# Heat Conduction, 3rd Edition

# Heat Conduction, 3rd Edition

ISBN: 978-1-118-41128-5

Aug 2012

744 pages

## Description

**The long-awaited revision of the bestseller on heat conduction**

*Heat Conduction, Third Edition* is an update of the classic text on heat conduction, replacing some of the coverage of numerical methods with content on micro- and nanoscale heat transfer. With an emphasis on the mathematics and underlying physics, this new edition has considerable depth and analytical rigor, providing a systematic framework for each solution scheme with attention to boundary conditions and energy conservation. Chapter coverage includes:

- Heat conduction fundamentals
- Orthogonal functions, boundary value problems, and the Fourier Series
- The separation of variables in the rectangular coordinate system
- The separation of variables in the cylindrical coordinate system
- The separation of variables in the spherical coordinate system
- Solution of the heat equation for semi-infinite and infinite domains
- The use of Duhamel's theorem
- The use of Green's function for solution of heat conduction
- The use of the Laplace transform
- One-dimensional composite medium
- Moving heat source problems
- Phase-change problems
- Approximate analytic methods
- Integral-transform technique
- Heat conduction in anisotropic solids
- Introduction to microscale heat conduction

In addition, new capstone examples are included in this edition and extensive problems, cases, and examples have been thoroughly updated. A solutions manual is also available.

*Heat Conduction* is appropriate reading for students in mainstream courses of conduction heat transfer, students in mechanical engineering, and engineers in research and design functions throughout industry.

## Related Resources

### Student

Preface xiii

Preface to Second Edition xvii

1 Heat Conduction Fundamentals 1

1-1 The Heat Flux, 2

1-2 Thermal Conductivity, 4

1-3 Differential Equation of Heat Conduction, 6

1-4 Fourier’s Law and the Heat Equation in Cylindrical and Spherical Coordinate Systems, 14

1-5 General Boundary Conditions and Initial Condition for the Heat Equation, 16

1-6 Nondimensional Analysis of the Heat Conduction Equation, 25

1-7 Heat Conduction Equation for Anisotropic Medium, 27

1-8 Lumped and Partially Lumped Formulation, 29

References, 36

Problems, 37

2 Orthogonal Functions, Boundary Value Problems, and the Fourier Series 40

2-1 Orthogonal Functions, 40

2-2 Boundary Value Problems, 41

2-3 The Fourier Series, 60

2-4 Computation of Eigenvalues, 63

2-5 Fourier Integrals, 67

References, 73

Problems, 73

3 Separation of Variables in the Rectangular Coordinate System 75

3-1 Basic Concepts in the Separation of Variables Method, 75

3-2 Generalization to Multidimensional Problems, 85

3-3 Solution of Multidimensional Homogenous Problems, 86

3-4 Multidimensional Nonhomogeneous Problems: Method of Superposition, 98

3-5 Product Solution, 112

3-6 Capstone Problem, 116

References, 123

Problems, 124

4 Separation of Variables in the Cylindrical Coordinate System 128

4-1 Separation of Heat Conduction Equation in the Cylindrical Coordinate System, 128

4-2 Solution of Steady-State Problems, 131

4-3 Solution of Transient Problems, 151

4-4 Capstone Problem, 167

References, 179

Problems, 179

5 Separation of Variables in the Spherical Coordinate System 183

5-1 Separation of Heat Conduction Equation in the Spherical Coordinate System, 183

5-2 Solution of Steady-State Problems, 188

5-3 Solution of Transient Problems, 194

5-4 Capstone Problem, 221

References, 233

Problems, 233

Notes, 235

6 Solution of the Heat Equation for Semi-Infinite and Infinite Domains 236

6-1 One-Dimensional Homogeneous Problems in a Semi-Infinite Medium for the Cartesian Coordinate System, 236

6-2 Multidimensional Homogeneous Problems in a Semi-Infinite Medium for the Cartesian Coordinate System, 247

6-3 One-Dimensional Homogeneous Problems in An Infinite Medium for the Cartesian Coordinate System, 255

6-4 One-Dimensional homogeneous Problems in a Semi-Infinite Medium for the Cylindrical Coordinate System, 260

6-5 Two-Dimensional Homogeneous Problems in a Semi-Infinite Medium for the Cylindrical Coordinate System, 265

6-6 One-Dimensional Homogeneous Problems in a Semi-Infinite Medium for the Spherical Coordinate System, 268

References, 271

Problems, 271

7 Use of Duhamel’s Theorem 273

7-1 Development of Duhamel’s Theorem for Continuous Time-Dependent Boundary Conditions, 273

7-2 Treatment of Discontinuities, 276

7-3 General Statement of Duhamel’s Theorem, 278

7-4 Applications of Duhamel’s Theorem, 281

7-5 Applications of Duhamel’s Theorem for Internal Energy Generation, 294

References, 296

Problems, 297

8 Use of Green’s Function for Solution of Heat Conduction Problems 300

8-1 Green’s Function Approach for Solving Nonhomogeneous Transient Heat Conduction, 300

8-2 Determination of Green’s Functions, 306

8-3 Representation of Point, Line, and Surface Heat Sources with Delta Functions, 312

8-4 Applications of Green’s Function in the Rectangular Coordinate System, 317

8-5 Applications of Green’s Function in the Cylindrical Coordinate System, 329

8-6 Applications of Green’s Function in the Spherical Coordinate System, 335

8-7 Products of Green’s Functions, 344

References, 349

Problems, 349

9 Use of the Laplace Transform 355

9-1 Definition of Laplace Transformation, 356

9-2 Properties of Laplace Transform, 357

9-3 Inversion of Laplace Transform Using the Inversion Tables, 365

9-4 Application of the Laplace Transform in the Solution of Time-Dependent Heat Conduction Problems, 372

9-5 Approximations for Small Times, 382

References, 390

Problems, 390

10 One-Dimensional Composite Medium 393

10-1 Mathematical Formulation of One-Dimensional Transient Heat Conduction in a Composite Medium, 393

10-2 Transformation of Nonhomogeneous Boundary Conditions into Homogeneous Ones, 395

10-3 Orthogonal Expansion Technique for Solving M-Layer Homogeneous Problems, 401

10-4 Determination of Eigenfunctions and Eigenvalues, 407

10-5 Applications of Orthogonal Expansion Technique, 410

10-6 Green’s Function Approach for Solving Nonhomogeneous Problems, 418

10-7 Use of Laplace Transform for Solving Semi-Infinite and Infinite Medium Problems, 424

References, 429

Problems, 430

11 Moving Heat Source Problems 433

11-1 Mathematical Modeling of Moving Heat Source Problems, 434

11-2 One-Dimensional Quasi-Stationary Plane Heat Source Problem, 439

11-3 Two-Dimensional Quasi-Stationary Line Heat Source Problem, 443

11-4 Two-Dimensional Quasi-Stationary Ring Heat Source Problem, 445

References, 449

Problems, 450

12 Phase-Change Problems 452

12-1 Mathematical Formulation of Phase-Change Problems, 454

12-2 Exact Solution of Phase-Change Problems, 461

12-3 Integral Method of Solution of Phase-Change Problems, 474

12-4 Variable Time Step Method for Solving Phase-Change Problems: A Numerical Solution, 478

12-5 Enthalpy Method for Solution of Phase-Change Problems: A Numerical Solution, 484

References, 490

Problems, 493

Note, 495

13 Approximate Analytic Methods 496

13-1 Integral Method: Basic Concepts, 496

13-2 Integral Method: Application to Linear Transient Heat Conduction in a Semi-Infinite Medium, 498

13-3 Integral Method: Application to Nonlinear Transient Heat Conduction, 508

13-4 Integral Method: Application to a Finite Region, 512

13-5 Approximate Analytic Methods of Residuals, 516

13-6 The Galerkin Method, 521

13-7 Partial Integration, 533

13-8 Application to Transient Problems, 538

References, 542

Problems, 544

14 Integral Transform Technique 547

14-1 Use of Integral Transform in the Solution of Heat Conduction Problems, 548

14-2 Applications in the Rectangular Coordinate System, 556

14-3 Applications in the Cylindrical Coordinate System, 572

14-4 Applications in the Spherical Coordinate System, 589

14-5 Applications in the Solution of Steady-state problems, 599

References, 602

Problems, 603

Notes, 607

15 Heat Conduction in Anisotropic Solids 614

15-1 Heat Flux for Anisotropic Solids, 615

15-2 Heat Conduction Equation for Anisotropic Solids, 617

15-3 Boundary Conditions, 618

15-4 Thermal Resistivity Coefficients, 620

15-5 Determination of Principal Conductivities and Principal Axes, 621

15-6 Conductivity Matrix for Crystal Systems, 623

15-7 Transformation of Heat Conduction Equation for Orthotropic Medium, 624

15-8 Some Special Cases, 625

15-9 Heat Conduction in an Orthotropic Medium, 628

15-10 Multidimensional Heat Conduction in an Anisotropic Medium, 637

References, 645

Problems, 647

Notes, 649

16 Introduction to Microscale Heat Conduction 651

16-1 Microstructure and Relevant Length Scales, 652

16-2 Physics of Energy Carriers, 656

16-3 Energy Storage and Transport, 661

16-4 Limitations of Fourier’s Law and the First Regime of Microscale Heat Transfer, 667

16-5 Solutions and Approximations for the First Regime of Microscale Heat Transfer, 672

16-6 Second and Third Regimes of Microscale Heat Transfer, 676

16-7 Summary Remarks, 676

References, 676

APPENDIXES 679

Appendix I Physical Properties 681

Table I-1 Physical Properties of Metals, 681

Table I-2 Physical Properties of Nonmetals, 683

Table I-3 Physical Properties of Insulating Materials, 684

Appendix II Roots of Transcendental Equations 685

Appendix III Error Functions 688

Appendix IV Bessel Functions 691

Table IV-1 Numerical Values of Bessel Functions, 696

Table IV-2 First 10 Roots of Jn(z) = 0, n = 0, 1, 2, 3, 4, 5, 704

Table IV-3 First Six Roots of βJ1(β) − cJ0(β) = 0, 705

Table IV-4 First Five Roots of J0(β)Y0(cβ) − Y0(β)J0(cβ) = 0, 706

Appendix V Numerical Values of Legendre Polynomials of the

First Kind 707

Appendix VI Properties of Delta Functions 710

Index 713