Fundamentals of Heat and Mass Transfer 7th Edition

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SEVENTH EDITION

Fundamentals of Heat and Mass 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

VICE PRESIDENT & PUBLISHER EXECUTIVE EDITOR EDITORIAL ASSISTANT MARKETING MANAGER PRODUCTION MANAGER PRODUCTION EDITOR DESIGNER EXECUTIVE MEDIA EDITOR PRODUCTION MANAGEMENT SERVICES

Don Fowley Linda Ratts Renata Marchione Christopher Ruel Dorothy Sinclair Sandra Dumas Wendy Lai Thomas Kulesa MPS Ltd.

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 13 978-0470-50197-9 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

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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, 1.26 ) are entirely computer based.

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v

What’s New in the 7th Edition
Chapter-by-Chapter Content Changes 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

vi

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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. Chapter 14 Diffusion Mass Transfer was revised extensively for the previous edition, and only modest changes have been made in this edition.
Problem Sets Approximately 250 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

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vii

(b) optional material that instructors can use to address speciﬁc interests or that can be covered in a second, intermediate heat transfer course. To assist instructors in preparing a syllabus for a rst course in heat transfer , we have several recommendations. ﬁ 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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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. The material in Chapter 14 Diffusion Mass Transfer is relevant to many contemporary technologies, particularly those involving materials synthesis, chemical processing, and energy conversion. Emerging applications in biotechnology also exhibit strong diffusion mass transfer effects. Time permitting, we encourage coverage of Chapter 14. However, if only problems involving stationary media are of interest, Section 14.2 may be omitted or included in a follow-on 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 (tberg@engr.uconn.edu) Storrs, Connecticut Adrienne S. Lavine (lavine@seas.ucla.edu) Los Angeles, California Frank P. Incropera (fpi@nd.edu) 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 identies Supplemental Sections and is used throughout the text . ﬁ 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.
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Contents

Symbols
CHAPTER

xxi

1 Introduction
1.1 1.2 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

1
2 3

1.3

12

1.4 1.5

36 38

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Contents 1.6 1.7 Relevance of Heat Transfer Summary References Problems 41 45 48 49

CHAPTER

2 Introduction to Conduction
2.1 2.2 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

67
68 70

2.3 2.4 2.5

82 90 94 95 95

CHAPTER

3 One-Dimensional, Steady-State Conduction
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 3.1

111
112

132 136

142 142

154

178 182 189

190 193 193

Contents

xiii
229
230 231 235 241

CHAPTER

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

4.5

250

4.6

256 257 257 W-1

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

W-5 W-9 W-10

5 Transient Conduction
5.1 5.2 5.3 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

279
280 283 287

289 298 299

5.4 5.5

5.6

303

5.7 5.8

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 330 5.10.2 Discretization of the Heat Equation: The Implicit Method 337 5.11 Summary References Problems 5S.1 Graphical Representation of One-Dimensional, Transient Conduction in the Plane Wall, Long Cylinder, and Sphere 5S.2 Analytical Solutions of Multidimensional Effects References Problems 327 330

345 346 346 W-12 W-16 W-22 W-22

CHAPTER

6 Introduction to Convection
6.1 The Convection Boundary Layers 6.1.1 The Velocity Boundary Layer 378 6.1.2 The Thermal Boundary Layer 379 6.1.3 The Concentration Boundary Layer 380 6.1.4 Signiﬁcance of the Boundary Layers 382 Local and Average Convection Coefﬁcients 6.2.1 Heat Transfer 382 6.2.2 Mass Transfer 383 6.2.3 The Problem of Convection 385 Laminar and Turbulent Flow 6.3.1 Laminar and Turbulent Velocity Boundary Layers 389 6.3.2 Laminar and Turbulent Thermal and Species Concentration Boundary Layers 391 The Boundary Layer Equations 6.4.1 Boundary Layer Equations for Laminar Flow 394 6.4.2 Compressible Flow 397 Boundary Layer Similarity: The Normalized Boundary Layer Equations 6.5.1 Boundary Layer Similarity Parameters 398 6.5.2 Functional Form of the Solutions 400 Physical Interpretation of the Dimensionless Parameters Boundary Layer Analogies 6.7.1 The Heat and Mass Transfer Analogy 410 6.7.2 Evaporative Cooling 413 6.7.3 The Reynolds Analogy 416 Summary References Problems

377
378

6.2

382

6.3

389

6.4

394

6.5

398

6.6 6.7

407 409

6.8

417 418 419 W-25

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 6S.1.4 Conservation of Species W-32 References Problems

W-36 W-36

Contents

xv
433
435 436 437

CHAPTER

7 External Flow
7.1 7.2 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 443 7.2.3 Mixed Boundary Layer Conditions 444 7.2.4 Unheated Starting Length 445 7.2.5 Flat Plates with Constant Heat Flux Conditions 446 7.2.6 Limitations on Use of Convection Coefﬁcients 446 Methodology for a Convection Calculation The Cylinder in Cross Flow 7.4.1 Flow Considerations 455 7.4.2 Convection Heat and Mass Transfer 457 The Sphere Flow Across Banks of Tubes Impinging Jets 7.7.1 Hydrodynamic and Geometric Considerations 477 7.7.2 Convection Heat and Mass Transfer 478 Packed Beds Summary References Problems

7.3 7.4

447 455

7.5 7.6 7.7

465 468 477

7.8 7.9

482 483 486 486

CHAPTER

8 Internal Flow
8.1 Hydrodynamic Considerations 8.1.1 Flow Conditions 518 8.1.2 The Mean Velocity 519 8.1.3 Velocity Proﬁle in the Fully Developed Region 520 8.1.4 Pressure Gradient and Friction Factor in Fully Developed Flow 522 Thermal Considerations 8.2.1 The Mean Temperature 524 8.2.2 Newton’s Law of Cooling 525 8.2.3 Fully Developed Conditions 525 The Energy Balance 8.3.1 General Considerations 529 8.3.2 Constant Surface Heat Flux 530 8.3.3 Constant Surface Temperature 533 Laminar Flow in Circular Tubes: Thermal Analysis and Convection Correlations 8.4.1 The Fully Developed Region 537 8.4.2 The Entry Region 542 8.4.3 Temperature-Dependent Properties 544 Convection Correlations: Turbulent Flow in Circular Tubes Convection Correlations: Noncircular Tubes and the Concentric Tube Annulus Heat Transfer Enhancement

517
518

8.2

523

8.3

529

8.4

537

8.5 8.6 8.7

544 552 555

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Contents 8.8 Flow in Small Channels 8.8.1 Microscale Convection in Gases (0.1 m Dh 8.8.2 Microscale Convection in Liquids 559 8.8.3 Nanoscale Convection (Dh 100 nm) 560 8.9 Convection Mass Transfer 8.10 Summary References Problems 558 100 m) 558

563 565 568 569

CHAPTER

9 Free Convection
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 605 9.6.2 Inclined and Horizontal Plates 608 9.6.3 The Long Horizontal Cylinder 613 9.6.4 Spheres 617 9.7 Free Convection Within Parallel Plate Channels 9.7.1 Vertical Channels 619 9.7.2 Inclined Channels 621 9.8 Empirical Correlations: Enclosures 9.8.1 Rectangular Cavities 621 9.8.2 Concentric Cylinders 624 9.8.3 Concentric Spheres 625 9.9 Combined Free and Forced Convection 9.10 Convection Mass Transfer 9.11 Summary References Problems 9.1 9.2 9.3 9.4 9.5 9.6

593
594 597 598 599 602 604

618

621

627 628 629 630 631

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 656 10.3.2 Modes of Pool Boiling 657 10.4 Pool Boiling Correlations 10.4.1 Nucleate Pool Boiling 660 10.4.2 Critical Heat Flux for Nucleate Pool Boiling 662 10.4.3 Minimum Heat Flux 663 10.4.4 Film Pool Boiling 663 10.4.5 Parametric Effects on Pool Boiling 664

653
654 655 656

660

Contents 10.5 Forced Convection Boiling 10.5.1 External Forced Convection Boiling 670 10.5.2 Two-Phase Flow 670 10.5.3 Two-Phase Flow in Microchannels 673 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 10.12 Summary References Problems
CHAPTER

xvii
669

673 675 679 684 689 690 691 691 693

11 Heat Exchangers
11.1 11.2 11.3 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 712 11.3.2 The Counterﬂow Heat Exchanger 714 11.3.3 Special Operating Conditions 715 Heat Exchanger Analysis: The Effectiveness–NTU Method 11.4.1 Deﬁnitions 722 11.4.2 Effectiveness–NTU Relations 723 Heat Exchanger Design and Performance Calculations Additional Considerations Summary References Problems

705
706 708 711

11.4

722

11.5 11.6 11.7

730 739 747 748 748 W-40 W-44 W-49 W-50

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

12 Radiation: Processes and Properties
12.1 12.2 12.3 Fundamental Concepts Radiation Heat Fluxes Radiation Intensity 12.3.1 Mathematical Deﬁnitions 773 12.3.2 Radiation Intensity and Its Relation to Emission 774 12.3.3 Relation to Irradiation 779 12.3.4 Relation to Radiosity for an Opaque Surface 781 12.3.5 Relation to the Net Radiative Flux for an Opaque Surface

767
768 771 773

782

xviii

Contents 12.4 Blackbody Radiation 12.4.1 The Planck Distribution 783 12.4.2 Wien’s Displacement Law 784 12.4.3 The Stefan–Boltzmann Law 784 12.4.4 Band Emission 785 12.5 Emission from Real Surfaces 12.6 Absorption, Reﬂection, and Transmission by Real Surfaces 12.6.1 Absorptivity 802 12.6.2 Reﬂectivity 803 12.6.3 Transmissivity 805 12.6.4 Special Considerations 805 12.7 Kirchhoff’s Law 12.8 The Gray Surface 12.9 Environmental Radiation 12.9.1 Solar Radiation 819 12.9.2 The Atmospheric Radiation Balance 821 12.9.3 Terrestrial Solar Irradiation 823 12.10 Summary References Problems 782

792 801

810 812 818

826 830 830

CHAPTER

13 Radiation Exchange Between Surfaces
13.1 The View Factor 13.1.1 The View Factor Integral 862 13.1.2 View Factor Relations 863 Blackbody Radiation Exchange Radiation Exchange Between Opaque, Diffuse, Gray Surfaces in an Enclosure 13.3.1 Net Radiation Exchange at a Surface 877 13.3.2 Radiation Exchange Between Surfaces 878 13.3.3 The Two-Surface Enclosure 884 13.3.4 Radiation Shields 886 13.3.5 The Reradiating Surface 888 Multimode Heat Transfer Implications of the Simplifying Assumptions Radiation Exchange with Participating Media 13.6.1 Volumetric Absorption 896 13.6.2 Gaseous Emission and Absorption 897 Summary References Problems

861
862

13.2 13.3

872 876

13.4 13.5 13.6

893 896 896

13.7

901 902 903

CHAPTER

14 Diffusion Mass Transfer
14.1 Physical Origins and Rate Equations 14.1.1 Physical Origins 934 14.1.2 Mixture Composition 935 14.1.3 Fick’s Law of Diffusion 936 14.1.4 Mass Diffusivity 937

933
934

Contents 14.2 Mass Transfer in Nonstationary Media 14.2.1 Absolute and Diffusive Species Fluxes 939 14.2.2 Evaporation in a Column 942 The Stationary Medium Approximation Conservation of Species for a Stationary Medium 14.4.1 Conservation of Species for a Control Volume 948 14.4.2 The Mass Diffusion Equation 948 14.4.3 Stationary Media with Speciﬁed Surface Concentrations 950 Boundary Conditions and Discontinuous Concentrations at Interfaces 14.5.1 Evaporation and Sublimation 955 14.5.2 Solubility of Gases in Liquids and Solids 955 14.5.3 Catalytic Surface Reactions 960 Mass Diffusion with Homogeneous Chemical Reactions Transient Diffusion Summary References Problems

xix
939

14.3 14.4

947 947

14.5

954

14.6 14.7 14.8

962 965 971 972 972

APPENDIX

A Thermophysical Properties of Matter B Mathematical Relations and Functions C Thermal Conditions Associated with Uniform Energy

981

APPENDIX

1013

APPENDIX

Generation in One-Dimensional, Steady-State Systems
APPENDIX

1019

D The Gauss–Seidel Method E The Convection Transfer Equations
E.1 E.2 E.3 E.4 Conservation of Mass Newton’s Second Law of Motion Conservation of Energy Conservation of Species

1025

APPENDIX

1027
1028 1028 1029 1030

APPENDIX

F Boundary Layer Equations for Turbulent Flow

1031

G An Integral Laminar Boundary Layer Solution for Parallel Flow over a Flat Plate
APPENDIX

1035

Index

1039

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Symbols

A Ab Ac Ap Ar a Bi Bo C CD Cf Ct Co c cp cv D DAB Db Dh d E E tot Ec ˙ Eg ˙ Ein ˙ Eout ˙ Est e F

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 molar concentration, kmol/m3; heat capacity rate, W/K drag coefﬁcient friction coefﬁcient thermal capacitance, J/K Conﬁnement number speciﬁc heat, J/kg K; speed of light, m/s speciﬁc heat at constant pressure, J/kg K speciﬁc heat at constant volume, J/kg K diameter, m binary mass diffusivity, m2/s 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

Fo Fr f G Gr Gz g H h hfg h fg hsf hm hrad I i J Ja J* i ji jH jm k kB k0 k1 k1 L Le

Fourier number Froude number friction factor; similarity variable irradiation, W/m2; mass velocity, kg/s m2 Grashof number Graetz number gravitational acceleration, m/s2 nozzle height, m; Henry’s constant, bars convection heat transfer coefﬁcient, W/m2 K; Planck’s constant, J s latent heat of vaporization, J/kg modiﬁed heat of vaporization, J/kg latent heat of fusion, J/kg convection mass transfer coefﬁcient, m/s radiation heat transfer coefﬁcient, W/m2 K electric current, A; radiation intensity, W/m2 sr electric current density, A/m2; enthalpy per unit mass, J/kg radiosity, W/m2 Jakob number diffusive molar ﬂux of species i relative to the mixture molar average velocity, kmol/s m2 diffusive mass ﬂux of species i relative to the mixture mass average velocity, kg/s m2 Colburn j factor for heat transfer Colburn j factor for mass transfer thermal conductivity, W/m K Boltzmann’s constant, J/K zero-order, homogeneous reaction rate constant, kmol/s m3 ﬁrst-order, homogeneous reaction rate constant, s 1 ﬁrst-order, surface reaction rate constant, m/s length, m Lewis number

xxii

Symbols

M ˙ Mi ˙ Mi,g ˙ Min ˙ Mout ˙ Mst i Ma m ˙ m mi N NL, NT Nu NTU Ni Ni ˙ Ni ˙ Ni

ni ˙ ni

P PL , PT Pe Pr p Q q ˙ q q q q* R Ra Re Re Rf Rm Rm,n Rt Rt,c Rt,f

mass, kg rate of transfer of mass for species, i, kg/s rate of increase of mass of species i due to chemical reactions, kg/s 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 mass fraction of species i, i / integer number number of tubes in longitudinal and transverse directions Nusselt number number of transfer units molar transfer rate of species i relative to ﬁxed coordinates, kmol/s molar ﬂux of species i relative to ﬁxed coordinates, kmol/s m2 molar rate of increase of species i per unit volume due to chemical reactions, kmol/s m3 surface reaction rate of species i, kmol/s m2 Avogadro’s number mass ﬂux of species i relative to ﬁxed coordinates, kg/s m2 mass rate of increase of species i per unit volume due to chemical reactions, kg/s m3 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 K universal gas constant, J/kmol K Rayleigh number Reynolds number electric resistance, fouling factor, m2 K/W mass transfer resistance, s/m3 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 solubility, kmol/m3 atm; shape factor for two-dimensional conduction, m; nozzle pitch, m; plate spacing, m; Seebeck coefﬁcient, V/K Sc solar constant, W/m2 SD, SL, ST diagonal, longitudinal, and transverse pitch of a tube bank, m Sc Schmidt number Sh Sherwood number St Stanton number T temperature, K t time, s U overall heat transfer coefﬁcient, W/m2 K; internal energy, J u, v, w mass average ﬂuid velocity components, m/s u*, v*, w* molar average velocity components, m/s V volume, m3; ﬂuid velocity, m/s v speciﬁc volume, m3/kg W width of a slot nozzle, m ˙ W rate at which work is performed, W We Weber number X vapor quality Xtt Martinelli parameter X, Y, Z components of the body force per unit volume, N/m3 x, y, z rectangular coordinates, m xc critical location for transition to turbulence, m xfd,c concentration entry length, m xfd,h hydrodynamic entry length, m xfd,t thermal entry length, m xi mole fraction of species i, Ci /C Z thermoelectric material property, K 1 Rt,o ro r, , z r, , S Greek Letters thermal diffusivity, m2/s; accommodation coefﬁcient; absorptivity volumetric thermal expansion coefﬁcient, K 1 mass ﬂow rate per unit width in ﬁlm condensation, kg/s m ratio of speciﬁc heats hydrodynamic boundary layer thickness, m concentration boundary layer thickness, m c thermal penetration depth, m p thermal boundary layer thickness, m t emissivity; porosity; heat exchanger effectiveness ﬁn effectiveness f thermodynamic efﬁciency; similarity variable ﬁn efﬁciency f overall efﬁciency of ﬁn array o zenith angle, rad; temperature difference, K absorption coefﬁcient, m 1 wavelength, m mean free path length, nm mfp

Symbols

xxiii h i hydrodynamic; hot ﬂuid; helical general species designation; inner surface of an annulus; initial condition; tube inlet condition; incident radiation based on characteristic length saturated liquid conditions latent energy log mean condition 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 steady state surroundings thermal transmitted saturated vapor conditions local conditions on a surface spectral free stream conditions

e

viscosity, kg/s m kinematic viscosity, m2/s; frequency of radiation, s 1 mass density, kg/m3; reﬂectivity electric resistivity, /m Stefan–Boltzmann constant, W/m2 K4; electrical conductivity, 1/ 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

L l lat lm m max o p ph R r, ref rad S s sat sens sky ss sur t tr v x

Subscripts A, B species in a binary mixture abs absorbed am arithmetic mean atm atmospheric b base of an extended surface; blackbody C carnot c cross-sectional; concentration; cold ﬂuid; critical cr critical insulation thickness cond conduction conv convection CF counterﬂow D diameter; drag dif diffusion e excess; emission; electron evap evaporation f ﬂuid properties; ﬁn conditions; saturated liquid conditions fc forced convection fd fully developed conditions g saturated vapor conditions H heat transfer conditions

Superscripts * molar average; dimensionless quantity Overbar surface average conditions; time mean

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C H A P T E R

Introduction

1

2

Chapter 1

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

Convection from a surface to a moving fluid

Net radiation heat exchange between two surfaces

T1

T 1 > T2

T2

Ts > T∞
Moving fluid, T∞

Surface, T1 Surface, T2

q"

q" Ts

q" 1 q" 2

FIGURE 1.1

Conduction, convection, and radiation heat transfer modes.

1.2

Physical Origins and Rate Equations

3

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 T1 > T2

xo

q" x

q" x

x

T2

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

4

Chapter 1

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 Fouriers law. For the one-dimensional ’ plane wall shown in Figure 1.3, having a temperature distribution T(x), the rate equation is expressed as qx k dT dx (1.1)

The heat ﬂux qx (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 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 dx and the heat ﬂux is then qx or qx k T1 L T2 k T L (1.2) k T2 L T1 T2 L T1

Note that this equation provides a heat ux , that is, the rate of heat transfer per unit area. ﬂ 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 A.

1.2

Physical Origins and Rate Equations

5

* 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 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 T1 = 1400 K T2 = 1150 K qx qx ''

x

Wall area, A

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
T 1.7 W/m 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 k qx (HW) qx (0.5 m 1.2 m) 2833 W/m2 1700 W

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

Introduction

1.2.2

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 u 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 T 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 T , 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

u∞

Fluid

y

T∞

Velocity distribution u(y)

q"

Temperature distribution T(y) Ts x

u(y)

Heated surface

T(y)

FIGURE 1.4 Boundary layer development in convection heat transfer.

1.2

Physical Origins and Rate Equations

7

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''
Air Hot components on printed circuit boards

q''

Air (a) (b)

Moist air

q''
Cold water Vapor bubbles

Water droplets

q"
Water

Hot plate (c) (d)

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

8

Chapter 1

Introduction

TABLE 1.1 Typical values of the convection heat transfer coefﬁcient
Process Free convection Gases Liquids Forced convection Gases Liquids Convection with phase change Boiling or condensation h (W/m2 K) 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 T ) (1.3a)

where q , the convective heat ux (W/m2), is proportional to the difference between the surﬂ face and ﬂuid temperatures, Ts and T , respectively. This expression is known as Newtons ’ law of cooling, and the parameter h (W/m2 K) is termed the convection heat transfer coefﬁ cient. 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 T ) and negative if heat is transferred to the surface (T Ts). However, nothing precludes us from expressing Newton’s law of cooling as q h(T 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

Physical Origins and Rate Equations

9

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 StefanBoltzmann law – Eb
4 Ts

(1.4)

where Ts is the absolute temperature (K) of the surface and is the Stefan– 5.67 10 8 W/m2 K4). Such a surface is called an ideal radiator Boltzmann constant ( 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 T4 s (1.5)

where is a radiative property of the surface termed the emissivity. With values in the 1, this property provides a measure of how efﬁciently a surface emits energy range 0 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) 1. If 1 and the surface is opaque, portions of the irradiation are where 0 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.

Gas

Gas

T, h G E q" conv Surroundings at Tsur q" rad

T, h q" conv

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

Surface of emissivity = , area A, and temperature Ts (b)

Ts > Tsur, Ts > T

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

10

Chapter 1

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 a blackbody at Tsur, in which case G T sur. If the surface is assumed to be one for which (a gray surface), the net rate of radiation heat transfer from the surface, expressed per unit area of the surface, is qrad q A Eb(Ts ) G (T 4 s
4 Tsur)

(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 coefcient h ﬁ hr (Ts Tsur)(Ts2
2 Tsur) r

is (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 T ) 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 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

Physical Origins and Rate Equations

11

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

Air

q' E L Ts = 200°C ε = 0.8 G

T∞ = 25°C h = 15 W/m2•K

D = 70 mm

Tsur = 25°C

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 G Ts4
4 T sur

0.8(5.67 5.67 10

10
8

8

W/m2 K4)(473 K)4

2270 W/m2

W/m2 K4 (298 K)4

447 W/m2

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, q q L q h( DL)(Ts T ) ( DL) (T 4 s
4 Tsur)

The heat loss per unit length of pipe is then q 15 W/m2 K( 0.8( 577 W/m 0.07 m)(200 10
8

25) C W/m2 K4 (4734 2984) K4

0.07 m) 5.67 421 W/m

998 W/m

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

Introduction

2. The net rate of radiation heat transfer from the pipe may be expressed as q rad q rad D (E G) 0.8 447) W/m2 421 W/m 0.07 m (2270

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 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 qA 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 rst law of thermodynamics (the law of conservation of energy) provides ﬁ 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

Relationship to Thermodynamics

13

(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: tot Est

Q

W

(1.12a)

tot where Est 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
(b)
•

(a)

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

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: ˙ Est dEst dt ˙ Ein ˙ Eout ˙ Eg (1.12c)

1.3

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 (solidication, freezing ), the latent energy decreases. ﬁ 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

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 ˙ and the speciﬁc volume of the ﬂuid (pv). The symbol W is traditionally used for the rate at 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: ˙ m (ut pv
1

⁄2 V 2

gz)in

˙ m (ut

pv

1

⁄2 V 2

gz)out

q

˙ W

0

(1.12d)

Terms within the parentheses are expressed for a unit mass of ﬂuid at the inﬂow and out˙ ﬂow 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 specic heats , the difference in enthalpies (per unit mass) between ﬁ the inlet and outlet ﬂows may then be expressed as (iin iout) cp(Tin Tout), where cp is

q zout (ut , pv, V)out

(ut , pv, V)in

zin

W

•

Reference height

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

1.3

Relationship to Thermodynamics

17

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 ow rate m of the ﬂuid may be expressed as m ﬂ VAc, where is the ﬂuid density and Ac is the cross-sectional area of the channel through which the ﬂuid ﬂows. The ˙ volumetric ow rate is simply ˙ VAc m/ . ﬂ

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 T 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

Introduction

and the convective heat transfer coefﬁcient is h 35 W/m2 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 L 35 W/m2·K 6m D 3m

Air T∞ 25°C Ts ,ε

0.83 0.95 0.93

Generator, ηgen Gearbox, ηgb Hub Nacelle

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:
Air 20°C T∞ 25°C h 35 W/m2·K L
• Eg

Tsur

qrad qconv

6m D 3m

Ts ε 0.83 ηgen ηgb 0.95 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

Relationship to Thermodynamics

19

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: tot dEst dt

˙ E tot in

tot ˙ Eout

(1)

tot ˙ ˙ ˙ Under steady-state conditions, this reduces to E tot Eout 0. The E tot term corresponds to in in tot ˙ ˙ the mechanical work entering the nacelle W, and the Eout 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
˙ Ein ˙ Eout ˙ Eg 0 (3) ˙ ˙ ˙ Here, Ein once again corresponds to the mechanical work W. However, Eout 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 ˙ P. in the generator, giving rise to a negative source of mechanical energy. That is, Eg 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 Equation (5) can therefore be written as q P 1 gb gen

˙ W

gb gen

(6)

1

2.5

106 W

1 0.93 0.95

1

0.33

106 W

(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

Introduction

q

qrad DL

qconv A[qrad D2 4

qconv]
4 Tsur)

[

(Ts4

h(Ts

T )]

0.33

106 W

or 3m 6m 5.67 10 (3 m)2 4
8

[0.83

W/m2 K4 (Ts4 25)K)]

(273 0.33

20)4)K4 106 W

35 W/m2 K (Ts

(273

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.

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 Re 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

Relationship to Thermodynamics

21

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

T∞, h

E out T
• •

•

Tsur

I

E g, E st
L

Diameter D

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
˙ Eg ˙ Eg ˙ Eout ˙ Est

where thermal energy generation is due to the electric resistance heating, 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 Eg qV, where q I 2Re /( D2/4). Energy outﬂow is due to convection and net radiation from the surface, Equations 1.3a and 1.7, respectively, ˙ Eout h( DL)(T T ) dUt dt ( DL)(T 4
4 Tsur)

and the change in energy storage is due to the temperature change, ˙ Est d ( VcT) dt

˙ The term Est 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,

22

Chapter 1

Introduction

and V is the volume of the rod, V energy balance, it follows that I 2Re L Hence dT dt I 2Re Dh(T h( DL)(T T )

( D2/4)L. Substituting the rate equations into the ( DL)(T 4
4 T sur)

c

D2 L dT 4 dt

T ) D c( D2/4)

(T 4

4 Tsur)

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
Dh(T T ) D (T 4
4 Tsur)

I 2Re

2. For ﬁxed environmental conditions (h, T , 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 K, T 300 K). Substituting these values into the foregoing equation, the rod temperature has been computed for operating currents in the range 0 I 10 A, and the following results were obtained:
150 125 100

T ( C)

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 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

Relationship to Thermodynamics

23

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 4e ); at the cathode, the protons and electrons recomelectrons are generated (2H2 l 4H bine 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

H2

H2 e

e

O2

O2

Eg
H2 e q

•

H e

H2O O2

Tc q

H H2 e H e

H2O O2

Tsur

H2O

H2O

O2

Porous anode

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 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 Eg 11.25 W must be removed from the fuel cell by convection and radiation. The ambient and surrounding

24

Chapter 1

Introduction

temperatures are T Tsur 25 C, and the relationship between the cooling air velocity and the convection heat transfer coefﬁcient h is h 10.9 W s0.8/m2.8 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 Eg

•

Tsur = 25 C

Tc = 56.4 C ε = 0.88 T∞ = 25 C h

Air

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

Relationship to Thermodynamics

25

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 Ein 0 and Eout Eg. This yields qconv where qrad A (Tc4 0.88 0.97 W Therefore, we may ﬁnd qconv 11.25 W hA(Tc which may be rearranged to yield V V 10.9 W . s0.8 m2.8 . K 9.4 m/s 10.28 W (2 0.05 m 0.05 m)
1.25 4 Tsur)

qrad

˙ Eg

11.25 W

(2

0.05 m

0.05 m)

5.67

10

8

W/m2 K4

(329.44

2984) K4

0.97 W T )

10.28 W V 0.8 A(Tc T )

10.9 W s0.8/m2.8 K

(56.4

25oC)

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

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

k T1

A

A

Ein

∆ Est

W

Ice-water mixture (Tf )

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 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 and the amount of energy inﬂow is Ein k(6W 2) T1 L Tf tm k(6W 2) T1 L Tf

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

Relationship to Thermodynamics

27

By substituting into the ﬁrst law expression, it follows that tm Mhsf L 6W k(T1
2

Tf)

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 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 tm

30 C, the time required to melt the ice is 12,430 s 207 min

0.67 kg 334,000 J/kg 0.005 m 6(0.100 m)2 0.05 W/m 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.

The Surface Energy Balance 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

Surroundings

Tsur q" rad T1 q" cond
Fluid

q" conv T x T2 T∞
Control surfaces

u∞, T∞

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

28

Chapter 1

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 ˙ Ein ˙ Eout 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 (qcond), convection from the surface to a ﬂuid (qconv), and net radiation exchange from the surface to the surroundings (qrad). The energy balance then takes the form. qcond qconv 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 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 T 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 K. 2. When the person is in water at T 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 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

Relationship to Thermodynamics

29

Schematic:
Ti = 308 K
Skin/fat

Ts ε = 0.95

q" rad q" cond q" conv k = 0.3 W/m•K L = 3 mm
Air or water

Tsur = 297 K

T∞ = 297 K h = 2 W/m2•K (Air) h = 200 W/m2•K (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, ˙ ˙ Ein Eout 0 It follows that, on a unit area basis, qcond Ti q conv q rad 0 or, rearranging and substituting from Equations 1.2, 1.3a, and 1.7, Ts 4 (T s4 Tsur) h(Ts T ) 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 T ) hr (Ts Tsur) L Solving for Ts, with Tsur T , we have k Ts kTi L k L (h (h hr)T hr)

30

Chapter 1

Introduction

We estimate hr using Equation 1.9 with a guessed value of Ts 305 K and T 297 K, to yield hr 5.9 W/m2 K. Then, substituting numerical values into the preceding equation, we ﬁnd 0.3 W/m K 308 K 3 10 3 m 0.3 W/m K 3 10 3 m (2 (2 5.9) W/m2 K 5.9) W/m2 K 297 K 307.2 K

Ts

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. The rate of heat loss can be found by evaluating the conduction through the skin/fat layer: T Ts (308 307.2) K qs kA i 0.3 W/m K 1.8 m2 146 W 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 Ts and qs kA Ti L Ts 0.3 W/m K 1.8 m2 (308 300.7) K 3 10 3 m 1320 W 0.3 W/m K 308 K 3 10 3 m 0.3 W/m K 3 10 3 m 200 W/m2 K 200 W/m2 K 297 K 300.7 K

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

Relationship to Thermodynamics

31

Application of the Conservation Laws: Methodology 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.

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 nite control volume or ﬁ 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.

Relationship to the Second Law of Thermodynamics and the Efﬁciency of Heat Engines
1.3.2
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 KelvinPlanck state– ment 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

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 W Qin Qin Qout Qin 1 Qout 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 efciency C, is given by ﬁ
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 reservoir Q in Th

High-temperature side resistance

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

Relationship to Thermodynamics

33

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

Qout Qin

1

qout qin

1

Tc,i Th,i

(1.17)

where the ratio of heat ows over a time interval, Qout /Qin, has been replaced by the correﬂ sponding 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 qout (Th (Tc,i Th,i)/Rt,h Tc)/Rt,c (1.18a) (1.18b)

Equations 1.18 can be solved for the internal temperatures, to yield Th,i Tc,i Th Tc qin Rt,h qoutRt,c Tc qin(1 m, m)Rt,c

(1.19a) (1.19b)

In Equation 1.19b, qout has been related to qin and tic, modiﬁed efﬁciency can then be expressed as m using Equation 1.17. The more realis-

1

Tc,i Th,i

1

Tc

qin(1 m)Rt,c Th qinRt,h

(1.20)

Solving for

m

results in 1 Tc qin Rtot (1.21)

m

Th

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 qin m qin 1

Th

Tc 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 0. However, under these circumstances C) if qin

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

Introduction

˙ ˙ the power output W is zero according to Equation 1.22. To increase W, qin must be 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 8 10 8 K/W and Rt,c 2 10 8 K/W, and, under design conditions, they are Rt,h 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

Relationship to Thermodynamics

35

Schematic:
Rt,h Rt,h 8 9 10 10
8 8

Products of combustion qin 2500 MW K/W (clean) K/W (fouled)

Th

1000 K

Th,i
Power plant

W Tc,i

•

Rt,c Rt,c

2 10 K/W (clean) 8 2.2 10 K/W (fouled)

8

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

Th

Tc qinRtot

where, for clean conditions Rtot Thus 1 Tc qin Rtot 1 300 K 106 W 0.60 60% Rt,h Rt,c 8 10
8

K/W

2

10

8

K/W

1.0

10

7

K/W

m

Th

1000 K

2500

1.0

10

7

K/W

The power output is given by ˙ W qin m 2500 MW

0.60

1500 MW

2. Under fouled conditions, the preceding calculations are repeated to ﬁnd m 0.583

˙ 58.3% and W

1460 MW

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

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 correspond˙ ing power output would be W qin C 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 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) l l l l Unit 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 (Systme International dUnits) system of units was dened by è ’ é ﬁ 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

Units and Dimensions

37

kilogram-mole. For oxygen, as an example, the molecular weight 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 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 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
Unit and Symbol meter (m) kilogram (kg) mole (mol) second (s) ampere (A) kelvin (K) radian (rad) steradian (sr)

Quantity and Symbol Length (L) Mass (M) Amount of substance Time (t) Electric current (I ) Thermodynamic temperature (T) Plane anglea ( ) Solid anglea ( ) a Supplementary unit.

TABLE 1.3
Quantity

SI derived units for selected quantities
Name and Symbol newton (N) pascal (Pa) joule (J) watt (W) Formula m kg/s2 N/m2 N m J/s Expression in SI Base Units m kg/s2 kg/m s2 m2 kg/s2 m2 kg/s3

Force Pressure and stress Energy Power

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

Introduction

TABLE 1.4
Prex ﬁ femto pico nano micro milli centi hecto kilo mega giga tera peta exa

Multiplying preﬁxes
Abbreviation f p n m c h k M G T P E Multiplier 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

Analysis of Heat Tranfer Problems: Methodology

39

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 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 h 200 W/m2 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 K. 2. Effect of airﬂow on the cure temperature for 2 which the cure temperature is 50 C. Schematic:
Tsur = 30°C Glamp = 2000 W/m2 T∞ = 20°C 2 ≤ h ≤ 200 W/m2•K
Air T Coating, α = 0.8, ε = 0.5

h

200 W/m2 K. Value of h for

q" conv

q" rad

α Glamp

40

Chapter 1

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 0.5 with respect to irradiation from the surroundings. sur 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 (Eg 0), Equation 1.12c reduces to ˙ ˙ Ein Eout 0
˙ where Est 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 ( G)lamp Substituting numerical values 0.8 2000 W/m2 15 W/m2 K (T 0.5 293) K 5.67 T 10
8

qconv T )

qrad (T 4

0
4 Tsur)

Substituting from Equations 1.3a and 1.7, we obtain h(T 0

W/m2 K4 (T 4 104 C

3034) K4

0

and solving by trial-and-error, we obtain 377 K 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)

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 K

1.6

Relevance of Heat Tranfer

41

Comments: 1. The coating (plate) temperature may be reduced by decreasing T 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 K, T 204 C and radiation dominates (qrad 1232 W/m2, qconv 368 W/m2). Conversely, for h 200 W/m2 K, T 28 C and convection dominates (qconv 1606 W/m2, qrad 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

Introduction

Nuclear power 8.3%

Alternative sources 6.8%

Petroleum 39.3%

Natural gas 23.3%

Coal 22.9%

Electricity generation 35.4%

Transportation 25.4%

Industrial 21.7%

Residential/ commercial 17.4%

68.6%

19.9% 19.9% Waste heat 57.6%

78.5%

Useful power 42.4%

(a)

Petroleum 43.2%

Natural gas 20.7%

Coal 36.1%

Electricity generation 40.6%

Transportation 33.5%

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

Relevance of Heat Tranfer

43

(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

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 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

Summary

45

Keratin Epidermal layer Epidermis Basal cell layer

Sebaceous gland Sensory receptor Sweat gland Nerve fiber Hair follicle Dermis

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

Chapter 1

Introduction

TABLE 1.5 Summary of heat transfer processes
Equation Number dT dx T ) (1.1) (1.3a) Transport Property or Coefcient ﬁ k (W/m K) h (W/m2 K)

Mode Conduction Convection

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) q (W/m2) h(Ts k

Radiation

q (W/m2) (Ts4 or q(W) hr A(Ts

4 Tsur) Tsur)

(1.7) (1.8)

hr (W/m2 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 ux and a heat rate? What are their units? ﬂ • 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 Fouriers 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

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 Newtons law of cooling ? Can you write the equation from memory? ’ • What role is played by the convection heat transfer coefcient in Newton’s law of ﬁ 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 coefcient ? ﬁ • 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

Introduction

Schematic:

Hot coffee

q5 q1 q2 q3
Coffee Cover Air space Plastic flask Plastic flask Air space Cover

q8 q6 q4 q7
Room air Surroundings

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.

Problems

49

Problems
Conduction
1.1 The thermal conductivity of a sheet of rigid, extruded insulation is reported to be k 0.029 W/m 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 the insulation? 2 m sheet of 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 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 K.
Combustion chamber Compressor Tc Shaft Th

(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 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 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 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 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 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?

Turbine d

P
• mout

• min

L

(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 L 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 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 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

Introduction

insulated. What is the minimum thickness of styrofoam insulation (k 0.030 W/m 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 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 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 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 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 K) or copper (k 390 W/m 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 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 t Circuits

Convection
1.17 For a boiling process such as shown in Figure 1.5c, the ambient temperature T 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 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 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 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 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).

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 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 K. 1.23 A transmission case measures W and receives a power input of Pi engine. 0.30 m on a side 150 hp from the

51
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 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 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 K, respectively. In an application involving air at a temperature of T 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 T 15 C. From reliability considerations, the chip temperature must not exceed T 85 C.
Coolant

T∞, h w Air

Transmission case, η, Ts

Chip

T∞, h Pi

W

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

If the coolant is air and the corresponding convection coefﬁcient is h 200 W/m2 K, what is the maximum allowable chip power? If the coolant is a dielectric liquid for which h 3000 W/m2 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

Introduction

Dryer wall

Pe
Tset = 70°C

Insulation pad Electrical heater Bimetallic switch Air T∞, h

range 40 T 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 4 T3

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 K, how much heater power Pe is required when the desired dryer air temperature is T 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 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 K. 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

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 (T Tsur), where h(W/m2 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 Tsur 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

Liquid-nitrogen filled shroud

LN2

Electrical heater Baseplate

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?

53
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 qin ˙ W qin qout qin Tc,i Th,i Tc,i

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

Battery

Resistor Air

where qin is the refrigerator cooling rate, qout is the heat ˙ rejection rate, and W is the power input. Show that COPm 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 COPm Tc Th Tc qin Rtot qin Rtot

V = 24 V

T• = 25C

Lead wire

where Rtot Rt,c Rt,h. Also, show that the power input may be expressed as Th ˙ W qin
High-temperature reservoir Q out Tc Tc

(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 ˙ ˙ ˙ ˙ Ein(W), Eg(W), Eout(W), and Est(W). If a control surface is placed about the entire system, what are ˙ ˙ ˙ ˙ the values of Ein, Eg, Eout, and Est? (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)? (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.
W

qin Rtot qin Rtot
Th

Th,i
Internally reversible refrigerator

High-temperature side resistance Low-temperature side resistance

Tc,i Qin
Low-temperature reservoir

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 ˙ W under (a) clean and (b) dusty coil conditions.

54

Chapter 1

Introduction exposed surface is h 8 W/m2 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 T 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.
Enclosure, Tsur
Tsur

Solid silicon powder

Vsi Ts,o Ts, ε s Solid silicon sheet

Vsi tsi

H

Substrate

Air T∞, h •
•

Solid silicon sheet

Molten silicon String

Air T∞, h

Pelec

•

Molten silicon Crucible D

Chip (Ts, ε)

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 T )1/4, where C 4.2 W/m2 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 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 qsol 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 T ) and q conv,gen h(Tgen T ), respectively, where the ambient temperature T Ts (which is the nacelle temperature) and h 40 W/m2 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 qo[1 ˙ (r/ro)2], where q is the local rate of energy generation per ˙ unit volume, qo 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 T and provides a uniform convection coefﬁcient h.

Problems

55 estimate the magnitudes of kinetic and potential energy changes. Assume the blood’s properties are similar to those of water. 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 T 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 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 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 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

ro

T∞, h
• • q = qo [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 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 T 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 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 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,

56

Chapter 1

Introduction (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 volumetric ﬂow rate ˙ should the fan operate? Heat loss 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 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 K and the temperature of the air and the surroundings is T 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 K, and 0.8. The bottom surface of the oven is also at s 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

10 W/m2 K, respectively, compute the oxygen evaporation rate (kg/s) as a function of surface emissivity over the range 0.2 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 0.65. Would the neighborhood cat be comfortable S walking on a roof constructed of the material on a day when GS 750 W/m2, T 16 C, and h 7 W/m2 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 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 T 300 K may be approximated as h 0.70 (Ts T )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.
Surroundings, Tsur Air T∞, h
Electric resistor

T∞, h

Ts, εs Lo
Steel sheet

ts Vs Ts
Concrete pad

Tb

Fan Inlet, ∀, Ti
•

Discharge

To, Vo

D Pelec

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

Problems pan is hfr 3 W/m2 K. The oven air and wall are at temperatures T 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 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 4 4 heat ﬂux may be expressed as q rad (Th Tw). 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 K. The thermal conductivity of the gas in the gap is 0.0436 W/m K.
D
Hot plate, Th Stagnant gas, k Silicon wafer, Tw, i Gap, L

57 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 K, and an emissivity that equals its absorptivity to the radiant ﬂux ( 0.65). For l 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 T 700 K, convection coefﬁcients at the upper and lower surfaces of the wafer are 8 and 4 W/m2 K, respectively. The silicon wafer has a density of 2700 kg/m3 and a speciﬁc heat of 875 J/kg K.

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 Th 1300 C. Display your results graphically. Comment on the relative importance of the two heat

58

Chapter 1

Introduction

Lstack
SiC chamber

Estack

e

Gas, T• Wafer, Tw, ε hu hl Elevator

Heating zone

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

Mounting pin holder

Water channel

Bipolar plate Membrane

Hydrogen flow channel

Airflow channel

(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 ow channels within each bipolar plate, as ﬂ 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 Ec,g 45 W of thermal energy when operating at T 80 C. The external surfaces of the stack are exposed to air at T 25 C and surroundings at Tsur 30 C, with 0.88 and h 150 W/m2 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 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 K. Would variations in the surface emissivity have a signiﬁcant effect on the rod temperature?

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 T 30 C. The electrical resistivity, e( m), of the bar material is a function of temperature, e,o e [1 (T To)], where e,o 0.0171 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 T )0.25, where C 1.21 W m 1.75 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 I 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 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 K. The room air and wall temperatures are T 22 C and Tsur 20 C, respectively. If electricity costs C $0.18/kW h, determine the daily cost of leaving the charger plugged in when not in use.

59

Tsur w 20 mm

Wall Charger

L

50 mm 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 qi to its inner surface. The canister is suddenly submerged in a liquid bath of temperature T Ti, where Ti is the initial temperature of the canister wall.
Canister Reacting chemicals

ro = 0.6 m Ti = 500 K p 3 ρ = 8055 kg/m c = 510 J/kg•K

q" i
Bath

T∞ = 300 K h = 500 W/m2•K

ri = 0.5 m

(a) Assuming negligible temperature gradients in the canister wall and a constant heat ﬂux qi , 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 qi 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 h 10,000 W/m2 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 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

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 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 T 20 C and h 15 W/m2 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 T 35 C, h 10 W/m2 K, and (b) a breezy winter day, for which Tsur T 15 C, h 30 W/m2 K. The panel emissivity is 0.90.

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

Air

Heat sink, Ts A s, ε

T∞ = 27°C

GS

Air T∞, h

Tsur

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 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

Problems (a) If the temperature rise of the airﬂow, (To Ti), is not to exceed 15 C, what is the minimum allowable volumetric ﬂow rate ˙of the air? The density and speciﬁc heat of the air may be approximated as 1.161 kg/m3 and cp 1007 J/kg 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 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

61
(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 h 200 W/m2 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 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 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 T 5 C, and the emissivity is 0.95? Express your results in units of W/m2 K and W/m2 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 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

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 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 K, compute and plot the maximum allowable heat dissipation for variations in h over the range 50 h 250 W/m2 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 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.

62

Chapter 1

Introduction (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 h 200 W/m2 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 T 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 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 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 h 20 W/m 2 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 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 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

h 20 W/m2 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 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 K, h 20 W/m2 K, and 0.5. The suggested ranges of the independent variables are 0.1 k 400 W/m K, 2 h 200 W/m2 K, and 0.05 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 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 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 K. What heat ﬂux q o is required to maintain the inner surface of the duct at Ti 85 C?

Problems

63
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 K.
GS
Cover glass Air space Absorber plate Water tubing Insulation

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 T 20 C and h 15 W/m2 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. 1.83 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 K, may be approximated by an expression of the form h 0.80(Ts T )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( m), of the bar material is a function of temperature, e (T To)], where e,o e,o[1 0.0828 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 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 h 100 W/m2 K. What minimum convection coefﬁcient is required to maintain a safe-operating temperature below 120 C? Will increasing the emissivity signiﬁcantly affect this result?

(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 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

Introduction (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.

(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 T 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

Cold, dry night air Exterior pane Air gap Interior pane Insulation

Warm, moist room air

Insulation on inner surface

Cold, dry night air Exterior pane Air gap Interior pane Insulation

Warm, moist room air

Insulation on outer surface

Tsur
Water

tt tins h, T∞

(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.

Problems

65
(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

Surface slabs

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

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.

(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

Double pane Ambient air Single pane Room air

Air

(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

Introduction 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.

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.

Solar radiation Evacuated tubes Reflecting panel

Doublepaned cover

Styrofoam Absorber plates
Evacuated space

Working fluid Transparent outer tube Inner tube

C H A P T E R

Introduction to Conduction

2

68

Chapter 2

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 Fouriers 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

The Conduction Rate Equation

69

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 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 or for the heat ux ﬂ qx qx A k dT dx (2.2) kA dT dx (2.1)

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 (Fouriers law ) as follows: ’ q k T k i T x j T y k T z (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
T(x)

qn n

k

T n n

(2.4)

T1 q'' x T2 x

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

70

Chapter 2

Introduction to Conduction

'' qy

'' qn

'' qx

n
Isotherm

y 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 denes ﬁ 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

The Thermal Properties of Matter

71

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.
The Solid State 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]:

k

1 Cc 3

mfp

(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.

72

Chapter 2

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].
The Solid State: Micro- and Nanoscale Effects 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

500 400 300 200 Silver Copper Gold Aluminum Aluminum alloy 2024 Tungsten Platinum 50 Iron

100 Thermal conductivity (W/m•K)

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

The Thermal Properties of Matter

73

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 lms of the same material having thicknesses L1 and L2 are shown in ﬁ 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]: kx k ky k

1 1

mfp

3L 2 mfp (2.9a) (2.9b)

3 L

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 L Lcrit, kx and ky are reduced from the bulk value as indicated in Equations 2.9a,b. mfp

y L1
(a)
L2 < L1

x

(b)

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

1

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.

74

Chapter 2

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 300 K [3,4]
Material Aluminum oxide Diamond (IIa) Gallium arsenide Gold Silicon Silicon dioxide Yttria-stabilized zirconia mfp (nm)

Lcrit, x (nm) 36 2200 160 220 290 4 170

Lcrit,y (nm) 22 1400 100 140 180 3 110

5.08 315 23 31 43 0.6 25

2.5 L = 98 nm 2 Thermal conductivity (W/m•K) L = 55 nm L = 32 nm

1.5

L = 23 nm

1

L = 10 nm

0.5 λmfp (T = 300 K) = 25 nm 0 0 100 200 300 400 500

Temperature (K)

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

2.2

The Thermal Properties of Matter

75

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

1c c 3 v

mfp

(2.10)

For an ideal gas, the mean free path may be expressed as mfp kBT 2 d 2p

(2.11)

where kB is Boltzmann’s constant, kB 1.381 10 23 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

Helium 4.003, 0.219

Thermal conductivity (W/m•K)

0.2

0.1

Water (steam, 1 atm) 18.02, 0.458

Air 28.97, 0.372

Carbon dioxide 44.01, 0.464

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 ( ) of the gases are also shown.

76

Chapter 2

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 d2 kBT (2.12)

where the parameter is the ratio of speciﬁc heats, cp /cv, and is Avogadro’s number, 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)

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

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 ber- , ﬁ powder-, and ake -type insulations, the solid material is ﬁnely dispersed throughout an air ﬂ 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. Reective insulations are ﬂ 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

78

Chapter 2

Introduction to Conduction

0.014 Effective thermal conductivity (W/m•K) 0.012 0.01 0.008 0.006 0.004 0.002 0 10

3

10

2

10 Pressure (atm)

1

100

FIGURE 2.10 Measured thermal conductivity of carbon-doped silica aerogel as a function of pressure at T 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 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 K). Because of their very small densities, however, gases are poorly suited for thermal energy storage ( cp 1 kJ/m3 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 rcp 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

The Thermal Properties of Matter

79

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): cp k 2702 kg/m3 903 J/kg K 237 W/m K k cp 97.1 Table A.1, pure aluminum (700 K): 2702 kg/m3 1090 J/kg K 225 W/m K k cp at 300 K at 700 K (by linear interpolation) at 700 K (by linear interpolation) 237 W/m K 2702 kg/m3 903 J/kg K 10
6

m2/s

cp k
Hence

225 W/m K 2702 kg/m3 1090 J/kg K

76

10

6

m2/s

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

cp k

3160 kg/m3 at 300 K 1195 J/kg K at 1000 K 87 W/m K at 1000 K

87 W/m K 3160 kg/m3 1195 J/kg K 23 10
6

m2/s

80

Chapter 2

Introduction to Conduction

Table A.3, parafﬁn (300 K):

cp k

900 kg/m3 2890 J/kg K 0.24 W/m K

k cp 9.2

0.24 W/m K 900 kg/m3 2890 J/kg K 10
8

m2/s

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 kp 2kbf 2kbf 2 (kp (kp kbf) kbf) kbf

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 K, p 3970 kg/m3, and cp,p 0.765 kJ/kg K.

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

2.2

The Thermal Properties of Matter

81

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

cp,p

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 cp,bf 4.179 kJ/kg K, bf 855 10 6 N s/m2. Analysis: From the problem statement, knf kp kp 2kbf 2kbf 2 (kp (kp kbf) kbf) kbf 0.613 W/m K, bf 997 kg/m3,

36.0 W/m K 2 0.613 W/m K 2 0.05(36.0 0.613) W/m K 36.0 W/m K 2 0.613 W/m K 0.05(36.0 0.613) W/m K 0.613 W/m K 0.705 W/m K 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

Similarly, the conservation of energy principle yields, nfVcp,nf T bfV(1

)cp,bf T

pV

cp,p T nf Dividing by the volume V, temperature T, and nanoﬂuid density cp,nf bf cp,bf (1 nf

yields

)
3

pcp,p

997 kg/m

4.179 kJ/kg K

(1

0.05) 3970 kg/m3 1146 kg/m3

0.765 kJ/kg K

(0.05)

3.587 kJ/kg K

82

Chapter 2

Introduction to Conduction

From the problem statement, the dynamic viscosity of the nanoﬂuid is nf 855

10

6

N s/m2

(1

2.5

0.05)

962

10

6

N s/m2

The nanoﬂuid’s thermal diffusivity is knf 0.705 W/m K nf nf cp,nf 1146 kg/m3 3587 J/kg K

171

10

9

m2/s

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

1.150 0.858 171 147

nf bf

1146 kg/m3 997 kg/m3 962 855 10 10 1.166
6

1.149 N s/m2 6 N s/m2 1.130

nf bf

10 9 m2 /s 10 9 m2/s

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 eld in a ﬁ 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

The Heat Diffusion Equation

83

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 dy 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 qy qz
T(x, y, z) dx qx qy qz

dy

qx dx x qy dy y qz dz z qz + dz qy + dy

(2.13a) (2.13b) (2.13c)

dz

dz qx z y x qy dx qz dy Eg E st
• •

qx + dx

FIGURE 2.11 coordinates.

Differential control volume, dx dy dz, for conduction analysis in Cartesian

84

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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 ˙ Eg ˙ 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 ˙ Est 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 Eg and Est represent different physical ˙ processes. The energy generation term Eg 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 Est 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 ˙ Ein ˙ Eg ˙ Eout ˙ Est (1.12c)

˙ Hence, recognizing that the conduction rates constitute the energy inﬂow Ein and outﬂow ˙ , and substituting Equations 2.14 and 2.15, we obtain Eout T ˙ qx qy qz q dx dy dz qx dx qy dy qz dz cp dx dy dz (2.16) t Substituting from Equations 2.13, it follows that qx dx x qy dy y qz dz z ˙ q dx dy dz cp T dx dy dz t (2.17)

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

The Heat Diffusion Equation

85

Equations 2.18 into Equation 2.17 and dividing out the dimensions of the control volume (dx dy dz), we obtain x k T x y k T y z k T z ˙ q cp T 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, x k T dx x qx qx dx (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
2

T x2

2

T y2

2

T z2

˙ q 1 T t k

(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 x k T x y k T y z k T z ˙ q 0 (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 d k dT dx dx 0 (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.

86

Chapter 2

Introduction to Conduction

qz + dz

qr

rdφ qφ + dφ dz

z r T(r,φ ,z) r x y φ qφ dr qr + dr

qz

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

dz, for conduction

Cylindrical Coordinates 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

q where qr k

k T

k i

T r

j1 r

T

k

T z

(2.24)

T r

q

k T r

qz

k

T z

(2.25)

qθ + dθ r sin θ dφ qr rdθ z θ qφ + dφ

qφ T(r, φ , θ) y dr qr + dr

r x φ qθ

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

r d , for

2.3

The Heat Diffusion Equation

87

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: T 1 r r kr r
Spherical Coordinates and Fourier’s law is

1 r2

k

T z

k

T z

˙ q

cp

T t

(2.26)

In spherical coordinates, the general form of the heat ﬂux vector T r j1 r T 1 r sin T

q where qr

k T

k i

k

(2.27)

k

T r

q

k T r

q

k r sin

T

(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: T 1 kr 2 2 r r r 1 r 2 sin 1 r sin2
2

k T ˙ q

T T t

k sin

cp

(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 (k/r)( T/ ), not q k( T/ ). of a cylindrical coordinate system is q

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 10 m2 having the properties 1600 kg/m3, k 40 W/m K, and cp 4 kJ/kg K.

88

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1. Determine the rate of heat transfer entering the wall (x 0) and leaving the wall (x 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.

1 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, Est. 3. Time rate of temperature change at x 0, 0.25, and 0.5 m. Schematic:
A = 10 m2
• q = 1000 W/m3 k = 40 W/m•K ρ = 1600 kg/m3 cp = 4 kJ/kg•K

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 qin qx(0) bkA kA T xx
0

kA(b

2cx)x

0

300 C/m

40 W/m K

10 m2

120 kW

2.3

The Heat Diffusion Equation

89

Similarly, qout qout qx(L) (b kA 2cL)kA T xx
L

kA(b

2cx)x

L

[ 300 C/m 1 m] 40 W/m K 10 m2 160 kW

2( 50 C/m2)

˙ 2. The rate of change of energy storage in the wall Est may be determined by applying an overall energy balance to the wall. Using Equation 1.12c for a control volume about the wall, ˙ Ein ˙ where Eg ˙ qAL, it follows that ˙ Est ˙ Est ˙ Est ˙ Ein ˙ Eg ˙ Eout qin ˙ qAL qout 1m 160 kW ˙ Eg ˙ Eout ˙ Est

120 kW 30 kW

1000 W/m3

10 m2

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 t k 2T cp x2 ˙ q cp

From the prescribed temperature distribution, it follows that
2

T x2

x x (b

T x 2cx) 2c 2( 50 C/m2) 100 C/m2

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 t 40 W/m K 1600 kg/m3 4 kJ/kg K 1000 W/m3 1600 kg/m3 4 kJ/kg K T t 6.25 4.69 10 10
4

( 100 C/m2)

C/s C/s

1.56

10

4

C/s

4

90

Chapter 2

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

1 (2.30) c t 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].

mfp

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 rst kind. It is closely approximated, for ﬁ 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 x x 0

qs

2.4

Boundary and Initial Conditions

91

TABLE 2.2 Boundary conditions for the heat diffusion equation at the surface (x 0)
1. Constant surface temperature T(0, t) Ts (2.31) x Ts T(x, t)

2. Constant surface heat ﬂux (a) Finite heat ﬂux T k x x 0

qs''

qs

(2.32) x T(x, t)

(b) Adiabatic or insulated surface T x x 0

0

(2.33) x T(x, t)

3. Convection surface condition T k x x 0

T(0, t)

h[T

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 x 0 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 T 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.

92

Chapter 2

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:
T(x, y, z, t)
Air Copper bar (k, α) T(x, t)

y x w

z

T∞, h
Air

T∞, h

T(L, t) L x

L I

Heat sink

To

q
•

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
2

T x2

˙ q 1 T t k

(1)

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)

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

h[T(L, t)

T ]

(3)

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)

2.4

Boundary and Initial Conditions

93

. If To, T , 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 To and that the electric current is sufﬁciently large to heat the bar to temperatures T in excess of T , the following distributions would correspond to the initial condition (t 0), the ﬁnal (steady-state) condition (t l ), and two intermediate times.
T(x, ∞), Steady-state condition T(x, t) T∞ To
0 Distance, x

b

a

T∞ T(x, 0), Initial condition

L

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 q x t coordinates, the transient variations are as follows.
+

q"(L, t) x q"(x, t) x

0

q"(0, t) x

– 0 Time, t

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

94

Chapter 2

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 Fouriers 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 Fouriers 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 Fouriers 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, Eg, 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?

Problems

95

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 L

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

If the properties remain constant and no internal heat generation occurs, sketch the heat ﬂux distribution, q x (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?

96

Chapter 2

Introduction to Conduction

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. 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.

r

T1, A1

x

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 k ko aT

qx

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 T1 < T2 T2

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 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 T 20 C. If the steady-state temperature distribution

A

B

x

0.5 L

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.

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

97
Insulation 1m B, TB = 100°C

k = 10 W/m•K y
2m

x
A, TA = 0°C

T1 x L

T2

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 K, a 10 3 W/m K2, and T is in kelvins. The gradient at surface B is T/ x 30 K/m. What is T/ y at surface 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

Determine the heat ﬂux and the unknown quantity for each case and sketch the temperature distribution, indicating the direction of the heat ﬂux. Case 1 2 3 4 5 T1( C) 50 30 70 T2( C) 20 10 40 30 160 80 200 dT/dx (K/m)

Ao qx(x)

2.12 Consider a plane wall 100 mm thick and of thermal conductivity 100 W/m 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.
T(x) T2 T1 x
(a)

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 qo exp( ax), where qo is a constant. Obtain an expression for qx(x) when the left face (x 0) is well insulated.

T(x) T2 T1 x
(b)

T(x) T2

T1 x
( c)

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.

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 T . 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?

98

Chapter 2

Introduction to Conduction (a) Explain why the apparatus of Problem 2.17 cannot be used to obtain an accurate measurement of the aerogel’s thermal conductivity. (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. (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 K and kpc 0.21 W/m 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 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.

Plate, To

Sample Heater leads Sample

∆T1 Insulation ∆T2

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.
Tc,i Heater leads Coolant in

t D

Aerogel sample

Heater x Aluminum plate T2 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

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. 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 To(t) Ti 2qo t cpk
1/ 2

99

To(t)

Sample 1, D, L, ρ Heater leads Sample 2, D, L, ρ

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, q x . 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, T T (t). In Stage 2, the water has come to a boil, and its temperature remains at a ﬁxed value, T 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

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( ) 33.50 C. The density of the sample materials, determined by measurement of volume and mass, is 3965 kg/m3.

100

Chapter 2

Introduction to Conduction 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 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 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 T 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?

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 T , 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 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.

ξ q
•

Tc

(c) Determine the surface heat ﬂuxes, qx( L) and q x( 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? (e) Obtain an expression for the heat ﬂux distribution q x(x). Is the heat ﬂux zero at any location? Explain any signiﬁcant features of the distribution. (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? (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 K, respectively.

x

(a) Using the appropriate form of the heat equation, derive an expression for the x-dependence of the steady-state heat ﬂux q (x). (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). 2.30 The steady-state temperature distribution in a onedimensional wall of thermal conductivity 50 W/m K and

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

101
(a) Applying an overall energy balance to the wall, cal˙ culate 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 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 e ka2 ax t x (a) L

0 x (b) L

t→∞

t→∞

t→∞

t x (c) L

0 x (d) L

t

0

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 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 T 20 C and h 500 W/m2 K. The surface at x L is well insulated.

Bx

C

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
•

(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

Fluid

x

L = 50 mm

102

Chapter 2

Introduction to Conduction (b) With the temperature at x 0 and the ﬂuid temperature ﬁxed at T(0) 0 C and T 20 C, respectively, compute and plot the temperature at x L, T(L), as a function of h for 10 h 100 W/m2 K. Brieﬂy explain your results. 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 K and T 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 0.95. S

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 r2 r

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.
120 100 80

Ambient air T∞, h

GS L

Ts

x

Coal, • k, q

(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 ˙ qL2 1 2k x2 L2

From this distribution, what can you say about conditions at the bottom surface (x 0)? Sketch the temperature distribution and label key features. (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. x q = 0, k = 4.5 W/m•K T∞ = 20°C h = 30 W/m2•K
0.18 m Air
•

T( C)

60 40 20 0

(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 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 K. 2.43 The cylindrical system illustrated has negligible variation of temperature in the r- and z-directions. Assume

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

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

103
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

φ

Conducting • rod, q, kr

(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 T(r) r C1 ln r o C2

ri ro
Cladding, kc

T∞, h

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 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, q r(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 o

Fundamentals of Heat and Mass Transfer 7th Edition

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