitance analysis.

An approximate, finite-difference solution may be obtained by discretizing the time derivative (Section 5.10) and marching the solution out in time.

288

Chapter 5

5.3.1

䊏

Transient Conduction

Radiation Only

If there is no imposed heat flux or generation and convection is either nonexistent (a vacuum) or negligible relative to radiation, Equation 5.15 reduces to 4 Vc dT As,r (T 4 T sur ) dt

(5.16)

Separating variables and integrating from the initial condition to any time t, it follows that As,r Vc

冕 dt 冕 T t

T

0

Ti

dT T4

(5.17)

4 sur

Evaluating both integrals and rearranging, the time required to reach the temperature T becomes t

冏

冦

冏 冏

T T T Ti Vc ln sur ln sur 3 T T Tsur Ti 4As,r Tsur sur

冤 冢TT 冣 tan 冢TT 冣冥冧

2 tan1

1

sur

i

冏 (5.18)

sur

This expression cannot be used to evaluate T explicitly in terms of t, Ti, and Tsur, nor does it readily reduce to the limiting result for Tsur 0 (radiation to deep space). However, returning to Equation 5.17, its solution for Tsur 0 yields t

5.3.2

冢

Vc 1 1 3As,r T 3 T 3i

冣

(5.19)

Negligible Radiation

An exact solution to Equation 5.15 may also be obtained if radiation may be neglected and all quantities (other than T, of course) are independent of time. Introducing a temperature difference ⬅ T T앝, where d/dt dT/dt, Equation 5.15 reduces to a linear, first-order, nonhomogeneous differential equation of the form d

a b 0 dt

(5.20)

where a ⬅ (hAs,c /Vc) and b ⬅ [(qs As,h E˙ g)/Vc]. Although Equation 5.20 may be solved by summing its homogeneous and particular solutions, an alternative approach is to eliminate the nonhomogeneity by introducing the transformation ⬅ ba

(5.21)

5.3

䊏

289

General Lumped Capacitance Analysis

Recognizing that d/dt d/dt, Equation 5.21 may be substituted into (5.20) to yield d

a 0 dt

(5.22)

Separating variables and integrating from 0 to t (i to ), it follows that exp(at) i

(5.23)

T T앝 (b/a) exp(at) Ti T앝 (b/a)

(5.24)

T T exp(at) b/a [1 exp(at)] Ti T Ti T

(5.25)

or substituting for and ,

Hence

As it must, Equation 5.25 reduces to Equation 5.6 when b 0 and yields T Ti at t 0. As t l 앝, Equation 5.25 reduces to (T T앝) (b/a), which could also be obtained by performing an energy balance on the control surface of Figure 5.5 for steady-state conditions.

5.3.3

Convection Only with Variable Convection Coefficient

In some cases, such as those involving free convection or boiling, the convection coefficient h varies with the temperature difference between the object and the fluid. In these situations, the convection coefficient can often be approximated with an expression of the form h C(T T앝)n

(5.26)

where n is a constant and the parameter C has units of W/m2 䡠 K(1 n). If radiation, surface heating, and volumetric generation are negligible, Equation 5.15 may be written as C(T T앝)nAs,c(T T앝) CAs,c(T T앝)1 n Vc dT dt

(5.27)

Substituting and d/dt dT/dt into the preceding expression, separating variables and integrating yields

冤

冥

nCAs,cni t 1 i Vc

1/n

(5.28)

It can be shown that Equation 5.28 reduces to Equation 5.6 if the heat transfer coefficient is independent of temperature, n 0.

5.3.4

Additional Considerations

In some cases the ambient or surroundings temperature may vary with time. For example, if the container of Figure 5.1 is insulated and of finite volume, the liquid temperature will

290

Chapter 5

䊏

Transient Conduction

increase as the metal forging is cooled. An analytical solution for the time-varying solid (and liquid) temperature is presented in Example 11.8. As evident in Examples 5.2 through 5.4, the heat equation can be solved numerically for a wide variety of situations involving variable properties or time-varying boundary conditions, internal energy generation rates, or surface heating or cooling.

EXAMPLE 5.2 Consider the thermocouple and convection conditions of Example 5.1, but now allow for radiation exchange with the walls of a duct that encloses the gas stream. If the duct walls are at 400 C and the emissivity of the thermocouple bead is 0.9, calculate the steady-state temperature of the junction. Also, determine the time for the junction temperature to increase from an initial condition of 25 C to a temperature that is within 1 C of its steady-state value.

SOLUTION Known: Thermophysical properties and diameter of the thermocouple junction used to measure temperature of a gas stream passing through a duct with hot walls. Find: 1. Steady-state temperature of the junction. 2. Time required for the thermocouple to reach a temperature that is within 1 C of its steady-state value. Schematic: Hot duct wall, Tsur = 400°C Gas stream

Junction, T(t) Ti = 25°C, D = 0.7 mm ρ = 8500 kg/m3 c = 400 J/kg•K ε = 0.9

T∞ = 200°C

h = 400 W/m2•K

Assumptions: Same as Example 5.1, but radiation transfer is no longer treated as negligible and is approximated as exchange between a small surface and large surroundings. Analysis: 1. For steady-state conditions, the energy balance on the thermocouple junction has the form E˙ in E˙ out 0 Recognizing that net radiation to the junction must be balanced by convection from the junction to the gas, the energy balance may be expressed as 4 T 4) h(T T앝)]As 0 [(T sur

5.3

䊏

291

General Lumped Capacitance Analysis

Substituting numerical values, we obtain T 218.7 C

䉰

2. The temperature-time history, T(t), for the junction, initially at T(0) Ti 25 C, follows from the energy balance for transient conditions, E˙ in E˙ out E˙ st From Equation 5.15, the energy balance may be expressed as 4 )]As Vc dT [h(T T앝) (T 4 T sur dt

The solution to this first-order differential equation can be obtained by numerical integration, giving the result, T(4.9 s) 217.7 C. Hence, the time required to reach a temperature that is within 1 C of the steady-state value is t 4.9 s.

䉰

Comments: 1. The effect of radiation exchange with the hot duct walls is to increase the junction temperature, such that the thermocouple indicates an erroneous gas stream temperature that exceeds the actual temperature by 18.7 C. The time required to reach a temperature that is within 1 C of the steady-state value is slightly less than the result of Example 5.l, which only considers convection heat transfer. Why is this so? 2. The response of the thermocouple and the indicated gas stream temperature depend on the velocity of the gas stream, which in turn affects the magnitude of the convection coefficient. Temperature–time histories for the thermocouple junction are shown in the following graph for values of h 200, 400, and 800 W/m2 䡠 K. Junction temperature, T (°C)

260 220 180

800 400 200 h (W/m2•K)

140 100 60 20

0

2

6 4 Elapsed time, t (s)

8

10

The effect of increasing the convection coefficient is to cause the junction to indicate a temperature closer to that of the gas stream. Further, the effect is to reduce the time required for the junction to reach the near-steady-state condition. What physical explanation can you give for these results? 3. The IHT software includes an integral function, Der(T, t), that can be used to represent the temperature–time derivative and to integrate first-order differential equations.

292

Chapter 5

䊏

Transient Conduction

EXAMPLE 5.3 A 3-mm-thick panel of aluminum alloy (k 177 W/m 䡠 K, c 875 J/kg 䡠 K, and 2770 kg/m3) is finished on both sides with an epoxy coating that must be cured at or above Tc 150 C for at least 5 min. The production line for the curing operation involves two steps: (1) heating in a large oven with air at T앝,o 175 C and a convection coefficient of ho 40 W/m2 䡠 K, and (2) cooling in a large chamber with air at T앝,c 25 C and a convection coefficient of hc 10 W/m2 䡠 K. The heating portion of the process is conducted over a time interval te, which exceeds the time tc required to reach 150 C by 5 min (te tc 300 s). The coating has an emissivity of 0.8, and the temperatures of the oven and chamber walls are 175 and 25 C, respectively. If the panel is placed in the oven at an initial temperature of 25 C and removed from the chamber at a safe-to-touch temperature of 37 C, what is the total elapsed time for the two-step curing operation?

SOLUTION Known: Operating conditions for a two-step heating/cooling process in which a coated aluminum panel is maintained at or above a temperature of 150 C for at least 5 min. Find: Total time tt required for the two-step process. Schematic: Tsur,o = 175°C

Tsur,c = 25°C

2L = 3 mm

As

ho, T∞,o = 175°C

Epoxy, ε = 0.8

Aluminum, T(0) = Ti,o = 25°C Step 1: Heating (0 ≤ t ≤ tc)

hc, T∞,c = 25°C

T(tt) = 37°C Step 2: Cooling (tc< t ≤ tt)

Assumptions: 1. Panel temperature is uniform at any instant. 2. Thermal resistance of epoxy is negligible. 3. Constant properties. Analysis: To assess the validity of the lumped capacitance approximation, we begin by calculating Biot numbers for the heating and cooling processes. Bih

ho L (40 W/m2 䡠 K)(0.0015 m) 3.4 104 k 177 W/m 䡠 K

Bic

hc L (10 W/m2 䡠 K)(0.0015 m) 8.5 105 k 177 W/m 䡠 K

5.3

䊏

293

General Lumped Capacitance Analysis

Hence the lumped capacitance approximation is excellent. To determine whether radiation exchange between the panel and its surroundings should be considered, the radiation heat transfer coefficient is determined from Equation 1.9. A representative value of hr for the heating process is associated with the cure condition, in which case hr,o (Tc Tsur,o)(T 2c T 2sur,o) 0.8 5.67 108 W/m2 䡠 K4(423 448)K(4232 4482)K2 15 W/m2 䡠 K Using Tc 150 C with Tsur,c 25 C for the cooling process, we also obtain hr,c 8.8 W/m2 䡠 K. Since the values of hr,o and hr,c are comparable to those of ho and hc, respectively, radiation effects must be considered. With V 2LAs and As,c As,r 2As, Equation 5.15 may be expressed as 4 [h(T T앝 ) (T 4 T sur )] cL dT dt

Selecting a suitable time increment, t, the equation may be integrated numerically to obtain the panel temperature at t t, 2t, 3t, and so on. Selecting t 10 s, calculations for the heating process are extended to te tc 300 s, which is 5 min beyond the time required for the panel to reach Tc 150 C. At te the cooling process is initiated and continued until the panel temperature reaches 37 C at t tt. The integration was performed using IHT, and results of the calculations are plotted as follows: 200 175 150

∆t(T >150°C)

T (°C)

125 Cooling Heating

100 75 50 25

0

tc

300

te

600 t (s)

900 tt

1200

The total time for the two-step process is tt 989 s

䉰

with intermediate times of tc 124 s and te 424 s.

Comments: 1. The duration of the two-step process may be reduced by increasing the convection coefficients and/or by reducing the period of extended heating. The second option is made possible by the fact that, during a portion of the cooling period, the panel

294

Chapter 5

䊏

Transient Conduction

temperature remains above 150 C. Hence, to satisfy the cure requirement, it is not necessary to extend heating for as much as 5 min from t tc. If the convection coefficients are increased to ho hc 100 W/m2 䡠 K and an extended heating period of 300 s is maintained, the numerical integration yields tc 58 s and tt 445 s. The corresponding time interval over which the panel temperature exceeds 150 C is t(T 150 C) 306 s (58 s t 364 s). If the extended heating period is reduced to 294 s, the numerical integration yields tc 58 s, tt 439 s, and t(T 150 C) 300 s. Hence the total process time is reduced, while the curing requirement is still satisfied. 2. Generally, the accuracy of a numerical integration improves with decreasing t, but at the expense of increased computation time. In this case, however, results obtained for t 1 s are virtually identical to those obtained for t 10 s, indicating that the larger time interval is sufficient to accurately depict the temperature history. 3. The complete solution for this example is provided as a ready-to-solve model in the Advanced section of IHT, using Models, Lumped Capacitance. The model can be used to check the results of Comment 1 or to independently explore modifications of the cure process. 4. If the Biot numbers were not small, it would be inappropriate to apply the lumped capacitance method. For moderate or large Biot numbers, temperatures near the solid’s centerline would continue to increase for some time after the conclusion of heating, as thermal energy near the solid’s surface propagates inward. The temperatures near the centerline would subsequently reach a maximum and would then decrease to the steady-state value. Correlations for the maximum temperature experienced at the panel’s centerline, along with the time at which these maximum temperatures are reached, have been correlated for a broad range of Bih and Bic values [1].

EXAMPLE 5.4 Air to be supplied to a hospital operating room is first purified by forcing it through a singlestage compressor. As it travels through the compressor, the air temperature initially increases due to compression, then decreases as it is returned to atmospheric pressure. Harmful pathogen particles in the air will also be heated and subsequently cooled, and they will be destroyed if their maximum temperature exceeds a lethal temperature Td. Consider spherical pathogen particles (D 10 m, 900 kg/m3, c 1100 J/kg 䡠 K, and k 0.2 W/m 䡠 K) that are dispersed in unpurified air. During the process, the air temperature may be described by an expression of the form T앝(t) 125 C 100 C 䡠 cos(2t/tp), where tp is the process time associated with flow through the compressor. If tp 0.004 s, and the initial and lethal pathogen temperatures are Ti 25 C and Td 220 C, respectively, will the pathogens be destroyed? The value of the convection heat transfer coefficient associated with the pathogen particles is h 4600 W/m2 䡠 K.

SOLUTION Known: Air temperature versus time, convection heat transfer coefficient, pathogen geometry, size, and properties.

5.3

䊏

295

General Lumped Capacitance Analysis

Find: Whether the pathogens are destroyed for tp 0.004 s. Schematic: Airstream

T∞(t) 125°C 100°C •cos(2πt/t π p) h 4600 W/m2 •K

Pathogen k 0.2 W/m •K c 1100 J/kg •K ρ 900 kg/m3

D 10 µm

Td 220°C

Assumptions: 1. Constant properties. 2. Negligible radiation. Analysis: The Biot number associated with a spherical pathogen particle is Bi

h(D/6) 4600 W/m2 䡠 K (10 106 m/6) 0.038 k 0.2 W/m 䡠 K

Hence, the lumped capacitance approximation is valid and we may apply Equation 5.2. dT hAs [T T (t)] 6h [T 125 C 100 C 䡠 cos(2t/t )] 앝 p dt Vc cD

(1)

The solution to this first-order differential equation may be obtained analytically, or by numerical integration.

Pathogen and air temperature, T (°C)

Numerical Integration A numerical solution of Equation 1 may be obtained by specifying the initial particle temperature, Ti, and using IHT or an equivalent numerical solver to integrate the equation. The plot of the numerical solution follows. 250 200 150 Pathogen 100 Air 50 0

0

0.001

0.002 Elapsed time, t (s)

0.003

0.004

296

Chapter 5

䊏

Transient Conduction

Inspection of the predicted pathogen temperatures yields Tmax 212 C 220 C 䉰

Hence, the pathogen is not destroyed.

Analytical Solution Equation 1 is a linear nonhomogeneous differential equation, therefore the solution can be found as the sum of a homogeneous and a particular solution, T Th Tp. The homogeneous part, Th, corresponds to the homogeneous differential equation, dTh /dt (6h/cD)Th, which has the familiar solution, Th c0 exp(6ht/cD). The particular solution, Tp, can then be found using the method of undetermined coefficients; for a nonhomogeneous term that includes a cosine function and a constant term, the particular solution is assumed to be of the form Tp c1 cos(2t/tp) c2 sin(2t/tp) c3. Substituting this expression into Equation 1 yields values for the coefficients, resulting in

冤 冢 冣

冢 冣冥

2cD 2t Tp 125 C 100 C A cos 2t tp 6htp sin tp

(2)

where A

(6h/cD)2 (6h/cD)2 (2/tp)2

The initial condition, T(0) Ti, is then applied to the complete solution, T Th Tp, to yield c0 100 C(A 1). Thus, the particle temperature is

冢

冦

冣 冤 冢 冣

冢 冣冥冧 (3)

2cD

sin 2t T(t) 125 C 100 C (A 1) exp 6ht A cos 2t t tp cD 6htp p

To find the maximum pathogen temperature, we could differentiate Equation 3 and set the result equal to zero. This yields a lengthy, implicit equation for the critical time tcrit at which the maximum temperature is reached. The maximum temperature may then be found by substituting t tcrit into Equation 3. Alternatively, Equation 3 can be plotted or T(t) may be tabulated to find Tmax 212 C 220 C Hence, the pathogen is not destroyed.

䉰

Comments: 1. The analytical and numerical solutions agree, as they must. 2. As evident in the previous plot, the air and pathogen particles initially have the same temperature, Ti 25 C. The pathogen thermal response lags that of the air since a temperature difference must exist between the air and the particle in order for the pathogen to be heated or cooled. As required by Equation 1 and as evident in the plot, the maximum particle temperature is reached when there is no temperature difference between the air and the pathogen.

5.3

䊏

297

General Lumped Capacitance Analysis

Pathogen and air temperature, T (°C)

3. The maximum pathogen temperature may be increased by extending the duration of the process. For a process time of tp 0.008 s, the air and pathogen particle temperatures are as follows. 250 200 150 Pathogen 100 Air 50 0

0

0.002

0.004

0.006

0.008

Elapsed time, t (s)

The maximum particle temperature is now Tmax 221 C Td 220 C, and the pathogen would be killed. However, because the duration of the cycle is twice as long as originally specified, approximately half of the air could be supplied to the operating room compared to the tp 0.004 s case. A trade-off exists between the amount of air that can be delivered to the operating room and its purity. 4. The maximum possible radiation heat transfer coefficient may be calculated based on the extreme temperatures of the problem and by assuming a particle emissivity of unity. Hence, hr,max (Tmax Tmin )(T 2max T 2min ) 5.67 108 W/m2 䡠 K4 (498 298)K (4982 2982)K2 15.2 W/m2 䡠 K Since hr,max h, radiation heat transfer is negligible. 5. The Der(T, t) function of the IHT software was used to generate the numerical solution for this problem. See Comment 3 of Example 5.2. If one is familiar with a numerical solver such as IHT, it is often much faster to obtain a numerical solution than an analytical solution, as is the case in this example. Moreover, if one seeks maximum or minimum values of the dependent variable or variables, such as the pathogen temperature in this example, it is often faster to determine the maxima or minima by inspection, rather than with an analytical solution. However, analytical solutions often explicitly show parameter dependencies and can provide insights that numerical solutions might obscure. 6. A time increment of t 0.00001 s was used to generate the numerical solutions. Generally, the accuracy of a numerical integration improves with decreasing t, but at the expense of increased computation time. For this example, results for t 0.000005 s are virtually identical to those obtained for the larger time increment, indicating that either increment is sufficient to accurately depict the temperature history and to determine the maximum particle temperature. 7. Assumption of instantaneous pathogen death at the lethal temperature is an approximation. Pathogen destruction also depends on the duration of exposure to the high temperatures [2].

298

Chapter 5

䊏

Transient Conduction

5.4 Spatial Effects Situations frequently arise for which the Biot number is not small, and we must cope with the fact that temperature gradients within the medium are no longer negligible. Use of the lumped capacitance method would yield incorrect results, so alternative approaches, presented in the remainder of this chapter, must be utilized. In their most general form, transient conduction problems are described by the heat equation, Equation 2.19, for rectangular coordinates or Equations 2.26 and 2.29, respectively, for cylindrical and spherical coordinates. The solutions to these partial differential equations provide the variation of temperature with both time and the spatial coordinates. However, in many problems, such as the plane wall of Figure 5.4, only one spatial coordinate is needed to describe the internal temperature distribution. With no internal generation and the assumption of constant thermal conductivity, Equation 2.19 then reduces to ⭸2T 1 ⭸T ⭸x2 ␣ ⭸t

(5.29)

To solve Equation 5.29 for the temperature distribution T(x, t), it is necessary to specify an initial condition and two boundary conditions. For the typical transient conduction problem of Figure 5.4, the initial condition is T(x, 0) Ti and the boundary conditions are ⭸T ⭸x and k

⭸T ⭸x

冏

xL

冏

0

(5.30)

(5.31)

x0

h[T(L, t) T앝]

(5.32)

Equation 5.30 presumes a uniform temperature distribution at time t 0; Equation 5.31 reflects the symmetry requirement for the midplane of the wall; and Equation 5.32 describes the surface condition experienced for time t 0. From Equations 5.29 through 5.32, it is evident that, in addition to depending on x and t, temperatures in the wall also depend on a number of physical parameters. In particular T T(x, t, Ti, T앝, L, k, ␣, h)

(5.33)

The foregoing problem may be solved analytically or numerically. These methods will be considered in subsequent sections, but first it is important to note the advantages that may be obtained by nondimensionalizing the governing equations. This may be done by arranging the relevant variables into suitable groups. Consider the dependent variable T. If the temperature difference ⬅ T T앝 is divided by the maximum possible temperature difference i ⬅ Ti T앝, a dimensionless form of the dependent variable may be defined as T T앝 * ⬅ i Ti T앝

(5.34)

5.5

䊏

299

The Plane Wall with Convection

Accordingly, * must lie in the range 0 * 1. A dimensionless spatial coordinate may be defined as x* ⬅ x L

(5.35)

where L is the half-thickness of the plane wall, and a dimensionless time may be defined as t* ⬅ ␣t2 ⬅ Fo L

(5.36)

where t* is equivalent to the dimensionless Fourier number, Equation 5.12. Substituting the definitions of Equations 5.34 through 5.36 into Equations 5.29 through 5.32, the heat equation becomes ⭸2* ⭸* (5.37) ⭸x*2 ⭸Fo and the initial and boundary conditions become

and ⭸* ⭸x*

冏

*(x*, 0) 1

(5.38)

⭸* ⭸x*

0

(5.39)

Bi *(1, t*)

(5.40)

x*1

冏

x*0

where the Biot number is Bi ⬅ hL/k. In dimensionless form the functional dependence may now be expressed as * f(x*, Fo, Bi)

(5.41)

Recall that a similar functional dependence, without the x* variation, was obtained for the lumped capacitance method, as shown in Equation 5.13. Comparing Equations 5.33 and 5.41, the considerable advantage associated with casting the problem in dimensionless form becomes apparent. Equation 5.41 implies that for a prescribed geometry, the transient temperature distribution is a universal function of x*, Fo, and Bi. That is, the dimensionless solution has a prescribed form that does not depend on the particular value of Ti, T앝, L, k, ␣, or h. Since this generalization greatly simplifies the presentation and utilization of transient solutions, the dimensionless variables are used extensively in subsequent sections.

5.5 The Plane Wall with Convection Exact, analytical solutions to transient conduction problems have been obtained for many simplified geometries and boundary conditions and are well documented [3–6]. Several mathematical techniques, including the method of separation of variables (Section 4.2), may be used for this purpose, and typically the solution for the dimensionless temperature distribution, Equation 5.41, is in the form of an infinite series. However, except for very small values of the Fourier number, this series may be approximated by a single term, considerably simplifying its evaluation.

300

Chapter 5

5.5.1

䊏

Transient Conduction

Exact Solution

Consider the plane wall of thickness 2L (Figure 5.6a). If the thickness is small relative to the width and height of the wall, it is reasonable to assume that conduction occurs exclusively in the x-direction. If the wall is initially at a uniform temperature, T(x, 0) Ti, and is suddenly immersed in a fluid of T앝 Ti, the resulting temperatures may be obtained by solving Equation 5.37 subject to the conditions of Equations 5.38 through 5.40. Since the convection conditions for the surfaces at x* 1 are the same, the temperature distribution at any instant must be symmetrical about the midplane (x* 0). An exact solution to this problem is of the form [4]

*

兺C

n

exp (2n Fo) cos (n x*)

(5.42a)

n1

where Fo ␣t/L2, the coefficient Cn is Cn

4 sin n 2n sin (2n)

(5.42b)

and the discrete values of n (eigenvalues) are positive roots of the transcendental equation n tan n Bi

(5.42c)

The first four roots of this equation are given in Appendix B.3. The exact solution given by Equation 5.42a is valid for any time, 0 Fo 앝.

5.5.2

Approximate Solution

It can be shown (Problem 5.43) that for values of Fo 0.2, the infinite series solution, Equation 5.42a, can be approximated by the first term of the series, n 1. Invoking this approximation, the dimensionless form of the temperature distribution becomes * C1 exp (21 Fo) cos (1x*)

(5.43a)

* * o cos (1x*)

(5.43b)

or where *o ⬅ (To T앝)/(Ti T앝) represents the midplane (x* 0) temperature 2 * o C1 exp (1 Fo)

T(x, 0) = Ti

T∞, h

r r* = __ ro

(5.44)

T(r, 0) = Ti

T∞, h

T∞, h ro

L

L x* = _x L (a)

(b)

FIGURE 5.6 One-dimensional systems with an initial uniform temperature subjected to sudden convection conditions: (a) Plane wall. (b) Infinite cylinder or sphere.

Graphical representations of the one-term approximations are presented in Section 5S.1.

5.5

䊏

301

The Plane Wall with Convection

An important implication of Equation 5.43b is that the time dependence of the temperature at any location within the wall is the same as that of the midplane temperature. The coefficients C1 and 1 are evaluated from Equations 5.42b and 5.42c, respectively, and are given in Table 5.1 for a range of Biot numbers.

TABLE 5.1 Coefficients used in the one-term approximation to the series solutions for transient one-dimensional conduction Plane Wall

Infinite Cylinder

Sphere

Bia

1 (rad)

C1

1 (rad)

C1

1 (rad)

C1

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.0998 0.1410 0.1723 0.1987 0.2218 0.2425 0.2615 0.2791 0.2956 0.3111

1.0017 1.0033 1.0049 1.0066 1.0082 1.0098 1.0114 1.0130 1.0145 1.0161

0.1412 0.1995 0.2440 0.2814 0.3143 0.3438 0.3709 0.3960 0.4195 0.4417

1.0025 1.0050 1.0075 1.0099 1.0124 1.0148 1.0173 1.0197 1.0222 1.0246

0.1730 0.2445 0.2991 0.3450 0.3854 0.4217 0.4551 0.4860 0.5150 0.5423

1.0030 1.0060 1.0090 1.0120 1.0149 1.0179 1.0209 1.0239 1.0268 1.0298

0.15 0.20 0.25 0.30 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.3779 0.4328 0.4801 0.5218 0.5932 0.6533 0.7051 0.7506 0.7910 0.8274 0.8603

1.0237 1.0311 1.0382 1.0450 1.0580 1.0701 1.0814 1.0919 1.1016 1.1107 1.1191

0.5376 0.6170 0.6856 0.7465 0.8516 0.9408 1.0184 1.0873 1.1490 1.2048 1.2558

1.0365 1.0483 1.0598 1.0712 1.0932 1.1143 1.1345 1.1539 1.1724 1.1902 1.2071

0.6609 0.7593 0.8447 0.9208 1.0528 1.1656 1.2644 1.3525 1.4320 1.5044 1.5708

1.0445 1.0592 1.0737 1.0880 1.1164 1.1441 1.1713 1.1978 1.2236 1.2488 1.2732

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

1.0769 1.1925 1.2646 1.3138 1.3496 1.3766 1.3978 1.4149 1.4289

1.1785 1.2102 1.2287 1.2402 1.2479 1.2532 1.2570 1.2598 1.2620

1.5994 1.7887 1.9081 1.9898 2.0490 2.0937 2.1286 2.1566 2.1795

1.3384 1.4191 1.4698 1.5029 1.5253 1.5411 1.5526 1.5611 1.5677

2.0288 2.2889 2.4556 2.5704 2.6537 2.7165 1.7654 2.8044 2.8363

1.4793 1.6227 1.7202 1.7870 1.8338 1.8673 1.8920 1.9106 1.9249

20.0 30.0 40.0 50.0 100.0 앝

1.4961 1.5202 1.5325 1.5400 1.5552 1.5708

1.2699 1.2717 1.2723 1.2727 1.2731 1.2733

2.2881 2.3261 2.3455 2.3572 2.3809 2.4050

1.5919 1.5973 1.5993 1.6002 1.6015 1.6018

2.9857 3.0372 3.0632 3.0788 3.1102 3.1415

1.9781 1.9898 1.9942 1.9962 1.9990 2.0000

Bi hL/k for the plane wall and hro /k for the infinite cylinder and sphere. See Figure 5.6.

a

302

Chapter 5

5.5.3

䊏

Transient Conduction

Total Energy Transfer

In many situations it is useful to know the total energy that has left (or entered) the wall up to any time t in the transient process. The conservation of energy requirement, Equation 1.12b, may be applied for the time interval bounded by the initial condition (t 0) and any time t 0 Ein Eout Est

(5.45)

Equating the energy transferred from the wall Q to Eout and setting Ein 0 and Est E(t) E(0), it follows that Q [E(t) E(0)] or

冕

Q c[T(x, t) Ti ]dV

(5.46a)

(5.46b)

where the integration is performed over the volume of the wall. It is convenient to nondimensionalize this result by introducing the quantity Qo cV(Ti T앝)

(5.47)

which may be interpreted as the initial internal energy of the wall relative to the fluid temperature. It is also the maximum amount of energy transfer that could occur if the process were continued to time t 앝. Hence, assuming constant properties, the ratio of the total energy transferred from the wall over the time interval t to the maximum possible transfer is Q Qo

t) T ] dV 1 冕(1 *)dV 冕 [T(x, T T V V i

i

(5.48)

앝

Employing the approximate form of the temperature distribution for the plane wall, Equation 5.43b, the integration prescribed by Equation 5.48 can be performed to obtain Q sin 1 1 * 1 o Qo

(5.49)

where *o can be determined from Equation 5.44, using Table 5.1 for values of the coefficients C1 and 1.

5.5.4

Additional Considerations

Because the mathematical problem is precisely the same, the foregoing results may also be applied to a plane wall of thickness L that is insulated on one side (x* 0) and experiences convective transport on the other side (x* 1). This equivalence is a consequence of the fact that, regardless of whether a symmetrical or an adiabatic requirement is prescribed at x* 0, the boundary condition is of the form ⭸*/⭸x* 0. Also note that the foregoing results may be used to determine the transient response of a plane wall to a sudden change in surface temperature. The process is equivalent to having

5.6

䊏

303

Radial Systems with Convection

an infinite convection coefficient, in which case the Biot number is infinite (Bi 앝) and the fluid temperature T앝 is replaced by the prescribed surface temperature Ts.

5.6 Radial Systems with Convection For an infinite cylinder or sphere of radius ro (Figure 5.6b), which is at an initial uniform temperature and experiences a change in convective conditions, results similar to those of Section 5.5 may be developed. That is, an exact series solution may be obtained for the time dependence of the radial temperature distribution, and a one-term approximation may be used for most conditions. The infinite cylinder is an idealization that permits the assumption of one-dimensional conduction in the radial direction. It is a reasonable approximation for cylinders having L/ro 10.

5.6.1

Exact Solutions

For a uniform initial temperature and convective boundary conditions, the exact solutions [4], applicable at any time (Fo 0), are as follows. Infinite Cylinder

In dimensionless form, the temperature is *

兺C

n

exp (2nFo)J0(nr*)

(5.50a)

n1

where Fo ␣t/r2o, J ( ) Cn 2 2 1 n 2 n J 0 (n) J 1 (n)

(5.50b)

and the discrete values of n are positive roots of the transcendental equation n

J1(n) Bi J0(n)

(5.50c)

where Bi hro /k. The quantities J1 and J0 are Bessel functions of the first kind, and their values are tabulated in Appendix B.4. Roots of the transcendental equation (5.50c) are tabulated by Schneider [4]. Sphere

Similarly, for the sphere *

兺C

n

n1

exp (2nFo) 1 sin (nr*) nr*

(5.51a)

where Fo ␣t/ro2, Cn

4[sin (n) n cos (n)] 2n sin (2n)

(5.51b)

304

Chapter 5

䊏

Transient Conduction

and the discrete values of n are positive roots of the transcendental equation 1 n cot n Bi

(5.51c)

where Bi hro /k. Roots of the transcendental equation are tabulated by Schneider [4].

5.6.2

Approximate Solutions

For the infinite cylinder and sphere, the foregoing series solutions can again be approximated by a single term, n 1, for Fo 0.2. Hence, as for the case of the plane wall, the time dependence of the temperature at any location within the radial system is the same as that of the centerline or centerpoint. Infinite Cylinder

The one-term approximation to Equation 5.50a is * C1 exp (21Fo)J0(1r*)

(5.52a)

* * o J0(1r*)

(5.52b)

or

where *o represents the centerline temperature and is of the form 2 * o C1 exp (1Fo)

(5.52c)

Values of the coefficients C1 and 1 have been determined and are listed in Table 5.1 for a range of Biot numbers. Sphere

From Equation 5.51a, the one-term approximation is * C1 exp (21Fo) 1 sin (1r*) 1r*

(5.53a)

1 sin ( r*) * * o 1 1r*

(5.53b)

or

where *o represents the center temperature and is of the form 2 * o C1 exp (1Fo)

(5.53c)

Values of the coefficients C1 and 1 have been determined and are listed in Table 5.1 for a range of Biot numbers.

5.6.3

Total Energy Transfer

As in Section 5.5.3, an energy balance may be performed to determine the total energy transfer from the infinite cylinder or sphere over the time interval t t. Substituting from

Graphical representations of the one-term approximations are presented in Section 5S.1.

5.6

䊏

Radial Systems with Convection

305

the approximate solutions, Equations 5.52b and 5.53b, and introducing Qo from Equation 5.47, the results are as follows. Infinite Cylinder

Q 2* 1 o J1(1) 1 Qo

(5.54)

Q 3* 1 3o [sin (1) 1 cos (1)] Qo 1

(5.55)

Sphere

Values of the center temperature *o are determined from Equation 5.52c or 5.53c, using the coefficients of Table 5.1 for the appropriate system.

5.6.4

Additional Considerations

As for the plane wall, the foregoing results may be used to predict the transient response of long cylinders and spheres subjected to a sudden change in surface temperature. Namely, an infinite Biot number would be prescribed, and the fluid temperature T앝 would be replaced by the constant surface temperature Ts.

EXAMPLE 5.5 Consider a steel pipeline (AISI 1010) that is 1 m in diameter and has a wall thickness of 40 mm. The pipe is heavily insulated on the outside, and, before the initiation of flow, the walls of the pipe are at a uniform temperature of 20 C. With the initiation of flow, hot oil at 60 C is pumped through the pipe, creating a convective condition corresponding to h 500 W/m2 䡠 K at the inner surface of the pipe. 1. What are the appropriate Biot and Fourier numbers 8 min after the initiation of flow? 2. At t 8 min, what is the temperature of the exterior pipe surface covered by the insulation? 3. What is the heat flux q(W/m2) to the pipe from the oil at t 8 min? 4. How much energy per meter of pipe length has been transferred from the oil to the pipe at t 8 min?

SOLUTION Known: Wall subjected to sudden change in convective surface condition. Find: 1. Biot and Fourier numbers after 8 min. 2. Temperature of exterior pipe surface after 8 min. 3. Heat flux to the wall at 8 min. 4. Energy transferred to pipe per unit length after 8 min.

306

Chapter 5

䊏

Transient Conduction

Schematic: T(x, 0) = Ti = –20°C

T(L, t)

T(0, t)

T∞ = 60°C h = 500 W/m2•K

Insulation Steel, AISI 1010

Oil

L = 40 mm x

Assumptions: 1. Pipe wall can be approximated as plane wall, since thickness is much less than diameter. 2. Constant properties. 3. Outer surface of pipe is adiabatic. Properties: Table A.1, steel type AISI 1010 [T (20 60) C/2 ⬇ 300 K]: 7832 kg/m3, c 434 J/kg 䡠 K, k 63.9 W/m 䡠 K, ␣ 18.8 106 m2/s. Analysis: 1. At t 8 min, the Biot and Fourier numbers are computed from Equations 5.10 and 5.12, respectively, with Lc L. Hence 2 Bi hL 500 W/m 䡠 K 0.04 m 0.313 k 63.9 W/m 䡠 K

Fo ␣t2 L

18.8 106 m2s 8 min 60 s/min 5.64 (0.04 m)2

䉰 䉰

2. With Bi 0.313, use of the lumped capacitance method is inappropriate. However, since Fo 0.2 and transient conditions in the insulated pipe wall of thickness L correspond to those in a plane wall of thickness 2L experiencing the same surface condition, the desired results may be obtained from the one-term approximation for a plane wall. The midplane temperature can be determined from Equation 5.44 * o

To T앝 C1 exp (21Fo) Ti T앝

where, with Bi 0.313, C1 1.047 and 1 0.531 rad from Table 5.1. With Fo 5.64, 2 * o 1.047 exp [(0.531 rad) 5.64] 0.214

Hence after 8 min, the temperature of the exterior pipe surface, which corresponds to the midplane temperature of a plane wall, is T(0, 8 min) T앝 * o (Ti T앝) 60 C 0.214(20 60) C 42.9 C

䉰

5.6

䊏

Radial Systems with Convection

307

3. Heat transfer to the inner surface at x L is by convection, and at any time t the heat flux may be obtained from Newton’s law of cooling. Hence at t 480 s, qx(L, 480 s) ⬅ qL h[T(L, 480 s) T앝] Using the one-term approximation for the surface temperature, Equation 5.43b with x* 1 has the form * *o cos (1) T(L, t) T앝 (Ti T앝)*o cos (1) T(L, 8 min) 60 C (20 60) C 0.214 cos(0.531 rad) T(L, 8 min) 45.2 C The heat flux at t 8 min is then qL 500 W/m2 䡠 K (45.2 60) C 7400 W/m2

䉰

4. The energy transfer to the pipe wall over the 8-min interval may be obtained from Equations 5.47 and 5.49. With Q sin(1) 1 *o 1 Qo Q sin(0.531 rad) 1

0.214 0.80 Qo 0.531 rad it follows that Q 0.80 cV(Ti T) or with a volume per unit pipe length of V DL, Q 0.80 cDL(Ti T앝) Q 0.80 7832 kg/m3 434 J/kg 䡠 K

1 m 0.04 m (20 60) C Q 2.73 107 J/m

䉰

Comments: 1. The minus sign associated with q and Q simply implies that the direction of heat transfer is from the oil to the pipe (into the pipe wall). 2. The solution for this example is provided as a ready-to-solve model in the Advanced section of IHT, which uses the Models, Transient Conduction, Plane Wall option. Since the IHT model uses a multiple-term approximation to the series solution, the results are more accurate than those obtained from the foregoing one-term approximation. IHT Models for Transient Conduction are also provided for the radial systems treated in Section 5.6.

308

Chapter 5

䊏

Transient Conduction

EXAMPLE 5.6 A new process for treatment of a special material is to be evaluated. The material, a sphere of radius ro 5 mm, is initially in equilibrium at 400 C in a furnace. It is suddenly removed from the furnace and subjected to a two-step cooling process. Step 1 Cooling in air at 20 C for a period of time ta until the center temperature reaches a critical value, Ta(0, ta ) 335 C. For this situation, the convection heat transfer coefficient is ha 10 W/m2 䡠 K. After the sphere has reached this critical temperature, the second step is initiated. Step 2 Cooling in a well-stirred water bath at 20 C, with a convection heat transfer coefficient of hw 6000 W/m2 䡠 K. The thermophysical properties of the material are 3000 kg/m3, k 20 W/m 䡠 K, c 1000 J/kg 䡠 K, and ␣ 6.66 106 m2/s. 1. Calculate the time ta required for step 1 of the cooling process to be completed. 2. Calculate the time tw required during step 2 of the process for the center of the sphere to cool from 335 C (the condition at the completion of step 1) to 50 C.

SOLUTION Known: Temperature requirements for cooling a sphere. Find: 1. Time ta required to accomplish desired cooling in air. 2. Time tw required to complete cooling in water bath. Schematic: T∞ = 20°C ha = 10 W/m2•K

T∞ = 20°C hw = 6000 W/m2•K

Air

Ti = 400°C Ta(0, ta) = 335°C

Water

Sphere, ro = 5 mm ρ = 3000 kg/m3 c = 1 kJ/kg•K α = 6.66 × 10–6 m2/s k = 20 W/m•K

Step 1

Assumptions: 1. One-dimensional conduction in r. 2. Constant properties.

Ti = 335°C Tw(0, tw) = 50°C Step 2

5.6

䊏

309

Radial Systems with Convection

Analysis: 1. To determine whether the lumped capacitance method can be used, the Biot number is calculated. From Equation 5.10, with Lc ro /3, Bi

ha ro 10 W/m2 䡠 K 0.005 m 8.33 104 3k 3 20 W/m 䡠 K

Accordingly, the lumped capacitance method may be used, and the temperature is nearly uniform throughout the sphere. From Equation 5.5 it follows that ta

Vc i roc Ti T앝 ln ln ha As a 3ha Ta T앝

where V (4/3)r 3o and As 4r 2o. Hence ta

3000 kg/m3 0.005 m 1000 J/kg 䡠 K 400 20 ln 94 s 335 20 3 10 W/m2 䡠 K

䉰

2. To determine whether the lumped capacitance method may also be used for the second step of the cooling process, the Biot number is again calculated. In this case Bi

h w ro 6000 W/m2 䡠 K 0.005 m 0.50 3k 3 20 W/m 䡠 K

and the lumped capacitance method is not appropriate. However, to an excellent approximation, the temperature of the sphere is uniform at t ta and the one-term approximation may be used for the calculations. The time tw at which the center temperature reaches 50 C, that is, T(0, tw) 50 C, can be obtained by rearranging Equation 5.53c Fo 12 ln 1

冤 C 冥 1 ln 冤 C1 T(0,T t) T T 冥 *o

w

2 1

1

1

i

where tw Fo r o2 /␣. With the Biot number now defined as Bi

h w ro 6000 W/m2 䡠 K 0.005 m 1.50 k 20 W/m 䡠 K

Table 5.1 yields C1 1.376 and 1 1.800 rad. It follows that Fo

冤

冥

(50 20) C 1 ln 1

0.82 2 1.376 (335 20) C (1.800 rad)

and r2 (0.005 m)2 3.1s tw Fo ␣o 0.82 6.66 106 m2/s Note that, with Fo 0.82, use of the one-term approximation is justified.

䉰

310

Chapter 5

䊏

Transient Conduction

Comments: 1. If the temperature distribution in the sphere at the conclusion of step 1 were not uniform, the one-term approximation could not be used for the calculations of step 2. 2. The surface temperature of the sphere at the conclusion of step 2 may be obtained from Equation 5.53b. With o* 0.095 and r* 1, *(ro)

T(ro) T앝 0.095 sin (1.800 rad) 0.0514 Ti T앝 1.800 rad

and T(ro) 20 C 0.0514(335 20) C 36 C The infinite series, Equation 5.51a, and its one-term approximation, Equation 5.53b, may be used to compute the temperature at any location in the sphere and at any time t ta. For (t ta ) 0.2(0.005 m)2/6.66 106 m2/s 0.75 s, a sufficient number of terms must be retained to ensure convergence of the series. For (t ta ) 0.75 s, satisfactory convergence is provided by the one-term approximation. Computing and plotting the temperature histories for r 0 and r ro, we obtain the following results for 0 (t ta ) 5 s: 400

300

T (°C)

r* = 1 200

r* = 0

100 50 0

0

1

2

3

4

5

t – ta (s)

3. The IHT Models, Transient Conduction, Sphere option could be used to analyze the cooling processes experienced by the sphere in air and water, steps 1 and 2. The IHT Models, Lumped Capacitance option may only be used to analyze the air-cooling process, step 1.

5.7 The Semi-Infinite Solid An important simple geometry for which analytical solutions may be obtained is the semiinfinite solid. Since, in principle, such a solid extends to infinity in all but one direction, it is characterized by a single identifiable surface (Figure 5.7). If a sudden change of conditions is imposed at this surface, transient, one-dimensional conduction will occur within the

5.7

䊏

311

The Semi-Infinite Solid

Case (1)

Case (2)

Case (3)

T(x, 0) = Ti T(0, t) = Ts

T(x, 0) = Ti –k ∂ T/∂ x⎥x = 0 = q"o

T(x, 0) = Ti –k ∂ T/∂ x⎥x = 0 = h[T∞ – T(0, t)]

Ts

T∞, h q"o

x

x

x

T(x, t) T∞

Ts t

t

Ti

Ti x

t

Ti x

x

FIGURE 5.7 Transient temperature distributions in a semi-infinite solid for three surface conditions: constant surface temperature, constant surface heat flux, and surface convection.

solid. The semi-infinite solid provides a useful idealization for many practical problems. It may be used to determine transient heat transfer near the surface of the earth or to approximate the transient response of a finite solid, such as a thick slab. For this second situation the approximation would be reasonable for the early portion of the transient, during which temperatures in the slab interior (well removed from the surface) are essentially uninfluenced by the change in surface conditions. These early portions of the transient might correspond to very small Fourier numbers, and the approximate solutions of Sections 5.5 and 5.6 would not be valid. Although the exact solutions of the preceding sections could be used to determine the temperature distributions, many terms might be required to evaluate the infinite series expressions. The following semi-infinite solid solutions often eliminate the need to evaluate the cumbersome infinite series exact solutions at small Fo. It will be shown that a plane wall of thickness 2L can be accurately approximated as a semi-infinite solid for Fo ␣t/L2 0.2. The heat equation for transient conduction in a semi-infinite solid is given by Equation 5.29. The initial condition is prescribed by Equation 5.30, and the interior boundary condition is of the form T(x l 앝, t) Ti

(5.56)

Closed-form solutions have been obtained for three important surface conditions, instantaneously applied at t 0 [3, 4]. These conditions are shown in Figure 5.7. They include application of a constant surface temperature Ts Ti, application of a constant surface heat flux qo, and exposure of the surface to a fluid characterized by T앝 Ti and the convection coefficient h. The solution for case 1 may be obtained by recognizing the existence of a similarity variable , through which the heat equation may be transformed from a partial differential equation, involving two independent variables (x and t), to an ordinary differential equation expressed in terms of the single similarity variable. To confirm that such a

312

Chapter 5

䊏

Transient Conduction

requirement is satisfied by ⬅ x/(4␣t)1/2, we first transform the pertinent differential operators, such that ⭸T dT ⭸ 1 dT ⭸x d ⭸x (4␣t)1/2 d

冤 冥

2 ⭸2T ⭸T ⭸ 1 d T2 d 2 d ⭸x ⭸x 4␣t d ⭸x ⭸T dT ⭸ x dT 1/2 d ⭸t d ⭸t 2t(4␣t)

Substituting into Equation 5.29, the heat equation becomes d 2T 2 dT d d2

(5.57)

With x 0 corresponding to 0, the surface condition may be expressed as T( 0) Ts

(5.58)

and with x l 앝, as well as t 0, corresponding to l 앝, both the initial condition and the interior boundary condition correspond to the single requirement that T( l 앝) Ti

(5.59)

Since the transformed heat equation and the initial/boundary conditions are independent of x and t, ⬅ x/(4␣t)1/2 is, indeed, a similarity variable. Its existence implies that, irrespective of the values of x and t, the temperature may be represented as a unique function of . The specific form of the temperature dependence, T(), may be obtained by separating variables in Equation 5.57, such that d(dT/d) 2 d (dT/d) Integrating, it follows that ln(dT/d) 2 C1 or dT C exp (2) 1 d Integrating a second time, we obtain

冕 exp(u ) du C

T C1

2

2

0

where u is a dummy variable. Applying the boundary condition at 0, Equation 5.58, it follows that C2 Ts and

冕 exp(u ) du T

T C1

2

s

0

5.7

䊏

313

The Semi-Infinite Solid

From the second boundary condition, Equation 5.59, we obtain Ti C1

冕 exp(u ) du T 앝

2

s

0

or, evaluating the definite integral, C1

2(Ti Ts) 1/2

Hence the temperature distribution may be expressed as T Ts (2/1/2) Ti Ts

冕 exp (u ) du ⬅ erf

2

(5.60)

0

where the Gaussian error function, erf , is a standard mathematical function that is tabulated in Appendix B. Note that erf() asymptotically approaches unity as becomes infinite. Thus, at any nonzero time, temperatures everywhere are predicted to have changed from Ti (become closer to Ts). The infinite speed at which boundary-condition information propagates into the semi-infinite solid is physically unrealistic, but this limitation of Fourier’s law is not important except at extremely small time scales, as discussed in Section 2.3. The surface heat flux may be obtained by applying Fourier’s law at x 0, in which case qs k

⭸T ⭸x

冏

k(Ti Ts)

x0

d(erf ) ⭸ d ⭸x

冏

0

qs k(Ts Ti)(2/1/2)exp(2)(4␣t)1/2 兩0 qs

k(Ts Ti) (␣t)1/2

(5.61)

Analytical solutions may also be obtained for the case 2 and case 3 surface conditions, and results for all three cases are summarized as follows. Case 1

Constant Surface Temperature: T(0, t) Ts

冢

T(x, t) Ts x erf Ti Ts 2兹␣t qs(t) Case 2

冣

(5.60)

k(Ts Ti)

(5.61)

兹␣t

Constant Surface Heat Flux: qs qo T(x, t) Ti

冢 冣

冢

2 q x 2qo(␣t/)1/2 x exp x o erfc k 4␣t k 2兹␣t

冣

(5.62)

Chapter 5

䊏

Transient Conduction

Surface Convection: k

Case 3

冢

T(x, t) Ti x erfc T앝 Ti 2兹␣t

⭸T ⭸x

冏

x0

h[T앝 T(0, t)]

冣

冤 冢

2 exp hx h ␣t k k2

h兹␣t x

冣冥冤erfc 冢2兹␣t k 冣冥

(5.63)

The complementary error function, erfc w, is defined as erfc w ⬅ 1 erf w. Temperature histories for the three cases are shown in Figure 5.7, and distinguishing features should be noted. With a step change in the surface temperature, case 1, temperatures within the medium monotonically approach Ts with increasing t, while the magnitude of the surface temperature gradient, and hence the surface heat flux, decreases as t1/2. A thermal penetration depth ␦p can be defined as the depth to which significant temperature effects propagate within a medium. For example, defining ␦p as the x-location at which (T – Ts)/ (Ti – Ts) 0.90, Equation 5.60 results in ␦p 2.3兹␣t.2 Hence, the penetration depth increases as t1/2 and is larger for materials with higher thermal diffusivity. For a fixed surface heat flux (case 2), Equation 5.62 reveals that T(0, t) Ts(t) increases monotonically as t1/2. For surface convection (case 3), the surface temperature and temperatures within the medium approach the fluid temperature T앝 with increasing time. As Ts approaches T앝, there is, of course, a reduction in the surface heat flux, qs(t) h[T Ts(t)]. Specific temperature histories computed from Equation 5.63 are plotted in Figure 5.8. The result corresponding to h 앝 is equivalent to that associated with a sudden change in surface temperature, case 1. That is, for h 앝, the surface instantaneously achieves the imposed fluid temperature (Ts T앝), and with the second term on the right-hand side of Equation 5.63 reducing to zero, the result is equivalent to Equation 5.60. An interesting permutation of case 1 occurs when two semi-infinite solids, initially at uniform temperatures TA,i and TB,i, are placed in contact at their free surfaces (Figure 5.9). 1.0 0.5

T – Ti ______ T∞ – Ti

314

T∞ T(x, t) h

∞ 3 0.4 0.5

1

x 2

0.1 0.3 0.2

0.05

0.1

h √α t = 0.05 _____ k

0.01 0

0.5

1.0

x _____ 2 √ αt

1.5

FIGURE 5.8 Temperature histories in a semi-infinite solid with surface convection [4]. (Adapted with permission.)

To apply the semi-infinite approximation to a plane wall of thickness 2L, it is necessary that ␦p L. Substituting ␦p L into the expression for the thermal penetration depth yields Fo 0.19 ⬇ 0.2. Hence, a plane wall of thickness 2L can be accurately approximated as a semi-infinite solid for Fo ␣t/L2 0.2. This restriction will also be demonstrated in Section 5.8. 2

5.7

䊏

315

The Semi-Infinite Solid

T B

TA, i

kB, ρB, cB q"s, B t Ts

t

q"s, A A

TB, i

kA, ρA, cA

FIGURE 5.9 Interfacial contact between two semiinfinite solids at different initial temperatures.

x

If the contact resistance is negligible, the requirement of thermal equilibrium dictates that, at the instant of contact (t 0), both surfaces must assume the same temperature Ts, for which TB,i Ts TA,i. Since Ts does not change with increasing time, it follows that the transient thermal response and the surface heat flux of each of the solids are determined by Equations 5.60 and 5.61, respectively. The equilibrium surface temperature of Figure 5.9 may be determined from a surface energy balance, which requires that qs,A qs,B

(5.64)

Substituting from Equation 5.61 for qs,A and qs,B and recognizing that the x-coordinate of Figure 5.9 requires a sign change for qs,A, it follows that kA(Ts TA,i) 1/2

(␣At)

kB(Ts TB,i) (␣Bt)1/2

(5.65)

or, solving for Ts, Ts

1/2 (kc)1/2 A TA,i (kc)B TB,i 1/2 (kc)1/2 A (kc)B

(5.66)

Hence the quantity m ⬅ (kc)1/2 is a weighting factor that determines whether Ts will more closely approach TA,i (mA mB) or TB,i (mB mA).

EXAMPLE 5.7 On a hot and sunny day, the concrete deck surrounding a swimming pool is at a temperature of Td 55 C. A swimmer walks across the dry deck to the pool. The soles of the swimmer’s dry feet are characterized by an Lsf 3-mm-thick skin/fat layer of thermal conductivity ksf 0.3 W/m 䡠 K. Consider two types of concrete decking; (i) a dense stone mix and (ii) a lightweight aggregate characterized by density, specific heat, and thermal conductivity of lw 1495 kg/m3, cp,lw 880 J/kg 䡠 K, and klw 0.28 W/m 䡠 K, respectively. The density and specific heat of the skin/fat layer may be approximated to be those of liquid water, and the skin/fat layer is at an initial temperature of Tsf,i 37 C. What is the temperature of the bottom of the swimmer’s feet after an elapsed time of t 1 s?

316

Chapter 5

Transient Conduction

䊏

SOLUTION Known: Concrete temperature, initial foot temperature, and thickness of skin/fat layer on the sole of the foot. Skin/fat and lightweight aggregate concrete properties. Find: The temperature of the bottom of the swimmer’s feet after 1 s. Schematic: Tsf,i = 37°C Lsf ⫽ 3 mm x

Foot Skin/fat Ts

Concrete deck Td,i = 55°C

Assumptions: 1. One-dimensional conduction in the x-direction. 2. Constant and uniform properties. 3. Negligible contact resistance. Properties: Table A.3 stone mix concrete (T 300 K): sm 2300 kg/m3, ksm 1.4 W/m 䡠 K, csm 880 J/kg 䡠 K. Table A.6 water (T 310 K): sf 993 kg/m3, csf 4178 J/kg 䡠 K. Analysis: If the skin/fat layer and the deck are both semi-infinite media, from Equation 5.66 the surface temperature Ts is constant when the swimmer’s foot is in contact with the deck. For the lightweight aggregate concrete decking, the thermal penetration depth at t 1 s is K 1s 冪k ct 2.3冪14950.28kg/mW/m 䡠880 J/kg 䡠 K

␦p,lw 2.3兹␣lwt 2.3

lw

3

lw lw

1.06 103 m 1.06 mm

Since the thermal penetration depth is relatively small, it is reasonable to assume that the lightweight aggregate deck behaves as a semi-infinite medium. Similarly, the thermal penetration depth in the stone mix concrete is ␦p,sm 1.91 mm, and the thermal penetration depth associated with the skin/fat layer of the foot is ␦p,sf 0.62 mm. Hence, it is reasonable to assume that the stone mix concrete deck responds as a semi-infinite medium, and, since ␦p,sf Lsf, it is also correct to assume that the skin/fat layer behaves as a semi-infinite medium. Therefore, Equation 5.66 may be used to determine the surface temperature of the swimmer’s foot for exposure to the two types of concrete decking. For the lightweight aggregate, Ts,lw

1/2 (kc)1/2 lw Td,i (kc)sf Tsf,i 1/2 (kc)1/2 lw (kc)sf

W/m 䡠 K 1495 kg/m 880 J/kg 䡠 K) 55 C 冤(0.28

(0.3 W/m 䡠 K 993 kg/m 4178 J/kg 䡠 K) 37 C冥 43.3 C (0.28 W/m 䡠 K 1495 kg/m 880 J/kg 䡠 K) 冤 (0.3 W/m 䡠 K 993 kg/m 4178 J/kg 䡠 K) 冥 3

1/2

3

1/2

3

3

1/2

1/2

䉰

5.8

䊏

317

Objects with Constant Surface Temperatures or Surface Heat Fluxes

Repeating the calculation for the stone mix concrete gives Ts,sm 47.8 C.

䉰

Comments: 1. The lightweight aggregate concrete feels cooler to the swimmer, relative to the stone mix concrete. Specifically, the temperature rise from the initial skin/fat temperature that is associated with the stone mix concrete is Tsm Tsm – Tsf,i 47.8 C – 37 C 10.8 C, whereas the temperature rise associated with the lightweight aggregate is Tlw Tlw – Tsf,i 43.3 C – 37 C 6.3 C. 2. The thermal penetration depths associated with an exposure time of t 1 s are small. Stones and air pockets within the concrete may be of the same size as the thermal penetration depth, making the uniform property assumption somewhat questionable. The predicted foot temperatures should be viewed as representative values.

5.8 Objects with Constant Surface Temperatures or Surface Heat Fluxes In Sections 5.5 and 5.6, the transient thermal response of plane walls, cylinders, and spheres to an applied convection boundary condition was considered in detail. It was pointed out that the solutions in those sections may be used for cases involving a step change in surface temperature by allowing the Biot number to be infinite. In Section 5.7, the response of a semi-infinite solid to a step change in surface temperature, or to an applied constant heat flux, was determined. This section will conclude our discussion of transient heat transfer in one-dimensional objects experiencing constant surface temperature or constant surface heat flux boundary conditions. A variety of approximate solutions are presented.

5.8.1

Constant Temperature Boundary Conditions

In the following discussion, the transient thermal response of objects to a step change in surface temperature is considered. Insight into the thermal response of objects to an applied constant temperature boundary condition may be obtained by casting the heat flux in Equation 5.61 into the nondimensional form

Semi-Infinite Solid

q* ⬅

qs Lc k(Ts Ti)

(5.67)

where Lc is a characteristic length and q* is the dimensionless conduction heat rate that was introduced in Section 4.3. Substituting Equation 5.67 into Equation 5.61 yields q*

1 兹Fo

(5.68)

Chapter 5

䊏

Transient Conduction

where the Fourier number is defined as Fo ⬅ ␣t/L2c. Note that the value of qs is independent of the choice of the characteristic length, as it must be for a semi-infinite solid. Equation 5.68 is plotted in Figure 5.10a, and since q* Fo1/2, the slope of the line is 1/2 on the log-log plot. Interior Heat Transfer: Plane Wall, Cylinder, and Sphere Results for heat transfer to the interior of a plane wall, cylinder, and sphere are also shown in Figure 5.10a. These results are generated by using Fourier’s law in conjunction with Equations 5.42, 5.50, and 5.51 for Bi l 앝. As in Sections 5.5 and 5.6, the characteristic length is Lc L or ro for a plane wall of thickness 2L or a cylinder (or sphere) of radius ro, respectively. For each geometry, q* initially follows the semi-infinite solid solution but at some point decreases rapidly as the objects approach their equilibrium temperature and qs (t l 앝) l 0. The value of q* is expected to decrease more rapidly for geometries that possess large surface area to volume ratios, and this trend is evident in Figure 5.10a.

100

Exterior objects, Lc = (As/4)1/2 Semi-infinite solid

q*

10

1

Interior, Lc = L or ro sphere 0.1

infinite cylinder plane wall

0.01

0.0001

0.001

0.01

0.1

1

10

Fo = ␣t/L2c (a) 100

Exterior objects, Lc = (As/4)1/2 Semi-infinite solid 10

q*

318

1

Interior, Lc = L or ro sphere

0.1

infinite cylinder plane wall

0.01 0.0001

0.001

0.01

0.1

Fo = ␣t/L2c (b)

1

10

FIGURE 5.10 Transient dimensionless conduction heat rates for a variety of geometries. (a) Constant surface temperature. Results for the geometries of Table 4.1 lie within the shaded region and are from Yovanovich [7]. (b) Constant surface heat flux.

5.8

䊏

Objects with Constant Surface Temperatures or Surface Heat Fluxes

319

Additional results are shown in Figure 5.10a for objects that are embedded in an exterior (surrounding) medium of infinite extent. The infinite medium is initially at temperature Ti, and the surface temperature of the object is suddenly changed to Ts. For the exterior cases, Lc is the characteristic length used in Section 4.3, namely Lc (As /4)1/2. For the sphere in a surrounding infinite medium, the exact solution for q*(Fo) is [7] Exterior Heat Transfer: Various Geometries

1 (5.69)

1 兹Fo As seen in the figure, for all of the exterior cases q* closely mimics that of the sphere when the appropriate length scale is used in its definition, regardless of the object’s shape. Moreover, in a manner consistent with the interior cases, q* initially follows the semi-infinite solid solution. In contrast to the interior cases, q* eventually reaches the nonzero, steady-state value of q*ss that is listed in Table 4.1. Note that qs in Equation 5.67 is the average surface heat flux for geometries that have nonuniform surface heat flux. As seen in Figure 5.10a, all of the thermal responses collapse to that of the semiinfinite solid for early times, that is, for Fo less than approximately 103. This remarkable consistency reflects the fact that temperature variations are confined to thin layers adjacent to the surface of any object at early times, regardless of whether internal or external heat transfer is of interest. At early times, therefore, Equations 5.60 and 5.61 may be used to predict the temperatures and heat transfer rates within the thin regions adjacent to the boundaries of any object. For example, predicted local heat fluxes and local dimensionless temperatures using the semi-infinite solid solutions are within approximately 5% of the predictions obtained from the exact solutions for the interior and exterior heat transfer cases involving spheres when Fo 103. q*

5.8.2

Constant Heat Flux Boundary Conditions

When a constant surface heat flux is applied to an object, the resulting surface temperature history is often of interest. In this case, the heat flux in the numerator of Equation 5.67 is now constant, and the temperature difference in the denominator, Ts Ti, increases with time. Semi-Infinite Solid In the case of a semi-infinite solid, the surface temperature history can be found by evaluating Equation 5.62 at x 0, which may be rearranged and combined with Equation 5.67 to yield

q* 1 2

冪Fo

(5.70)

As for the constant temperature case, q* Fo1/2, but with a different coefficient. Equation 5.70 is presented in Figure 5.10b. A second set of results is shown in Figure 5.10b for the interior cases of the plane wall, cylinder, and sphere. As for the constant surface temperature results of Figure 5.10a, q* initially follows the semiinfinite solid solution and subsequently decreases more rapidly, with the decrease occurring first for the sphere, then the cylinder, and finally the plane wall. Compared to the constant surface temperature case, the rate at which q* decreases is not as dramatic, since steadystate conditions are never reached; the surface temperature must continue to increase with

Interior Heat Transfer: Plane Wall, Cylinder, and Sphere

320

Chapter 5

䊏

Transient Conduction

time. At late times (large Fo), the surface temperature increases linearly with time, yielding q* Fo1, with a slope of 1 on the log-log plot. Results for heat transfer between a sphere and an exterior infinite medium are also presented in Figure 5.10b. The exact solution for the embedded sphere is

Exterior Heat Transfer: Various Geometries

q* [1 exp(Fo) erfc(Fo1/2)]1

(5.71)

As in the constant surface temperature case of Figure 5.10a, this solution approaches steady-state conditions, with qss 1. For objects of other shapes that are embedded within an infinite medium, q* would follow the semi-infinite solid solution at small Fo. At larger Fo, q* must asymptotically approach the value of qss given in Table 4.1 where Ts in Equation 5.67 is the average surface temperature for geometries that have nonuniform surface temperatures.

5.8.3

Approximate Solutions

Simple expressions have been developed for q*(Fo) [8]. These expressions may be used to approximate all the results included in Figure 5.10 over the entire range of Fo. These expressions are listed in Table 5.2, along with the corresponding exact solutions. Table 5.2a is for the constant surface temperature case, while Table 5.2b is for the constant surface heat flux situation. For each of the geometries listed in the left-hand column, the tables provide the length scale to be used in the definition of both Fo and q*, the exact solution for q*(Fo), the approximation solutions for early times (Fo 0.2) and late times (Fo 0.2), and the maximum percentage error associated with use of the approximations (which occurs at Fo ⬇ 0.2 for all results except the external sphere with constant heat flux).

EXAMPLE 5.8 Derive an expression for the ratio of the total energy transferred from the isothermal surfaces of a plane wall to the interior of the plane wall, Q/Qo, that is valid for Fo 0.2. Express your results in terms of the Fourier number Fo.

SOLUTION Known: Plane wall with constant surface temperatures. Find: Expression for Q/Qo as a function of Fo ␣t/L2. Schematic: Ti

Ts L

L x

5.8 䊏

TABLE 5.2a Summary of transient heat transfer results for constant surface temperature casesa [8]

Geometry Semi-infinite

Length Scale, Lc

Exact Solutions

L (arbitrary)

1 兹Fo

Interior Cases

앝

Plane wall of thickness 2L

L

Infinite cylinder

ro

2

兺 exp( 앝

2

Various shapes (Table 4.1, cases 12–15)

None

1 兹Fo

2 exp(21 Fo)

1 /2

1.7

2 n

Fo)

J0(n) 0

1 0.50 0.65 Fo 兹Fo

2 exp(21 Fo)

1 2.4050

0.8

Fo)

n n

1 1 兹Fo

2 exp(21 Fo)

1

6.3

兺 exp(

2 n

n1

Exterior Cases Sphere

Use exact solution.

Maximum Error (%)

n (n 12)

兺 exp( 앝

ro

Fo ⱖ 0.2

Fo)

n1

Sphere

Fo ⬍ 0.2

2 n

n1

2

Approximate Solutions

ro

1

1 兹Fo

(As /4)1/2

None

Use exact solution. 1

q* ss, 兹 Fo

q* ss from Table 4.1

None 7.1

q* ⬅ q⬙s Lc /k(Ts Ti) and Fo ⬅ ␣t/L2c , where Lc is the length scale given in the table, Ts is the object surface temperature, and Ti is (a) the initial object temperature for the interior cases and (b) the temperature of the infinite medium for the exterior cases. a

Objects with Constant Surface Temperatures or Surface Heat Fluxes

q*(Fo)

321

322 Chapter 5

TABLE 5.2b Summary of transient heat transfer results for constant surface heat flux casesa [8] q*(Fo)

Semi-infinite Interior Cases Plane wall of thickness 2L

1 2

L (arbitrary)

冪Fo 1

2 n

n1

2 n

兺 冤 冥 冤3Fo 15 2 兺 exp( Fo)冥 앝 exp( Fo) n 1 2Fo 2 4 2n n1

1

앝

1

2

Infinite cylinder

ro

Sphere

ro

n1

Exterior Cases Sphere Various shapes (Table 4.1, cases 12–15)

2 n

2 n

Maximum Error (%)

Fo ⱖ 0.2

Use exact solution.

冤Fo 13 2 兺 exp( Fo)冥 앝

L

Fo ⬍ 0.2

n n J1(n) 0 tan(n) n

None

Fo

冤Fo 13冥

冪Fo 8 1 2冪Fo 4

冤2Fo 14冥 冤3Fo 15冥

1 2

冪

1 2

1

5.3 1

2.1

1

ro

[1 exp(Fo)erfc(Fo 1/2)]1

1 2

冪Fo 4

0.77

1 兹Fo

(As /4)1/2

None

1 2

冪Fo 4

0.77

q*ss 兹Fo

4.5

3.2

Unknown

q* ⬅ qs Lc /k(Ts Ti) and Fo ⬅ ␣t/L2c, where Lc is the length scale given in the table, Ts is the object surface temperature, and Ti is (a) the initial object temperature for the interior cases and (b) the temperature of the infinite medium for the exterior cases.

a

Transient Conduction

Geometry

Exact Solutions

䊏

Approximate Solutions Length Scale, Lc

5.8

䊏

Objects with Constant Surface Temperatures or Surface Heat Fluxes

323

Assumptions: 1. One-dimensional conduction. 2. Constant properties. 3. Validity of the approximate solution of Table 5.2a. Analysis: From Table 5.2a for a plane wall of thickness 2L and Fo 0.2, q*

qs L 1 where Fo ␣t2 k(Ts Ti ) 兹Fo L

Combining the preceding equations yields qs

k(Ts Ti) 兹␣t

Recognizing that Q is the accumulated heat that has entered the wall up to time t, we can write

冕 t

qs dt Q t0 ␣ Qo Lc(Ts Ti) L兹␣

冕t t

1/2

t0

dt 2 兹Fo 兹

䉰

Comments: 1. The exact solution for Q/Qo at small Fourier number involves many terms that would need to be evaluated in the infinite series expression. Use of the approximate solution simplifies the evaluation of Q/Qo considerably. 2. At Fo 0.2, Q/Qo ⬇ 0.5. Approximately half of the total possible change in thermal energy of the plane wall occurs during Fo 0.2. 3. Although the Fourier number may be viewed as a dimensionless time, it has an important physical interpretation for problems involving heat transfer by conduction through a solid concurrent with thermal energy storage in the solid. Specifically, as suggested by the solution, the Fourier number provides a measure of the amount of energy stored in the solid at any time.

EXAMPLE 5.9 A proposed cancer treatment utilizes small, composite nanoshells whose size and composition are carefully specified so that the particles efficiently absorb laser irradiation at particular wavelengths [9]. Prior to treatment, antibodies are attached to the nanoscale particles. The doped particles are then injected into the patient’s bloodstream and are distributed throughout the body. The antibodies are attracted to malignant sites, and therefore carry and adhere the nanoshells only to cancerous tissue. After the particles have come to rest within the tumor, a laser beam penetrates through the tissue between the skin and the cancer, is absorbed by the nanoshells, and, in turn, heats and destroys the cancerous tissues.

324

Chapter 5

䊏

Transient Conduction

Consider an approximately spherical tumor of diameter Dt 3 mm that is uniformly infiltrated with nanoshells that are highly absorptive of incident radiation from a laser located outside the patient’s body. Mirror Laser Nanoshell impregnated tumor

1. Estimate the heat transfer rate from the tumor to the surrounding healthy tissue for a steady-state treatment temperature of Tt,ss 55 C at the surface of the tumor. The thermal conductivity of healthy tissue is approximately k 0.5 W/m 䡠 K, and the body temperature is Tb 37 C. 2. Find the laser power necessary to sustain the tumor surface temperature at Tt,ss 55 C if the tumor is located d 20 mm beneath the surface of the skin, and the laser heat flux decays exponentially, ql (x) ql,o(1 ) ex, between the surface of the body and the tumor. In the preceding expression, ql,o is the laser heat flux outside the body, 0.05 is the reflectivity of the skin surface, and 0.02 mm1 is the extinction coefficient of the tissue between the tumor and the surface of the skin. The laser beam has a diameter of Dl 5 mm. 3. Neglecting heat transfer to the surrounding tissue, estimate the time at which the tumor temperature is within 3 C of Tt,ss 55 C for the laser power found in part 2. Assume the tissue’s density and specific heat are that of water. 4. Neglecting the thermal mass of the tumor but accounting for heat transfer to the surrounding tissue, estimate the time needed for the surface temperature of the tumor to reach Tt 52 C.

SOLUTION Known: Size of a small sphere; thermal conductivity, reflectivity, and extinction coefficient of tissue; depth of sphere below the surface of the skin. Find: 1. Heat transferred from the tumor to maintain its surface temperature at Tt,ss 55 C. 2. Laser power needed to sustain the tumor surface temperature at Tt,ss 55 C. 3. Time for the tumor to reach Tt 52 C when heat transfer to the surrounding tissue is neglected. 4. Time for the tumor to reach Tt 52 C when heat transfer to the surrounding tissue is considered and the thermal mass of the tumor is neglected.

5.8

䊏

Objects with Constant Surface Temperatures or Surface Heat Fluxes

325

Schematic: Laser beam, q"l,o

Dl = 5 mm

Skin, = 0.05 x Tumor

d = 20 mm

Healthy tissue Tb = 37°C k = 0.5 W/m•K κ = 0.02 mm1 Dt = 3 mm

Assumptions: 1. One-dimensional conduction in the radial direction. 2. Constant properties. 3. Healthy tissue can be treated as an infinite medium. 4. The treated tumor absorbs all irradiation incident from the laser. 5. Lumped capacitance behavior for the tumor. 6. Neglect potential nanoscale heat transfer effects. 7. Neglect the effect of perfusion. 3 Properties: Table A.6, water (320 K, assumed): v1 f 989.1 kg/m , cp 4180 J/kg 䡠 K.

Analysis: 1. The steady-state heat loss from the spherical tumor may be determined by evaluating the dimensionless heat rate from the expression for case 12 of Table 4.1: q 2kDt(Tt,ss Tb) 2 0.5 W/m 䡠 K 3 103 m (55 37) C 0.170 W

䉰

2. The laser irradiation will be absorbed over the projected area of the tumor, D2t/4. To determine the laser power corresponding to q 0.170 W, we first write an energy balance for the sphere. For a control surface about the sphere, the energy absorbed from the laser irradiation is offset by heat conduction to the healthy tissue, q 0.170 W ⬇ ql(x d)Dt2/4, where, ql(x d) ql,o (1 )e⫺d and the laser power is Pl ql,oD2l /4. Hence, Pl qD2l ed/[(1 )D2t ] 1 0.170 W (5 103 m)2 e(0.02 mm 20 mm)/[(1 0.05) (3 103 m)2] 0.74 W 䉰 3. The general lumped capacitance energy balance, Equation 5.14, may be written ql (x d)D2t /4 q Vcp dT dt

Chapter 5

䊏

Transient Conduction

Separating variables and integrating between appropriate limits, q Vc

冕 dt 冕dT t

Tt

t0

Tb

yields Vcp 989.1 kg/m3 (/6) (3 103 m)3 4180 J/kg 䡠 K t q (Tt Tb) 0.170 W

(52 C 37 C) or t 5.16 s

䉰

4. Using Equation 5.71, q/2kDt(Tt Tb) q* [1 exp(Fo)erfc(Fo1/2)]1 which may be solved by trial-and-error to yield Fo 10.3 4␣t/D2t. Then, with ␣ k/cp 0.50 W/m 䡠 K/(989.1 kg/m3 4180 J/kg 䡠 K) 1.21 107 m2/s, we find t FoD2t /4␣ 10.3 (3 103 m)2 /(4 1.21 107 m2/s) 192 s

䉰

Comments: 1. The analysis does not account for blood perfusion. The flow of blood would lead to advection of warmed fluid away from the tumor (and relatively cool blood to the vicinity of the tumor), increasing the power needed to reach the desired treatment temperature. 2. The laser power needed to treat various-sized tumors, calculated as in parts 1 and 2 of the problem solution, is shown below. Note that as the tumor becomes smaller, a higher-powered laser is needed, which may seem counterintuitive. The power required to heat the tumor, which is the same as the heat loss calculated in part 1, increases in direct proportion to the diameter, as might be expected. However, since the laser power flux remains constant, a smaller tumor cannot absorb as much energy (the energy absorbed has a D2t dependence). Less of the overall laser power is utilized to heat the tumor, and the required laser power increases for smaller tumors. 2.5

2 Laser power, Pl (W)

326

1.5

1

0.5

1

2 3 Tumor diameter, Dt (mm)

4

5.9

䊏

327

Periodic Heating

3. To determine the actual time needed for the tumor temperature to approach steadystate conditions, a numerical solution of the heat diffusion equation applied to the surrounding tissue, coupled with a solution for the temperature history within the tumor, would be required. However, we see that significantly more time is needed for the surrounding tissue to reach steady-state conditions than to increase the temperature of the isolated spherical tumor. This is due to the fact that higher temperatures propagate into a large volume when heating of the surrounding tissue is considered, while in contrast the thermal mass of the tumor is limited by the tumor’s size. Hence, the actual time to heat both the tumor and the surrounding tissue will be slightly greater than 192 s. 4. Since temperatures are likely to increase at a considerable distance from the tumor, the assumption that the surroundings are of infinite size would need to be checked by inspecting results of the proposed numerical solution described in Comment 3.

5.9 Periodic Heating In the preceding discussion of transient heat transfer, we have considered objects that experience constant surface temperature or constant surface heat flux boundary conditions. In many practical applications the boundary conditions are not constant, and analytical solutions have been obtained for situations where the conditions vary with time. One situation involving nonconstant boundary conditions is periodic heating, which describes various applications, such as thermal processing of materials using pulsed lasers, and occurs naturally in situations such as those involving the collection of solar energy. Consider, for example, the semi-infinite solid of Figure 5.11a. For a surface temperature history described by T(0, t) Ti T sin t, the solution of Equation 5.29 subject to the interior boundary condition given by Equation 5.56 is T(x, t) Ti exp[x兹/2␣] sin[t x兹/2␣] T

(5.72)

This solution applies after sufficient time has passed to yield a quasi-steady state for which all temperatures fluctuate periodically about a time-invariant mean value. At locations in the solid, the fluctuations have a time lag relative to the surface temperature.

T(0, t) = Ti ∆Tsin(t)

Ti

∆T

␦p

x

(a)

qs(0, t) = ∆qs ∆qssin(t)

y

w x

␦p

(b)

FIGURE 5.11 Schematic of (a) a periodically heated, onedimensional semi-infinite solid and (b) a periodically heated strip attached to a semi-infinite solid.

328

Chapter 5

䊏

Transient Conduction

In addition, the amplitude of the fluctuations within the material decays exponentially with distance from the surface. Consistent with the earlier definition of the thermal penetration depth, ␦p can be defined as the x-location at which the amplitude of the temperature fluctuation is reduced by approximately 90% relative to that of the surface. This ␣/. The heat flux at the surface may be determined by applying results in ␦p 4 兹苶 Fourier’s law at x 0, yielding qs(t) kT兹/␣ sin(t /4)

(5.73)

Equation 5.73 reveals that the surface heat flux is periodic, with a time-averaged value of zero. Periodic heating can also occur in two- or three-dimensional arrangements, as shown in Figure 5.11b. Recall that for this geometry, a steady state can be attained with constant heating of the strip placed upon a semi-infinite solid (Table 4.1, case 13). In a similar manner, a quasi-steady state may be achieved when sinusoidal heating (qs qs qs sin t) is applied to the strip. Again, a quasi-steady state is achieved for which all temperatures fluctuate about a time-invariant mean value. The solution of the two-dimensional, transient heat diffusion equation for the twodimensional configuration shown in Figure 5.11b has been obtained, and the relationship between the amplitude of the applied sinusoidal heating and the amplitude of the temperature response of the heated strip can be approximated as [10] T 艐

冤

冥

冤

冥

qs qs 1 ln(/2) ln(w2/4␣) C1 1 ln(/2) C2 Lk 2 Lk 2

(5.74)

where the constant C1 depends on the thermal contact resistance at the interface between the heated strip and the underlying material. Note that the amplitude of the temperature fluctuation, T, corresponds to the spatially averaged temperature of the rectangular strip of length L and width w. The heat flux from the strip to the semi-infinite medium is assumed to be spatially uniform. The approximation is valid for L w. For the system of Figure 5.11b, the thermal penetration depth is smaller than that of Figure 5.11a because of the lateral spreading of thermal energy and is ␦p 艐兹␣/.

EXAMPLE 5.10 A nanostructured dielectric material has been fabricated, and the following method is used to measure its thermal conductivity. A long metal strip 3000 angstroms thick, w 100 m wide, and L 3.5 mm long is deposited by a photolithography technique on the top surface of a d 300-m-thick sample of the new material. The strip is heated periodically by an electric current supplied through two connector pads. The heating rate is qs(t) qs qs sin(t), where qs is 3.5 mW. The instantaneous, spatially averaged temperature of the metal strip is found experimentally by measuring the time variation of its electrical resistance, R(t) E(t)/I(t), and by knowing how the electrical resistance of the metal varies with temperature. The measured temperature of the metal strip is periodic; it has an amplitude of T 1.37 K at a relatively low heating frequency of 2 rad/s and 0.71 K at a frequency of 200 rad/s. Determine the thermal conductivity of the nanostructured dielectric material. The density and specific heats of the conventional version of the material are 3100 kg/m3 and 820 J/kg 䡠 K, respectively.

5.9

䊏

329

Periodic Heating

SOLUTION Known: Dimensions of a thin metal strip, the frequency and amplitude of the electric power dissipated within the strip, the amplitude of the induced oscillating strip temperature, and the thickness of the underlying nanostructured material. Find: The thermal conductivity of the nanostructured material. Schematic: Heated metal strip

Connector pad

L I⫹

E⫹

E⫺

x

I⫺

y d

Sample z

Assumptions: 1. Two-dimensional transient conduction in the x- and z-directions. 2. Constant properties. 3. Negligible radiation and convection losses from the metal strip and top surface of the sample. 4. The nanostructured material sample is a semi-infinite solid. 5. Uniform heat flux at the interface between the heated strip and the nanostructured material. Analysis: Substitution of T 1.37 K at 2 rad/s and T 0.71 K at 200 rad/s into Equation 5.74 results in two equations that may be solved simultaneously to yield C2 5.35

k 1.11 W/m 䡠 K

䉰

The thermal diffusivity is ␣ 4.37 107 m2/s, while the thermal penetration depths ␣/, resulting in ␦p 260 m and ␦p 26 m at 2 rad/s are estimated by ␦p ⬇ 兹苶 and 200 rad/s, respectively.

Comments: 1. The foregoing experimental technique, which is widely used to measure the thermal conductivity of microscale devices and nanostructured materials, is referred to as the 3 method [10]. 2. Because this technique is based on measurement of a temperature that fluctuates about a mean value that is approximately the same as the temperature of the surroundings, the measured value of k is relatively insensitive to radiation heat transfer losses from the top of the metal strip. Likewise, the technique is insensitive to thermal contact resistances that may exist at the interface between the sensing strip and the underlying material since these effects cancel when measurements are made at two different excitation frequencies [10].

330

Chapter 5

䊏

Transient Conduction

3. The specific heat and density are not strongly dependent on the nanostructure of most solids, and properties of conventional material may be used. 4. The thermal penetration depth is less than the sample thickness. Therefore, treating the sample as a semi-infinite solid is a valid approach. Thinner samples could be used if higher heating frequencies were employed.

5.10 Finite-Difference Methods Analytical solutions to transient problems are restricted to simple geometries and boundary conditions, such as the one-dimensional cases considered in the preceding sections. For some simple two- and three-dimensional geometries, analytical solutions are still possible. However, in many cases the geometry and/or boundary conditions preclude the use of analytical techniques, and recourse must be made to finite-difference (or finite-element) methods. Such methods, introduced in Section 4.4 for steady-state conditions, are readily extended to transient problems. In this section we consider explicit and implicit forms of finite-difference solutions to transient conduction problems.

5.10.1

Discretization of the Heat Equation: The Explicit Method

Once again consider the two-dimensional system of Figure 4.4. Under transient conditions with constant properties and no internal generation, the appropriate form of the heat equation, Equation 2.21, is 1 ⭸T ⭸2T ⭸2T ␣ ⭸t ⭸x2 ⭸y2

(5.75)

To obtain the finite-difference form of this equation, we may use the central-difference approximations to the spatial derivatives prescribed by Equations 4.27 and 4.28. Once again the m and n subscripts may be used to designate the x- and y-locations of discrete nodal points. However, in addition to being discretized in space, the problem must be discretized in time. The integer p is introduced for this purpose, where t pt

(5.76)

and the finite-difference approximation to the time derivative in Equation 5.75 is expressed as ⭸T ⭸t

冏

艐 m, n

p 1 p T m, n T m, n t

(5.77)

The superscript p is used to denote the time dependence of T, and the time derivative is expressed in terms of the difference in temperatures associated with the new (p 1) and previous ( p) times. Hence calculations must be performed at successive times separated by the interval t, and just as a finite-difference solution restricts temperature determination to discrete points in space, it also restricts it to discrete points in time.

Analytical solutions for some simple two- and three-dimensional geometries are found in Section 5S.2.

5.10

䊏

331

Finite-Difference Methods

If Equation 5.77 is substituted into Equation 5.75, the nature of the finite-difference solution will depend on the specific time at which temperatures are evaluated in the finite-difference approximations to the spatial derivatives. In the explicit method of solution, these temperatures are evaluated at the previous ( p) time. Hence Equation 5.77 is considered to be a forward-difference approximation to the time derivative. Evaluating terms on the right-hand side of Equations 4.27 and 4.28 at p and substituting into Equation 5.75, the explicit form of the finite-difference equation for the interior node (m, n) is p 1 p p p p p p p 1 T m, n T m, n T m 1, n T m1, n 2T m, n T m, n 1 T m, n1 2T m,n ␣ t (x)2 (y)2

(5.78)

Solving for the nodal temperature at the new (p 1) time and assuming that x y, it follows that p 1 p p p p p T m, n Fo(T m 1, n T m1, n T m, n 1 T m, n1) (1 4Fo)T m, n

(5.79)

where Fo is a finite-difference form of the Fourier number Fo ␣ t2 (x)

(5.80)

This approach can easily be extended to one- or three-dimensional systems. If the system is one-dimensional in x, the explicit form of the finite-difference equation for an interior node m reduces to p p

T m1 ) (1 2Fo)T mp T mp 1 Fo(T m 1

(5.81)

Equations 5.79 and 5.81 are explicit because unknown nodal temperatures for the new time are determined exclusively by known nodal temperatures at the previous time. Hence calculation of the unknown temperatures is straightforward. Since the temperature of each interior node is known at t 0 ( p 0) from prescribed initial conditions, the calculations begin at t t ( p 1), where Equation 5.79 or 5.81 is applied to each interior node to determine its temperature. With temperatures known for t t, the appropriate finite-difference equation is then applied at each node to determine its temperature at t 2 t ( p 2). In this way, the transient temperature distribution is obtained by marching out in time, using intervals of t. The accuracy of the finite-difference solution may be improved by decreasing the values of x and t. Of course, the number of interior nodal points that must be considered increases with decreasing x, and the number of time intervals required to carry the solution to a prescribed final time increases with decreasing t. Hence the computation time increases with decreasing x and t. The choice of x is typically based on a compromise between accuracy and computational requirements. Once this selection has been made, however, the value of t may not be chosen independently. It is, instead, determined by stability requirements. An undesirable feature of the explicit method is that it is not unconditionally stable. In a transient problem, the solution for the nodal temperatures should continuously approach final (steady-state) values with increasing time. However, with the explicit method, this solution may be characterized by numerically induced oscillations, which are physically impossible. The oscillations may become unstable, causing the solution to diverge from the actual steady-state conditions. To prevent such erroneous results, the prescribed value of t must be maintained below a certain limit, which depends on x and other parameters of the system. This dependence is termed a stability criterion, which may be obtained mathematically or demonstrated from a thermodynamic argument (see Problem 5.108). For the problems of interest in this text, the criterion is determined by requiring that the coefficient associated with the node of interest at the previous time is greater than or equal to zero.

332

Chapter 5

䊏

Transient Conduction

p In general, this is done by collecting all terms involving T m,n to obtain the form of the coefficient. This result is then used to obtain a limiting relation involving Fo, from which the maximum allowable value of t may be determined. For example, with Equations 5.79 and 5.81 already expressed in the desired form, it follows that the stability criterion for a onedimensional interior node is (1 2Fo) 0, or

Fo 1 2

(5.82)

and for a two-dimensional node, it is (1 4Fo) 0, or Fo 1 4

(5.83)

For prescribed values of x and ␣, these criteria may be used to determine upper limits to the value of t. Equations 5.79 and 5.81 may also be derived by applying the energy balance method of Section 4.4.3 to a control volume about the interior node. Accounting for changes in thermal energy storage, a general form of the energy balance equation may be expressed as E˙ in E˙ g E˙ st

(5.84)

In the interest of adopting a consistent methodology, it is again assumed that all heat flow is into the node. To illustrate application of Equation 5.84, consider the surface node of the onedimensional system shown in Figure 5.12. To more accurately determine thermal conditions near the surface, this node has been assigned a thickness that is one-half that of the interior nodes. Assuming convection transfer from an adjoining fluid and no generation, it follows from Equation 5.84 that hA(T앝 T 0p ) kA (T 1p T 0p ) cA x x 2

T 0p 1 T 0p t

or, solving for the surface temperature at t t, (T 1p T 0p ) T 0p T 0p 1 2h t (T앝 T 0p ) 2␣ t 2 c x x x A T∞, h T0

T1

T2

T3

•

qconv

E st

qcond

∆x ___ 2

∆x

FIGURE 5.12 Surface node with convection and one-dimensional transient conduction.

5.10

䊏

333

Finite-Difference Methods

Recognizing that (2ht/cx) 2(hx/k)(␣t/x2) 2 Bi Fo and grouping terms involving T 0p, it follows that T 0p 1 2Fo(T 1p Bi T앝) (1 2Fo 2Bi Fo)T 0p

(5.85)

The finite-difference form of the Biot number is Bi h x k

(5.86)

Recalling the procedure for determining the stability criterion, we require that the coefficient for T 0p be greater than or equal to zero. Hence 1 2Fo 2Bi Fo 0 or Fo(1 Bi) 1 2

(5.87)

Since the complete finite-difference solution requires the use of Equation 5.81 for the interior nodes, as well as Equation 5.85 for the surface node, Equation 5.87 must be contrasted with Equation 5.82 to determine which requirement is more stringent. Since Bi 0, it is apparent that the limiting value of Fo for Equation 5.87 is less than that for Equation 5.82. To ensure stability for all nodes, Equation 5.87 should therefore be used to select the maximum allowable value of Fo, and hence t, to be used in the calculations. Forms of the explicit finite-difference equation for several common geometries are presented in Table 5.3a. Each equation may be derived by applying the energy balance method to a control volume about the corresponding node. To develop confidence in your ability to apply this method, you should attempt to verify at least one of these equations.

EXAMPLE 5.11 A fuel element of a nuclear reactor is in the shape of a plane wall of thickness 2L 20 mm and is convectively cooled at both surfaces, with h 1100 W/m2 䡠 K and T앝 250 C. At normal operating power, heat is generated uniformly within the element at a volumetric rate of q· 1 107 W/m3. A departure from the steady-state conditions associated with normal operation will occur if there is a change in the generation rate. Consider a sudden change to q· 2 2 107 W/m3, and use the explicit finite-difference method to determine the fuel element temperature distribution after 1.5 s. The fuel element thermal properties are k 30 W/m 䡠 K and ␣ 5 106 m2/s.

SOLUTION Known: Conditions associated with heat generation in a rectangular fuel element with surface cooling. Find: Temperature distribution 1.5 s after a change in operating power.

334

Transient, two-dimensional finite-difference equations (x y)

Chapter 5

TABLE 5.3

(a) Explicit Method Configuration

Finite-Difference Equation

Stability Criterion

(b) Implicit Method

䊏

∆y

m, n m – 1, n

m + 1, n

∆x

p 1 p p Fo(Tm 1,n

Tm1,n Tm,n p p

Tm,n 1 Tm,n1) p

(1 4Fo)Tm,n

m, n – 1

1. Interior node

m, n + 1

p 1 p p 3Fo(Tm 1,n

2Tm1,n Tm,n p p

2Tm,n 1 Tm,n1 2Bi T앝)

Fo

1 4

(5.83)

p 1 p 1 p 1 (1 4Fo)Tm,n Fo(Tm 1,n

Tm1,n p 1 p 1 p

Tm,n 1

Tm,n1 ) Tm,n

(5.95)

(5.79)

∆x

m – 1, n

2

m, n m + 1, n

∆y

T∞, h m, n – 1

m, n + 1 ∆y

T∞, h m, n

m – 1, n

p

(1 4Fo 43 Bi Fo)T m,n

(5.89)

p 1 3Fo 䡠 (1 4Fo(1 3Bi))Tm,n p 1 p 1 p 1 p 1 (Tm 1,n 2T m1,n 2Tm,n 1

Tm,n1 ) 4 p Tm,n 3 Bi Fo T앝 (5.98)

(5.91)

p 1 (1 2Fo(2 Bi))Tm,n p 1 p 1 p 1 Fo(2Tm1,n Tm,n 1

Tm,n1 ) p

2Bi Fo T앝 Tm,n

1

Fo(3 Bi)

3 4

Fo(2 Bi)

1 2

(5.88)

2

2. Node at interior corner with convection p 1 p p Fo(2Tm1,n

Tm,n 1 Tm,n p

Tm,n1 2Bi T앝) p

(1 4Fo2Bi Fo)Tm,n

(5.90)

(5.99)

m, n – 1

3. Node at plane surface with convectiona

∆x

T∞, h

m – 1, n

m, n ∆y

p 1 p p Tm,n 2Fo(Tm1,n

Tm,n1

2Bi T앝) p

(1 4Fo 4Bi Fo)Tm,n (5.92)

m, n – 1 ∆x

a

4. Node at exterior corner with convection

Fo(1 Bi)

1 4

(5.93)

p 1 (1 4Fo(1 Bi))Tm,n p 1 p 1 2Fo(Tm1,n

Tm,n1 ) p Tm,n 4Bi Fo T앝

To obtain the finite-difference equation and/or stability criterion for an adiabatic surface (or surface of symmetry), simply set Bi equal to zero.

(5.100)

Transient Conduction

m, n + 1

5.10

䊏

335

Finite-Difference Methods

Schematic: Fuel element q•1 = 1 × 107 W/m3 q•2 = 2 × 107 W/m3 α = 5 × 10–6 m2/s k = 30 W/m•K

m–1

m

T∞ = 250°C h = 1100 W/m2•K Coolant

Symmetry adiabat

1

m+1

2

3

4 5

0 •

qcond

5

4 •

•

E g, E st

Eg, qcond

qcond

•

Est

L = 10 mm

qconv

x ∆ x = _L_ 5

L ∆ x = __ ___ 2 10

Assumptions: 1. One-dimensional conduction in x. 2. Uniform generation. 3. Constant properties. Analysis: A numerical solution will be obtained using a space increment of x 2 mm. Since there is symmetry about the midplane, the nodal network yields six unknown nodal temperatures. Using the energy balance method, Equation 5.84, an explicit finite-difference equation may be derived for any interior node m. kA

p T m1 T mp T p T mp T p 1 Tmp

q˙ A x A x c m

kA m 1 x x t

Solving for T p 1 and rearranging, m

冤

p p T mp 1 Fo T m1

T m 1

冥

q˙ (x)2

(1 2Fo)T mp k

(1)

This equation may be used for node 0, with T pm1 T pm 1, as well as for nodes 1, 2, 3, and 4. Applying energy conservation to a control volume about node 5, hA(T앝 T 5p ) k A

T 4p T 5p T p 1 T 5p

q˙ A x A x c 5 x 2 2 t

or

冤

T 5p 1 2Fo T 4p Bi T앝

冥

q˙ (x)2

(1 2Fo 2Bi Fo)T 5p 2k

(2)

Since the most restrictive stability criterion is associated with Equation 2, we select Fo from the requirement that Fo(1 Bi) 1 2

336

Chapter 5

䊏

Transient Conduction

Hence, with Bi h x k

1100 W/m2 䡠 K (0.002 m) 0.0733 30 W/m 䡠 K

it follows that Fo 0.466 or t

Fo(x)2 0.466(2 103 m)2 0.373 s ␣ 5 106 m2/s

To be well within the stability limit, we select t 0.3 s, which corresponds to Fo

5 106 m2/s(0.3 s) 0.375 (2 103 m)2

Substituting numerical values, including q˙ q˙ 2 2 107 W/m3, the nodal equations become T 0p 1 0.375(2T 1p 2.67) 0.250T 0p T 1p 1 0.375(T 0p T 2p 2.67) 0.250T 1p T 2p 1 0.375(T 1p T 3p 2.67) 0.250T 2p T 3p 1 0.375(T 2p T 4p 2.67) 0.250T 3p T 4p 1 0.375(T 3p T 5p 2.67) 0.250T 4p T 5p 1 0.750(T 4p 19.67) 0.195T 5p To begin the marching solution, the initial temperature distribution must be known. This distribution is given by Equation 3.47, with q˙ q˙ 1. Obtaining Ts T5 from Equation 3.51, T5 T앝

7 3 q˙ L m 340.91 C 250 C 10 W/m 0.01 h 1100 W/m2 䡠 K

it follows that

冢

2

冣

T(x) 16.67 1 x 2 340.91 C L Computed temperatures for the nodal points of interest are shown in the first row of the accompanying table. Using the finite-difference equations, the nodal temperatures may be sequentially calculated with a time increment of 0.3 s until the desired final time is reached. The results are illustrated in rows 2 through 6 of the table and may be contrasted with the new steady-state condition (row 7), which was obtained by using Equations 3.47 and 3.51 with q˙ q˙2:

5.10

䊏

337

Finite-Difference Methods

Tabulated Nodal Temperatures p

t(s)

T0

T1

T2

T3

T4

T5

0 1 2 3 4 5 앝

0 0.3 0.6 0.9 1.2 1.5 앝

357.58 358.08 358.58 359.08 359.58 360.08 465.15

356.91 357.41 357.91 358.41 358.91 359.41 463.82

354.91 355.41 355.91 356.41 356.91 357.41 459.82

351.58 352.08 352.58 353.08 353.58 354.07 453.15

346.91 347.41 347.91 348.41 348.89 349.37 443.82

340.91 341.41 341.88 342.35 342.82 343.27 431.82

Comments: 1. It is evident that, at 1.5 s, the wall is in the early stages of the transient process and that many additional calculations would have to be made to reach steady-state conditions with the finite-difference solution. The computation time could be reduced slightly by using the maximum allowable time increment (t 0.373 s), but with some loss of accuracy. In the interest of maximizing accuracy, the time interval should be reduced until the computed results become independent of further reductions in t. Extending the finite-difference solution, the time required to achieve the new steady-state condition may be determined, with temperature histories computed for the midplane (0) and surface (5) nodes having the following forms: 480 465.1

T0

T (°C)

440 431.8

T5 400

360

320

0

100

200 t (s)

300

400

With steady-state temperatures of T0 465.15 C and T5 431.82 C, it is evident that the new equilibrium condition is reached within 250 s of the step change in operating power. 2. This problem can be solved using Tools, Finite-Difference Equations, One-Dimensional, Transient in the Advanced section of IHT. The problem may also be solved using FiniteElement Heat Transfer (FEHT).

5.10.2

Discretization of the Heat Equation: The Implicit Method

In the explicit finite-difference scheme, the temperature of any node at t t may be calculated from knowledge of temperatures at the same and neighboring nodes for the preceding time t. Hence determination of a nodal temperature at some time is independent of

338

Chapter 5

䊏

Transient Conduction

temperatures at other nodes for the same time. Although the method offers computational convenience, it suffers from limitations on the selection of t. For a given space increment, the time interval must be compatible with stability requirements. Frequently, this dictates the use of extremely small values of t, and a very large number of time intervals may be necessary to obtain a solution. A reduction in the amount of computation time may often be realized by employing an implicit, rather than explicit, finite-difference scheme. The implicit form of a finite-difference equation may be derived by using Equation 5.77 to approximate the time derivative, while evaluating all other temperatures at the new (p 1) time, instead of the previous (p) time. Equation 5.77 is then considered to provide a backward-difference approximation to the time derivative. In contrast to Equation 5.78, the implicit form of the finite-difference equation for the interior node of a two-dimensional system is then p 1 p p 1 p 1 p 1 1 T m, n T m, n T m 1, n T m1, n 2T m, n ␣ t (x)2

p 1 p 1 p 1 T m, n 1 T m, n1 2T m, n

(y)2

(5.94)

Rearranging and assuming x y, it follows that p 1 p 1 p 1 p 1 p 1 p (1 4Fo)T m, n Fo(T m 1, n T m1, n T m, n 1 T m, n1) T m, n

(5.95)

From Equation 5.95 it is evident that the new temperature of the (m, n) node depends on the new temperatures of its adjoining nodes, which are, in general, unknown. Hence, to determine the unknown nodal temperatures at t t, the corresponding nodal equations must be solved simultaneously. Such a solution may be effected by using Gauss–Seidel iteration or matrix inversion, as discussed in Section 4.5 and Appendix D. The marching solution would then involve simultaneously solving the nodal equations at each time t t, 2t, . . . , until the desired final time was reached. Relative to the explicit method, the implicit formulation has the important advantage of being unconditionally stable. That is, the solution remains stable for all space and time intervals, in which case there are no restrictions on x and t. Since larger values of t may therefore be used with an implicit method, computation times may often be reduced, with little loss of accuracy. Nevertheless, to maximize accuracy, t should be sufficiently small to ensure that the results are independent of further reductions in its value. The implicit form of a finite-difference equation may also be derived from the energy balance method. For the surface node of Figure 5.12, it is readily shown that (1 2Fo 2Fo Bi)T 0p 1 2Fo T 1p 1 2Fo Bi T앝 T 0p

(5.96)

For any interior node of Figure 5.12, it may also be shown that p 1 p 1

T m 1 ) T mp (1 2Fo)T mp 1 Fo (T m1

(5.97)

Forms of the implicit finite-difference equation for other common geometries are presented in Table 5.3b. Each equation may be derived by applying the energy balance method.

5.10

䊏

339

Finite-Difference Methods

EXAMPLE 5.12 A thick slab of copper initially at a uniform temperature of 20 C is suddenly exposed to radiation at one surface such that the net heat flux is maintained at a constant value of 3 105 W/m2. Using the explicit and implicit finite-difference techniques with a space increment of x 75 mm, determine the temperature at the irradiated surface and at an interior point that is 150 mm from the surface after 2 min have elapsed. Compare the results with those obtained from an appropriate analytical solution.

SOLUTION Known: Thick slab of copper, initially at a uniform temperature, is subjected to a constant net heat flux at one surface. Find: 1. Using the explicit finite-difference method, determine temperatures at the surface and 150 mm from the surface after an elapsed time of 2 min. 2. Repeat the calculations using the implicit finite-difference method. 3. Determine the same temperatures analytically. Schematic: q"o = 3 × 105 W/m2

0

q"o

m–1

1

q"cond

q"cond

∆x ___ 2

x

m

m+1 q"cond

∆x = 75 mm

Assumptions: 1. One-dimensional conduction in x. 2. For the analytical solution, the thick slab may be approximated as a semi-infinite medium with constant surface heat flux. For the finite-difference solutions, implementation of the boundary condition T(x l 앝) Ti will be discussed below in this example. 3. Constant properties. Properties: Table A.1, copper (300 K): k 401 W/m 䡠 K, ␣ 117 106 m2/s. Analysis: 1. An explicit form of the finite-difference equation for the surface node may be obtained by applying an energy balance to a control volume about the node. qo A kA

T 1p T 0p T p 1 T 0p A x c 0 x 2 t

340

Chapter 5

䊏

Transient Conduction

or T 0p 1 2Fo

冢qkx T 冣 (1 2Fo)T 0

p 1

p 0

The finite-difference equation for any interior node is given by Equation 5.81. Both the surface and interior nodes are governed by the stability criterion Fo 1 2 Noting that the finite-difference equations are simplified by choosing the maxi1 mum allowable value of Fo, we select Fo 2. Hence (x)2 (0.075 m)2 24 s t Fo ␣ 1 2 117 106 m2/s With qo x 3 105 W/m2 (0.075 m) 56.1 C k 401 W/m 䡠 K the finite-difference equations become T 0p 1 56.1 C T 1p

and

T mp 1

p p T m 1

T m1 2

for the surface and interior nodes, respectively. Performing the calculations, the results are tabulated as follows:

Explicit Finite-Difference Solution for Fo 2 1

p

t(s)

T0

T1

T2

T3

T4

0 1 2 3 4 5

0 24 48 72 96 120

20 76.1 76.1 104.2 104.2 125.2

20 20 48.1 48.1 69.1 69.1

20 20 20 34.0 34.0 48.1

20 20 20 20 27.0 27.0

20 20 20 20 20 23.5

After 2 min, the surface temperature and the desired interior temperature are T0 125.2 C and T2 48.1 C. It can be seen from the explicit finite-difference solution that, with each successive time step, one more nodal temperature changes from its initial condition. For this reason, it is not necessary to formally implement the second boundary condition T(x l 앝) T. Also note that calculation of identical temperatures at successive times for the same node is an idiosyncrasy of using the maximum allowable value of Fo with the explicit finitedifference technique. The actual physical condition is, of course, one in which the temperature changes continuously with time. The idiosyncrasy is diminished and the accuracy of the calculations is improved by reducing the value of Fo.

5.10

䊏

341

Finite-Difference Methods

To determine the extent to which the accuracy may be improved by reducing Fo, 1 let us redo the calculations for Fo 4 (t 12 s). The finite-difference equations are then of the form T 0p 1 1 (56.1 C T 1p) 1T 0p 2 2 p p T mp 1 1(T m 1

T m1 ) 1T mp 4 2 and the results of the calculations are tabulated as follows:

Explicit Finite-Difference Solution for Fo 4 1

p

t(s)

T0

T1

T2

0 1 2 3 4 5 6 7 8 9 10

0 12 24 36 48 60 72 84 96 108 120

20 48.1 62.1 72.6 81.4 89.0 95.9 102.3 108.1 113.6 118.8

20 20 27.0 34.0 40.6 46.7 52.5 57.9 63.1 67.9 72.6

20 20 20 21.8 24.4 27.5 30.7 34.1 37.6 41.0 44.4

T3

T4

20 20 20 20 2