Finite Difference Methods in Heat Transfer

Cover
Title Page
Copyright Page
Contents
Preface
Preface—First Edition
Chapter 1: Basic Relations
	1.1. Classification of Second-Order Partial Differential Equations
		1.1.1. Physical Significance of Parabolic, Elliptic, and Hyperbolic Systems
	1.2. Parabolic Systems
	1.3. Elliptic Systems
		1.3.1. Steady-State Diffusion
		1.3.2. Steady-State Advection–Diffusion
		1.3.3. Fluid Flow
	1.4. Hyperbolic Systems
	1.5. Systems of Equations
		1.5.1. Characterization of System of Equations
		1.5.2. Wave Equation
	1.6. Boundary Conditions
	1.7. Uniqueness of the Solution
	Problems
Chapter 2: Discrete Approximation of Derivatives
	2.1. Taylor Series Formulation
		2.1.1. Finite Difference Approximation of First Derivative
		2.1.2. Finite Difference Approximation of Second Derivative
		2.1.3. Differencing via Polynomial Fitting
		2.1.4. Finite Difference Approximation of Mixed Partial Derivatives
		2.1.5. Changing the Mesh Size
		2.1.6. Finite Difference Operators
	2.2. Control Volume Approach
	2.3. Boundary and Initial Conditions
		2.3.1. Discretization of Boundary Conditions with Taylor Series
			2.3.1.1. Boundary Condition of the First Kind
			2.3.1.2. Boundary Conditions of the Second and Third Kinds
		2.3.2. Discretization of Boundary Conditions with Control Volumes
			2.3.2.1. Boundary Condition of the First Kind
			2.3.2.2. Boundary Condition of the Second Kind
			2.3.2.3. Boundary Condition of the Third Kind
	2.4. Errors Involved in Numerical Solutions
		2.4.1. Round-Off Errors
		2.4.2. Truncation Error
		2.4.3. Discretization Error
		2.4.4. Total Error
		2.4.5. Stability
		2.4.6. Consistency
	2.5. Verification and Validation
		2.5.1. Code Verification
		2.5.2. Solution Verification
	Problems
	Notes
Chapter 3: Methods of Solving Systems of Algebraic Equations
	3.1. Reduction to Algebraic Equations
	3.2. Direct Methods
		3.2.1. Gauss Elimination Method
		3.2.2. Thomas Algorithm
	3.3. Iterative Methods
		3.3.1. Gauss–Seidel Iteration
		3.3.2. Successive Overrelaxation
		3.3.3. Red-Black Ordering Scheme
		3.3.4. LU Decomposition with Iterative Improvement
		3.3.5. Biconjugate Gradient Method
	3.4. Nonlinear Systems
	Problems
Chapter 4: One-Dimensional Steady-State Systems
	4.1. Diffusive Systems
		4.1.1. Slab
		4.1.2. Solid Cylinder and Sphere
		4.1.3. Hollow Cylinder and Sphere
		4.1.4. Heat Conduction through Fins
			4.1.4.1. Fin of Uniform Cross Section
			4.1.4.2. Finite Difference Solution
	4.2. Diffusive–Advective Systems
		4.2.1. Stability for Steady-State Systems
		4.2.2. Finite Volume Method
			4.2.2.1. Interpolation Functions
	Problems
Chapter 5: One-Dimensional Transient Systems
	5.1. Diffusive Systems
		5.1.1. Simple Explicit Method
			5.1.1.1. Prescribed Potential at the Boundaries
			5.1.1.2. Convection Boundary Conditions
			5.1.1.3. Prescribed Flux Boundary Condition
			5.1.1.4. Stability Considerations
			5.1.1.5. Effects of Boundary Conditions on Stability
			5.1.1.6. Effects of r on Truncation Error
			5.1.1.7. Fourier Method of Stability Analysis
		5.1.2. Simple Implicit Method
			5.1.2.1. Stability Analysis
		5.1.3. Crank–Nicolson Method
			5.1.3.1. Prescribed Heat Flux Boundary Condition
		5.1.4. Combined Method
			5.1.4.1. Stability of Combined Method
		5.1.5. Cylindrical and Spherical Symmetry
		5.1.6. Application of Simple Explicit Method
			5.1.6.1. Solid Cylinder and Sphere
			5.1.6.2. Stability of Solution
			5.1.6.3. Hollow Cylinder and Sphere
		5.1.7. Application of Simple Implicit Scheme
			5.1.7.1. Solid Cylinder and Sphere
			5.1.7.2. Hollow Cylinder and Sphere
		5.1.8. Application of Crank–Nicolson Method
	5.2. Advective–Diffusive Systems
		5.2.1. Purely Advective (Wave) Equation
			5.2.1.1. Upwind Method
			5.2.1.2. MacCormack’s Method
			5.2.1.3. Warming and Beam’s Method
		5.2.2. Advection–Diffusion Equation
			5.2.2.1. Simple Explicit Scheme
			5.2.2.2. Implicit Finite Volume Method
	5.3. Hyperbolic Heat Conduction Equation
		5.3.1. Finite Difference Representation of Hyperbolic Heat Conduction Equation
	Problems
Chapter 6: Transient Multidimensional Systems
	6.1. Simple Explicit Method
		6.1.1. Two-Dimensional Diffusion
		6.1.2. Two-Dimensional Transient Convection–Diffusion
			6.1.2.1. FTCS Differencing
			6.1.2.2. Upwind Differencing
			6.1.2.3. Control Volume Approach
	6.2. Combined Method
	6.3. ADI Method
	6.4. ADE Method
	6.5. An Application Related to the Hyperthermia Treatment of Cancer
	Problems
Chapter 7: Nonlinear Diffusion
	7.1. Lagging Properties by One Time Step
	7.2. Use of Three-Time-Level Implicit Scheme
		7.2.1. Internal Nodes
		7.2.2. Limiting Case R = 0 for Cylinder and Sphere
		7.2.3. Boundary Nodes
	7.3. Linearization
		7.3.1. Stability Criterion
	7.4. False Transient
		7.4.1. Simple Explicit Scheme
		7.4.2. Simple Implicit Scheme
		7.4.3. A Set of Diffusion Equations
	7.5. Applications in Coupled Conduction and Radiation in Participating Media
		7.5.1. One-Dimensional Problem with Diffusion Approximation
		7.5.2. Solution of the Three-Dimensional Equation of Radiative Transfer
	Problems
Chapter 8: Multidimensional Incompressible Laminar Flow
	8.1. Vorticity-Stream Function Formulation
		8.1.1. Vorticity and Stream Function
		8.1.2. Finite Difference Representation of Vorticity-Stream Function Formulation
			8.1.2.1. Vorticity Transport Equation
			8.1.2.2. Poisson’s Equation for Stream Function
			8.1.2.3. Poisson’s Equation for Pressure
		8.1.3. Method of Solution for Omega and Psi
			8.1.3.1. Solution for a Transient Problem
			8.1.3.2. Solution for a Steady-State Problem
		8.1.4. Method of Solution for Pressure
		8.1.5. Treatment of Boundary Conditions
			8.1.5.1. Boundary Conditions on Velocity
			8.1.5.2. Boundary Conditions on Psi
			8.1.5.3. Boundary Condition on Omega
			8.1.5.4. Boundary Conditions on Pressure
			8.1.5.5. Initial Condition
		8.1.6. Energy Equation
	8.2. Primitive Variables Formulation
		8.2.1. Determination of the Velocity Field: The SIMPLEC Method
		8.2.2. Treatment of Boundary Conditions
			8.2.2.1. Pressure
			8.2.2.2. Momentum and Energy Equations
	8.3. Two-Dimensional Steady Laminar Boundary Layer Flow
	Problems
Chapter 9: Compressible Flow
	9.1. Quasi-One-Dimensional Compressible Flow
		9.1.1. Solution with MacCormack’s Method
		9.1.2. Solution with WAF-TVD Method
	9.2. Two-Dimensional Compressible Flow
	Problems
Chapter 10: Phase Change Problems
	10.1. Mathematical Formulation of Phase Change Problems
		10.1.1. Interface Condition
		10.1.2. Generalization to Multidimensions
		10.1.3. Dimensionless Variables
		10.1.4. Mathematical Formulation
	10.2. Variable Time Step Approach for Single-Phase Solidification
		10.2.1. Finite Difference Approximation
			10.2.1.1. Differential Equation
			10.2.1.2. Boundary Condition at x = 0
			10.2.1.3. Interface Conditions
		10.2.2. Determination of Time Steps
			10.2.2.1. Starting Time Step Δt0
			10.2.2.2. Time Step Δt1
			10.2.2.3. Time Step Δtn
	10.3. Variable Time Step Approach for Two-Phase Solidification
		10.3.1. Finite Difference Approximation
			10.3.1.1. Equation for the Solid Phase
			10.3.1.2. Boundary Condition at x = 0
			10.3.1.3. Equation for the Liquid Phase
			10.3.1.4. Interface Conditions
		10.3.2. Determination of Time Steps
			10.3.2.1. Starting Time Step Δt0
			10.3.2.2. Time Step Δt1
			10.3.2.3. Time Steps Δtn, (2 ≤ n ≤ N – 4)
			10.3.2.4. Time Step ΔtN-3
			10.3.2.5. Time Step ΔtN–2
			10.3.2.6. Time Step ΔtN–1
	10.4. Enthalpy Method
		10.4.1. Explicit Enthalpy Method: Phase Change with Single Melting Temperature
			10.4.1.1. Algorithm for Explicit Method
			10.4.1.2. Interpretation of Enthalpy Results
			10.4.1.3. Improved Algorithm for Explicit Method
		10.4.2. Implicit Enthalpy Method: Phase Change with Single Melting Temperature
			10.4.2.1. Algorithm for Implicit Method
		10.4.3. Explicit Enthalpy Method: Phase Change over a Temperature Range
	10.5. Phase Change Model for Convective–Diffusive Problems
		10.5.1. Model for the Passive Scalar Transport Equation
		10.5.2. Model for the Energy Equation
	Problems
Chapter 11: Numerical Grid Generation
	11.1. Coordinate Transformation Relations
		11.1.1. Gradient
		11.1.2. Divergence
		11.1.3. Laplacian
		11.1.4. Normal Derivatives
		11.1.5. Tangential Derivatives
	11.2. Basic Ideas in Simple Transformations
	11.3. Basic Ideas in Numerical Grid Generation and Mapping
	11.4. Boundary Value Problem of Numerical Grid Generation
	11.5. Finite Difference Representation of Boundary Value Problem of Numerical Grid Generation
	11.6. Steady-State Heat Conduction in Irregular Geometry
	11.7. Steady-State Laminar Free Convection in Irregular Enclosures—Vorticity-Stream Function Formulation
		11.7.1. The Nusselt Number
		11.7.2. Results
	11.8. Transient Laminar Free Convection in Irregular Enclosures—Primitive Variables Formulation
	11.9. Computational Aspects for the Evaluation of Metrics
		11.9.1. One-Dimensional Advection–Diffusion Equation
		11.9.2. Two-Dimensional Heat Conduction in a Hollow Sphere
	Problems
	Notes
Chapter 12: Hybrid Numerical–Analytical Solutions
	12.1. Combining Finite Differences and Integral Transforms
		12.1.1. The Hybrid Approach
		12.1.2. Hybrid Approach Application: Transient Forced Convection in Channels
	12.2. Unified Integral Transforms
		12.2.1. Total Transformation
		12.2.2. Partial Transformation
		12.2.3. Computational Algorithm
		12.2.4. Test Case
	12.3. Convective Eigenvalue Problem
	Problems
Appendix A. Subroutine Gauss
Appendix B. Subroutine Trisol
Appendix C. Subroutine SOR
Appendix D. Subroutine BICGM2
Appendix E. Program to Solve Example 10.1
Bibliography
Index
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