Heat transfer is the area of engineering science which describes the energy transport between material bodies due to a difference in temperature. The three different modes of heat transport are conduction, convection and radiation. In most problems, these three modes exist simultaneously. However, the significance of these modes depends on the problems studied and often, insignificant modes are neglected. Very often books published on Computational Fluid Dynamics using the Finite Element Method give very little or no significance to thermal or heat transfer problems. From the research point of view, it is important to explain the handling of various types of heat transfer problems with different types of complex boundary conditions. Problems with slow fluid motion and heat transfer can be difficult problems to handle. Therefore, the complexity of combined fluid flow and heat transfer problems should not be underestimated and should be dealt with carefully. This book: Is ideal for teaching senior undergraduates the fundamentals of how to use the Finite Element Method to solve heat transfer and fluid dynamics problems Explains how to solve various heat transfer problems with different types of boundary conditions Uses recent computational methods and codes to handle complex fluid motion and heat transfer problems Includes a large number of examples and exercises on heat transfer problems In an era of parallel computing, computational efficiency and easy to handle codes play a major part. Bearing all these points in mind, the topics covered on combined flow and heat transfer in this book will be an asset for practising engineers and postgraduate students. Other topics of interest for the heat transfer community, such as heat exchangers and radiation heat transfer, are also included.
Author(s): R. W. Lewis Perumal Nithiarasu Kankanhalli Seetharamu
Edition: 1
Year: 2004
Language: English
Pages: 356
Fundamentals of the Finite Element Method for Heat and Fluid Flow......Page 4
Contents......Page 10
Preface......Page 16
1.1 Importance of Heat Transfer......Page 18
1.2 Heat Transfer Modes......Page 19
1.3 The Laws of Heat Transfer......Page 20
1.4.1 Heat transfer from a plate exposed to solar heat flux......Page 22
1.4.2 Incandescent lamp......Page 24
1.4.3 Systems with a relative motion and internal heat generation......Page 25
1.5 Heat Conduction Equation......Page 27
1.6 Boundary and Initial Conditions......Page 30
1.7 Solution Methodology......Page 31
1.9 Exercise......Page 32
Bibliography......Page 34
2.1 Introduction......Page 35
2.2.1 Heat flow in a composite slab......Page 36
2.2.2 Fluid flow network......Page 39
2.2.3 Heat transfer in heat sinks (combined conduction–convection)......Page 42
2.2.4 Analysis of a heat exchanger......Page 44
2.3 Transient Heat Transfer Problem (Propagation Problem)......Page 46
2.5 Exercise......Page 48
Bibliography......Page 54
3.1 Introduction......Page 55
3.2 Elements and Shape Functions......Page 58
3.2.1 One-dimensional linear element......Page 59
3.2.2 One-dimensional quadratic element......Page 62
3.2.3 Two-dimensional linear triangular elements......Page 65
3.2.4 Area coordinates......Page 69
3.2.5 Quadratic triangular elements......Page 71
3.2.6 Two-dimensional quadrilateral elements......Page 74
3.2.7 Isoparametric elements......Page 79
3.2.8 Three-dimensional elements......Page 87
3.3 Formulation (Element Characteristics)......Page 92
3.3.1 Ritz method (Heat balance integral method—Goodman’s method)......Page 93
3.3.2 Rayleigh–Ritz method (Variational method)......Page 95
3.3.3 The method of weighted residuals......Page 97
3.3.4 Galerkin finite element method......Page 102
3.4 Formulation for the Heat Conduction Equation......Page 104
3.4.1 Variational approach......Page 105
3.4.2 The Galerkin method......Page 108
3.5 Requirements for Interpolation Functions......Page 109
3.7 Exercise......Page 115
Bibliography......Page 117
4.2.1 Homogeneous wall......Page 119
4.2.2 Composite wall......Page 120
4.2.3 Finite element discretization......Page 122
4.2.4 Wall with varying cross-sectional area......Page 124
4.2.5 Plane wall with a heat source: solution by linear elements......Page 125
4.2.6 Plane wall with a heat source: solution by quadratic elements......Page 129
4.2.7 Plane wall with a heat source: solution by modified quadratic equations (static condensation)......Page 131
4.3 Radial Heat Flow in a Cylinder......Page 132
4.3.1 Cylinder with heat source......Page 134
4.4 Conduction–Convection Systems......Page 137
4.6 Exercise......Page 140
Bibliography......Page 142
5.1 Introduction......Page 143
5.2.1 Triangular elements......Page 144
5.3 Rectangular Elements......Page 153
5.4 Plate with Variable Thickness......Page 156
5.5 Three-dimensional Problems......Page 158
5.6 Axisymmetric Problems......Page 159
5.6.1 Galerkin’s method for linear triangular axisymmetric elements......Page 162
5.8 Exercise......Page 164
Bibliography......Page 166
6.2 Lumped Heat Capacity System......Page 167
6.3.1 Transient governing equations and boundary and initial conditions......Page 169
6.3.2 The Galerkin method......Page 170
6.4 One-dimensional Transient State Problem......Page 171
6.4.1 Time discretization using the Finite Difference Method (FDM)......Page 173
6.4.2 Time discretization using the Finite Element Method (FEM)......Page 177
6.5 Stability......Page 178
6.6 Multi-dimensional Transient Heat Conduction......Page 179
6.7.1 The governing equations......Page 181
6.7.2 Enthalpy formulation......Page 182
6.8.1 One-dimensional heat conduction......Page 185
6.10 Exercise......Page 187
Bibliography......Page 189
7.1 Introduction......Page 190
7.1.1 Types of fluid-motion-assisted heat transport......Page 191
7.2.1 Conservation of mass or continuity equation......Page 192
7.2.2 Conservation of momentum......Page 194
7.2.3 Energy equation......Page 198
7.3 Non-dimensional Form of the Governing Equations......Page 200
7.3.1 Forced convection......Page 201
7.3.2 Natural convection (Buoyancy-driven convection)......Page 202
7.4 The Transient Convection–diffusion Problem......Page 204
7.4.1 Finite element solution to convection–diffusion equation......Page 205
7.4.2 Extension to multi-dimensions......Page 212
7.5 Stability Conditions......Page 217
7.6 Characteristic-based Split (CBS) Scheme......Page 218
7.6.1 Spatial discretization......Page 223
7.6.2 Time-step calculation......Page 227
7.6.3 Boundary and initial conditions......Page 228
7.6.4 Steady and transient solution methods......Page 229
7.8 Nusselt Number, Drag and Stream Function......Page 230
7.8.1 Nusselt number......Page 231
7.8.2 Drag calculation......Page 232
7.8.3 Stream function......Page 233
7.9 Mesh Convergence......Page 234
7.10.1 Geometry, boundary and initial conditions......Page 235
7.10.2 Solution......Page 236
7.11.1 Forced convection heat transfer......Page 237
7.11.2 Buoyancy-driven convection heat transfer......Page 240
7.11.3 Mixed convection heat transfer......Page 244
7.12 Introduction to Turbulent Flow......Page 247
7.12.1 Solution procedure and result......Page 250
7.13 Extension to Axisymmetric Problems......Page 251
7.14 Summary......Page 252
Bibliography......Page 253
8.1 Introduction......Page 257
8.2 Generalized Porous Medium Flow Approach......Page 260
8.2.1 Non-dimensional scales......Page 262
8.3.1 Temporal discretization......Page 264
8.3.2 Spatial discretization......Page 266
8.3.3 Semi- and quasi-implicit forms......Page 269
8.4 Non-isothermal Flows......Page 271
8.5 Forced Convection......Page 272
8.6 Natural Convection......Page 273
8.6.1 Constant porosity medium......Page 275
Bibliography......Page 279
9.2.1 Steady state problems......Page 282
9.2.2 Transient flow......Page 294
9.3 Non-isothermal Benchmark Flow Problem......Page 297
9.3.1 Backward-facing step......Page 298
9.4 Thermal Conduction in an Electronic Package......Page 300
9.5 Forced Convection Heat Transfer From Heat Sources......Page 303
9.7 Exercise......Page 311
Bibliography......Page 313
10.1 Introduction......Page 316
10.2.1 Mesh generation......Page 317
10.2.2 Linear triangular element data......Page 319
10.2.3 Element size calculation......Page 320
10.2.4 Shape functions and their derivatives......Page 321
10.2.5 Boundary normal calculation......Page 322
10.2.6 Mass matrix and mass lumping......Page 323
10.2.7 Implicit pressure or heat conduction matrix......Page 324
10.3 Main Unit......Page 326
10.3.1 Time-step calculation......Page 327
10.3.2 Element loop and assembly......Page 330
10.3.3 Updating solution......Page 331
10.3.4 Boundary conditions......Page 332
10.3.5 Monitoring steady state......Page 333
Bibliography......Page 334
A Green’s Lemma......Page 336
B.2 Linear Tetrahedron......Page 338
C Finite Element Assembly Procedure......Page 340
D Simplified Form of the Navier–Stokes Equations......Page 343
Index......Page 346
