Page 4 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

ix
CONTENTS
Three-dimensional Problems .. ... .. .. .. .. .. .. ... .. .. .. 141
5.5
Axisymmetric Problems .. .. ... .. .. .. .. .. .. ... .. .. .. 142
5.6
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 .. ... .. .. .. .. .. .. ... .. .. .. 206