Fundamentals of the Finite Element Method for Heat and Mass Transfer

Perumal Nithiarasu, DSc, PhD, is currently the head of Zienkiewicz Centre for Computational Engineering at Swansea University. He has more than twenty years of teaching and research experience in the areas of finite element method, heat and mass transfer, fluid dynamics and biomedical engineering. He is a founding co-chair of the international conference series, Computational Methods for Thermal Problems (ThermaCOMP, www.thermacomp.com). Professor Nithiarasu is a winner of ECCOMAS young investigator award in 2004 and he was awarded the Zienkiewicz Silver Medal of the Institution of Civil Engineers, UK in 2002. Professor Nithiarasu has published more than 300 articles and two textbooks in the areas of heat and fluid flow.?Professor Nithiarasu is the founding editor of the International Journal for Numerical Methods in Biomedical Engineering, published by Wiley. He serves on the editorial boards of several international journals. Roland Lewis, DSc, PhD, FREng, has more than forty years of experience in teaching, research and administration in the area of heat transfer. Previously, Professor Lewis served as the head of Mechanical Engineering department in Swansea University.?His contributions in the areas of solidification and porous media are very well known. Until recently, he was the editor in chief of the International Journal for Numerical Methods in Engineering and Communications in Numerical Methods in Engineering. Although retired, he is actively editing the International Journal of Numerical Methods for Heat & Fluid Flow. He has also been serving as the honorary chair of the international conference series, Computational Methods in Thermal Problems (ThermaCOMP, www.thermacomp.com).?Author of nearly 400 articles, Professor Lewis was honoured with IACM Computational Mechanics award and fellowship. He is also a fellow of the Royal Academy of Engineering, UK. K.N. Seetharamu currently holds a prestigious chair professor position in the Mechanical Engineering department of the PES University, Bangalore. He has more than forty years of teaching and research experience in the areas of heat transfer and finite element method. Previously he was a professor of Thermal Engineering in Institute of Technology Madras.?Professor Seetharamu also has spent more than ten years in University of Sains Malaysia, carrying out research in the areas of heat transfer, energy and electronics packaging.?Author of more than 300 publications, Professor Seetharamu is one of the top heat transfer engineers in India. He is a fellow of the Indian National Academy. Recently, the Indian Society for Heat and Mass Transfer has established a biennial award in his name to honour Professor Seetharmau's achievements.

• Preface to the Second Edition xiiSeries Editor’s Preface xiv1 Introduction 11.1 Importance of Heat and Mass Transfer 11.2 Heat Transfer Modes 21.3 The Laws of Heat Transfer 31.4 Mathematical Formulation of Some Heat Transfer Problems 51.4.1 Heat Transfer from a Plate Exposed to Solar Heat Flux 51.4.2 Incandescent Lamp 71.4.3 Systems with a Relative Motion and Internal Heat Generation 81.5 Heat Conduction Equation 101.6 Mass Transfer 131.7 Boundary and Initial Conditions 131.8 Solution Methodology 151.9 Summary 151.10 Exercises 16References 172 Some Basic Discrete Systems 192.1 Introduction 192.2 Steady-state Problems 202.2.1 Heat Flow in a Composite Slab 202.2.2 Fluid Flow Network 232.2.3 Heat Transfer in Heat Sinks 262.3 Transient Heat Transfer Problem 282.4 Summary 312.5 Exercises 31References 363 The Finite Element Method 393.1 Introduction 393.2 Elements and Shape Functions 423.2.1 One-dimensional Linear Element 433.2.2 One-dimensional Quadratic Element 463.2.3 Two-dimensional Linear Triangular Element 493.2.4 Area Coordinates 533.2.5 Quadratic Triangular Element 553.2.6 Two-dimensional Quadrilateral Elements 583.2.7 Isoparametric Elements 633.2.8 Three-dimensional Elements 723.3 Formulation (Element Characteristics) 763.3.1 Ritz Method (Heat Balance Integral Method – Goodman’s Method) 783.3.2 Rayleigh–Ritz Method (Variational Method) 793.3.3 The Method of Weighted Residuals 823.3.4 Galerkin Finite Element Method 863.4 Formulation for the Heat Conduction Equation 893.4.1 Variational Approach 903.4.2 The Galerkin Method 933.5 Requirements for Interpolation Functions 943.6 Summary 1003.7 Exercises 100References 1024 Steady-State Heat Conduction in One-dimension 1054.1 Introduction 1054.2 Plane Walls 1054.2.1 Homogeneous Wall 1054.2.2 Composite Wall 1074.2.3 Finite Element Discretization 1084.2.4 Wall with Varying Cross-sectional Area 1104.2.5 Plane Wall with a Heat Source: Solution by Linear Elements 1124.2.6 Plane Wall with Heat Source: Solution by Quadratic Elements 1154.2.7 Plane Wall with a Heat Source: Solution by Modified Quadratic Equations (Static Condensation) 1174.3 Radial Heat Conduction in a Cylinder Wall 1184.4 Solid Cylinder with Heat Source 1204.5 Conduction – Convection Systems 1234.6 Summary 1264.7 Exercises 127References 1295 Steady-state Heat Conduction in Multi-dimensions 1315.1 Introduction 1315.2 Two-dimensional Plane Problems 1325.2.1 Triangular Elements 1325.3 Rectangular Elements 1425.4 Plate with Variable Thickness 1455.5 Three-dimensional Problems 1465.6 Axisymmetric Problems 1485.6.1 Galerkin Method for Linear Triangular Axisymmetric Elements 1505.7 Summary 1535.8 Exercises 153References 1556 Transient Heat Conduction Analysis 1576.1 Introduction 1576.2 Lumped Heat Capacity System 1576.3 Numerical Solution 1596.3.1 Transient Governing Equations and Boundary and Initial Conditions 1596.3.2 The Galerkin Method 1606.4 One-dimensional Transient State Problem 1626.4.1 Time Discretization-Finite Difference Method (FDM) 1636.4.2 Time Discretization-Finite Element Method(FEM) 1686.5 Stability 1696.6 Multi-dimensional Transient Heat Conduction 1696.7 Summary 1716.8 Exercises 171References 1737 Laminar Convection Heat Transfer 1757.1 Introduction 1757.1.1 Types of Fluid Motion Assisted Heat Transport 1767.2 Navier-Stokes Equations 1777.2.1 Conservation of Massor Continuity Equation 1777.2.2 Conservation of Momentum 1797.2.3 Energy Equation 1837.3 Nondimensional Form of the Governing Equations 1847.4 The Transient Convection-Diffusion Problem 1887.4.1 Finite Element Solution to the Convection-Diffusion Equation 1897.4.2 A Simple Characteristic Galerkin Method for Convection-Diffusion Equation 1917.4.3 Extension to Multi-dimensions 1977.5 Stability Conditions 2027.6 Characteristic Based Split (CBS) Scheme 2027.6.1 Spatial Discretization 2087.6.2 Time-step Calculation 2117.6.3 Boundary and Initial Conditions 2117.6.4 Steady and Transient Solution Methods 2137.7 Artificial Compressibility Scheme 2147.8 Nusselt Number, Drag and Stream Function 2157.8.1 Nusselt Number 2157.8.2 Drag Calculation 2167.8.3 Stream Function 2177.9 Mesh Convergence 2187.10 Laminar Isothermal Flow 2197.11 Laminar Nonisothermal Flow 2317.11.1 Forced Convection Heat Transfer 2327.11.2 Buoyancy-driven Convection Heat Transfer 2387.11.3 Mixed Convection Heat Transfer 2407.12 Extension to Axisymmetric Problems 2437.13 Summary 2467.14 Exercises 247References 2498 Turbulent Flow and Heat Transfer 2538.1 Introduction 2538.1.1 Time Averaging 2548.1.2 Relationship between κ, ε, νT and αT 2568.2 TreatmentofTurbulentFlows 2578.2.1 Reynolds Averaged Navier-Stokes (RANS) 2578.2.2 One-equation Models 2588.2.3 Two-equation Models 2598.2.4 Nondimensional Form of the Governing Equations 2608.3 Solution Procedure 2628.4 Forced Convective Flow and Heat Transfer 2638.5 Buoyancy-driven Flow 2728.6 Other Methods for Turbulence 2758.6.1 Large Eddy Simulation(LES) 2758.7 Detached Eddy Simulation (DES) and Monotonically Integrated LES (miles) 2788.8 Direct Numerica lSimulation(DNS) 2788.9 Summary 279References 2799 Heat Exchangers 2819.1 Introduction 2819.2 LMTD and Effectiveness-NTU Methods 2839.2.1 LMTD Method 2839.2.2 Effectiveness – NTU Method 2859.3 Computational Approaches 2869.3.1 System Analysis 2869.3.2 Finite Element Solution to Differential Equations 2899.4 Analysis of Heat Exchanger Passages 2899.5 Challenges 2979.6 Summary 299References 29910 Mass Transfer 30110.1 Introduction 30110.2 Conservation of Species 30210.2.1 Nondimensional Form 30410.2.2 Buoyancy-driven Mass Transfer 30510.2.3 Double-diffusive Natural Convection 30610.3 Numerical Solution 30710.4 Turbulent Mass Transport 31710.5 Summary 319References 31911 Convection Heat and Mass Transfer in Porous Media 32111.1 Introduction 32111.2 Generalized Porous Medium Flow Approach 32411.2.1 Nondimensional Scales 32711.2.2 Limiting Cases 32911.3 Discretization Procedure 32911.3.1 Temporal Discretization 33011.3.2 Spatial Discretization 33111.3.3 Semi- and Quasi-Implicit Forms 33211.4 Nonisothermal Flows 33311.5 Porous Medium-Fluid Interface 34211.6 Double-diffusive Convection 34711.7 Summary 349References 34912 Solidification 35312.1 Introduction 35312.2 Solidification via Heat Conduction 35412.2.1 The Governing Equations 35412.2.2 Enthalpy Formulation 35412.3 Convection During Solidification 35612.3.1 Governing Equations and Discretization 35812.4 Summary 363References 36413 Heat and Mass Transfer in Fuel Cells 36513.1 Introduction 36513.1.1 Fuel Cell Types 36713.2 Mathematical Model 36813.2.1 Anodic and Cathodic Compartments 37113.2.2 Electrolyte Compartment 37313.3 Numerical Solution Algorithms 37313.3.1 Finite Element Modeling of SOFC 37413.4 Summary 378References 37814 An Introduction to Mesh Generation and Adaptive Finite Element Methods 37914.1 Introduction 37914.2 Mesh Generation 38014.2.1 Advancing Front Technique (AFT) 38114.2.2 Delaunay Triangulation 38214.2.3 Mesh Cosmetics 38714.3 Boundary Grid Generation 39014.3.1 Boundary Grid for a Planar Domain 39014.3.2 NURBS Patches 39114.4 Adaptive Refinement Methods 39214.5 Simple Error Estimation and Mesh Refinement 39314.5.1 Heat Conduction 39414.6 Interpolation Error Based Refinement 39714.6.1 Anisotropic Adaptive Procedure 39814.6.2 Choice of Variables and Adaptivity 39914.7 Summary 401References 40215 Implementation of Computer Code 40515.1 Introduction 40515.2 Preprocessing 40615.2.1 Mesh Generation 40615.2.2 Linear Triangular Element Data 40815.2.3 Element Area Calculation 40915.2.4 Shape Functions and Their Derivatives 41015.2.5 Boundary Normal Calculation 41115.2.6 Mass Matrix and Mass Lumping 41215.2.7 Implicit Pressure or Heat Conduction Matrix 41415.3 Main Unit 41615.3.1 Time-step Calculation 41615.3.2 Element Loop and Assembly 41915.3.3 Updating Solution 42015.3.4 Boundary Conditions 42115.3.5 Monitoring Steady State 42215.4 Postprocessing 42315.4.1 Interpolation of Data 42415.5 Summary 424References 424A Gaussian Elimination 425Reference 426B Green’s Lemma 427C Integration Formulae 429C.1 Linear Triangles 429C.2 Linear Tetrahedron 429D Finite Element Assembly Procedure 431E Simplified Form of the Navier–Stokes Equations 435F Calculating Nodal Values of Second Derivatives 437Index 439