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تذكير بمساهمة فاتح الموضوع : أخواني في الله أحضرت لكم كتاب Fundamentals of the Finite Element Method for Heat and Fluid Flow Roland W. Lewis, Perumal Nithiarasu, Kankanhally N. Seetharamu
![كتاب Fundamentals of the Finite Element Method for Heat and Fluid Flow - صفحة 2 F_o_f_10](https://i.servimg.com/u/f61/15/36/74/14/f_o_f_10.jpg) و المحتوى كما يلي :
Contents Preface xiii 1 Introduction 1 1.1 Importance of Heat Transfer 1 1.2 Heat Transfer Modes 2 1.3 The Laws of Heat Transfer . 3 1.4 Formulation of Heat Transfer Problems 5 1.4.1 Heat transfer from a plate exposed to solar heat flux . 5 1.4.2 Incandescent lamp . 7 1.4.3 Systems with a relative motion and internal heat generation . 8 1.5 Heat Conduction Equation . 10 1.6 Boundary and Initial Conditions 13 1.7 Solution Methodology . 14 1.8 Summary 15 1.9 Exercise . 15 Bibliography . 17 2 Some Basic Discrete Systems 18 2.1 Introduction . 18 2.2 Steady State Problems . 19 2.2.1 Heat flow in a composite slab . 19 2.2.2 Fluid flow network . 22 2.2.3 Heat transfer in heat sinks (combined conduction–convection) . 25 2.2.4 Analysis of a heat exchanger . 27 2.3 Transient Heat Transfer Problem (Propagation Problem) . 29 2.4 Summary 31 2.5 Exercise . 31 Bibliography . 37 3 The Finite Element Method 38 3.1 Introduction . 38 3.2 Elements and Shape Functions . 41 3.2.1 One-dimensional linear element 42 3.2.2 One-dimensional quadratic element 45viii CONTENTS 3.2.3 Two-dimensional linear triangular elements 48 3.2.4 Area coordinates 52 3.2.5 Quadratic triangular elements . 54 3.2.6 Two-dimensional quadrilateral elements . 57 3.2.7 Isoparametric elements . 62 3.2.8 Three-dimensional elements 70 3.3 Formulation (Element Characteristics) . 75 3.3.1 Ritz method (Heat balance integral method—Goodman’s method) . 76 3.3.2 Rayleigh–Ritz method (Variational method) . 78 3.3.3 The method of weighted residuals . 80 3.3.4 Galerkin finite element method 85 3.4 Formulation for the Heat Conduction Equation 87 3.4.1 Variational approach 88 3.4.2 The Galerkin method 91 3.5 Requirements for Interpolation Functions . 92 3.6 Summary 98 3.7 Exercise . 98 Bibliography . 100 4 Steady State Heat Conduction in One Dimension 102 4.1 Introduction . 102 4.2 Plane Walls . 102 4.2.1 Homogeneous wall . 102 4.2.2 Composite wall . 103 4.2.3 Finite element discretization 105 4.2.4 Wall with varying cross-sectional area 107 4.2.5 Plane wall with a heat source: solution by linear elements . 108 4.2.6 Plane wall with a heat source: solution by quadratic elements 112 4.2.7 Plane wall with a heat source: solution by modified quadratic equations (static condensation) 114 4.3 Radial Heat Flow in a Cylinder 115 4.3.1 Cylinder with heat source . 117 4.4 Conduction–Convection Systems . 120 4.5 Summary 123 4.6 Exercise . 123 Bibliography . 125 5 Steady State Heat Conduction in Multi-dimensions 126 5.1 Introduction . 126 5.2 Two-dimensional Plane Problems . 127 5.2.1 Triangular elements 127 5.3 Rectangular Elements 136 5.4 Plate with Variable Thickness . 139CONTENTS ix 5.5 Three-dimensional Problems 141 5.6 Axisymmetric Problems 142 5.6.1 Galerkin’s method for linear triangular axisymmetric elements . 145 5.7 Summary 147 5.8 Exercise . 147 Bibliography . 149 6 Transient Heat Conduction Analysis 150 6.1 Introduction . 150 6.2 Lumped Heat Capacity System 150 6.3 Numerical Solution . 152 6.3.1 Transient governing equations and boundary and initial conditions . 152 6.3.2 The Galerkin method 153 6.4 One-dimensional Transient State Problem . 154 6.4.1 Time discretization using the Finite Difference Method (FDM) . 156 6.4.2 Time discretization using the Finite Element Method (FEM) 160 6.5 Stability . 161 6.6 Multi-dimensional Transient Heat Conduction 162 6.7 Phase Change Problems—Solidification and Melting . 164 6.7.1 The governing equations 164 6.7.2 Enthalpy formulation 165 6.8 Inverse Heat Conduction Problems 168 6.8.1 One-dimensional heat conduction . 168 6.9 Summary 170 6.10 Exercise . 170 Bibliography . 172 7 Convection Heat Transfer 173 7.1 Introduction . 173 7.1.1 Types of fluid-motion-assisted heat transport . 174 7.2 Navier–Stokes Equations 175 7.2.1 Conservation of mass or continuity equation . 175 7.2.2 Conservation of momentum 177 7.2.3 Energy equation 181 7.3 Non-dimensional Form of the Governing Equations . 183 7.3.1 Forced convection . 184 7.3.2 Natural convection (Buoyancy-driven convection) 185 7.3.3 Mixed convection . 187 7.4 The Transient Convection–diffusion Problem . 187 7.4.1 Finite element solution to convection–diffusion equation 188 7.4.2 Extension to multi-dimensions . 195 7.5 Stability Conditions . 200 7.6 Characteristic-based Split (CBS) Scheme . 201 7.6.1 Spatial discretization 206x CONTENTS 7.6.2 Time-step calculation 210 7.6.3 Boundary and initial conditions 211 7.6.4 Steady and transient solution methods 212 7.7 Artificial Compressibility Scheme . 213 7.8 Nusselt Number, Drag and Stream Function . 213 7.8.1 Nusselt number . 214 7.8.2 Drag calculation 215 7.8.3 Stream function . 216 7.9 Mesh Convergence . 217 7.10 Laminar Isothermal Flow 218 7.10.1 Geometry, boundary and initial conditions 218 7.10.2 Solution . 219 7.11 Laminar Non-isothermal Flow . 220 7.11.1 Forced convection heat transfer 220 7.11.2 Buoyancy-driven convection heat transfer 223 7.11.3 Mixed convection heat transfer 227 7.12 Introduction to Turbulent Flow . 230 7.12.1 Solution procedure and result . 233 7.13 Extension to Axisymmetric Problems . 234 7.14 Summary 235 7.15 Exercise . 236 Bibliography . 236 8 Convection in Porous Media 240 8.1 Introduction . 240 8.2 Generalized Porous Medium Flow Approach . 243 8.2.1 Non-dimensional scales 245 8.2.2 Limiting cases . 247 8.3 Discretization Procedure 247 8.3.1 Temporal discretization 247 8.3.2 Spatial discretization 249 8.3.3 Semi- and quasi-implicit forms 252 8.4 Non-isothermal Flows . 254 8.5 Forced Convection . 255 8.6 Natural Convection . 256 8.6.1 Constant porosity medium . 258 8.7 Summary 262 8.8 Exercise . 262 Bibliography . 262 9 Some Examples of Fluid Flow and Heat Transfer Problems 265 9.1 Introduction . 265 9.2 Isothermal Flow Problems . 265 9.2.1 Steady state problems . 265 9.2.2 Transient flow . 277CONTENTS xi 9.3 Non-isothermal Benchmark Flow Problem 280 9.3.1 Backward-facing step . 281 9.4 Thermal Conduction in an Electronic Package 283 9.5 Forced Convection Heat Transfer From Heat Sources 286 9.6 Summary 294 9.7 Exercise . 294 Bibliography . 296 10 Implementation of Computer Code 299 10.1 Introduction . 299 10.2 Preprocessing 300 10.2.1 Mesh generation 300 10.2.2 Linear triangular element data . 302 10.2.3 Element size calculation 303 10.2.4 Shape functions and their derivatives . 304 10.2.5 Boundary normal calculation . 305 10.2.6 Mass matrix and mass lumping 306 10.2.7 Implicit pressure or heat conduction matrix 307 10.3 Main Unit 309 10.3.1 Time-step calculation 310 10.3.2 Element loop and assembly 313 10.3.3 Updating solution . 314 10.3.4 Boundary conditions 315 10.3.5 Monitoring steady state 316 10.4 Postprocessing . 317 10.4.1 Interpolation of data 317 10.5 Summary 317 Bibliography . 317 A Green’s Lemma 319 B Integration Formulae 321 B.1 Linear Triangles . 321 B.2 Linear Tetrahedron . 321 C Finite Element Assembly Procedure 323 D Simplified Form of the Navier–Stokes Equations 326 Index Index Note: Figures and Tables are indicated by italic page numbers advancing front method for generation of unstructured meshes 301 air, dry, thermal conductivity 4 aircraft structures, heat transfer in 126 aluminium alloy(s), thermal conductivity 4 analytical solution(s) compared with FEM plane homogeneous wall 112 two-dimensional square plate 131–2 mixed convection heat transfer 228, 230 procedure 112n(1) in transient heat conduction analysis 159 anisotropic materials, heat conduction equation(s) 11–12 annular enclosure, natural convection in fluid-saturated porous media 261–2 area coordinates, for triangular element 52–4 artificial compressibility-based CBS scheme 205, 213 assembly of finite element equations 41 for one-dimensional problems 86, 107 procedure 323–5 axisymmetric problems convection heat transfer in 234–5 Galerkin method 145–6 example calculations 146–7 steady-state heat conduction in 126–7, 142–7 exercises on 148 Babuska–Brezzi condition 202 backward Euler scheme 161 backward-facing step forced convection heat transfer after 281–3 isothermal steady-state flow over 270, 272–4 non-isothermal flow over 281–3 basis functions 41 see also shape functions benchmark problems natural convection in square cavity 224–6 with porous media 256–62 non-isothermal flow problem 280–3 steady-state isothermal flow backward-facing step 270, 272–4 in double-driven cavity 274–6, 277, 278 in lid-driven cavity 266–70, 271330 INDEX benchmark problems (continued) transient isothermal flow past cylinder 276–80 Berenati–Brosilow correlation 255 Bernard convection, transient solution for convection heat transfer 212 Biot number 152 boundary conditions application of in one-dimensional problems 19–20 in two-dimensional problems 136 in CBS scheme 211 computer code for 315 conduction equation 13–14 convection heat transfer 211, 212 Boussinesq approximation 185, 247, 257 Boussinesq hypothesis 232 brick see hexahedron element Brinkman extension to Darcy’s law 242 forced convection in porous media 257 buoyancy-driven convection 2, 174, 185, 223–4 examples 224 heat transfer 224–6 non-dimensional form of governing equations 185–7 in two-dimensional square enclosure 224–6 with porous media 258–62 C0 elements 47 C1 elements 47 CBS scheme see characteristic based split scheme CBSflow code interface(s) to graphical package(s) 317 main unit 309–16 boundary conditions 315 element loop and assembly 313–14 monitoring of steady state 316 solution updating 314 time-step calculation 310–13 overall procedure 300 postprocessing unit 317 preprocessing unit 300–9 boundary normal calculation 305–6 element size calculation 303–4 heat conduction calculations 307–9 linear triangular element data 302 mass lumping 307 mass matrix calculation 306–7 mesh generation subsection 300–2 pressure calculations 307–9 shape functions and derivatives, calculations 304–5 central difference scheme 162 central heating system, pipe network, exercise on 31, 33 CG scheme see characteristic Galerkin scheme characteristic based split (CBS) scheme 201–12 advantages over CG procedure 201–2 artificial compressibility form 205, 213, 230 axisymmetric convection heat transfer problems 235 boundary conditions 211, 212 implementation steps for convection in porous media 250 intermediate velocity calculation 202–3, 205–6, 250 pressure calculation 203–5, 206, 250 temperature calculation 205, 206 velocity/momentum correction 205, 206, 250INDEX 331 initial conditions 212 isothermal flow problems 218–20, 265–80 laminar non-isothermal flow problems, mixed convection 226–30 non-isothermal flow problems 220–30, 280–3 buoyancy-driven/natural convection 223–6 forced convection 220–3, 281–3 porous medium flow equations solved using 247–53 quasi-implicit form 253 semi-implicit form 252–3, 266 spatial discretization 206–10 for convection in porous media 249–52 steady-state solution method 212 temporal discretization, for convection in porous media 247–9 time-step calculation 210–11 transient solution method 212 characteristic Galerkin (CG) scheme 188–95 extension to multi-dimensions 195–200 combined conduction–convection, steady-state problem, discrete system 25–7 composite slab heat flow in 19–21 exercise(s) 31, 32, 34 composite wall steady-state heat conduction in 103–4 exercises on 123, 124 computational fluid dynamics (CFD) 173 books on 173 examples of applications 173 computer code implementation 299–319 see also CBSflow code conduction–convection systems 120–3 conduction heat transfer 2 conduction heat transfer equation(s) 11–12 boundary conditions 13–14 for composite slab 20–1 initial conditions 13 conduction heat transfer problems examples 5–10 methodology 14–15 analytical solutions 14 numerical methods 14–15 conduction resistance, ratio to convection resistance 152 conservation of energy equation see energy-conservation equation conservation of mass equation see continuity equation; mass-conservation equation conservation of momentum equation see momentum-conservation equation continuity equation 177–8, 183, 245 non-dimensional form convection in porous media 245 forced convection 184 natural convection 186, 246 continuous/continuum system 18 convection–diffusion equation(s) 187–8 characteristic Galerkin (CG) approach 188–95 extension to multi-dimensions 195–200 finite element solutions 188–200 one-dimensional problems 189–95 stability conditions 200–1 time-step restrictions 200 two-dimensional problems 195–200 convection heat transfer 2–3, 173–239 axisymmetric problems 234–5 boundary condition 13 characteristic-based split (CBS) scheme 201–12332 INDEX convection heat transfer (continued) coefficient 3 exercises on 236 Navier–Stokes equations 175–83 non-dimensional form of governing equations 183–7, 218 in porous media 240–64 stability conditions 200–1 see also buoyancy-driven convection; forced convection; mixed convection; natural convection coordinate transformation 63 Jacobian(s) of 64, 66, 68 counterflow heat exchanger, exercise 32, 33 Crank–Nicolson method 162 application 157 cross-flow heat exchanger, exercise on 294–5 crystal growth, phase changes during 164 cubic triangular element, shape functions for 56–7 cylinders isothermal flow past, with vortex shedding 276–80 radial heat flow in 115–20 example calculations 117, 118–20 with heat source 117–20 cylindrical coordinate system axisymmetric convection heat transfer 2305 heat conduction equation 12, 115, 144 Darcy’s law 240–1 Brinkman’s extension 242, 257 Ergun’s correlation 242, 244 Forchheimer’s extension 241, 257 Darcy number 246 Darcy–Rayleigh number 247 Darcy–Weisbach formula 24 Delaunay mesh generator 288, 301 direct current circuit, exercise 35 Direct Numerical Simulation (DNS) turbulence modelling approach 230–1 Dirichlet (boundary) conditions 13, 211, 220 discrete systems 18–37 meaning of term 18 steady-state problems 19–29 fluid flow network 22–5 heat exchangers 27–9 heat flow in composite slab 19–21 heat sinks (combined conduction–convection) 25–7 steps in analysis 19 transient/propagation heat transfer problem 29–31 double-driven cavity, isothermal flow past 274–6, 277, 278 double-glazed window, exercise on 33–4 drag calculation 215–16 drag coefficient 215 values, for forced convection flow past a sphere 223 drag force 215 Forchheimer relationship 241 on porous medium particle 241 drawing of wires, fibres, etc 8–10, 14 edges, in finite element method 40 effective heat capacity method phase change problems 166 example calculations 166–7 electronic packages thermal conduction in 283–6 see also plastic ball grid array packagesINDEX 333 electroslag melting, phase changes during 164 elements (in finite element method) 40, 41–74 meaning of term 40, 41 see also one-dimensional elements; three-dimensional elements; two-dimensional elements emissivity 4 energy-conservation equation moving bodies/systems 9 in Navier–Stokes equations 181–3, 184 non-dimensional form convection in porous media 246 forced convection 184 natural convection 186, 247 phase change problems 164–5 enthalpy method, phase change problems 165–7 Ergun’s correlation for Darcy’s law 242, 244 Euler–Lagrange equation 78 explicit time-stepping scheme 157, 161 extrusion of plastics, metals, etc 8–10, 14 fin array, in heat sink 25 one-dimensional 75–6 rectangular example calculations 93–8 exercise on 100 tapered 120–2 example calculations 122–3 types 120 finite difference method (FDM) 38–9 compared with FEM, for two-dimensional plane problem 132 time discretization in transient heat conduction analysis 156–60 finite element discretization 39–40 composite wall 106–7 homogeneous wall 105–6, 110, 114 with convection 111 one-dimensional problems 85, 105–7 tapered fin 122 two-dimensional plane problems 130, 135 finite element method (FEM) 38–102 elements 41–74 isoparametric elements 62–70 one-dimensional linear element 42–5 one-dimensional quadratic element 42, 45–8 three-dimensional elements 70–4 two-dimensional linear triangular element 48–52 two-dimensional quadratic triangular element 54–7 two-dimensional quadrilateral elements 57–62 example calculations, for rectangular fin 93–8 steps in solution of continuum problem 39–41 assembly of element equations 41, 86, 323–5 calculation of secondary quantities 41 discretization of continuum 39–40, 85 formulation of element equations 41, 86 selection of interpolation or shape functions 40, 41–74 solving system of equations 41 time discretization in transient heat conduction analysis 160–1 finite volume method 39 first law of thermodynamics, in heat transfer terms 5334 INDEX fluid dynamics 173 computer-based analysis 173 Navier–Stokes equations 175–83 time-step restrictions 200–1 fluid flow, benchmark problems 265–80 fluid flow network discrete system, steady-state problem 22–5 exercise 31, 33 fluid resistance 22 fluid-motion-assisted heat transport, types 2–3, 174 forced convection 2–3, 174 heat transfer 220–3 backward-facing step 281–3 from heat sources 286–94 non-dimensional form of governing equations 184–5 three-dimensional flow over sphere 221–3 two-dimensional channel problem 220–1 in porous media 255–6 Forchheimer extension to Darcy’s law 241 forced convection in porous media 257 forcing vector(s) convection heat transfer 194, 209 in porous media 251–2 elemental 41 for plane composite wall 106 for plane homogeneous wall, with internal heat source 110, 113 for rectangular fin 95 for tapered fin 122 transient heat transfer 158 for two-dimensional square plate 138–9 forward Euler scheme 161 Fourier analysis 161 Fourier’s law of heat conduction 3 heat flux calculated by 10, 182 spatial variation of temperature 7 free convection 2, 174 see also natural convection Galerkin method 83, 85–7, 91–2 axisymmetric problems 145–6 example calculations 146–7 compared with exact solution 84, 87 transient heat conduction analysis 153–4, 161 generalized porous medium flow approach 243–7 see also porous medium flow equations Goodman’s method 76–7 gradient matrix after spatial discretization of CBS steps 208 one-dimensional elements 44, 47, 94 two-dimensional elements 50, 60, 128, 137 Grashof number 187, 246 Green’s lemma 319–20 applications 91, 191, 208 grid of nodal points 14–15 heat balance integral method, Goodman’s method 76–7 heat conduction analysis 10–12 differential control volume for 10 heat conduction equation(s) 11–12 boundary conditions 13–14 for composite slab 20–1 formulation of finite element equations for 87–92 by Galerkin method 91–2 by variational approach 88–91 initial conditions 13 heat convection 2–3, 173 types 2–3, 174 see also convection heat transfer heat exchangers calculation of effectiveness 27–9INDEX 335 exercises on 32, 33, 35–6, 100, 294–5 heat sinks exercise 35, 36 heat transfer in 25–7 heat transfer benchmark problems 280–3 coefficient, typical values 4 importance 1–2 laws 3–5 modes 2–3 problems 5–10, 283–94 incandescent lamp 7–8 moving systems 6–10 plate exposed to solar heat flux 5–7 heat treatment chamber, heat transfer processes associated 29–31 Hermite polynomials 47 hexahedron element 70, 73–4 linear 73 quadratic (20-node) 73–4 human body, exercise on 34 implicit pressure calculations in CBS scheme 203–5, 206, 250 computer code for 307–9 implicit time-stepping scheme(s) 157, 161, 162 incandescent lamp, energy balance in 7–8 insulating material, heat transfer through, exercise 31, 32 integrated circuit (IC) carriers, thermal conduction in 283–6 integration formulae 321–2 linear tetrahedron 321–2 linear triangle 321 internal heat source, plane wall with, one-dimensional steady-state heat conduction 108–15 interpolation functions 41 requirements for 92–3 see also shape functions inverse heat conduction problems 168–70 one-dimensional problem 168–70 inverse modelling 168 isoparametric elements 62–70 isothermal flow problems 218–20, 265–80 steady-state flow 265–76 transient flow 276–80 isotherm(s) linear triangular element 51–2 quadrilateral elements 61–2 isotropic materials, heat conduction equation(s) 12 Jacobian matrix 64 kinematic viscosity 184 Kroneker delta 180 Lagrangian interpolation 47 laminar flow in pipe network 22–4 Reynolds number criterion 174 laminar isothermal flow 218–20 boundary conditions 218–19 geometry of example 218 initial conditions 219 solution 219–20 laminar non-isothermal flow 220–30 buoyancy-driven convection heat transfer 223–6 forced convection heat transfer 220–3 mixed convection heat transfer 227–30 natural/free convection heat transfer 223–6 Large Eddy Simulation (LES) turbulence modelling approach 230, 231 lid-driven cavity, isothermal flow past 266–70 linear element 42–5, 42 in convection–diffusion problems
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أخواني في الله أحضرت لكم كتاب Fundamentals of the Finite Element Method for Heat and Fluid Flow Roland W. Lewis, Perumal Nithiarasu, Kankanhally N. Seetharamu
![كتاب Fundamentals of the Finite Element Method for Heat and Fluid Flow - صفحة 2 F_o_f_10](https://i.servimg.com/u/f61/15/36/74/14/f_o_f_10.jpg) و المحتوى كما يلي :
Contents Preface xiii 1 Introduction 1 1.1 Importance of Heat Transfer 1 1.2 Heat Transfer Modes 2 1.3 The Laws of Heat Transfer . 3 1.4 Formulation of Heat Transfer Problems 5 1.4.1 Heat transfer from a plate exposed to solar heat flux . 5 1.4.2 Incandescent lamp . 7 1.4.3 Systems with a relative motion and internal heat generation . 8 1.5 Heat Conduction Equation . 10 1.6 Boundary and Initial Conditions 13 1.7 Solution Methodology . 14 1.8 Summary 15 1.9 Exercise . 15 Bibliography . 17 2 Some Basic Discrete Systems 18 2.1 Introduction . 18 2.2 Steady State Problems . 19 2.2.1 Heat flow in a composite slab . 19 2.2.2 Fluid flow network . 22 2.2.3 Heat transfer in heat sinks (combined conduction–convection) . 25 2.2.4 Analysis of a heat exchanger . 27 2.3 Transient Heat Transfer Problem (Propagation Problem) . 29 2.4 Summary 31 2.5 Exercise . 31 Bibliography . 37 3 The Finite Element Method 38 3.1 Introduction . 38 3.2 Elements and Shape Functions . 41 3.2.1 One-dimensional linear element 42 3.2.2 One-dimensional quadratic element 45viii CONTENTS 3.2.3 Two-dimensional linear triangular elements 48 3.2.4 Area coordinates 52 3.2.5 Quadratic triangular elements . 54 3.2.6 Two-dimensional quadrilateral elements . 57 3.2.7 Isoparametric elements . 62 3.2.8 Three-dimensional elements 70 3.3 Formulation (Element Characteristics) . 75 3.3.1 Ritz method (Heat balance integral method—Goodman’s method) . 76 3.3.2 Rayleigh–Ritz method (Variational method) . 78 3.3.3 The method of weighted residuals . 80 3.3.4 Galerkin finite element method 85 3.4 Formulation for the Heat Conduction Equation 87 3.4.1 Variational approach 88 3.4.2 The Galerkin method 91 3.5 Requirements for Interpolation Functions . 92 3.6 Summary 98 3.7 Exercise . 98 Bibliography . 100 4 Steady State Heat Conduction in One Dimension 102 4.1 Introduction . 102 4.2 Plane Walls . 102 4.2.1 Homogeneous wall . 102 4.2.2 Composite wall . 103 4.2.3 Finite element discretization 105 4.2.4 Wall with varying cross-sectional area 107 4.2.5 Plane wall with a heat source: solution by linear elements . 108 4.2.6 Plane wall with a heat source: solution by quadratic elements 112 4.2.7 Plane wall with a heat source: solution by modified quadratic equations (static condensation) 114 4.3 Radial Heat Flow in a Cylinder 115 4.3.1 Cylinder with heat source . 117 4.4 Conduction–Convection Systems . 120 4.5 Summary 123 4.6 Exercise . 123 Bibliography . 125 5 Steady State Heat Conduction in Multi-dimensions 126 5.1 Introduction . 126 5.2 Two-dimensional Plane Problems . 127 5.2.1 Triangular elements 127 5.3 Rectangular Elements 136 5.4 Plate with Variable Thickness . 139CONTENTS ix 5.5 Three-dimensional Problems 141 5.6 Axisymmetric Problems 142 5.6.1 Galerkin’s method for linear triangular axisymmetric elements . 145 5.7 Summary 147 5.8 Exercise . 147 Bibliography . 149 6 Transient Heat Conduction Analysis 150 6.1 Introduction . 150 6.2 Lumped Heat Capacity System 150 6.3 Numerical Solution . 152 6.3.1 Transient governing equations and boundary and initial conditions . 152 6.3.2 The Galerkin method 153 6.4 One-dimensional Transient State Problem . 154 6.4.1 Time discretization using the Finite Difference Method (FDM) . 156 6.4.2 Time discretization using the Finite Element Method (FEM) 160 6.5 Stability . 161 6.6 Multi-dimensional Transient Heat Conduction 162 6.7 Phase Change Problems—Solidification and Melting . 164 6.7.1 The governing equations 164 6.7.2 Enthalpy formulation 165 6.8 Inverse Heat Conduction Problems 168 6.8.1 One-dimensional heat conduction . 168 6.9 Summary 170 6.10 Exercise . 170 Bibliography . 172 7 Convection Heat Transfer 173 7.1 Introduction . 173 7.1.1 Types of fluid-motion-assisted heat transport . 174 7.2 Navier–Stokes Equations 175 7.2.1 Conservation of mass or continuity equation . 175 7.2.2 Conservation of momentum 177 7.2.3 Energy equation 181 7.3 Non-dimensional Form of the Governing Equations . 183 7.3.1 Forced convection . 184 7.3.2 Natural convection (Buoyancy-driven convection) 185 7.3.3 Mixed convection . 187 7.4 The Transient Convection–diffusion Problem . 187 7.4.1 Finite element solution to convection–diffusion equation 188 7.4.2 Extension to multi-dimensions . 195 7.5 Stability Conditions . 200 7.6 Characteristic-based Split (CBS) Scheme . 201 7.6.1 Spatial discretization 206x CONTENTS 7.6.2 Time-step calculation 210 7.6.3 Boundary and initial conditions 211 7.6.4 Steady and transient solution methods 212 7.7 Artificial Compressibility Scheme . 213 7.8 Nusselt Number, Drag and Stream Function . 213 7.8.1 Nusselt number . 214 7.8.2 Drag calculation 215 7.8.3 Stream function . 216 7.9 Mesh Convergence . 217 7.10 Laminar Isothermal Flow 218 7.10.1 Geometry, boundary and initial conditions 218 7.10.2 Solution . 219 7.11 Laminar Non-isothermal Flow . 220 7.11.1 Forced convection heat transfer 220 7.11.2 Buoyancy-driven convection heat transfer 223 7.11.3 Mixed convection heat transfer 227 7.12 Introduction to Turbulent Flow . 230 7.12.1 Solution procedure and result . 233 7.13 Extension to Axisymmetric Problems . 234 7.14 Summary 235 7.15 Exercise . 236 Bibliography . 236 8 Convection in Porous Media 240 8.1 Introduction . 240 8.2 Generalized Porous Medium Flow Approach . 243 8.2.1 Non-dimensional scales 245 8.2.2 Limiting cases . 247 8.3 Discretization Procedure 247 8.3.1 Temporal discretization 247 8.3.2 Spatial discretization 249 8.3.3 Semi- and quasi-implicit forms 252 8.4 Non-isothermal Flows . 254 8.5 Forced Convection . 255 8.6 Natural Convection . 256 8.6.1 Constant porosity medium . 258 8.7 Summary 262 8.8 Exercise . 262 Bibliography . 262 9 Some Examples of Fluid Flow and Heat Transfer Problems 265 9.1 Introduction . 265 9.2 Isothermal Flow Problems . 265 9.2.1 Steady state problems . 265 9.2.2 Transient flow . 277CONTENTS xi 9.3 Non-isothermal Benchmark Flow Problem 280 9.3.1 Backward-facing step . 281 9.4 Thermal Conduction in an Electronic Package 283 9.5 Forced Convection Heat Transfer From Heat Sources 286 9.6 Summary 294 9.7 Exercise . 294 Bibliography . 296 10 Implementation of Computer Code 299 10.1 Introduction . 299 10.2 Preprocessing 300 10.2.1 Mesh generation 300 10.2.2 Linear triangular element data . 302 10.2.3 Element size calculation 303 10.2.4 Shape functions and their derivatives . 304 10.2.5 Boundary normal calculation . 305 10.2.6 Mass matrix and mass lumping 306 10.2.7 Implicit pressure or heat conduction matrix 307 10.3 Main Unit 309 10.3.1 Time-step calculation 310 10.3.2 Element loop and assembly 313 10.3.3 Updating solution . 314 10.3.4 Boundary conditions 315 10.3.5 Monitoring steady state 316 10.4 Postprocessing . 317 10.4.1 Interpolation of data 317 10.5 Summary 317 Bibliography . 317 A Green’s Lemma 319 B Integration Formulae 321 B.1 Linear Triangles . 321 B.2 Linear Tetrahedron . 321 C Finite Element Assembly Procedure 323 D Simplified Form of the Navier–Stokes Equations 326 Index Index Note: Figures and Tables are indicated by italic page numbers advancing front method for generation of unstructured meshes 301 air, dry, thermal conductivity 4 aircraft structures, heat transfer in 126 aluminium alloy(s), thermal conductivity 4 analytical solution(s) compared with FEM plane homogeneous wall 112 two-dimensional square plate 131–2 mixed convection heat transfer 228, 230 procedure 112n(1) in transient heat conduction analysis 159 anisotropic materials, heat conduction equation(s) 11–12 annular enclosure, natural convection in fluid-saturated porous media 261–2 area coordinates, for triangular element 52–4 artificial compressibility-based CBS scheme 205, 213 assembly of finite element equations 41 for one-dimensional problems 86, 107 procedure 323–5 axisymmetric problems convection heat transfer in 234–5 Galerkin method 145–6 example calculations 146–7 steady-state heat conduction in 126–7, 142–7 exercises on 148 Babuska–Brezzi condition 202 backward Euler scheme 161 backward-facing step forced convection heat transfer after 281–3 isothermal steady-state flow over 270, 272–4 non-isothermal flow over 281–3 basis functions 41 see also shape functions benchmark problems natural convection in square cavity 224–6 with porous media 256–62 non-isothermal flow problem 280–3 steady-state isothermal flow backward-facing step 270, 272–4 in double-driven cavity 274–6, 277, 278 in lid-driven cavity 266–70, 271330 INDEX benchmark problems (continued) transient isothermal flow past cylinder 276–80 Berenati–Brosilow correlation 255 Bernard convection, transient solution for convection heat transfer 212 Biot number 152 boundary conditions application of in one-dimensional problems 19–20 in two-dimensional problems 136 in CBS scheme 211 computer code for 315 conduction equation 13–14 convection heat transfer 211, 212 Boussinesq approximation 185, 247, 257 Boussinesq hypothesis 232 brick see hexahedron element Brinkman extension to Darcy’s law 242 forced convection in porous media 257 buoyancy-driven convection 2, 174, 185, 223–4 examples 224 heat transfer 224–6 non-dimensional form of governing equations 185–7 in two-dimensional square enclosure 224–6 with porous media 258–62 C0 elements 47 C1 elements 47 CBS scheme see characteristic based split scheme CBSflow code interface(s) to graphical package(s) 317 main unit 309–16 boundary conditions 315 element loop and assembly 313–14 monitoring of steady state 316 solution updating 314 time-step calculation 310–13 overall procedure 300 postprocessing unit 317 preprocessing unit 300–9 boundary normal calculation 305–6 element size calculation 303–4 heat conduction calculations 307–9 linear triangular element data 302 mass lumping 307 mass matrix calculation 306–7 mesh generation subsection 300–2 pressure calculations 307–9 shape functions and derivatives, calculations 304–5 central difference scheme 162 central heating system, pipe network, exercise on 31, 33 CG scheme see characteristic Galerkin scheme characteristic based split (CBS) scheme 201–12 advantages over CG procedure 201–2 artificial compressibility form 205, 213, 230 axisymmetric convection heat transfer problems 235 boundary conditions 211, 212 implementation steps for convection in porous media 250 intermediate velocity calculation 202–3, 205–6, 250 pressure calculation 203–5, 206, 250 temperature calculation 205, 206 velocity/momentum correction 205, 206, 250INDEX 331 initial conditions 212 isothermal flow problems 218–20, 265–80 laminar non-isothermal flow problems, mixed convection 226–30 non-isothermal flow problems 220–30, 280–3 buoyancy-driven/natural convection 223–6 forced convection 220–3, 281–3 porous medium flow equations solved using 247–53 quasi-implicit form 253 semi-implicit form 252–3, 266 spatial discretization 206–10 for convection in porous media 249–52 steady-state solution method 212 temporal discretization, for convection in porous media 247–9 time-step calculation 210–11 transient solution method 212 characteristic Galerkin (CG) scheme 188–95 extension to multi-dimensions 195–200 combined conduction–convection, steady-state problem, discrete system 25–7 composite slab heat flow in 19–21 exercise(s) 31, 32, 34 composite wall steady-state heat conduction in 103–4 exercises on 123, 124 computational fluid dynamics (CFD) 173 books on 173 examples of applications 173 computer code implementation 299–319 see also CBSflow code conduction–convection systems 120–3 conduction heat transfer 2 conduction heat transfer equation(s) 11–12 boundary conditions 13–14 for composite slab 20–1 initial conditions 13 conduction heat transfer problems examples 5–10 methodology 14–15 analytical solutions 14 numerical methods 14–15 conduction resistance, ratio to convection resistance 152 conservation of energy equation see energy-conservation equation conservation of mass equation see continuity equation; mass-conservation equation conservation of momentum equation see momentum-conservation equation continuity equation 177–8, 183, 245 non-dimensional form convection in porous media 245 forced convection 184 natural convection 186, 246 continuous/continuum system 18 convection–diffusion equation(s) 187–8 characteristic Galerkin (CG) approach 188–95 extension to multi-dimensions 195–200 finite element solutions 188–200 one-dimensional problems 189–95 stability conditions 200–1 time-step restrictions 200 two-dimensional problems 195–200 convection heat transfer 2–3, 173–239 axisymmetric problems 234–5 boundary condition 13 characteristic-based split (CBS) scheme 201–12332 INDEX convection heat transfer (continued) coefficient 3 exercises on 236 Navier–Stokes equations 175–83 non-dimensional form of governing equations 183–7, 218 in porous media 240–64 stability conditions 200–1 see also buoyancy-driven convection; forced convection; mixed convection; natural convection coordinate transformation 63 Jacobian(s) of 64, 66, 68 counterflow heat exchanger, exercise 32, 33 Crank–Nicolson method 162 application 157 cross-flow heat exchanger, exercise on 294–5 crystal growth, phase changes during 164 cubic triangular element, shape functions for 56–7 cylinders isothermal flow past, with vortex shedding 276–80 radial heat flow in 115–20 example calculations 117, 118–20 with heat source 117–20 cylindrical coordinate system axisymmetric convection heat transfer 2305 heat conduction equation 12, 115, 144 Darcy’s law 240–1 Brinkman’s extension 242, 257 Ergun’s correlation 242, 244 Forchheimer’s extension 241, 257 Darcy number 246 Darcy–Rayleigh number 247 Darcy–Weisbach formula 24 Delaunay mesh generator 288, 301 direct current circuit, exercise 35 Direct Numerical Simulation (DNS) turbulence modelling approach 230–1 Dirichlet (boundary) conditions 13, 211, 220 discrete systems 18–37 meaning of term 18 steady-state problems 19–29 fluid flow network 22–5 heat exchangers 27–9 heat flow in composite slab 19–21 heat sinks (combined conduction–convection) 25–7 steps in analysis 19 transient/propagation heat transfer problem 29–31 double-driven cavity, isothermal flow past 274–6, 277, 278 double-glazed window, exercise on 33–4 drag calculation 215–16 drag coefficient 215 values, for forced convection flow past a sphere 223 drag force 215 Forchheimer relationship 241 on porous medium particle 241 drawing of wires, fibres, etc 8–10, 14 edges, in finite element method 40 effective heat capacity method phase change problems 166 example calculations 166–7 electronic packages thermal conduction in 283–6 see also plastic ball grid array packagesINDEX 333 electroslag melting, phase changes during 164 elements (in finite element method) 40, 41–74 meaning of term 40, 41 see also one-dimensional elements; three-dimensional elements; two-dimensional elements emissivity 4 energy-conservation equation moving bodies/systems 9 in Navier–Stokes equations 181–3, 184 non-dimensional form convection in porous media 246 forced convection 184 natural convection 186, 247 phase change problems 164–5 enthalpy method, phase change problems 165–7 Ergun’s correlation for Darcy’s law 242, 244 Euler–Lagrange equation 78 explicit time-stepping scheme 157, 161 extrusion of plastics, metals, etc 8–10, 14 fin array, in heat sink 25 one-dimensional 75–6 rectangular example calculations 93–8 exercise on 100 tapered 120–2 example calculations 122–3 types 120 finite difference method (FDM) 38–9 compared with FEM, for two-dimensional plane problem 132 time discretization in transient heat conduction analysis 156–60 finite element discretization 39–40 composite wall 106–7 homogeneous wall 105–6, 110, 114 with convection 111 one-dimensional problems 85, 105–7 tapered fin 122 two-dimensional plane problems 130, 135 finite element method (FEM) 38–102 elements 41–74 isoparametric elements 62–70 one-dimensional linear element 42–5 one-dimensional quadratic element 42, 45–8 three-dimensional elements 70–4 two-dimensional linear triangular element 48–52 two-dimensional quadratic triangular element 54–7 two-dimensional quadrilateral elements 57–62 example calculations, for rectangular fin 93–8 steps in solution of continuum problem 39–41 assembly of element equations 41, 86, 323–5 calculation of secondary quantities 41 discretization of continuum 39–40, 85 formulation of element equations 41, 86 selection of interpolation or shape functions 40, 41–74 solving system of equations 41 time discretization in transient heat conduction analysis 160–1 finite volume method 39 first law of thermodynamics, in heat transfer terms 5334 INDEX fluid dynamics 173 computer-based analysis 173 Navier–Stokes equations 175–83 time-step restrictions 200–1 fluid flow, benchmark problems 265–80 fluid flow network discrete system, steady-state problem 22–5 exercise 31, 33 fluid resistance 22 fluid-motion-assisted heat transport, types 2–3, 174 forced convection 2–3, 174 heat transfer 220–3 backward-facing step 281–3 from heat sources 286–94 non-dimensional form of governing equations 184–5 three-dimensional flow over sphere 221–3 two-dimensional channel problem 220–1 in porous media 255–6 Forchheimer extension to Darcy’s law 241 forced convection in porous media 257 forcing vector(s) convection heat transfer 194, 209 in porous media 251–2 elemental 41 for plane composite wall 106 for plane homogeneous wall, with internal heat source 110, 113 for rectangular fin 95 for tapered fin 122 transient heat transfer 158 for two-dimensional square plate 138–9 forward Euler scheme 161 Fourier analysis 161 Fourier’s law of heat conduction 3 heat flux calculated by 10, 182 spatial variation of temperature 7 free convection 2, 174 see also natural convection Galerkin method 83, 85–7, 91–2 axisymmetric problems 145–6 example calculations 146–7 compared with exact solution 84, 87 transient heat conduction analysis 153–4, 161 generalized porous medium flow approach 243–7 see also porous medium flow equations Goodman’s method 76–7 gradient matrix after spatial discretization of CBS steps 208 one-dimensional elements 44, 47, 94 two-dimensional elements 50, 60, 128, 137 Grashof number 187, 246 Green’s lemma 319–20 applications 91, 191, 208 grid of nodal points 14–15 heat balance integral method, Goodman’s method 76–7 heat conduction analysis 10–12 differential control volume for 10 heat conduction equation(s) 11–12 boundary conditions 13–14 for composite slab 20–1 formulation of finite element equations for 87–92 by Galerkin method 91–2 by variational approach 88–91 initial conditions 13 heat convection 2–3, 173 types 2–3, 174 see also convection heat transfer heat exchangers calculation of effectiveness 27–9INDEX 335 exercises on 32, 33, 35–6, 100, 294–5 heat sinks exercise 35, 36 heat transfer in 25–7 heat transfer benchmark problems 280–3 coefficient, typical values 4 importance 1–2 laws 3–5 modes 2–3 problems 5–10, 283–94 incandescent lamp 7–8 moving systems 6–10 plate exposed to solar heat flux 5–7 heat treatment chamber, heat transfer processes associated 29–31 Hermite polynomials 47 hexahedron element 70, 73–4 linear 73 quadratic (20-node) 73–4 human body, exercise on 34 implicit pressure calculations in CBS scheme 203–5, 206, 250 computer code for 307–9 implicit time-stepping scheme(s) 157, 161, 162 incandescent lamp, energy balance in 7–8 insulating material, heat transfer through, exercise 31, 32 integrated circuit (IC) carriers, thermal conduction in 283–6 integration formulae 321–2 linear tetrahedron 321–2 linear triangle 321 internal heat source, plane wall with, one-dimensional steady-state heat conduction 108–15 interpolation functions 41 requirements for 92–3 see also shape functions inverse heat conduction problems 168–70 one-dimensional problem 168–70 inverse modelling 168 isoparametric elements 62–70 isothermal flow problems 218–20, 265–80 steady-state flow 265–76 transient flow 276–80 isotherm(s) linear triangular element 51–2 quadrilateral elements 61–2 isotropic materials, heat conduction equation(s) 12 Jacobian matrix 64 kinematic viscosity 184 Kroneker delta 180 Lagrangian interpolation 47 laminar flow in pipe network 22–4 Reynolds number criterion 174 laminar isothermal flow 218–20 boundary conditions 218–19 geometry of example 218 initial conditions 219 solution 219–20 laminar non-isothermal flow 220–30 buoyancy-driven convection heat transfer 223–6 forced convection heat transfer 220–3 mixed convection heat transfer 227–30 natural/free convection heat transfer 223–6 Large Eddy Simulation (LES) turbulence modelling approach 230, 231 lid-driven cavity, isothermal flow past 266–70 linear element 42–5, 42 in convection–diffusion problems
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