This text focuses on finite difference methods and their application to the solution of heat transfer problems. Such methods are based on the discretization of governing equations, initial and boundary conditions, which then replace a continuous partial differential problem by a system of algebraic equations. Finite difference methods are a versatile tool for scientists and for engineers. This updated book serves university students taking graduate-level coursework in heat transfer, as well as being an important reference for researchers and engineering.
Table of Contents:
Basic Relations
Classification of Second-Order Partial Differential Equations
Parabolic Systems
Elliptic Systems
Hyperbolic Systems
Systems of Equations
Boundary Conditions
Uniqueness of the Solution Problems
Discrete Approximation of Derivatives
Taylor Series Formulation
Finite Difference Operators
Control-Volume Approach
Application of Control-Volume Approach
Boundary Conditions
Errors Involved in Numerical Solutions Problems
Methods of Solving Sets of Algebraic Equations
Reduction to Algebraic Equations
Direct Methods
Iterative Methods
Nonlinear Systems Problems
One-Dimensional Steady-State Systems
Diffusive Systems
Diffusive-Convective System
Diffusive-Convective System with Flow Problems
One-Dimensional Parabolic Systems
Simple Explicit Method
Simple Implicit Method
Crank-Nicolson Method
Combined Method
Cylindrical and Spherical Symmetry
A Summary of Finite-Difference Schemes Problems
Multidimensional Parabolic Systems
Simple Explicit Method
(i) Two-Dimensional Diffusion
(ii) Two-Dimensional Steady Laminar Boundary Layer Flow
(iii) Two-Dimensional Transient Convection-Diffusion
Combined Method
(i) Three-Dimensional Diffusion
Alternating Direction Implicit (ADI) Method
Alternating Direction Explicit (ADE) Method
(i) One-Dimensional Diffusion
(ii) Two-Dimensional Diffusion
Modified Upwind Method
(i) Transient Forced Convection Inside Ducts for Step Change in Fluid Inlet
Temperature
Pressure-Velocity Coupling Problems
Elliptic Systems
Steady-State Diffusion
Velocity Field for Incompressible, Constant Property, Two-Dimensional Flow
Vorticity – Stream Function Formulation
Problems
Hyperbolic Systems
Hyperbolic Convection (Wave) Equation
Hyperbolic Heat Conduction Equation
System of Vector Equations Problems
Nonlinear Diffusion
Lagging Properties by One Time Step
Use of Three-Time Level Implicit Scheme
Linearization
Method of False Transients for Solving Steady-State Diffusion
Simultaneous Conduction and Radiation in Participating Media – Diffusion
Approximation
Three-Dimensional Simultaneous Conduction and Radiation in Participating Media
Problems
Phase Change Problems
Mathematical Formulation of Phase Change Problems
Variable Time Step Approach for Single-Phase Solidification
Variable Time Step Approach for Two-Phase Solidification
Enthalpy Method
Phase Change Problems with Natural Convection
Problems
Numerical Grid Generation
Coordinate Transformation Relations
Basic Ideas in Simple Transformations
Basic Ideas in Numerical Grid Generation and Mapping
Boundary Value Problem of Numerical Grid Generation
Finite Difference Representation of Boundary Value Problem of Numerical Grid Generation
Steady State Heat Conduction in Irregular Geometry
Laminar Forced Convection in Irregular Channels
Laminar Free Convection in Irregular Enclosures
Problems
Hybrid Numerical-Analytic Solutions
The Classical (CITT) and the Generalized Integral Transform (GITT) Techniques
GITT with Partial Transformation
Unified Integral Transforms (UNIT) Algorithm
Applications in Heat Conduction
Applications in Heat Convection
Problems
References
Appendices
Appendix I Discretization Formulae
Index
