Finite Difference Methods in Heat Transfer, Second Edition PDF
Preview Finite Difference Methods in Heat Transfer, Second Edition
Finite Difference Methods in Heat Transfer Finite Difference Methods in Heat Transfer Second Edition M. Necati Özis¸ik Helcio R.B. Orlande Marcelo José Colaço Renato Machado Cotta CRCPress Taylor&FrancisGroup 6000BrokenSoundParkwayNW,Suite300 BocaRaton,FL33487-2742 ©2017byTaylor&FrancisGroup,LLC CRCPressisanimprintofTaylor&FrancisGroup,anInformabusiness NoclaimtooriginalU.S.Governmentworks Printedonacid-freepaper InternationalStandardBookNumber-13:978-1-4822-4345-1(Hardback) This book contains information obtained from authentic and highly regarded sources. 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Colaço, and Roberta Viegas (MJC) To Carolina, Bianca (in memorium), Victor, Clara, and Gabriel (RMC) Contents Preface....................................................................................................................xv Preface—First Edition..........................................................................................xix 1. Basic Relations................................................................................................1 1.1 Classification of Second-Order Partial Differential Equations......2 1.1.1 Physical Significance of Parabolic, Elliptic, and Hyperbolic Systems.................................................................4 1.2 Parabolic Systems.................................................................................5 1.3 Elliptic Systems.....................................................................................7 1.3.1 Steady-State Diffusion.............................................................7 1.3.2 Steady-State Advection–Diffusion........................................7 1.3.3 Fluid Flow.................................................................................8 1.4 Hyperbolic Systems..............................................................................8 1.5 Systems of Equations...........................................................................9 1.5.1 Characterization of System of Equations...........................10 1.5.2 Wave Equation.......................................................................11 1.6 Boundary Conditions.........................................................................12 1.7 Uniqueness of the Solution...............................................................15 Problems..........................................................................................................18 2. Discrete Approximation of Derivatives...................................................23 2.1 Taylor Series Formulation.................................................................24 2.1.1 Finite Difference Approximation of First Derivative.......25 2.1.2 Finite Difference Approximation of Second Derivative................................................................................27 2.1.3 Differencing via Polynomial Fitting...................................28 2.1.4 Finite Difference Approximation of Mixed Partial Derivatives..............................................................................29 2.1.5 Changing the Mesh Size.......................................................31 2.1.6 Finite Difference Operators..................................................33 2.2 Control Volume Approach................................................................34 2.3 Boundary and Initial Conditions.....................................................38 2.3.1 Discretization of Boundary Conditions with Taylor Series...........................................................................40 2.3.1.1 Boundary Condition of the First Kind...............41 2.3.1.2 Boundary Conditions of the Second and Third Kinds.............................................................41 vii viii Contents 2.3.2 Discretization of Boundary Conditions with Control Volumes....................................................................42 2.3.2.1 Boundary Condition of the First Kind...............43 2.3.2.2 Boundary Condition of the Second Kind..........44 2.3.2.3 Boundary Condition of the Third Kind.............44 2.4 Errors Involved in Numerical Solutions.........................................46 2.4.1 Round-Off Errors...................................................................46 2.4.2 Truncation Error....................................................................46 2.4.3 Discretization Error...............................................................47 2.4.4 Total Error..............................................................................47 2.4.5 Stability....................................................................................48 2.4.6 Consistency.............................................................................48 2.5 Verification and Validation...............................................................49 2.5.1 Code Verification...................................................................50 2.5.2 Solution Verification..............................................................54 Problems..........................................................................................................58 Notes................................................................................................................62 3. Methods of Solving Systems of Algebraic Equations..........................65 3.1 Reduction to Algebraic Equations...................................................65 3.2 Direct Methods....................................................................................70 3.2.1 Gauss Elimination Method..................................................71 3.2.2 Thomas Algorithm................................................................72 3.3 Iterative Methods................................................................................75 3.3.1 Gauss–Seidel Iteration...........................................................75 3.3.2 Successive Overrelaxation....................................................79 3.3.3 Red-Black Ordering Scheme................................................81 3.3.4 LU Decomposition with Iterative Improvement..............83 3.3.5 Biconjugate Gradient Method..............................................83 3.4 Nonlinear Systems..............................................................................84 Problems..........................................................................................................88 4. One-Dimensional Steady-State Systems.................................................97 4.1 Diffusive Systems................................................................................97 4.1.1 Slab...........................................................................................97 4.1.2 Solid Cylinder and Sphere...................................................98 4.1.3 Hollow Cylinder and Sphere.............................................105 4.1.4 Heat Conduction through Fins.........................................110 4.1.4.1 Fin of Uniform Cross Section............................112 4.1.4.2 Finite Difference Solution...................................113 4.2 Diffusive–Advective Systems ........................................................116 4.2.1 Stability for Steady-State Systems.....................................118 4.2.2 Finite Volume Method........................................................119 4.2.2.1 Interpolation Functions.......................................121 Problems........................................................................................................124 Contents ix 5. One-Dimensional Transient Systems.....................................................129 5.1 Diffusive Systems..............................................................................129 5.1.1 Simple Explicit Method......................................................130 5.1.1.1 Prescribed Potential at the Boundaries............131 5.1.1.2 Convection Boundary Conditions.....................132 5.1.1.3 Prescribed Flux Boundary Condition...............133 5.1.1.4 Stability Considerations......................................134 5.1.1.5 Effects of Boundary Conditions on Stability...136 5.1.1.6 Effects of r on Truncation Error........................137 5.1.1.7 Fourier Method of Stability Analysis...............138 5.1.2 Simple Implicit Method......................................................146 5.1.2.1 Stability Analysis.................................................147 5.1.3 Crank–Nicolson Method....................................................148 5.1.3.1 Prescribed Heat Flux Boundary Condition.....151 5.1.4 Combined Method..............................................................152 5.1.4.1 Stability of Combined Method..........................154 5.1.5 Cylindrical and Spherical Symmetry...............................155 5.1.6 Application of Simple Explicit Method...........................156 5.1.6.1 Solid Cylinder and Sphere.................................156 5.1.6.2 Stability of Solution.............................................160 5.1.6.3 Hollow Cylinder and Sphere.............................162 5.1.7 Application of Simple Implicit Scheme...........................164 5.1.7.1 Solid Cylinder and Sphere.................................164 5.1.7.2 Hollow Cylinder and Sphere.............................165 5.1.8 Application of Crank–Nicolson Method..........................166 5.2 Advective–Diffusive Systems..........................................................169 5.2.1 Purely Advective (Wave) Equation..................................169 5.2.1.1 Upwind Method..................................................170 5.2.1.2 MacCormack’s Method.......................................172 5.2.1.3 Warming and Beam’s Method...........................173 5.2.2 Advection–Diffusion Equation..........................................179 5.2.2.1 Simple Explicit Scheme.......................................179 5.2.2.2 Implicit Finite Volume Method.........................182 5.3 Hyperbolic Heat Conduction Equation.........................................185 5.3.1 Finite Difference Representation of Hyperbolic Heat Conduction Equation................................................186 Problems........................................................................................................190 6. Transient Multidimensional Systems....................................................207 6.1 Simple Explicit Method...................................................................207 6.1.1 Two-Dimensional Diffusion...............................................208 6.1.2 Two-Dimensional Transient Convection–Diffusion.......213 6.1.2.1 FTCS Differencing................................................213 6.1.2.2 Upwind Differencing...........................................214 6.1.2.3 Control Volume Approach.................................215
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