Heat Conduction

DAVID W. HAHN is the Knox T. Millsaps Professor of Mechanical and Aerospace Engineering at the University of Florida, Gainesville. His areas of specialization include both thermal sciences and biomedical engineering, including the development and application of laser-based diagnostic techniques and general laser-material interactions. The late M. NECATI ÖZISIK retired as Professor Emeritus of North Carolina State University’s Mechanical and Aerospace Engineering Department, where he spent most of his academic career. Professor ÖziŞik dedicated his life to education and research in heat transfer. His outstanding contributions earned him several awards, including the Outstanding Engineering Educator Award from the American Society for Engineering Education in 1992.

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