Heat Conduction

Latif M. Jiji

Heat ConductionThird Edition

ABC

Professor Latif M. JijiDepartment of Mechanical EngineeringGrove School of EngineeringThe City College ofThe City University of New YorkNew York, New York 10031USAE-Mail: jiji@ccny.cuny.edu

ISBN 978-3-642-01266-2 e-ISBN 978-3-642-01267-9

DOI 10.1007/978-3-642-01267-9

Library of Congress Control Number: Applied for

c© 2009 Springer-Verlag Berlin Heidelberg

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''Additional material to this book can be downloaded from http://extra.springer.com ''

This book is dedicated to my wife Vera for opening many possibilities and providing balance in my life.

PREFACE

This book is designed to:

Provide students with the tools to model, analyze and solve a wide range of engineering applications involving conduction heat transfer.

Introduce students to three topics not commonly covered in conduction heat transfer textbooks: perturbation methods, heat transfer in living tissue, and microscale conduction.

Take advantage of the mathematical simplicity of one-dimensional conduction to present and explore a variety of physical situations that are of practical interest.

Present textbook material in an efficient and concise manner to be covered in its entirety in a one semester graduate course.

Drill students in a systematic problem solving methodology with emphasis on thought process, logic, reasoning and verification.

To accomplish these objectives requires judgment and balance in the selection of topics and the level of details. Mathematical techniques are presented in simplified fashion to be used as tools in obtaining solutions. Examples are carefully selected to illustrate the application of principles and the construction of solutions. Solutions follow an orderly approach which is used in all examples. To provide consistency in solutions logic, I have prepared solutions to all problems included in the first ten chapters myself. Instructors are urged to make them available electronically rather than posting them or presenting them in class in an abridged form.

This edition adds a new chapter, “Microscale Conduction.” This is a new and emerging area in heat transfer. Very little is available on this subject as textbook material at an introductory level. Indeed the preparation of such a chapter is a challenging task. I am fortunate

viii PREFACE

that Professor Chris Dames of the University of California, Riverside, agreed to take on this responsibility and prepared all the material for chapter 11.

Now for the originality of the material in this book. Much that is here was inspired by publications on conduction. I would like to especially credit Conduction Heat Transfer by my friend Vedat S Arpaci. His book contains a wealth of interesting problems and applications. My original notes on conduction contained many examples and problems taken from the literature. Not having been careful in my early years about recording references, I tried to eliminate those that I knew were not my own. Nevertheless, a few may have been inadvertently included.

ACKNOWLEDGMENTS

First I would like to acknowledge the many teachers who directly or indirectly inspired and shaped my career. Among them I wish to single out Professors Ascher H. Shapiro of the Massachusetts Institute of Technology, Milton Van Dyke of Stanford University, D.W. Ver Planck of Carnegie Institute of Technology and Gordon J. Van Wylen and John A. Clark of the University of Michigan. I only wish that I had recognized their lasting contributions to my education decades earlier.

Chapter 11 was carefully reviewed by Professor Gang Chen of the Massachusetts Institute of Technology. The chapter author Chris Dames and I are grateful for his technical comments which strengthened the chapter.

My wife Vera read the entire manuscript and made constructive observations. I would like to thank her for being a supportive, patient and understanding partner throughout this project.

Latif M. Jiji New York, New YorkMarch, 2009

CONTENTS

Preface vii

CHAPTER 1: BASIC CONCEPTS 1

1.1 Examples of Conduction Problems 1 1.2 Focal Point in Conduction Heat Transfer 2 1.3 Fourier’s Law of Conduction 2 1.4 Conservation of Energy: Differential Formulation of

the Heat Conduction in Rectangular Coordinates 5 1.5 The Heat Conduction Equation in Cylindrical and

Spherical Coordinates 9 1.6 Boundary Conditions

1.6.1 Surface Convection: Newton’s Law of Cooling 1.6.2 Surface Radiation: Stefan-Boltzmann Law 1.6.3 Examples of Boundary Conditions

10 10 11 12

1.7 Problem Solving Format 151.8 Units 16

REFERENCES 17 PROBLEMS 18

CHAPTER 2: ONE-DIMENSIONAL STEADY-STATE CONDUCTION 24

2.1 Examples of One-dimensional Conduction 24 2.2 Extended Surfaces: Fins 34

2.2.1 The Function of Fins 34 2.2.2 Types of Fins 34 2.2.2 Heat Transfer and Temperature Distribution in Fins 35 2.2.4 The Fin Approximation 36 2.2.5 The Fin Heat Equation: Convection at Surface 37 2.2.6 Determination of dxdAs / 39

2.2.7 Boundary Conditions 40 2.2.8 Determination of Fin Heat Transfer Rate fq 40

2.2.9 Steady State Applications: Constant Area Fins with Surface Convection 41

x CONTENTS

2.2.10 Corrected Length cL 44

2.2.11 Fin Efficiency f 44

2.2.12 Moving Fins 45 2.2.13 Application of Moving Fins 47 2.2.14 Variable Area Fins 49

2.3 Bessel Differential Equations and Bessel Functions 52 2.3.1 General Form of Bessel Equations 52 2.3.2 Solutions: Bessel Functions 52 2.3.3 Forms of Bessel Functions 54 2.3.4 Special Closed-form Bessel Functions: n = odd

integer/2 54 2.3.5 Special Relations for n = 1, 2, 3, … 55 2.3.6 Derivatives and Integrals of Bessel Functions 56 2.3.7 Tabulation and Graphical Representation of

Selected Bessel Functions 56 2.4 Equidimensional (Euler) Equation 58 2.5 Graphically Presented Solutions to Fin Heat Transfer

Rate fq 59

REFERENCES 60 PROBLEMS 61

CHAPTER 3: TWO-DIMESIONAL STEADY STATE CONDUCTION 72

3.1 The Heat Conduction Equation 72 3.2 Method of Solution and Limitations 72 3.3 Homogeneous Differential Equations and Boundary

Conditions 72 3.4 Sturm-Liouville Boundary-Value Problem:

Orthogonality 74 3.5 Procedure for the Application of Separation of

Variables Method 76 3.6 Cartesian Coordinates: Examples 83 3.7 Cylindrical Coordinates: Examples 97 3.8 Integrals of Bessel Functions 1023.9 Non-homogeneous Differential Equations 1033.10 Non-homogeneous Boundary Conditions: The Method

of Superposition 109

REFERENCES 111

CONTENTS xi

PROBLEMS 111

CHAPTER 4: TRANSIENT CONDUCTION 119

4.1 Simplified Model: Lumped-Capacity Method 1194.1.1 Criterion for Neglecting Spatial Temperature

Variation 1194.1.2 Lumped-Capacity Analysis 121

4.2 Transient Conduction in Plates 1244.3 Non-homogeneous Equations and Boundary Conditions 1284.4 Transient Conduction in Cylinders 1324.5 Transient Conduction in Spheres 1384.6 Time Dependent Boundary Conditions: Duhamel’s

Superposition Integral 1414.6.1 Formulation of Duhamel’s Integral 1424.6.2 Extension to Discontinuous Boundary

Conditions 1444.6.3 Applications 145

4.7 Conduction in Semi-infinite Regions: The Similarity Transformation Method 150

REFERENCES 154PROBLEMS 154

CHAPTER 5: POROUS MEDIA 163

5.1 Examples of Conduction in Porous Media 1635.2 Simplified Heat Transfer Model 164

5.2.1 Porosity 1645.2.2 Heat Conduction Equation: Cartesian

Coordinates 1655.2.3 Boundary Conditions 1675.2.4 Heat Conduction Equation: Cylindrical

Coordinates 1685.3 Applications 168

REFERENCES 174PROBLEMS 175

xii CONTENTS

CHAPTER 6: CONDUCTION WITH PHASE CHANGE: MOVING BOUNDARY PROBLEMS 184

6.1 Introduction 1846.2 The Heat Equation 1856.3 Moving Interface Boundary Conditions 1856.4 Non-linearity of the Interface Energy Equation 1886.5 Non-dimensional Form of the Governing Equations:

Governing Parameters 1896.6 Simplified Model: Quasi-Steady Approximation 1906.7 Exact Solutions 197

6.7.1 Stefan’s Solution 1976.7.2 Neumann’s Solution: Solidification of Semi-

Infinite Region 2006.7.3 Neumann’s Solution: Melting of Semi-Infinite

Region 2036.8 Effect of Density Change on the Liquid Phase 2046.9 Radial Conduction with Phase Change 2056.10 Phase Change in Finite Regions 209

REFERENCES 210PROBLEMS 210

CHAPTER 7: NON-LINEAR CONDUCTION PROBLEMS 215

7.1 Introduction 2157.2 Sources of Non-linearity 215

7.2.1 Non-linear Differential Equations 2157.2.2 Non-linear Boundary Conditions 216

7.3 Taylor Series Method 2167.4 Kirchhoff Transformation 220

7.4.1 Transformation of Differential Equations 2207.4.2 Transformation of Boundary Conditions 221

7.5 Boltzmann Transformation 2247.6 Combining Boltzmann and Kirchhoff Transformations 2267.7 Exact Solutions 227

REFERENCES 230PROBLEMS 230

CONTENTS xiii

CHAPTER 8: APPROXIMATE SOLUTIONS: THE INTEGRAL METHOD 236

8.1 Integral Method Approximation: Mathematical Simplification 236

8.2 Procedure 2368.3 Accuracy of the Integral Method 2378.4 Application to Cartesian Coordinates 2388.5 Application to Cylindrical Coordinates 2468.6 Non-linear Problems 2518.7 Energy Generation 260

REFERENCES 264PROBLEMS 264

CHAPTER 9: PERTURBATION SOLUTIONS 269

9.1 Introduction 2699.2 Solution Procedure 2709.3 Examples of Perturbation Problems in Conduction 2719.4 Perturbation Solutions: Examples 2739.5 Useful Expansions 296

REFERENCES 296PROBLEMS 297

CHAPTER 10: HEAT TRANSFER IN LIVING TISSUE 302

10.1 Introduction 30210.2 Vascular Architecture and Blood Flow 30210.3 Blood Temperature Variation 30410.4 Mathematical Modeling of Vessels-Tissue Heat

Transfer 30510.4.1 Pennes Bioheat Equation 30510.4.2 Chen-Holmes Equation 31210.4.3 Three-Temperature Model for Peripheral Tissue 31310.4.4 Weinbaum-Jiji Simplified Bioheat Equation for

Peripheral Tissue 31510.4.5 The s-Vessel Tissue Cylinder Model 323

REFERENCES 332PROBLEMS 334

xiv CONTENTS

CHAPTER 11: MICROSCALE CONDUCTION 347

11.1 Introduction 34711.1.1 Categories of Microscale Phenomena 34811.1.2 Purpose and Scope of this Chapter 350

11.2 Understanding the Essential Physics of Thermal Conductivity Using the Kinetic Theory of Gases 35111.2.1 Determination of Fourier’s Law and Expression

for Thermal Conductivity 35111.3 Energy Carriers 355

11.3.1 Ideal Gas: Heat is Conducted by Gas Molecules 35511.3.2 Metals: Heat is Conducted by Electrons 35911.3.3 Electrical Insulators and Semiconductors: Heat is Conducted by Phonons (Sound Waves) 36111.3.4 Radiation: Heat is Carried by Photons (Light

Waves) 37211.4 Thermal Conductivity Reduction by Boundary

Scattering: The Classical Size Effect 37611.4.1 Accounting for Multiple Scattering Mechanisms:

Matthiessen’s Rule 37711.4.2 Boundary Scattering for Heat Flow Parallel to

Boundaries 37911.4.3 Boundary Scattering for Heat Flow

Perpendicular to Boundaries 38711.5 Closing Thoughts 391

REFERENCES 394PROBLEMS 397

APPENDIX A: Ordinary Differential Equations

(1) Second Order Differential Equations with Constant Coefficients

402

402

APPENDIX B:

APPENDIX C:

(2) First Order Ordinary Differential Equations with Variable Coefficients

Integrals of Bessel Functions

Values of Bessel Functions

404

405

406

APPENDIX D: Fundamental Physical Constants and Material Properties

D-1 Fundamental Physical Constants D-2 Unit conversions

412

412412

CONTENTS xv

D-3 Properties of Helium Gas D-4 Properties of Copper at 300 K D-5 Properties of Fused Silica

(Amorphous Silicon Dioxide, SiO2) at 300 K

D-6 Properties of Silicon D-7 Measured Thermal Conductivity of

a 56 nm Diameter Silicon Nanowire at Selected Temperatures

D-8 Calculated Thermal Conductivity of Single-Walled CarbonNanotubes, Selected Values

412412412

413413

414

414

INDEX 416