Fundamentals of the Finite Element Method for Heat and Mass Transfer 2e

Preface to the Second Edition xii
<p>Series Editor s Preface xiv</p>
<p>1 Introduction 1</p>
<p>1.1 Importance of Heat and Mass Transfer 1</p>
<p>1.2 Heat Transfer Modes 2</p>
<p>1.3 The Laws of Heat Transfer 3</p>
<p>1.4 Mathematical Formulation of Some Heat Transfer Problems 5</p>
<p>1.4.1 Heat Transfer from a Plate Exposed to Solar Heat Flux&nbsp; 5</p>
<p>1.4.2 Incandescent Lamp&nbsp; 7</p>
<p>1.4.3 Systems with a Relative Motion and Internal Heat Generation&nbsp; 8</p>
<p>1.5 Heat Conduction Equation&nbsp; 10</p>
<p>1.6 Mass Transfer 13</p>
<p>1.7 Boundary and Initial Conditions 13</p>
<p>1.8 Solution Methodology 15</p>
<p>1.9 Summary 15</p>
<p>1.10 Exercises 16</p>
<p>References&nbsp; 17</p>
<p>2 Some Basic Discrete Systems 19</p>
<p>2.1 Introduction 19</p>
<p>2.2 Steady–state Problems&nbsp; 20</p>
<p>2.2.1 Heat Flow in a Composite Slab 20</p>
<p>2.2.2 Fluid Flow Network 23</p>
<p>2.2.3 Heat Transfer in Heat Sinks 26</p>
<p>2.3 Transient Heat Transfer Problem 28</p>
<p>2.4 Summary 31</p>
<p>2.5 Exercises 31</p>
<p>References&nbsp; 36</p>
<p>3 The Finite Element Method 39</p>
<p>3.1 Introduction 39</p>
<p>3.2 Elements and Shape Functions&nbsp; 42</p>
<p>3.2.1 One–dimensional Linear Element&nbsp; 43</p>
<p>3.2.2 One–dimensional Quadratic Element 46</p>
<p>3.2.3 Two–dimensional Linear Triangular Element 49</p>
<p>3.2.4 Area Coordinates 53</p>
<p>3.2.5 Quadratic Triangular Element 55</p>
<p>3.2.6 Two–dimensional Quadrilateral Elements 58</p>
<p>3.2.7 Isoparametric Elements 63</p>
<p>3.2.8 Three–dimensional Elements 72</p>
<p>3.3 Formulation (Element Characteristics) 76</p>
<p>3.3.1 Ritz Method (Heat Balance Integral Method Goodman s Method) 78</p>
<p>3.3.2 Rayleigh Ritz Method (Variational Method) 79</p>
<p>3.3.3 The Method of Weighted Residuals 82</p>
<p>3.3.4 Galerkin Finite ElementMethod 86</p>
<p>3.4 Formulation for the Heat Conduction Equation 89</p>
<p>3.4.1 Variational Approach 90</p>
<p>3.4.2 The GalerkinMethod 93</p>
<p>3.5 Requirements for Interpolation Functions 94</p>
<p>3.6 Summary 100</p>
<p>3.7 Exercises 100</p>
<p>References 102</p>
<p>4 Steady–State Heat Conduction in One–dimension 105</p>
<p>4.1 Introduction 105</p>
<p>4.2 PlaneWalls 105</p>
<p>4.2.1 Homogeneous Wall 105</p>
<p>4.2.2 CompositeWall 107</p>
<p>4.2.3 Finite Element Discretization 108</p>
<p>4.2.4 Wall with Varying Cross–sectional Area 110</p>
<p>4.2.5 Plane Wall with a Heat Source: Solution by Linear Elements 112</p>
<p>4.2.6 Plane Wall with Heat Source: Solution by Quadratic Elements 115</p>
<p>4.2.7 Plane Wall with a Heat Source: Solution by Modified Quadratic Equations (Static Condensation) 117</p>
<p>4.3 Radial Heat Conduction in a Cylinder Wall 118</p>
<p>4.4 Solid Cylinder with Heat Source 120</p>
<p>4.5 Conduction Convection Systems 123</p>
<p>4.6 Summary 126</p>
<p>4.7 Exercises 127</p>
<p>References 129</p>
<p>5 Steady–state Heat Conduction in Multi–dimensions 131</p>
<p>5.1 Introduction 131</p>
<p>5.2 Two–dimensional Plane Problems 132</p>
<p>5.2.1 Triangular Elements 132</p>
<p>5.3 Rectangular Elements 142</p>
<p>5.4 Plate with Variable Thickness 145</p>
<p>5.5 Three–dimensional Problems 146</p>
<p>5.6 Axisymmetric Problems 148</p>
<p>5.6.1 Galerkin Method for Linear Triangular Axisymmetric Elements 150</p>
<p>5.7 Summary 153</p>
<p>5.8 Exercises 153</p>
<p>References 155</p>
<p>6 Transient Heat Conduction Analysis 157</p>
<p>6.1 Introduction 157</p>
<p>6.2 Lumped Heat Capacity System 157</p>
<p>6.3 Numerical Solution 159</p>
<p>6.3.1 Transient Governing Equations and Boundary and Initial Conditions 159</p>
<p>6.3.2 The GalerkinMethod 160</p>
<p>6.4 One–dimensional Transient State Problem 162</p>
<p>6.4.1 Time Discretization–Finite Difference Method (FDM) 163</p>
<p>6.4.2 Time Discretization–Finite ElementMethod (FEM) 168</p>
<p>6.5 Stability 169</p>
<p>6.6 Multi–dimensional Transient Heat Conduction 169</p>
<p>6.7 Summary 171</p>
<p>6.8 Exercises 171</p>
<p>References 173</p>
<p>7 Laminar Convection Heat Transfer 175</p>
<p>7.1 Introduction 175</p>
<p>7.1.1 Types of Fluid Motion Assisted Heat Transport 176</p>
<p>7.2 Navier–Stokes Equations 177</p>
<p>7.2.1 Conservation of Mass or Continuity Equation 177</p>
<p>7.2.2 Conservation ofMomentum 179</p>
<p>7.2.3 Energy Equation 183</p>
<p>7.3 Nondimensional Form of the Governing Equations 184</p>
<p>7.4 The Transient Convection–Diffusion Problem 188</p>
<p>7.4.1 Finite Element Solution to the Convection–Diffusion Equation 189</p>
<p>7.4.2 A Simple Characteristic Galerkin Method for Convection–Diffusion Equation 191</p>
<p>7.4.3 Extension to Multi–dimensions 197</p>
<p>7.5 Stability Conditions 202</p>
<p>7.6 Characteristic Based Split (CBS) Scheme 202</p>
<p>7.6.1 Spatial Discretization 208</p>
<p>7.6.2 Time–step Calculation 211</p>
<p>7.6.3 Boundary and Initial Conditions 211</p>
<p>7.6.4 Steady and Transient Solution Methods 213</p>
<p>7.7 Artificial Compressibility Scheme 214</p>
<p>7.8 Nusselt Number, Drag and Stream Function 215</p>
<p>7.8.1 Nusselt Number 215</p>
<p>7.8.2 Drag Calculation 216</p>
<p>7.8.3 Stream Function 217</p>
<p>7.9 Mesh Convergence 218</p>
<p>7.10 Laminar Isothermal Flow 219</p>
<p>7.11 Laminar Nonisothermal Flow 231</p>
<p>7.11.1 Forced Convection Heat Transfer 232</p>
<p>7.11.2 Buoyancy–driven Convection Heat Transfer 238</p>
<p>7.11.3 Mixed Convection Heat Transfer 240</p>
<p>7.12 Extension to Axisymmetric Problems 243</p>
<p>7.13 Summary 246</p>
<p>7.14 Exercises 247</p>
<p>References 249</p>
<p>8 Turbulent Flow and Heat Transfer 253</p>
<p>8.1 Introduction 253</p>
<p>8.1.1 Time Averaging 254</p>
<p>8.1.2 Relationship between , , T and T 256</p>
<p>8.2 Treatment of Turbulent Flows 257</p>
<p>8.2.1 Reynolds Averaged Navier–Stokes (RANS) 257</p>
<p>8.2.2 One–equation Models 258</p>
<p>8.2.3 Two–equation Models 259</p>
<p>8.2.4 Nondimensional Form of the Governing Equations 260</p>
<p>8.3 Solution Procedure 262</p>
<p>8.4 Forced Convective Flow and Heat Transfer 263</p>
<p>8.5 Buoyancy–driven Flow 272</p>
<p>8.6 Other Methods for Turbulence 275</p>
<p>8.6.1 Large Eddy Simulation (LES) 275</p>
<p>8.7 Detached Eddy Simulation (DES) and Monotonically Integrated LES (MILES)278</p>
<p>8.8 Direct Numerical Simulation (DNS) 278</p>
<p>8.9 Summary 279</p>
<p>References 279</p>
<p>9 Heat Exchangers 281</p>
<p>9.1 Introduction 281</p>
<p>9.2 LMTD and Effectiveness–NTU Methods 283</p>
<p>9.2.1 LMTD Method 283</p>
<p>9.2.2 Effectiveness NTU Method 285</p>
<p>9.3 Computational Approaches 286</p>
<p>9.3.1 System Analysis 286</p>
<p>9.3.2 Finite Element Solution to Differential Equations 289</p>
<p>9.4 Analysis of Heat Exchanger Passages . 289</p>
<p>9.5 Challenges 297</p>
<p>9.6 Summary 299</p>
<p>References 299</p>
<p>10 Mass Transfer 301</p>
<p>10.1 Introduction 301</p>
<p>10.2 Conservation of Species 302</p>
<p>10.2.1 Nondimensional Form 304</p>
<p>10.2.2 Buoyancy–driven Mass Transfer 305</p>
<p>10.2.3 Double–diffusive Natural Convection 306</p>
<p>10.3 Numerical Solution 307</p>
<p>10.4 TurbulentMass Transport 317</p>
<p>10.5 Summary&nbsp; 319</p>
<p>References 319</p>
<p>11 Convection Heat and Mass Transfer in Porous Media 321</p>
<p>11.1 Introduction 321</p>
<p>11.2 Generalized Porous Medium Flow Approach 324</p>
<p>11.2.1 Nondimensional Scales 327</p>
<p>11.2.2 Limiting Cases 329</p>
<p>11.3 Discretization Procedure 329</p>
<p>11.3.1 Temporal Discretization 330</p>
<p>11.3.2 Spatial Discretization 331</p>
<p>11.3.3 Semi– and Quasi–Implicit Forms 332</p>
<p>11.4 Nonisothermal Flows 333</p>
<p>11.5 PorousMedium–Fluid Interface 342</p>
<p>11.6 Double–diffusive Convection 347</p>
<p>11.7 Summary 349</p>
<p>References 349</p>
<p>12 Solidification 353</p>
<p>12.1 Introduction 353</p>
<p>12.2 Solidification via Heat Conduction 354</p>
<p>12.2.1 The Governing Equations 354</p>
<p>12.2.2 Enthalpy Formulation 354</p>
<p>12.3 Convection During Solidification 356</p>
<p>12.3.1 Governing Equations and Discretization 358</p>
<p>12.4 Summary 363</p>
<p>References 364</p>
<p>13 Heat and Mass Transfer in Fuel Cells 365</p>
<p>13.1 Introduction 365</p>
<p>13.1.1 Fuel Cell Types 367</p>
<p>13.2 Mathematical Model 368</p>
<p>13.2.1 Anodic and Cathodic Compartments 371</p>
<p>13.2.2 Electrolyte Compartment 373</p>
<p>13.3 Numerical Solution Algorithms 373</p>
<p>13.3.1 Finite ElementModeling of SOFC 374</p>
<p>13.4 Summary&nbsp; 378</p>
<p>References 378</p>
<p>14 An Introduction to Mesh Generation and Adaptive Finite Element Methods 379</p>
<p>14.1 Introduction 379</p>
<p>14.2 Mesh Generation 380</p>
<p>14.2.1 Advancing Front Technique (AFT) 381</p>
<p>14.2.2 Delaunay Triangulation 382</p>
<p>14.2.3 Mesh Cosmetics 387</p>
<p>14.3 Boundary Grid Generation 390</p>
<p>14.3.1 Boundary Grid for a Planar Domain 390</p>
<p>14.3.2 NURBS Patches 391</p>
<p>14.4 Adaptive Refinement Methods 392</p>
<p>14.5 Simple Error Estimation and Mesh Refinement 393</p>
<p>14.5.1 Heat Conduction 394</p>
<p>14.6 Interpolation Error Based Refinement 397</p>
<p>14.6.1 Anisotropic Adaptive Procedure 398</p>
<p>14.6.2 Choice of Variables and Adaptivity 399</p>
<p>14.7 Summary 401</p>
<p>References 402</p>
<p>15 Implementation of Computer Code 405</p>
<p>15.1 Introduction 405</p>
<p>15.2 Preprocessing 406</p>
<p>15.2.1 Mesh Generation 406</p>
<p>15.2.2 Linear Triangular Element Data 408</p>
<p>15.2.3 Element Area Calculation 409</p>
<p>15.2.4 Shape Functions and Their Derivatives 410</p>
<p>15.2.5 Boundary Normal Calculation 411</p>
<p>15.2.6 MassMatrix and Mass Lumping 412</p>
<p>15.2.7 Implicit Pressure or Heat Conduction Matrix 414</p>
<p>15.3 Main Unit 416</p>
<p>15.3.1 Time–step Calculation 416</p>
<p>15.3.2 Element Loop and Assembly 419</p>
<p>15.3.3 Updating Solution 420</p>
<p>15.3.4 Boundary Conditions 421</p>
<p>15.3.5 Monitoring Steady State 422</p>
<p>15.4 Postprocessing 423</p>
<p>15.4.1 Interpolation of Data 424</p>
<p>15.5 Summary 424</p>
<p>References 424</p>
<p>A Gaussian Elimination 425</p>
<p>Reference 426</p>
<p>B Green s Lemma 427</p>
<p>C Integration Formulae 429</p>
<p>C.1 Linear Triangles&nbsp; 429</p>
<p>C.2 Linear Tetrahedron 429</p>
<p>D Finite Element Assembly Procedure 431</p>
<p>E Simplified Form of the Navier Stokes Equations 435</p>
<p>F Calculating Nodal Values of Second Derivatives 437</p>
<p>Index 439</p>