Finite Element Analysis for Heat Transfer. Theory and Software

Transcription

1 Finite Element Analysis for Heat Transfer Theory and Software

2 Hou-Cheng Huang and Asif S. Usmani Finite Element Analysis for Heat Transfer Theory and Software With 62 Figures Springer-Verlag London Berlin Heidelberg New York Paris Tokyo Hong Kong Barcelona Budapest

3 Hou-Cheng Huang, PhD, BSc, MSc Oxford Instruments UK Ltd., NMR Instruments, Osney Mead, Oxford OX2 ODX, UK Asif S. Usmani, PhD, BE, MS Department of Civil Engineering and Building Science, The University of Edinburgh, The King's Buildings, Edinburgh EH9 3IL, UK Cover Ulustration: Chapter 7, Figure S. Sequence of automatically-generated meshes based on a specified target error of 20%. ISBN-13: e-isbn-13: DOl: / British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publisbers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publisbers. o Springer-Verlag London Limited 1994 Softcover reprint of the hardcover 1st edition 1994 The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Typesetting: Camera ready produced by authors using ~~ 69/3830-S43210 Printed on acid-free paper

4 To our families

5 Preface This text presents an introduction to the application of the finite element method to the analysis of heat transfer problems. The discussion has been limited to diffusion and convection type of heat transfer in solids and fluids. The main motivation of writing this book stems from two facts. Firstly, we have not come across any other text which provides an introduction to the finite element method (FEM) solely from a heat transfer perspective. Most introductory texts attempt to teach FEM from a structural engineering background, which may distract non-structural engineers from pursuing this important subject with full enthusiasm. We feel that our approach provides a better alternative for non-structural engineers. Secondly, for people who are interested in using FEM for heat transfer, we have attempted to cover a wide range of topics, presenting the essential theory and full implementational details including two FORTRAN programs. In addition to the basic FEM heat transfer concepts and implementation, we have also presented some modem techniques which are being used to enhance the accuracy and speed of the conventional method. In writing the text we have endeavoured to keep it accessible to persons with qualifications of no more than an engineering graduate. As mentioned earlier this book may be used to learn FEM by beginners, this may include undergraduate students and practicing engineers. However, there is enough advanced material to interest more experienced practitioners. The first four chapters include a review of the basic heat transfer concepts, the governing equations and a gradual introduction to FEM. The last three chapters present special topics, which are phase change, adaptivity and convection/advection. Finally two appendices give complete details of the programs HEAT2D and HADAPT, with full user instructions and documented examples. HEAT2D is a program capable of steady and transient, linear and nonlinear analyses of diffusive and convective heat transfer. HADAPT is an adaptive version of HEAT2D and includes subroutines for error estimation and unstructured mesh generation of triangular and quadrilateral elements in arbitrary geometries. The complete source code for both the programs and sample input data files are provided on a floppy disc included with the book. Swansea, UK Hou-Cheng Huang, Asif S. Usmani

6 Conditions for Program Usage 1. The disk may only be used by the book purchaser on his/her microcomputer. 2. A back-up copy of the disk may be made by the book purchaser. 3. The program may not be transferred onto any other machine without the written permission of the author, or in the case of his death, by the written permission of his estate. 4. The software is supplied without any warranties of any kind, whether implied or otherwise. 5. All queries regarding function, performance, or adjustments to the programs should be directed to the authors: Dr. H.C. Huang 5 Segsbury Road Wantage Oxfordshire OXI2 9XR UK Dr. A.S. Usmani 57 Russell Street Swansea SAI4HX UK Additional Note Print-outs of disk content shown in the Appendices of this book have been especially formatted for reproduction; users should not expect their print-outs to match those in this section.

7 Contents 1 Introduction Importance of Numerical Analysis of Heat Transfer Reliability of Finite Element Analysis for Heat Transfer Various Heat Transfer Problems Objectives and Layout 3 References Governing Differential Equations Introduction Conduction Fourier's Law in Isotropic Materials Fourier's Law in Non-isotropic Materials Governing Equations of Heat Conduction Initial and Boundary Conditions Convection Radiation 18 References Finite Element Method Introduction Variational Principle and Rayleigh-Ritz Method Galerkin Weighted Residual Method Finite Element Method in Two Dimensions Introduction Conductivity Matrix and Load Vector Triangular Elements Natural Coordinate System Numerical Integration 44 References Temporal Discretisation for Heat Conduction 4.1 Introduction Finite Element Discretisation of the Transient Conduction Equation Recurrence Relations

8 x Contents Generalised Trapezoidal and Mid-point Family of Methods Convergence Automatic Time Step Selection Benchmark Example 58 References Phase Transformation Introduction The Stefan Problem Numerical Methods for Modelling Phase Transformation Effective Heat Capacity Enthalpy Method Heat Source Method Benchmark Examples Analytical Solution Numerical Solution Conclusion. 79 References Adaptive Heat Transfer Analysis Introduction Error Estimation for Heat Conduction Higher Order Approximation Implementation of the Adaptive Procedure Steady State Benchmark Example Transient Analysis 96 References Effects of Convection in Heat Transfer Introduction Steady State Advection-diffusion SUPG Method for Steady Advection-diffusion Benchmark Test for the Petrov-Galerkin Method Adaptive Solution of the Benchmark Problem Transient Advection-diffusion Taylor-Galerkin Method for the Hyperbolic Equation Benchmark Test for the Taylor-Galerkin Method Taylor-Galerkin Method for Coupled Advection-diffusion References A Software Description for HEAT2D A.l Introduction.... A.2 Glossary of Variable Names

9 Contents Xl A.3 AA A.5 A.6 A.7 A.2.1 Main Variables.. A.2.2 Main Arrays.... A.2.3 Main Subroutines. Program Overview.... Input Instructions..... Element Stiffness Calculations Phase Change Calculations.. A.6.1 Nodal Latent Heat Calculations A.6.2 Latent Heat Release for Each Iteration A.6.3 Correction of Temperatures after Each Iteration Documented Examples A.7.1 I-D Solidification Example.... A.7.2 Steady State Forced Convection Example Using the SUPG Method References B Software Description for HADAPT 169 B.l Introduction... ~ B.2 Glossary of Variable Names B.2.1 Main Variables for the Geometry and Mesh Data 169 B.2.2 Additional Variables Used in This Program 170 B.2.3 Additional Arrays Used in This Program 171 B.2A New Subroutines Used in This Program 172 B.3 Program Overview BA Input Instructions BA.l Geometry Data BA.2 Mesh Density Data 180 BA.3 Example Data File 180 BAA Output Data BA.5 Problem Data B.5 Error Estimate Calculations 186 B.6 Documented Examples B D Heat Conduction with Convective Boundary Condition B.6.2 Convection Dominated 2-D Heat Transfer 192 B.6.3 I-D Solidification Example Subject Index 197