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1 Introduction1

1.1 Thermal Transport1

1.2 Mass Transfer and Fluid Flow3

1.3 An Example4

1.4 Importance of Analytical and Experimental Methods6

1.5 Numerical Approach9

1.6 Basic Considerations in a Numerical Solution12

1.7 Outline and Scope of the Book15

References17

Part 1 Mathematical Background19

2 Governing Equations21

2.1 Classification21

2.2 Representative Differential Equations from Heat Transfer and Fluid Flow23

2.3 Boundary and Initial Conditions26

2.4 Integral Forms28

2.5 Numerical Solution31

2.5.1 Basic Equations31

2.5.2 Different Approaches33

References34

Problems35

3 Finite Differences37

3.1 Basic Concepts39

3.1.1 Direct Approximation Approach39

3.1.2 Polynomial Representation41

3.1.3 Taylor Series Approach and Accuracy44

3.1.4 Control Volume Approach and Conservation48

3.1.5 Numerical Considerations50

3.1.5.1 Total Truncation Error51

3.1.5.2 Discretization and Roundoff Errors52

3.1.5.3 Convergence53

3.1.5.4 Numerical Stability and the Equivalence Theorem53

3.2 Steady-State Diffusion55

3.2.1 Discretization55

3.2.2 Solution of Simultaneous Equations58

3.2.2.1 Iterative Methods59

3.2.2.2 Direct Methods65

3.3 Transient Diffusion70

3.3.1 Two-Level Time Discretization70

3.3.2 Matrix Stability Analysis72

3.3.3 Fourier Series Stability Analysis76

3.3.4 An Example of Numerical Instability78

3.3.5 Other Explicit and Implicit Schemes80

References81

Problems82

4 Finite Elements85

4.1 Basic Concepts86

4.1.1 Discretization88

4.1.2 Interpolation Functions88

4.1.3 Integral Representations and Galerkin’s Method89

4.1.4 Assembly90

4.1.5 Elements90

4.1.6 Condensation and Substructuring91

4.1.7 Practical Implementation93

4.2 Steady-State Diffusion94

4.2.1 Matrix Equations with Boundary Conditions94

4.2.2 One-Dimensional Diffusion97

4.2.3 Two-Dimensional Diffusion99

4.2.4 Typical FEM Solutions102

4.3 Transient Diffusion108

4.3.1 The Matrix System108

4.3.2 Finite Differences in Time109

4.3.3 Diagonalization111

4.3.4 Transient One-Dimensional Diffusion111

4.3.5 Other Methods and Solutions112

References113

Problems114

Part 2 Simulation of Transport Processes119

5 Numerical Methods for Conduction Heat Transfer121

5.1 Governing Equations122

5.2 Numerical Solution of Steady-State Conduction124

5.2.1 One-Dimensional Conduction124

5.2.1.1 Basic Equations124

5.2.1.2 Finite Difference Approximation of the Boundary Conditions127

5.2.1.3 An Example：Numerical Solution of Heat Transfer in an Extended Surface130

5.2.1.4 Runge-Kutta Methods132

5.2.1.5 Finite Difference Method135

5.2.2 Multidimensional Steady-State Conduction137

5.2.2.1 Finite Difference Formulation139

5.2.2.2 Solution：Iterative and Direct Methods144

5.2.2.3 Improvement in Accuracy of Numerical Results149

5.2.2.4 Finite Element Formulation150

5.2.3 Variable Property and Other Considerations152

5.3 Numerical Solution of Unsteady-State Conduction166

5.3.1 One-Dimensional Unsteady-State Conduction168

5.3.1.1 FTCS Explicit Method169

5.3.1.2 Other Methods178

5.3.2 Numerical Approximation of Lumped Mass and Semi-infinite Solids180

5.3.3 Multidimensional Unsteady-State Conduction184

5.3.4 Numerical Methods for Time-Varying Boundary Conditions189

5.3.5 Property Variation195

5.3.6 Finite Element Solution198

5.4 Grid Generation203

5.5 Summary206

References207

Problems209

6 Numerical Methods for Convection Heat Transfer215

6.1 Governing Equations217

6.2 Computation of Forced Convection with Constant Fluid Properties220

6.2.1 Inviscid Flow：Introduction to Stream Function and Vorticity221

6.2.2 Equations for Viscous Flow：Primitive and Derived Variables228

6.2.3 Linear Viscous Flow（Creeping Flow）229

6.2.4 Computation of Boundary Layer Flows233

6.2.4.1 Similarity Solution：Ordinary Differential Equations234

6.2.4.2 Finite Difference Approach238

6.2.5 Numerical Solution of the Full Equations250

6.2.5.1 Central Differencing252

6.2.5.2 Upwind，Hybrid and Other Lower-Order Differencing Schemes253

6.2.5.3 Higher-Order Differencing Schemes for Convection256

6.2.5.4 Other Numerical Methods and Considerations259

6.2.5.5 Steady State Solution265

6.2.5.6 Primitive Variables Approach266

6.2.5.7 Simpler Algorithm269

6.2.6 Finite Difference Considerations of the Conservative Form274

6.2.7 Concluding Remarks on Flow Calculations279

6.2.8 Energy Equation280

6.2.8.1 Numerical Formulation280

6.2.8.2 Boundary Conditions284

6.2.8.3 Numerical Solution289

6.2.9 Numerical Solution of Turbulent Flows297

6.3 Computation of Natural Convection Flow and Transport308

6.3.1 Similarity Solutions310

6.3.2 Finite Difference Methods315

6.3.3 Additional Considerations324

6.4 Convection with Variable Fluid Properties325

6.5 Finite Element Methods330

6.5.1 Discretization and Interpolation Functions331

6.5.2 Integral Representation331

6.5.3 Element Equations and Assembly333

6.5.4 Solution335

6.5.5 Examples and Other Considerations335

6.5.6 Comparison of Finite Element and Finite Difference Methods337

6.6 Summary338

References339

Problems345

7 Numerical Methods for Radiation Heat Transfer353

7.1 Basic Concepts354

7.2 Numerical Techniques for Enclosures with Diffuse-Gray Surfaces359

7.2.1 Radiosity Method359

7.2.2 Absorption Factor Method364

7.2.3 Additional Considerations365

7.2.3.1 Computation of View Factors365

7.2.3.2 Temperature Dependence of Surface Properties366

7.2.3.3 Spectral Variation369

7.3 Nonuniform Irradiation and Emission：Discrete Integral Equations370

7.4 Numerical Solution of Radiation in the Presence of Other Modes379

7.4.1 Combined Modes at Boundaries：Nonparticipating Media380

7.4.2 Participating Media385

7.5 Other Methods For Participating Media394

7.6 Monte Carlo Method403

7.7 Summary406

References407

Problems409

Part 3 Combined Modes and Process Applications415

8 Applications of Computational Heat Transfer417

8.1 Numerical Simulation of Thermal Systems in Manufacturing418

8.1.1 Heat Treatment：Temperature Regulation418

8.1.2 Surface Treatment：Semi-infinite Approximation422

8.1.3 Continuously Moving Materials：Moving Boundary Effects424

8.1.4 Melting and Solidification：Phase Change Considerations427

8.1.5 Other Processes436

8.2 Numerical Simulation of Environmental Heat Transfer Problems442

8.2.1 Cooling Ponds：Periodic Processes442

8.2.2 Recirculating Flows in Enclosed Spaces447

8.2.3 Fire-Induced Flows in Partial Enclosures454

8.2.4 Free Boundary Flows and Other Problems457

8.2.5 Summary466

8.3 Computer Simulation and Computer-Aided Design of Thermal Systems468

8.3.1 General Approach468

8.3.2 Example of Computer Simulation of a Thermal System470

References475

Problems479

Appendices483

A Finite Difference Approximations483

B Sample Computer Programs487

B.1 Successive Over-Relaxation（SOR）Method488

B.2 Tridiagonal Matrix Algorithm（TDMA）or Thomas Algorithm491

B.3 Gauss-Jordan Elimination Method492

B.4 Forward-Time-Central-Space（FTCS）Method495

B.5 Crank-Nicolson Method497

B.6 Newton-Raphson Method503

B.7 Finite Difference Method for ODEs504

B.8 Runge-Kutta Method507

B.9 Alternating-Direction-Implicit（ADI）Method512

C Material Properties523

Nomenclature533

Index537