Computational Methods for Heat and Mass Transfer (Hardcover)

PrefaceNomenclaturePart I: Basic Equations and Numerical Analysis1. Review of Basic Laws and Equations1.1 Basic Equations1.2 Fluid Flows1.2.1 Fluid Properties1.2.2 Basic Equations in Integral Forms1.2.3 Differential Analysis of Fluid Motion 1.2.4 Boundary Conditions for Flow Field1.3 Heat Transfer1.3.1 Basic Modes and Transport Rate Equation1.3.2 The First Law of Thermodynamics and Heat Equations1.3.3 Thermal Initial and Boundary Conditions 1.4 Mass Transfer1.4.1 Basic Modes and Transport Rate Equation1.4.2 Conservation of Mass Species and Mass Concentration Equation1.4.3 Initial and Boundary Conditions for Mass Transfer1.5 Mathematical Classification of Governing EquationProblems2. Approximations and Errors2.1 Truncation Error2.2 Round off Error2.2.1 Significant Figures or Digits2.2.2 Computers Number System2.2.3 Machine Epsilon2.3 Error Definitions2.4 Approximate Error2.5 Convergence CriteriaProblems3. Numerical Solution of Systems of Equations 3.1 Mathematical Background3.1.1 Representation of the System of Equations3.1.2 The Cramer's Rule and the Elimination of Unknowns3.2 Direct Methods3.2.1 Gauss Elimination3.2.2 Gauss-Jordon Elimination Method3.2.3 Decomposition of Factorization Methods3.2.4 Banded Systems3.2.5 Error Equations and Iterative Refinement3.3 Iterative Methods3.3.1 Jacobi Method3.3.2 Gauss-Seidel Method3.3.3 Convergence Criterion for Iterative Methods3.3.4 Successive Over-Relaxation (SOR) method3.3.5 Conjugate Gradient Method3.3.6 Pre-conditioned Conjugate Gradient MethodProblems4. Numerical Integration 4.1 Newton - Cotes Integration Formulas4.1.1 The Trapezoidal Rule 4.1.2 Simpson's Integration Formula4.1.3 Summary of Newton-Cotes Integration Formulas4.2 Romberg Integration4.3 Gauss Quadrature4.3.1 Two-point Gauss-Legendre Formula4.3.2 Higher-Points Gauss-Legendre Formulas4.4 Multi-Dimensional Numerical IntegrationProblemsPart II: Finite Difference - Control Volume Method5. Basics of Finite Difference and Control Volume Method 5.1 Introduction and Basic steps in Finite Difference Method5.2 Discretization of the Domain5.3 Discretization of the Mathematical Model5.3.1 The Taylor Series Method5.3.2 Control Volume Method5.4 One-dimensional Steady State Diffusion5.5 Variable Source Term5.6 Boundary Conditions5.7 Grid Size Distribution5.8 Non-uniform Transport Property5.9 Nonlinearity5.10 Linearization of a Variable Source TermProblems6. Multi-Dimensional Problems 6.1 Two-dimensional Steady State Problems6.2 Boundary Conditions6.2.1 Corner Boundary Nodes6.3 Irregular Geometries6.4 Three Dimensional Steady State ProblemsProblems7. Diffusion Equation7.1 Time Discretization Procedure7.2 Explicit Scheme7.2.1 Discretization by Control Volume Approach7.2.2 Finite Difference Equations by Taylor Series Expansions7.2.3 Stability Consideration7.2.4 Other Explicit Scheme7.2.5 Boundary Conditions 7.3 Implicit Scheme7.3.1 Discretization Equation by Control Volume Approach7.3.2 Finite Difference Equation by Taylor Series Expansion7.3.3 A General Formulation of Fully-Implicit Scheme for One-dimensional Problems7.3.4 A General Formulation of Fully-Implicit Scheme for Two-dimensional Problems7.3.5 Solution Methods for Two-dimensional Implicit Scheme7.3.6 Boundary Conditions for Implicit Scheme7.4 Crank-Nicolson Scheme7.4.1 Solution Method for Crank-Nicolson Scheme7.5 Splitting Method7.5.1 ADI Method7.5.2 ADE MethodProblems8. Finite Difference-Control Volume Method: Convection Heat Transfer8.1 Spatial Discretization using Control Volume Method8.1.1 Central Difference Scheme8.1.2 Upwind Scheme8.1.3 Exponential Scheme8.1.4 Hybrid Scheme8.1.5 Power Law Scheme8.1.6 Generalized Convection-Diffusion Scheme8.2 Discretization of a General Transport Equations8.2.1 One-dimensional Unsteady State Problem8.2.2 Two-dimensional Unsteady State Problem8.2.3 Three-dimensional Unsteady State Problem8.3 Solution of Flow Field8.3.1 Stream Function/Vorticity-Based Method8.3.2 Direct Solution with the Primitive VariableProblemsPart III: Finite Element Method 9. Introduction and Basic Steps in Finite Element Method9.1 Comparison of Finite Difference/Control Volume Method and Finite Element Method9.2 Basic Steps in Finite Element Methods9.3 Integral Formulation9.3.1 Variational Formulations9.3.2 Method of Weighted Residuals9.4 Variational Methods9.4.1 The Rayleigh-Ritz Variational Method9.4.2 Weighted Residual Variational MethodsProblems10. Element Shape Functions10.1 One-dimensional Element10.1.1 One-dimensional Linear Element 10.1.2 One-dimensional Quadratic Line Element10.1.3 One-dimensional Cubic Element10.2 Two-dimensional Element10.2.1 Linear Triangular Element10.2.2 Quadratic Triangular Element10.2.3 Two-dimensional Quadrilateral Element10.3 Three-dimensional Element10.3.1 Three-dimensional Tetrahedron Element10.3.2 Three-dimensional Hexahedron ElementProblems 11. Finite Element Method: One-dimensional Steady State Problems11.1 Finite Element Formulation using Galerkin MethodDiscretization of the Solution DomainSelection of Approximation Solution FunctionFormation of Integral Statement of the Problem Formation of Element Characteristics Equation Assembly of Element Equations to Form the Global System Implementation of Boundary ConditionsSolution of Global System11.2 Finite Element Formulation using Variational ApproachFormation of the Integral Statement of Variational FormFormation of Element Characteristics Equation11.3 Boundary Conditions11.3.1 Boundary Conditions of the First Kind or Constant Surface Flux11.3.2 Mixed Boundary Conditions11.3.3 Variable Source Term11.3.4 Axisymmetric ProblemsProblems12. Finite Element Method: Multi-dimensional Steady State Problems12.1 Two-dimensional Steady State Diffusion EquationMesh Generation or Discretization of Solution DomainElement and node NumberingSelection of Approximation Solution FunctionFormulation of an Integral Statement using Galerkin ApproachFormation of Element Characteristics EquationsAssembly of Element Equations and Formation of Global System 12.2 Two-dimensional Axisymmetric Problems Selection of Approximate Solution FunctionFormation of an Integral StatementFormation of Element Characteristics EquationsImplementation of Boundary ConditionsAxisymmetric Element12.3 Three-dimensional Problems12.4 FE Formulation Using Variational Approach12.5 Point SourceProblems13. Finite Element Method: Unsteady State Problems13.1 Discretization Scheme13.2 One-dimensional Unsteady State Problems13.2.1 Semi-discrete Finite Element Formulations13.2.2 Time Approximations13.2.3 Stability Considerations13.3 Two-dimensional Unsteady State Diffusion Equation13.4 Three-dimensional Unsteady State Diffusion EquationProblems14. Finite Element Method: Convection Heat Transfer14.1 Classification of Finite Elements Methods for Convection Problems14.2 Velocity-Pressure or Mixed Method14.2.1 One dimensional Convection-Diffusion Problem 14.2.2 Two-dimensional Viscous Incompressible Flow14.2.3 Unsteady Two-dimensional Viscous Incompressible Flow14.2.4 Unsteady Two-dimensional Viscous Incompressible Flow14.2.5 Convection Problems14.3 Solution Methods14.3.1 Steady State Problems14.3.2 Unsteady State ProblemProblemsAppendix A. Review of Vectors and Matrices Appendix B. Integral Theorems BibliographyIndex