Fundamentals of the Finite Element Method for Heat and Fluid Flow

Fundamentals of the Finite Element Method for Heat and Fluid Flow
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Roland W. Lewis, Perumal Nithiarasu, Kankanhally N. Seetharamu
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1 سبتمبر 2021
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Fundamentals of the Finite Element Method for Heat and Fluid Flow
Roland W. Lewis, Perumal Nithiarasu, Kankanhally N. Seetharamu
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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