WILEY SERIES IN COMPUTATIONAL MECHANICS

Series Advisors:

René de Borst

Perumal Nithiarasu

Tayfun E. Tezduyar

Genki Yagawa

Tarek Zohdi

Fundamentals of the Finite Element	Nithiarasu, Lewis	January 2016
Method for Heat and Mass Transfer	and Seetharamu
Introduction to Computational Contact	Konyukhov	April 2015
Mechanics: A Geometrical Approach
Extended Finite Element Method:	Khoei	December 2014
Theory and Applications
Computational Fluid-Structure	Bazilevs, Takizawa and	January 2013
Interaction: Methods and Applications	Tezduyar
Introduction to Finite Strain Theory for	Hashiguchi and Yamakawa	November 2012
Continuum Elasto-Plasticity
Nonlinear Finite Element Analysis of	De Borst, Crisfield, Remmers	August 2012
Solids and Structures, Second Edition	and Verhoosel
An Introduction to Mathematical	Oden	November 2011
Modeling: A Course in Mechanics
Computational Mechanics of	Munjiza, Knight and Rougier	November 2011
Discontinua
Introduction to Finite Element Analysis:	Szabó and Babuška	March 2011
Formulation, Verification and Validation

Second Edition

P. Nithiarasu

Zienkiewicz Centre for Computational Engineering
College of Engineering, Swansea University, UK

R.W. Lewis

Zienkiewicz Centre for Computational Engineering
College of Engineering, Swansea University, UK

K. N. Seetharamu

Department of Mechanical Engineering
PESIT, Bangalore, Karnataka, India

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Library of Congress Cataloging-in-Publication Data

Names: Nithiarasu, Perumal. | Lewis, R. W. (Roland Wynne) | Seetharamu, K. N.

| Lewis, R. W. (Roland Wynne). Fundamentals of the finite element method for heat and fluid flow.

Title: Fundamentals of the finite element method for heat and mass transfer.

Description: Second edition / P. Nithiarasu, R.W. Lewis, K.N. Seetharamu. | Chichester, West Sussex : John Wiley & Sons, Inc., 2016. | First edition: Fundamentals of the finite element method for heat and fluid flow / Roland W. Lewis, Perumal Nithiarasu, Kankanhalli N. Seetharamu (Hoboken, NJ : Wiley, 2004). | Includes bibliographical references and index.

Identifiers: LCCN 2015034600 | ISBN 9780470756256 (cloth : alk. paper)

Subjects: LCSH: Finite element method. | Heat equation. | Heat--Transmission.

| Fluid dynamics. | Mass transfer.

Classification: LCC QC20.7.F56 L49 2016 | DDC 530.15/5353--dc23 LC record available at http://lccn.loc.gov/2015034600

A catalogue record for this book is available from the British Library.

Cover images: courtesy of the authors.

CONTENTS

Preface to the Second Edition

Series Editor's Preface

1 Introduction

1.1 Importance of Heat and Mass Transfer
1.2 Heat Transfer Modes
1.3 The Laws of Heat Transfer
1.4 Mathematical Formulation of Some Heat Transfer Problems
1.5 Heat Conduction Equation
1.6 Mass Transfer
1.7 Boundary and Initial Conditions
1.8 Solution Methodology
1.9 Summary
1.10 Exercises
References

2 Some Basic Discrete Systems

2.1 Introduction
2.2 Steady-state Problems
2.3 Transient Heat Transfer Problem
2.4 Summary
2.5 Exercises
References
Note

3 The Finite Element Method

3.1 Introduction
3.2 Elements and Shape Functions
3.3 Formulation (Element Characteristics)
3.4 Formulation for the Heat Conduction Equation
3.5 Requirements for Interpolation Functions
3.6 Summary
3.7 Exercises
References

4 Steady-State Heat Conduction in One-dimension

4.1 Introduction
4.2 Plane Walls
4.3 Radial Heat Conduction in a Cylinder Wall
4.4 Solid Cylinder with Heat Source
4.5 Conduction–convection Systems
4.6 Summary
4.7 Exercises
References
Note

5 Steady-state Heat Conduction in Multi-dimensions

5.1 Introduction
5.2 Two-dimensional Plane Problems
5.3 Rectangular Elements
5.4 Plate with Variable Thickness
5.5 Three-dimensional Problems
5.6 Axisymmetric Problems
5.7 Summary
5.8 Exercises
References

6 Transient Heat Conduction Analysis

6.1 Introduction
6.2 Lumped Heat Capacity System
6.3 Numerical Solution
6.4 One-dimensional Transient State Problem
6.5 Stability
6.6 Multi-dimensional Transient Heat Conduction
6.7 Summary
6.8 Exercises
References

7 Laminar Convection Heat Transfer

7.1 Introduction
7.2 Navier-Stokes Equations
7.3 Nondimensional Form of the Governing Equations
7.4 The Transient Convection-Diffusion Problem
7.5 Stability Conditions
7.6 Characteristic Based Split (CBS) Scheme
7.7 Artificial Compressibility Scheme
7.8 Nusselt Number, Drag and Stream Function
7.9 Mesh Convergence
7.10 Laminar Isothermal Flow
7.11 Laminar Nonisothermal Flow
7.12 Extension to Axisymmetric Problems
7.13 Summary
7.14 Exercises
References

8 Turbulent Flow and Heat Transfer

8.1 Introduction
8.2 Treatment of Turbulent Flows
8.3 Solution Procedure
8.4 Forced Convective Flow and Heat Transfer
8.5 Buoyancy-driven Flow
8.6 Other Methods for Turbulence
8.7 Detached Eddy Simulation (DES) and Monotonically Integrated LES (MILES)
8.8 Direct Numerical Simulation (DNS)
8.9 Summary
References

9 Heat Exchangers

9.1 Introduction
9.2 LMTD and Effectiveness-NTU Methods
9.3 Computational Approaches
9.4 Analysis of Heat Exchanger Passages
9.5 Challenges
9.6 Summary
References

10 Mass Transfer

10.1 Introduction
10.2 Conservation of Species
10.3 Numerical Solution
10.4 Turbulent Mass Transport
10.5 Summary
References

11 Convection Heat and Mass Transfer in Porous Media

11.1 Introduction
11.2 Generalized Porous Medium Flow Approach
11.3 Discretization Procedure
11.4 Nonisothermal flows
11.5 Porous Medium-Fluid Interface
11.6 Double-diffusive Convection
11.7 Summary
References

12 Solidification

12.1 Introduction
12.2 Solidification via Heat Conduction
12.3 Convection During Solidification
12.4 Summary
References

13 Heat and Mass Transfer in Fuel Cells

13.1 Introduction
13.2 Mathematical Model
13.3 Numerical Solution Algorithms
13.4 Summary
References

14 An Introduction to Mesh Generation and Adaptive Finite Element Methods

14.1 Introduction
14.2 Mesh Generation
14.3 Boundary Grid Generation
14.4 Adaptive Refinement Methods
14.5 Simple Error Estimation and Mesh Refinement
14.6 Interpolation Error Based Refinement
14.7 Summary
References

15 Implementation of Computer Code

15.1 Introduction
15.2 Preprocessing
15.3 Main Unit
15.4 Postprocessing
15.5 Summary
References
Note

Appendix A Gaussian Elimination

Reference

Appendix B Green’s Lemma

Appendix C Integration Formulae

C.1 Linear Triangles
C.2 Linear Tetrahedron

Appendix D Finite Element Assembly Procedure

Appendix E Simplified Form of the Navier–Stokes Equations

Appendix F Calculating Nodal Values of Second Derivatives

Index

EULA

List of Illustrations

Chapter 1

Figure 1.1 Heat transfer from a plate subjected to solar heat flux.
Figure 1.2 Energy balance in an incandescent light source.
Figure 1.3 Conservation of energy in a moving body.
Figure 1.4 A differential control volume for heat conduction analysis.
Figure 1.5 Boundary conditions.
Figure 1.6 Rectangular fin.

Chapter 2

Figure 2.1 Heat transfer through a composite slab.
Figure 2.2 Fluid flow network.
Figure 2.3 Array of thin rectangular fins.
Figure 2.4 A simplified model of the rectangular fins of Figure 2.3.
Figure 2.5 A typical component from the rectangular fin arrangement and conductive–convective heat transfer mechanism. k – thermal conductivity, h – heat transfer coefficient, T_a – atmospheric temperature.
Figure 2.6 Heat treatment chamber and associated heat transfer processes.
Figure 2.7 Heat transfer in a composite wall.
Figure 2.8 Heat transfer through an insulating material.
Figure 2.9 Pipe network for central heating.
Figure 2.10 Counter flow heat exchanger.
Figure 2.11 Counter flow heat exchanger.
Figure 2.12 Incompressible flow through a pipe network.
Figure 2.13 A direct current circuit.
Figure 2.14 A heat sink.
Figure 2.15 A printed circuit board.

Chapter 3

Figure 3.1 Numerical model for heat transfer calculations.
Figure 3.2 Typical finite element mesh. Elements, nodes and edges.
Figure 3.3 One-dimensional finite elements: (a) a linear element; (b) a quadratic element; (c) linear; and (d) quadratic variation of temperature over an element.
Figure 3.4 Variation of shape functions, temperature and derivatives within a linear element.
Figure 3.5 Variation of shape functions and their derivatives over a one-dimensional quadratic element.
Figure 3.6 A one-dimensional linear element represented by local coordinates.
Figure 3.7 A linear triangular element.
Figure 3.8 Isotherm within a linear triangular element.
Figure 3.9 Area coordinates of a triangular element.
Figure 3.10 A quadratic triangular element.
Figure 3.11 Shape function designations of a quadratic triangular element.
Figure 3.12 Ten-node cubic triangular element.
Figure 3.13 A typical quadrilateral element.
Figure 3.14 A simple rectangular element.
Figure 3.15 Nondimensional coordinates of a rectangular element.
Figure 3.16 Square plate.
Figure 3.17 Isoparametric mapping of triangular and quadrilateral elements.
Figure 3.18 Eight-node isoparametric element.
Figure 3.19 Isoparametric transformation of a single triangular element: (a) global; (b) local – linear; and (c) local – quadratic.
Figure 3.20 Triangular elements.
Figure 3.21 Three-dimensional elements.
Figure 3.22 Quadratic tetrahedral element.
Figure 3.23 Twenty-node hexahedral element.
Figure 3.24 A fin problem.
Figure 3.25 Comparison between Ritz method and exact solution.
Figure 3.26 Comparison between variational method and exact solution.
Figure 3.27 Comparison between various weighted residual methods and exact solution.
Figure 3.28 Heat dissipation from a fin (Figure 3.24). Spatial discretization. Nodes: 6; Elements: 5.
Figure 3.29 Heat transfer from a rectangular fin.
Figure 3.30 Heat transfer from a rectangular fin. One linear element.
Figure 3.31 Rectangular element.
Figure 3.32 Quadrilateral element.
Figure 3.33 Triangular element.

Chapter 4

Figure 4.1 Heat conduction through a homogeneous wall.
Figure 4.2 Heat conduction in a composite wall.
Figure 4.3 Heat conduction through a homogeneous wall subjected to heat convection on one side and constant temperature on the other side. Approximation using a single linear element.
Figure 4.4 Heat conduction through a composite wall subjected to heat convection on one side and constant heat flux on the other side. Approximation using three linear elements.
Figure 4.5 Heat conduction through a wall with linearly varying area of cross-section.
Figure 4.6 Plane wall with heat source.
Figure 4.7 Finite element discritization.
Figure 4.8 Finite element discritization for the example with convection.
Figure 4.9 Quadratic finite element discretization.
Figure 4.10 Radial heat conduction in an infinitely long cylinder.
Figure 4.11 Different types of fins.
Figure 4.12 Tapered fin.
Figure 4.13 Tapered fin. Locations i and j.
Figure 4.14 Tapered fin. Finite element discretization.
Figure 4.15 A composite wall.
Figure 4.16 A composite wall.

Chapter 5

Figure 5.1 Examples of heat conduction in two-dimensional, three-dimensional and axisymmetric geometries. (a) Two dimensional plane geometry; (b) Three dimensional domain; (c) Axisymmetric configuration.
Figure 5.2 Typical two-dimensional plane geometry and triangular element. Front view (left) and side view (right).
Figure 5.3 Typical two-dimensional triangular element with heat generation and heat flux and convection boundaries.
Figure 5.4 Square plate with different temperature boundary conditions.
Figure 5.5 Discretization using triangular elements.
Figure 5.6 Solution for Example NaN on an unstructured mesh. The temperature obtained at the center of the plate is 200.42 °C. (a) Finite element mesh; (b) Temperature contours. Temperature varies between 100 °C and 500 °C. Interval between two contours is 25 °C.
Figure 5.7 Fine structured mesh.
Figure 5.8 Interpolation into a triangular element.
Figure 5.9 A square domain with mixed boundary conditions.
Figure 5.10 Discretization using two triangular elements.
Figure 5.11 Rectangular element with different boundary conditions.
Figure 5.12 Heat conduction in a square plate. Approximated using a rectangular (square) element.
Figure 5.13 A triangular plate with linearly varying thickness.
Figure 5.14 A linear tetrahedral element.
Figure 5.15 Representation of Example 5.2.1 in three dimensions.
Figure 5.16 Solution for the Example 5.2.1 on a three-dimensional mesh, temperature at the center point, (0.5,0.5,0.5), of the cube is 200.66 °C. (a) Finite element mesh; (b) Temperature contours. Temperature varies between 100 °C and 500 °C. Interval between two contours is 25 °C.
Figure 5.17 An axisymmetric problem.
Figure 5.18 An axisymmetric triangular element.
Figure 5.19 An axisymmetric element.
Figure 5.20 Cylinder of an IC engine.

Chapter 6

Figure 6.1 Lumped heat capacity system. A hot metal body is immersed in a liquid maintained at a constant temperature.
Figure 6.2 One-dimensional linear element.
Figure 6.3 Temperature variation within a time step.
Figure 6.4 One-dimensional transient heat transfer. Two elements and three nodes.
Figure 6.5 One-dimensional transient heat conduction analysis in a rod.
Figure 6.6 Linear triangular element meshes. (a) Coarse finite element mesh, 122 nodes and 158 elements. (b) Fine finite element mesh, 2349 nodes and 4276 elements.
Figure 6.7 Temperature distribution at t = 1. (a) Temperature distribution on the coarse mesh, T_max$ = 1.12? > at the right-hand face. (b) Temperature distribution on the fine mesh, T_max = 1.128 at the left hand face.
Figure 6.8 Temperature distribution along the length of the rod at t = 1.
Figure 6.9 Time discretization between n^th and (n + 1)^th time levels (time element).
Figure 6.10 Square and cubical domains with thermal boundary conditions.
Figure 6.11 Transient temperature distribution in a 2D plane geometry. (a) Temperature distribution at t = 0.001 s, T(0.5, 0.5) = 0.0 °C. (b) Temperature distribution at t = 0.01 s, T(0.5, 0.5) = 0.0 °C. (c) Temperature distribution at t = 0.1 s, T(0.5, 0.5) = 155.38 °C. (d) Temperature distribution at t = 0.5 s, T(0.5, 0.5) = 200.40 °C.
Figure 6.12 Temperature distribution at the center of square domain (cube in 3D) with respect to time.
Figure 6.13 Plane wall discretization.
Figure 6.14 Stainless steel plate.

Chapter 7

Figure 7.1 Flow and heat transport in a channel.
Figure 7.2 Infinitesimal control volume. Derivation of conservation of mass in a flow field.
Figure 7.3 Forced flow over a flat plate and natural convection inside a room.
Figure 7.4 Infinitesimal control volume in a flow field. Derivation of conservation of momentum in x₁ direction. Rate of change of momentum.
Figure 7.5 Infinitesimal control volume in a flow field. Derivation of conservation of momentum in x₁ direction. Viscous and pressure forces.
Figure 7.6 Infinitesimal control volume in a flow field. Derivation of conservation of energy.
Figure 7.7 Natural convective flow near a hot vertical plate.
Figure 7.8 Characteristic in a space–time domain.
Figure 7.9 One-dimensional linear element.
Figure 7.10 Convection-diffusion of a scalar variable. Problem setup.
Figure 7.11 Steady-state spatial variation of a function,ϕ, in one-dimensional space for different element Peclet numbers.
Figure 7.12 Two-dimensional linear triangular element.
Figure 7.13 Convection of discontinuous inlet data at at angle θ^o skew to the horizontal axis. Problem statement and contours of the scalar variable distribution.
Figure 7.14 Two-dimensional linear triangular element.
Figure 7.15 Example of a convection boundary condition.
Figure 7.16 Normal gradient of velocity close to the wall.
Figure 7.17 Typical convergence study.
Figure 7.18 Flow through a two-dimensional rectangular channel. Geometry and boundary conditions.
Figure 7.19 Flow through a two-dimensional rectangular channel. Finite element mesh.
Figure 7.20 Flow through a two-dimensional rectangular channel. Velocity profiles at different sections.
Figure 7.21 Flow through a two-dimensional rectangular channel. Comparison of velocity profiles at various distances.
Figure 7.22 Incompressible isothermal flow in a lid-driven cavity. Geometry and boundary conditions.
Figure 7.23 Linear triangular element meshes, (a-e) unstructured meshes, (f) 100x100 structured mesh.
Figure 7.24 Isothermal flow in a lid-driven cavity. Pressure contours at Re = 5000.
Figure 7.25 Isothermal flow in a lid-driven cavity. Stream traces at Re = 5000.
Figure 7.26 Incompressible isothermal flow in a lid-driven cavity. u₁ velocity profile along the mid-vertical line. Comparison with the benchmark steady-state results of Ghia et al., (1982). Source: Data from Ghia et al. 1982.
Figure 7.27 Isothermal flow in a lid-driven cavity. Stream traces and pressure contours for different Reynolds numbers.
Figure 7.28 Isothermal flow in a lid-driven cavity. Comparison of mid-vertical plane u₁ velocity profiles for different Reynolds numbers with Ghia et al., (1982). Source: Data from Ghia et al. 1982.
Figure 7.29 Incompressible isothermal flow past a backward facing step. Problem definition and boundary conditions.
Figure 7.30 Incompressible isothermal flow past a backward facing step. Finite element mesh, Nodes:4656, Elements:8662.
Figure 7.31 Incompressible isothermal flow past a backward facing step: (a) velocity contours; (b) pressure contours; (c) comparison of velocity profiles with experimental data, Re = 229.
Figure 7.32 Isothermal flow past a circular cylinder. Geometry and boundary conditions.
Figure 7.33 Isothermal flow past a circular cylinder. Three dimensional finite element mesh and an instantaneous u₁ velocity contour, Re = 100.
Figure 7.34 Isothermal flow past a circular cylinder. Comparison of u₁ velocity variation at an exit point, Re = 100.
Figure 7.35 Isothermal flow past a circular cylinder. Drag and lift coefficients, Re = 100.
Figure 7.36 Forced convection flow through a two-dimensional rectangular channel. Temperature profiles at various distances.
Figure 7.37 Forced convection flow past a sphere.
Figure 7.38 Forced convection heat transfer from a sphere. Three dimensional mesh.
Figure 7.39 Forced convection heat transfer from a sphere. Temperature distribution in the vicinity of the sphere.
Figure 7.40 Forced convection flow past a sphere. Temperature contours, Re = 100.
Figure 7.41 Forced convection heat transfer downstream of a backward facing step. Geometry and boundary conditions.
Figure 7.42 Forced convection heat transfer downstream of a backward facing step. Unstructured meshes.
Figure 7.43 Forced convection heat transfer downstream of a backward facing step. Local Nusselt number distribution on the hot wall for a Reynolds number of 500 on different meshes.
Figure 7.44 Forced convection heat transfer downstream of a backward facing step. Temperature contours at different Reynolds numbers.
Figure 7.45 Forced convection heat transfer downstream of a backward facing step. Local Nusselt number distribution on the hot wall for different Reynolds numbers.
Figure 7.46 Buoyancy-driven flow in a square enclosure. Geometry and boundary conditions.
Figure 7.47 Buoyancy-driven flow in a square enclosure. Finite element mesh. Nodes: 2601, Elements: 5000.
Figure 7.48 Natural convection in a square enclosure. Streamlines and temperature contours for different Rayleigh numbers, Pr = 0.71.
Figure 7.49 Mixed convection in a vertical channel. Geometry and boundary conditions.
Figure 7.50 Mixed convection in a vertical channel. Unstructured finite element mesh.
Figure 7.51 Mixed convection in a vertical channel. Developing velocity profiles are various vertical vertical sections.
Figure 7.52 Mixed convection in a vertical channel. Comparison of velocity profile at exit with fully developed analytical solution. Source: Data from Aung and Worku (1986b).
Figure 7.53 Coordinate system for axisymmetric geometries.
Figure 7.54 Schematic diagram of a cross flow heat exchanger, d = 1, B = 8, D = 6, pitch = 3, L = 42.
Figure 7.55 Schematic diagram of water processing plant, inlet/exit channel height = 1, L₁ = 4, L₂ = 5, L₃ = 4, L₄ = 6, L₅ = 30.

Chapter 8

Figure 8.1 Random variation of velocity in a turbulent flow with respect to time.
Figure 8.2 Turbulent flow in a two-dimensional rectangular channel. Comparison of fully developed velocity profiles at Re = 12 300.
Figure 8.3 Turbulent incompressible flow in a rectangular channel using the matrix free CBS-AC scheme with the Spalart-Allmaras model at Re=12 300. Logarithmic representation of time-averaged velocity profile.
Figure 8.4 Forced convective turbulent flow and heat transfer in a rectangular channel. Horizontal velocity, SA model turbulent variable and temperature distribution, Re = 5000, Pr = 0.71. (a) Velocity vectors; (b) Horizontal velocity; (c) SA model variable; (d) Temperature.
Figure 8.5 Forced convective turbulent flow and heat transfer in a rectangular channel. Local Nusselt number distribution along the hot wall, Re = 5000, Pr = 0.71.
Figure 8.6 Turbulent flow past a two-dimensional backward facing step. Problem definition.
Figure 8.7 Incompressible turbulent flow past a backward-facing step. Velocity profiles at various downstream sections at Re = 3025 (a) One-equation model; (b) SA model; (c) Two-equation model.
Figure 8.8 Turbulent incompressible flow past a backward-facing step. Unstructured mesh used. Number of elements: 297 054; Number of nodes: 65 372.
Figure 8.9 Turbulent incompressible flow past a backward facing step. Comparison of the velocity profile in the recirculation region, Re = 3025.
Figure 8.10 Turbulent incompressible flow over a circular cylinder at Re=10 000 using the matrix free CBS-AC scheme with the Spalart-Allmaras model: (a) unstructured mesh1 (Elements: 606 769, Nodes: 115 035); (b) unstructured mesh1 of close to solid wall (0.038 distance); (c) hybrid mesh2 (Elements: 489 463, Nodes: 88 964); (d) hybrid mesh2 of close to solid wall (0.01 distance).
Figure 8.11 Turbulent incompressible flow over a circular cylinder at Re=10 000 using the matrix free CBS-AC scheme with the Spalart-Allmaras model: (a) drag coefficient variation with respect to real time; (b) lift coefficient variation with respect to real time; (c) pressure coefficient distribution along the cylinder surface at real time = 100.
Figure 8.12 Turbulent incompressible flow over a circular cylinder at Re = 10 000 using the matrix free CBS-AC scheme with the Spalart-Allmaras model on unstructured mesh1 (up) and hybrid mesh2 (down). Real nondimensional time = 100. (a) , ; (b) , ; (c) p_min = −1.090, p_max = 0.743; (d) , ; (e) , ; (f) p_min = −0.967, p_max = 0.704.
Figure 8.13 Turbulent incompressible flow over a circular cylinder at Re = 10 000 using the matrix free CBS-AC scheme with the Spalart-Allmaras model on unstructured mesh1 (left) and hybrid mesh2 (right). Real nondimensional time = 100. (a) (red) = 0.0, ; (b) (red) = 0.0, .
Figure 8.14 Turbulent natural convection in a square cavity. Structured finite element mesh.
Figure 8.15 Turbulent natural convection in a square cavity. Velocity vectors. (a) Ra = 10⁸, (b) Ra = 10⁹; (c) Ra = 10¹⁰.
Figure 8.16 Turbulent natural convection in a square cavity. Stream traces (top), temperature contours (middle) and SA turbulence variable at different Rayleigh numbers. Average Nusselt numbers at Ra = 10⁸ is 30.67; at Ra = 10⁸ is 55.21 and at Ra = 10¹⁰ is 103.84. (a) Ra = 10⁸, Stream traces; (b) Ra = 10⁹, Stream traces; (c) Ra = 10¹⁰, Stream traces; (d) Ra = 10⁸, temperature; (e) Ra = 10⁹, temperature; (f) Ra = 10¹⁰, temperature; (g) Ra = 10⁸, turbulence variable; (h) Ra = 10⁹, turbulence variable; (i) Ra = 10¹⁰, turbulence variable.

Chapter 9

Figure 9.1 Schematic diagram of a parallel flow, tubular heat exchanger.
Figure 9.2 Local heat exchange in a parallel flow tubular heat exchanger.
Figure 9.3 Schematic diagram of a shell and tube heat exchanger.
Figure 9.4 (a) Simplified model of a heat exchanger; (b) a single element within the mesh.
Figure 9.5 Flow through a corrugated compact heat exchanger passage. Contours of horizontal velocity component, vertical velocity component, pressure and turbulent kinetic energy at Re = 8280. (a) Horizontal velocity; (b) Vertical velocity; (c) Pressure; (d) Turbulent kinetic energy.
Figure 9.6 Flow through a corrugated compact heat exchanger passage. Actual passage and the simplified passage used in the analysis.
Figure 9.7 Flow through a corrugated compact heat exchanger passage. Mesh, stream lines, pressure and temperature at Re = 2000. (a) Mesh; (b) Stream lines; (c) Pressure; (d) Temperature.
Figure 9.8 Flow through a corrugated compact heat exchanger passage. Mesh and stream lines at Re = 5000. (a) Mesh; (b) Stream lines.
Figure 9.9 Flow through a corrugated compact heat exchanger passage. Local Nusselt number distribution for different Reynolds numbers. (a) Re = 2000; (b) Re = 5000.
Figure 9.10 Flow through a corrugated compact heat exchanger passage. Average Nusselt number and pressure drop distribution for different Reynolds numbers. (a) Average Nusselt number; (b) Nondimensional pressure drop.
Figure 9.11 Conjugate heat transfer in a model fin and tube heat exchanger. Part of the geometry.
Figure 9.12 Conjugate heat transfer in a model fin and tube heat exchanger. Unstructured mesh. (a) Full view; (b) Fins and surface of the solid wall.
Figure 9.13 Conjugate heat transfer in a model fin and tube heat exchanger. Allocating material code.
Figure 9.14 Conjugate heat transfer in a model fin and tube heat exchanger. Contours of material code. (a) Side view; (b) Cross-sectional view.
Figure 9.15 Conjugate heat transfer in a model fin and tube heat exchanger. Pressure, velocity and temperature contours. (a) Pressure; (b) u₁ velocity; (c) u₃ velocity; (d) Temperature contours.
Figure 9.16 Conjugate heat transfer in a model fin and tube heat exchanger. Temperature contours at different sections. (a) x₁ = 2; (b) x₁ = 6; (c) x₁ = 10; (d) x₁ = 14.
Figure 9.17 Conjugate heat transfer in a model fin and tube heat exchanger. Temperature distributions (a) Along the mid-horizontal line of the fin in the x₂ direction; (b) Along the x₃ direction, in the fluid section, at x₂ = 0.523; (c) Along x₃ = 0.407 below the fin in the x₂ direction; (d) Along x₃ = 1.064 above the fin in the x₂ direction.

Chapter 10

Figure 10.1 Infinitesimal control volume in a flow field. Derivation of conservation of species concentration.
Figure 10.2 Steady-state mass diffusion in a rectangular domain.
Figure 10.3 Steady-state diffusion of a species from the bottom side of a channel. Part of the mesh and concentration distributions.
Figure 10.4 Unsteady-state mass diffusion in a rectangular domain.
Figure 10.5 Unsteady-state mass diffusion in a rectangular domain.
Figure 10.6 Unsteady-state diffusion of gas H₂ into CH₄ with a diffusivity of 0.726 cm²/s at 298 K. Top and bottom sides with zero concentration and both vertical sides insulated. Concentration distribution at different time periods.
Figure 10.7 Unsteady-state diffusion of gas H₂ into CH₄ with a diffusivity of 0.726 cm²/s at 298 K. Top and bottom sides insulated and both vertical sides with zero concentration. Concentration distribution at different time periods.
Figure 10.8 Forced convective mass transfer in a channel. Geometry and boundary conditions.
Figure 10.9 Forced convective mass transfer in a channel. Concentration distribution at different Reynolds numbers, Sc = 0.4.
Figure 10.10 Forced convective mass transfer in a channel. Local Sherwood number distribution at different Reynolds numbers, Sc = 0.4.
Figure 10.11 Buoyancy-driven convection in a cavity. Domain and boundary conditions and meshes used.
Figure 10.12 Double diffusive natural convection in a square cavity. Concentration and velocity distributions at difference buoyancy rations, Ra = 10⁵, Pr = 0.72, Le = 1.00.
Figure 10.13 Double diffusive natural convection in a square cavity. Average Sherwood number distribution at different buoyancy ratios, Ra = 10⁵, Pr = 0.72, Le = 1.00.
Figure 10.14 Double diffusive natural convection in a rectangular cavity with aspect ratio 7. Stream traces, temperature and concentration distributions at, Ra = 10⁴, Pr = 0.72, Le = 0.857, B = 1.
Figure 10.15 Buoyancy ndriven convection in a cavity with aspect ratio 7. Average Sherwood number for different Rayleigh numbers, Le = 0.857.

Chapter 11

Figure 11.1 Drag force on a porous medium grain.
Figure 11.2 Viscous forces on a bounding wall of a porous medium.
Figure 11.3 Fluid saturated porous medium. Infinitesimal control volume.
Figure 11.4 Forced convection in a channel filled with a variable porosity medium. Geometry and boundary conditions.
Figure 11.5 Forced convection in a channel. Comparison of Nusselt number with experimental data for different particle Reynolds numbers. Points: experimental (Vafai et al. 1984); dashed line: numerical (Vafai et al. 1984); solid: CBS. Source: Data from Vafai et al. 1984.
Figure 11.6 Forced convection in a channel. Comparison between the generalized model, Forcheimmer and Brinkman extensions to Darcy’s law.
Figure 11.7 Natural convection in a fluid saturated variable porosity medium. Problem boundary conditions.
Figure 11.8 Buoyancy-driven flow in a fluid saturated porous medium. Finite element mesh. Nodes: 2601, Elements: 5000.
Figure 11.9 Natural convection in a fluid saturated porous, square enclosure. Vector plots and temperature contours for different Rayleigh and Darcy numbers, Pr = 0.71.
Figure 11.10 Natural convection in a fluid saturated constant porosity medium. Problem definition.
Figure 11.11 Natural convection in a fluid saturated constant porosity medium within an annular enclosure. Comparison of hot wall steady-state Nusselt number with the experimental and numerical data. Source: Data from Prasad et al. (1985).
Figure 11.12 Natural convective heat transfer in a square cavity equally and vertically divided into saturated porous medium and free fluid. Problem domain, boundary conditions and the finite element mesh used.
Figure 11.13 Natural convective heat transfer in a square cavity equally and vertically divided into saturated porous medium and free fluid. Streamlines, isotherms and temperature distribution.
Figure 11.14 Natural convective heat transfer in a square cavity equally and horizontally divided into saturated porous medium and free fluid. Problem domain, boundary conditions and the finite element mesh used.
Figure 11.15 Natural convective heat transfer in a square cavity equally and horizontally divided into saturated porous medium and free fluid. Temperature distribution.
Figure 11.16 Double-diffusive natural convection in an axisymmetric, fluid saturated porous, square enclosure. Geometry and boundary conditions.
Figure 11.17 Double-diffusive natural convection in an axisymmetric, fluid saturated porous, square enclosure. Stream lines, isotherms and isoconcentration lines, Pr = 0.71, Da = 10^{− 6}, Ra = 10⁸, ε = 0.6, Le = 1, B = 1.
Figure 11.18 Double-diffusive natural convection in an axisymmetric, fluid saturated porous, square enclosure. Average Sherwood number distribution against the radious ratio r*.

Chapter 12

Figure 12.1 Variation of effective heat capacity and enthalpy across the solid–liquid interface.
Figure 12.2 A one-dimensional solidification problem.
Figure 12.3 Solution for the phase change problem using effective heat capacity method. (a) Unstructured mesh, nodes: 202, elements: 328; (b) Temperature distribution at t = 4; (c) Temperature distribution at point (1,0.25) with respect to time.
Figure 12.4 Solidification in a square cavity. Geometry and boundary conditions.
Figure 12.5 Solidification in a square cavity. Flow and isotherm patterns at different nondimensional times (a) t = 0.0305; (b) t = 0.1945; (c) t = 0.4815; (d) t = 0.9275.
Figure 12.6 Solidification in a square cavity. Heat transfer rate variation with time.

Chapter 13

Figure 13.1 A schematic diagram of the typical proton exchange fuel cell principle.
Figure 13.2 Heat and mass transfer in a planar SOFC. Schematic diagram and typical boundary conditions for a single domain approach. Source: Mauro (2009). Reproduced with permission of Mauro.
Figure 13.3 A section of the three-dimensional finite element mesh. Source: Mauro (2009). Reproduced with permission of Mauro.
Figure 13.4 Concentration overpotentials for the H₂-H₂O-Ar fuel mixture, at i_avg = 0.3A/cm² and i_avg = 0.7A/cm²: comparison with the experimental and numerical data available in Yakabe et al., 2000. Source: Data from Yakabe et al. (2000).

Chapter 14

Figure 14.1 Different type of meshes.
Figure 14.2 Stages in the progression of an advancing front algorithm.
Figure 14.3 Delaunay triangulation: (a) Delaunay triangulation of a given set of points; (b) constrained Delaunay element t: bold lines are the boundary edges, points 1,2 are within its circumcircle but not seen from element t.
Figure 14.4 Edge swapping to obtain a Delaunay triangulation.
Figure 14.5 The Bowyer-Watson method of point insertion for generating Delaunay triangulation.
Figure 14.6 a: concave domain to be triangulated and all data points; b: surrounding box and initial mesh of 2 elements; c: one point is inserted; d: first 5 points are inserted; e: inserted all boundary points; f: all the points have been inserted (one boundary edge is not in the triangulation); g: boundary edge recovery by edge swapping; h: elements containing box vertices are removed (result: convex domain); i: final mesh—all out-of-domain elements are deleted.
Figure 14.7 (a) Quadtree grid structure; (b) Delaunay triangulation generated on its nodes.
Figure 14.8 Boundary recovery by swapping edge: (a) boundary edge AB is not in the triangulation, (b) swapping edge CD → AB we recover the boundary edge but the triangulation becomes not Delaunay (only constrained Delaunay).
Figure 14.9 Boundary recovery by inserting additional node F in the intersection of AB and CD (b) and forming boundary edges AF and FB. Contraction of edge AF (c) and obtaining configuration shown in Figure 14.8(b).
Figure 14.10 Mesh cosmetics methods: (a) mesh smoothing, (b) edge swapping (2D), (c) edge splitting, (d) edge contraction.
Figure 14.11 Dual Voronoi diagram (gray) to a Delaunay triangulation.
Figure 14.12 CVT: Lloyd’s iterations. Grey polygons indicate Voronoi cells for inner nodes, grey circles – their centroids. (a): innitial mesh (inner nodes are placed randomly), (b): 1st iteration, (c): 3rd iteration, (d): 20th iteration.
Figure 14.13 Heat conduction in a square plate. Initial and adapted meshes and respective solutions.
Figure 14.14 An example of heat conduction with convective heat transfer boundary conditions.
Figure 14.15 Heat conduction in a rectangular plate. Initial and adapted meshes.
Figure 14.16 Adaptive finite element meshes for natural convection in a differentially heated square cavity.

Chapter 15

Figure 15.1 A typical unstructured mesh.
Figure 15.2 A triangular element.
Figure 15.3 Outward normal from a boundary side.
Figure 15.4 Multiply connected domain. Outward normal.
Figure 15.5 Element size calculation.

Chapter b_02

Figure B.1 Domain, boundary and outward normals.

Chapter b_04

Figure D.1 A domain with two linear triangular elements.

Preface to the Second Edition

In this second and enhanced edition of the book, we provide the readers with a detailed step-by-step application of the finite element method to heat and mass transfer problems. In addition to the fundamentals of the finite element method and heat and mass transfer, we have attempted to take the readers through some advanced topics of heat and mass transfer. The first edition of the book covered only the application of the finite element method to heat conduction and flow aided laminar heat convection. The second edition of the book has been enhanced further with turbulent flow and heat transfer, and mass transfer, in addition to advanced topics such as fuel cells.We believe that the second edition provides a comprehensive text for students, engineers and scientists who would like to pursue a finite element based heat transfer analysis. This textbook is suitable for beginners, senior undergraduate students, postgraduate students, engineers and early career researchers.

The first three chapters of the book deal with the essential fundamentals of both the heat conduction and the finite element method. In the first chapter, the fundamentals of energy balance and the standard derivations of relevant equations for the heat conduction analysis are discussed. Chapter 2 deals with the basic discrete systems which provide a basis for the finite element method formulations in the following chapters. The discrete system analysis is demonstrated through a variety of simple heat transfer and fluid flow problems. The third chapter gives a comprehensive account of the finite element method formulations and relevant history. Several examples and exercises included in Chapter 3 give the readers a complete overview of the theory and practice associated with the finite element method.

The application of the finite element method to heat conduction problems are discussed in detail in Chapters 4, 5 and 6. The conduction analysis starts with a simple one-dimensional steady-state heat conduction in Chapter 4 and is extended to multi-dimensions in Chapter 5. Chapter 6 gives the transient solution procedures for heat conduction problems.

Chapters 7, 8 and 9 deal with heat transfer by convection. In Chapter 7, heat transfer aided by the laminar motion of a single phase flow is discussed in detail. All the relevant differential equations are derived from first principles. All the three types of convection modes; forced, mixed and natural convection, are discussed in detail. Several examples and comparisons are provided to support the accuracy and flexibility of the finite element procedures discussed. In Chapter 8 the turbulent flow and heat transfer are discussed in some detail. Some examples and comparisons provide the readers a chance to assess the accuracy of the methods employed. Chapter 9 utilizes the finite element method developed in Chapters 1, 7 and 8 to provide a solution approach to flow and heat transfer in compact heat exchangers. Chapter 10 provides an introduction to the application of the finite element to problems of mass transfer. A detailed description of heat and mass transfer in porous media is then provide in Chapter 11. Two important applications of the finite element method for heat and mass transfer are explained in Chapters 12 and 13. Chapter 12 briefly introduces solidification problems using both heat conduction and convection approaches. Simple examples of solidification in this chapter may serve as a reference for students and researchers working in the area of solidification. In Chapter 13, we introduced a finite element solution approach to studying heat and mass transfer in fuel cells. Although the approach is only explained for solid oxide fuel cells, the method can be easily generalized to other types of fuel cells. Chapter 14 gives the reader sufficient information to understand the process of mesh generation. The main focus of this chapter is automatic and unstructured mesh generation. Some aspects of the adaptive mesh generation are also covered in this chapter. Finally, Chapter 15 briefly introduces the topic of computer implementation. The readers will be able to download the two-dimensional source codes and documentations from the website: www.zetacomp.com

Many people have assisted the authors either directly or indirectly during the preparation of this textbook. In particular, the authors wish to thank Dr Alessandro Mauro, Universitá degli Studi di Napoli Parthenope, for proofreading Chapter 13 and Dr Igor Sazonov, Swansea University, for helping the authors to put together part of Chapter 14. We would also like thank all our students, postdoctoral researchers and colleagues for providing help and support.

P. Nithiarasu, Swansea

R. W. Lewis, Swansea

K. N. Seetharamu, Bangalore

Series Editor's Preface

It is known that heat transfer provides a good context for teaching finite element methods and other computational mechanics topics. Fundamental concepts can be explained with such simple examples as heat conduction in 1D, then in 2D and 3D, and convective terms can be added to describe the special methods needed to deal with that class of partial differential equations. This book in our series does that, and with its distinguished, experienced authors, does it well. It not only teaches how to solve heat and mass transfer problems with finite element methods, but it also serves the purpose of teaching many different concepts in finite element methods. Readers from very diverse backgrounds will be able to benefit from this book. The book can be used by engineering undergraduate students to learn the fundamentals of heat and mass transfer and numerical methods, by graduate students in engineering and sciences to learn the advanced topics they need to know, and by practicing engineers and scientists as a good source and guide for research and development work in heat and mass transfer.