Heat Conduction

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 ?J1(?) ? cJ0(?) = 0 705

Table IV-4 First Five Roots of J0(?)Y0(c?) ? Y0(?)J0(c?) = 0 706

Appendix V Numerical Values of Legendre Polynomials of the First Kind 707

Appendix VI Properties of Delta Functions 710

Index 713