of the temperature distribution Ts(x,tm) at the time when melting starts which is given by Goodman ...... enthalpies are corrected to account for the errors in the assumed heat flux densities. If Ci,}+l ... L.I. Rubinstein, Trans. Math. Monographs 27 ...
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APPLIED AND ENGINEERING MATHEMATICS SERIES Series Editor: Professor P. C. Kendall, Department of Electronic and Electrical
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Computational Moving Boundary Problems
Engineering, The University of Sheffield, England Convexity Methods in Variational Calculus
Peter Smith
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2.
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I.
Spherical Harmonics and Tensors for Classical Field Theory
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An Introduction to Fast Fourier Transform Methods for Partial Differential Equations, with Applications
Engineering Applications of Stochastic Processes Theory, Problems and Solutions
Structural Mechanics: Graph and Matrix Methods
A. Kaveh Mathematical Methods for Geo-electromagnetic Induction
J. T. Weaver Computational Moving Boundary Problems
M. Zerroukat and C. R. Chatwin
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8.
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7.
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6.
Lasers & Optical Systems Engineering Group (LOSEG) Department of Mechanical Engineering University of Glasgow Glasgow G12 8QQ, United Kingdom
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Alexander Zayezdny, Daniel Tabak and Dov WuJich
M. Zerroukat and C. R. Chatwin
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5.
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Morgan Pickering.
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M. N.Jones
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JOHN WILEY & SONS INC.
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RESEARCH STUDIES PRESS LTD. Taunton, Somerset, England
New York· Chichester· Toronto· Brisbane· Singapore
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RESEARCH STUDrES PRESS LTD. 24 Belvedere Road, Taunton, Somerset, England TA I IHD
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Copyright © 1994 by Research Studies Press Ltd.
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All rights reserved.
In most engineering applications the analysis of these problems is often
impossible without recourse to numerical techniques which utilise either finite difference, finite element or boundary element methods. The success of finite element and boundary element methods lies in their ability to handle complex geometries; however, they are acknowledged to be more time consuming and less amenable to vectorisation and parallelism than finite difference techniques. Owing to their simplicity in formulation and programming, where
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appropriate, ftnite difference techniques continue to be the most popular choice.
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The aim of this book is to present and compare some of the well known and recent numerical methods available to solve moving boundary problems. It has
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not been possible here to review all the existing methods; however, the book provides an extensive bibliography, at the end of chapter 2, to assist any reader
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who has a profound interest in the topic. Recent advances in the ftnite difference solution of partial differential equations are also presented and are Shown to give improved accuracy.
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Printed in Great Britain by SRI' Ltd., Exeter
Boundary Problems.
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ISBN 0 86380 1684 (Research Studie!> Pres!> Ltd.) ISBN 0471 951765 (John Wiley & Sons Inc.)
mathematical reasons, become known as either Stefan problems or Moving
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A catalogue record for this book is available from the British Library.
physical systems. Such dynamic multi-phase problems have, for historical and
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British Library Cataloguing in Publication Data
momentum, charge or matter which are fundamental to the behaviour of many
l, fu
erroukat, M. (Mohamed). 1964Computational moving boundary problems I M. Zerroukat and C.R. hatwin. p. cm. - (Applied and engineering mathematics series; 8) Includes bibliographical references and index. ISBN 0-86380-168-4. -ISBN 0-471-95176-5 (Wiley) I. Boundary value problems. 2. Finite differences. 3. HeatTransmission-Mathematical models. I. Chatwin, C. R. (Chris R.) II. Title. III. Series: Electronic & electrical engineering research studies: Applied and engineering mathematics series: 8. TA347.B69Z47 1994 620'.00 I '5 I 5353-dc20 94-10632 CIP
consequently any change in phase modifies the rate of transport of: energy,
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Library of Congress Cataloging-in-Publication Data
adjacent phases. Transport properties vary considerably between phases,
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5011/h Easl Asia: JOHN WILEY & SONS (SEA) PTE LTD. 37 Jalan Pemimpin 05-04 Block B Union Industrial Building, Singapore 2057
concentration, voltage or chemical potential a phase change may occur, which for dynamic processes will be separated by moving boundaries between the
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No,.,11 alld 5011111 America: JOHN WILEY & SONS INC. 605 Third Avenue, New York. NY 10158, USA
When matter is sl,lbjected to a gradient of temperature, pressure,
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allada: JOHN WILEY & SONS CANADA LIMITED 22 Worcester Road, Rexdale. Ontario, Canada
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Preface
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Marketing and Distribution:
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No part of this book may be reproduced by any means, nor transmitled. nor translated into a machine language without the written permission of the publisher.
Furthermore, algorithmic manipulations
vi
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which enhance the computational efficiency of these methods are incorporated in the solution of the moving boundary problems.
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Several finite difference schemes are available for the solution of moving boundary problems; however, the accuracy of these methods depends largely on
i th
the accuracy of the recurrence formulae such as Euler, Crank-Nicolson and page
the fully implicit equations. Whilst the refinement of mesh size can also be
s
1.
used to increase the accuracy of the solution, this is always at the expense of When
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greater cpu-time and computer memory size requirements.
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computations are to be performed for extended times, the use of large mesh
2.
sizes is desirable to reduce the array size and cpu-time; for these conditions,
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Engineers and Scientists alike are always concerned with the attendant loss of accuracy.
Introduction 1.1.
Introduction
1
1.2.
References
6
Numerical methods for moving boundary problems 2.1.
. 8
2.2.
The heat balance integral method (Goodman)
.. 9
2.2.1. Heat balance method for the single-phase ice
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melting problem
Our
thanks are also due to all the staff of the Department of Mechanical
14
2.4.
Fixed grid network finite difference methods......
2A
2.5.
Variable space grid finite difference methods
26
2.6.
Variable time step grid finite difference methods
'Zl
2.6.1. The extended Douglas and Gallie's (EDG) method
21
2.6.2. Goodling and Khader's (GK) method
31
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Science Department, for his valuable discussions on many occasions.
11
Neumann's method
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The authors would like to thank Professor D.C. Gilles, from the Computing
of a finite slab 2.3.
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partial differential equation rather than classically into the partial differential equation itself.
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the approach of making the finite difference substitution into the solution of the
9
2.2.2. Heat balance method for two-phase melting
time and array size requirements, the combination of real, virtual grid computational performance. The new finite difference equations are based on
7
Introduction
When account is taken of the computational parameters such as accuracy, cpunetworks and the new explicit finite difference equations offers a high
1
2.6.3. The modified variable time step (MVTS) method
2.7.
achieving this work. Finally, the financial assistance received by M. Zerroukat
33
2.7.1. Post-iterative (isothermal) 2.7.2. Post-iterative (mushy).............
3'!
2.7.3. Apparent heat capacity
3'!
3'!
2.7.4. Enthalpy method
35
2.7.5. Pham's method
35
2.7.6. Effective heat capacity
:E
2.7.7. Tacke's method
37
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from the Algerian government is gratefully acknowledged.
The enthalpy formulation techniques
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Engineering, especially the LOSEG group, for providing the necessary means of
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M. Zerroukat and C.R. Chatwin
2.7.8. Blanchard and Fremond's method
38
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University of Glasgow, UK, 1994.
32
38
2.8.
Conclusion
2.9.
References
39
viii
ix
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5.
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Computational performance of finite difference methods.... 3.1.
Introduction
3.2.
Methods of-solution
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3.
Introduction.......................
44
5.2.
Mathematical formulation of the problem........................... 117
46
5.3.
Numerical computation schemes
118
5.3.1. Fixed-grid finite difference methods
118
114
5.3.2. The integral method of Hansen and Hougaard........... 121
3.3.
Comparative performance of finite difference methods
51
5.3.3. Adaptive-mesh finite element methods
126
3.4. 3.5.
Test problems and results Finite difference solution for non-linear heat conduction
54
5.3.4. Variable-grid finite difference methods
130
5.4.
Numerical results and discussion
135
problems
65
5.5.
References
139
s
3.2.2. The weighted time step method................................ 50
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3.5.2. Three-dimensional problems.................. References
6.
Multiple moving boundary problems....
141
71
6.1.
143
Introduction.......................
81
6.2. . Description of multiphase Stefan problems
144
Explicit computation of moving boundary problems
85
6.2.1. Heating of the solid stage....
144
4.1.
Introduction............................................
86
6.2.2. Melting stage......................................................... 145
4.2.
The explicit variable time step (EVTS) method 4.2.1. Mathematical formulation of two-phase Stefan
88
6.4.
the liquid region..................................................... 94 4.2.4. Computation of temperature distribution in
7.
6.4.3. Vaporisation stage
157
Test problems
162
Numerical results and discussion
166
6.7.
References
173
Heat flow during laser heat treatment of materials
175 177
4.2.7. Numerical results and discussion
101
7.1.
Introduction.......................
The enthalpy technique..
100
7.2.
Theory of laser-beam interaction with materials
178
4.3.1. One-dimensional problems
100
7.2.1. Application to Silicon
180
7.2.2. Application to Nodular Cast-iron
180
112
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References
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4.3.2. Three-dimensional problems................................... 110 4.4.
155
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4.3.
154
6.4.1. Heating of solid stage.............
6.6.
6.5.
98 99
The finite difference method of Zerroukat and Chatwin
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4.2.6. Test problems
of Bonnerot and J amet...................................................... 147
6.4.2. Melting stage......................................................... 157
the solid region....................................................... 96 The special case of zero temperature at the moving boundary
The conservative finite element method
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4.2.5
6.3.
146
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90
6.2.3. Vaporisation stage
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4.2.2. Discretisation of the space-time domain........... 4.2.3. Computation of temperature distribution in
89
l, fu
problems
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3.6.
66
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3.5.1. One-dimensional problems
4.
5.1.
46
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3.2.1. The exponential finite difference equation
Implicit moving boundary problems (oygen diffusion problem) 113
43
Numerical computation scheme........................................ 181
7.4.
Numerical results and discussion
185
7.5.
References
189
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7.3.
x
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Appendices Appendix A......................................................... Appendix B Appendix C........
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Index
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Index
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Author Subject
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Appendix D...... References
Introduction
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Many physical processes can be modelled as boundary value problems, in
which the solution of partial differential equations has to satisfy certain conditions on the boundaries of a prescribed domain. However, in many
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important processes involving the changing states of matter, a boundary separating two different phases develops during the process. In these problems,
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the position of the boundary is not known a priori, but has to be determined as an integral part of the solution. The term "Moving Boundary Problems (MBPs)" is associated with time-dependent boundary problems, where the position of the
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Moving Boundary (MB) has to be determined as function of time and space.
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Moving boundary problems, also known as Stefan problems, were studied as early as 1831 by Lame and Clapeyron [1]. However, J. Stefan was given the
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major credit due to a sequence of papers [2,3] which resulted from his study of the melting of the polar ice cap around 1890.
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The formulation of MBPs requires not only the initial and boundary conditions to be known, as in boundary value problems, but two more conditions
2
Chapter I
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are needed on the moving boundary; one to determine the boundary itself and ,
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the other to complete the solution of the partial differential equations governing the process in each region (formulation examples are given in appendix A).
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Applications of MBPs are mainly but not exclusively concerned with fluid
3
Introduction
Chapter J
Introduction
..
I
I
I
I
I
I
I
I I •
I I I
I
I I I
------~------~---I I
flow in porous media, diffusion problems, heat transfer involving phase
i th
transformation, shock waves in gas dynamics and cracks in solid mechanics
N
[4]. Moving boundaries also occur in many processes associated with the metal,
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glass, plastic and oil industries; preservation of foodstuffs; statistical decision
1
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-~------~--I I
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t ._-----~-
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./---,
theory, heat treatment of semi-conductors; cryosurgery; astrophysics;
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meteorology; geophysics and plasma physics [5,6].
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In the early years, experimental analyses were the only means available to Figure 1.1: Position of the MB in a fixed grid network
give an understanding of physical processes. However, with the advent of high
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speed digital computers, mathematical modelling and computer simulation
FGMs, where the Moving Boundary (MB) is often located between two
real process. This permits the simulation of the performance of new product
neighbouring grid points (Figure 1.1), break down when the boundary moves a
designs and the assessment of quality even before production, resulting in great
distance larger than a space increment during a time step. This restriction,
savings of time and money. Due to the wide range of applicability in engineering and science, in the last engineers and scientists alike. As analytical solutions are frequently impossible numerical analysis.
performed for extended times. The possibility of break down of the scheme is avoided in VSSMs, by using variable space elements (as shown in Figure 1.2).
/1
I
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Finite Difference Methods (FDMs) are the most popular choice for numerical
size (memory) requirements and the cpu-time if computations are to be
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for most engineering and scientific applications, recourse must be made to
upon the velocity of the moving boundary, may considerably increase the array
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two decades MBPs have drawn considerable attention from mathematicians,
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are often the cheapest and fastest means to give a broad understanding of the
-----
solution of MBPs; however, in recent years, Finite Element Methods (FEMs)
require more cpu-time and are less suitable to vectorisation and parallel
Fixed Grid Methods (FGMs),
(ii)
Variable Space-Step Methods (VSSMs),
-
-
I
•
-1I
I I
I
I
------!-----:'----~:'-----I I
LIx
--- -- ,.'4__ I
I
.~ I
.. I
"
..'
,
./
~ I
ME
..
x
Figure 1.2: Position of the MB in a variable space grid network
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(iii) Variable Time-Step Methods (VTSMs).
-
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(i)
-
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Finite difference methods for solving MBPs can be classified into:
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'-:- -----'
-
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computation than FDMs.
1
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and Boundary Elements Methods (BEMs) have been introduced. The advantage of FEMs and BEMs is their ability to handle complex geometries; however, they
-
However, these methods present considerable computational difficulties (underflow and overflow) at the beginning of computation when the MB is too
4
Chapter I
Chapter I
Introduction
5
Introduction
If that the implicit methods, which are stable for any mesh size, are very
the enlargement of space elements which is a consequence of the displacement of the MB.
inaccurate when used with large time steps. The combination of the Virtual
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close to the fIxed boundaries; moreover, it loses accuracy at later times, due to
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,L1x
I
- - - -
-.-,4-
,
I
I
I
- - ,- -
I
Due to the inaccuracy of the implicit fInite difference equations when the
I
,
.-
Fourier number is relatively large, a new numerical scheme which will be
-- - -- -I - - I
I
-
referred to as the Explicit Variable Time Step (EVTS) method is developed for
- - - - - -,- - - - - - - r - - - - - - r - - -
solving two-phase Stefan problems. Numerical results show that the EVTS
-,- -
I
I
-
-
-
-
I
I
-
-
-
-
-
-.- -
I
-
-
-
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-
linear and non-linear partial differential equations.
- -
s
L1t
-
-
I
"
, -
- -
I
-- - -- -,- - - - - - -.- - - - - -
chapter also presents some recent advances in the fInite difference solution of
_,I
- - -
"
-
equations offers superior performance to that of the implicit equations. This
i th
- -- - - - ,- -
Sub-Interval Elimination Technique (VSIET) and the explicit finite difference
-
-.- -
-
I
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,
I
,
I
I
I ,
I ,
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method out-performs the implicit methods in current use. This is the subject
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techniques which increase the accuracy of the methods presented in chapter 2.
VSSMs are avoided; this makes VTSMs a very attractive approach. However, VTSMs depends on the accuracy of recurrence formulae such as CrankNicolson and fully implicit equations, which are very inaccurate when time solve MBPs where the moving boundary has a relatively slow velocity. developed techniques for solving moving boundary problems. Chapter 2 reviews some of the well known numerical methods frequently used to solve moving
which will assist the reader who has a profound interest in the detailed account
fInite element method of Bonnerot and Jamet [7], and the finite difference method of Zerroukat and Chatwin [8] are reviewed and compared. In chapter 7, the different techniques illustrated in previous chapters are used on a practical application. This consists of the simulation of the heat flow during laser heat treatment of materials, where heating of solid, melting and resolidifIcation occur in a very short time.
Numerical results show good
agreement with experimental results.
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A comparative study of the fInite difference equations, which represents the basis of any finite difference scheme, is given in chapter 3. The study shows
interfaces appear and disappear during the process. Both the conservative
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of the mathematical and computational aspects of moving boundary or Stefan problems.
In chapter 6, multiple moving boundary problems are dealt with. A typical
problem is the collapse of a solid wall, where both mel ting and vaporisation
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essence of each approach. This chapter also contains important bibliographies
problem to illustrate their application.
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boundary problems. The methods are briefly described in order to illustrate the
The chapter reviews and compares a number of
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This book reviews and compares some of the well known and recently
the moving boundary.
available methods to solve this type of problem and uses the oxygen diffusion
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step becomes relatively large; VTSMs may not be sufficiently accurate if used to
problem is different from the general moving boundary problems due to the absence, in its formulation, of an explicit expression containing the velocity of
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this approach is limited to one-dimensional problems only. As the accuracy of
This type of
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rather than a space variable grid, the problems associated with FGMs and
Chapter 5 deals with implicit moving boundary problems.
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By using VTSMs, where a variable time grid network is adopted (Figure 1.3)
Furthermore, the more accurate fInite difference solutions
developed in chapter 3, are combined with algorithmic manipulation
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Figure 1.3: Position of the MB in a variable time grid network
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x
of chapter 4.
6
Chapter i
introduction
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1.2. References
2.
M.M. Lame and B.P.E. Clapeyron,Ann. Chem. Phys. 47, 250-256 (1831). J. Stefan, Sber. Akad. Wiss. Wien. 98,473-484 (1889).
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l.
J. Stefan, Ann. Chem. Phys. 42,269-286 (1891).
4.
J. Crank, Free and moving boundary problems, Clarendon Press, Oxford (1984).
s
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3.
RM. Furzeland, J. Inst. Math. Appl. 26, 411-429 (1980).
6.
E. Magenes (ed.), Free boundary Problems vols. I and II, Instituto
so
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5.
Nazionale di Alta Matematica Francescon Severi, Rome (1980). 8.
M. Zerroukat and C.R Chatwin, J. Comput. Phys., (in press).
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R Bonnerot and P. Jamet, J. Comput. Phys. 41, 357-378 (1981).
Numerical Methods for Moving Boundary Problems
2
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7.
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Chapter 2 reviews some of the well known numerical methods frequently used to solve moving boundary problems. The methods are briefly described in order to illustrate the essence of each approach.
This chapter also contains an
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important bibliography which will assist any reader who has a profound interest in the topic.
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8
Chapter 2
Numerical methods for MBPs
Chapter 2
Numerical methods for MBPs
9
If Material thickness Specific heat
K
Thermal conductivity
Latent heat of fusion/unit volume
methods.
2.2. The heat balance integral method (Goodman)
L\t
Time step
L\x
Space element
ib
Subscripts Space step index Interface node index
By integrating the one-dimensional heat flow equation with respect to the
i th
Heat transfer coefficient Enthalpy
Material density
A.
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h H
are often used as a reference standard against which to validate the numerical
p
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a c
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Nomenclature
Position of the interface Latent heat offusion =IJp
r
Fourier number
j
Time step index
t
Time variable Temperature
m s
Melting
T
produced an integral equation which expresses the overall heat balance of the
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Superscript
a
Diffusivity
Iteration index
conditions, e.g. a polynomial relationship. (ii)
However, with the onset of the computer revolution, MBPs have been the focus large number of numerical schemes have been developed. It is not feasible to
change boundary and the time dependence of the temperature distribution.
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of intensive research during the seventies and eighties, during which time a
(iii) Solve the integral equation to obtain the motion of the phase
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Research on Moving Boundary Problems (MBPs) started as early as 1831.
assumed temperature distribution to obtain the heat balance integral.
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2.1. Introduction
Integrate the heat flow equation with respect to the space variable over the appropriate interval and substitute the
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Liquid
Space variable
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Temperature at the interface
on the space variable which is consistent with the boundary
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Tb
system. The successive steps in Goodman's method are: (i) Assume a particular form for the dependence of the temperature
so
Solid
x
k
space variable x, and inserting the boundary conditions, Goodman [15]
s
M L
2.2.1. Heat balance method for the single-phase ice melting problem The heat balance method can be conveniently illustrated by solving the
reports [1-14] are available which contain up to date accounts of the
single-phase melting-ice problem in one space dimension as defined by the
mathematical developments in this field, they illustrate wide ranging
following equations using dimensionless variables.
Finite difference methods have been used extensively for solving moving
methods compute, at each time step, the position of the moving boundary as well as the temperature at each grid point of the space-time domain. Before
, t>O
T(x,t) T(x,t)
=
(2.1)
1 , x=O , t > 0
(2.2)
, x>O , t = 0
(2.3)
= 0
M(O)
= 0
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Neumann's methods [16] are illustrated; these are analytical methods which
, O:=:;x:=:;M(t)
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presenting the numerical techniques, the well known Goodman's [15] and
at =
a 2T ax 2
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methods which can be used to solve these problems are reviewed. These
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boundary problems. In this chapter, some of the well known numerical
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applications in physics, biology and engineering.
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review all of them in this chapter; however, many surveys and conference
(2.4)
10
Chapter 2 Numerical methods for MBPs
Chapter 2 Numerical methods for MBPs
11
If =0
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T(x,t)
x
,
= M(t) , t > 0
u = dM dt
, x
= M(t)
, t>
left side of (2.7) gives
°
(2.6)
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- aT ax
Substitution of(2.8) incorporating get) and rp(t) from (2.11), and integrating the
(2.5)
dM -1( 1+.../3 )dM = ---(g -2rpM) 6 dt dt
(2.12)
i th
where T, t, x and M refer to temperature, time, spatial location and the position of the moving boundary, respectively.
Further elimination of
g and
rp from (2.12) gives
s
=
°to x = MCt), gives
= [3..Jt
(2.14)
By substitution of (2.14) and (2.11) into (2.8), the temperature distribution which represents the solution of the problem, defined by (2.1) to (2.6), is readily
obtained.
2.2.2. Heat balance method for two-phase melting of a finite slab
e as
dx
(2.13)
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ax
(2.9)
x = M(t)
M(t)
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(2.8)
which satisfies (2.5). It is convenient to modify the condition (2.6) to the form
2 ( ar)2 = d ; ,
[3 = ,I 7 +.../3
from which it follows that
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Assuming a temperature distribution in the water phase given by
T(x,t) = W){x - M(t)} + rp(t){x - M(t)}2
12(3- .../3)
where
(2.7)
ce
fT(X t) dx = _(dM + aT(O,t») dt' dt ax o
M dM = 6( 3 - .../3) _ 1 2 dt 7+.../3 -"2[3
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!!:...
so
M(t)
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Integration of (2.1) with respect to x, from x
The heat balance method was applied to solve the melting of a finite slab of
by using the standard formula
=
The problem is described by the following equations
° , x = M(t)
(2.10)
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dT _ aT dM + ar dt at
de - ax
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thickness a initially at a uniform temperature that is below the melting point.
'
°x $;
$;
M(t)
(2.15)
a2 ax
,
M(t)
$;
x
(2.16)
ar/- a _/ dt
ax
ar
at
(2.11)
/ ar/ ax -Ks ar ax
s
a
, x = M(t)
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=A dM) dt T/(x,t) = Ts(x,t) = T b
K
$;
bo
M(t)
gM+1 rp =---,;j2
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s a - T- s -= s 2
x = M(t), so that the substitution into (2.9) yields
g = 1-.../3
e
to replace (dM / dt) in (2.6). Taking into consideration that T(O,t) = 1, (2.8) gives: 1 = -gM + rpM 2 and on differentiation, aT / ax = g, a 2T / 2 = 2rp at
2
a T/2 ax
(2.17)
Chapter 2
12
13
Chapter 2 Numerical methods for MBPs
Numerical methods for MBPs
If yo
~l(t), ~s(t),
heat capacity, temperature at the moving boundary and latent heat of fusion
f{J1(t) and f{Js(t) are determined so that the temperature distributions, (2.22) and (2.23), satisfy the boundary conditions. For
per unit volume, respectively. The subscripts land s denote liquid phase and
example, with the following boundary conditions
where ex=KI(pc), K, p, C, T b and
A. are diffusivity,
The parameters
thermal conductivity, density,
u
d fin
solid phase respectively. The solution of the problem during the pre-melting
-Kl ( arl )
stage is not presented since standard solutions are readily available in [16].
i th
ax
Goodman and Shea [17] have also solved for this stage by the heat balance
s
method.
=F
x=O
re
from x=o to x=M(t) and (2.16) from x=M(t) to x=a to obtain the integral equations
ax
M(t)
dM }=!!:...fa Tsdx+Tb dt dt
(2.19)
M
ax) and (ar s I ax), at x = M(t),
from (2.18) and (2.19) and
making use of (2.17), yields
al dt
, Tb
=0
(2.24)
= F
(2.25)
as dt
_(a - M)2 dO
s _
°
3as
(2.26)
(2.27)
s
dt
Equation (2.25) is obtained by substitution of (2.24) into (2.20). The evaluation of
e pl
Elimination of (arl I
a
l, fu
ax
=0
01 = M2(~ __1_dOI) 2Kl 3al &
e us
as{(aTS ) _(ars)
(2.18)
ce
f
- (arl - )} = Mfarl - d x =d- M T l dx-Tb -dM ax 0 ax dt dt o 0
dt
ur
M(t)
x=a
-A. dM + Kl dOL + K s dO.
so
ax
, Ks(O:XS )
Goodman and Shea [17J arrive at the following differential equations
Following the three stages (i) to (iii), defined in section 2.2, (2.15) is integrated
al {( -arl )
= const.
~l(t)
and f{J1(t) using the conditions (2.24) and (2.18), and using the first
K s Tb _ K l T b _ A.) dM + K l dOL + K s dOs = Ks(aTs ) _ K ( aTI ) ( as al dt al dt as dt ax x=a l ax x=O
(2.20)
e as
integral of (2.21),gives (2.26). Equation (2.27) is derived by first evaluating
~.(t)
and f{Js (t) using (2.24) and (2.19) and finally using the second of (2.21).
fe re
To solve the equations (2.25) to (2.27), the following initial conditions are where
needed
Os = f Ts(x,t) dx
(2.21)
M
M(tm)
=0
-F a 2
°s(tm) = 3K s
Ol(tm) = 0
(2.28)
e
° 1 = f T/(x,t) dx o
a
nc re
M
e th
Os (t m ) of (2.28) is obtained by integration with respect to x, from x
Suitable temperature distributions are then chosen to be
= 0 to x = a,
(2.22)
Ts(x,t) = T b + ~.(t){ x - M(t)} + f{Js(t){ x - M(t)} 2
(2.23)
Ts(x,t m) = -aF {~(2-~)} 2K. a a
ok
Tl(x,t) = T b + ~1(t){X - M(t)} + f{J1(t){X - M(t)}2
bo
of the temperature distribution Ts(x,t m ) at the time when melting starts which is given by Goodman and Shea [17J as
(2.29)
Chapter 2 Numerical methods for MBPs
Chapter 2
If
14
aT. aT= dM l K -KlpL -
Numerical methods for MBPs
• ax
yo
Goodman and Shea obtained the solution of (2.25) to (2.27) by lengthy
ax
u
T.(x,t) = Tl(x,t)
(2.27), can be obtained easily using a standard method such as Runge-Kutta.
x= M( t )
(2.32)
= M(t)
(2.33)
dt'
mathematical manipulations. As an alternative method, solution of (2.25) to
=T b
,
x
15
d fin
Many authors [18-24] have applied the heat balance method to different where L is the latent heat of fusion.
problems, as well as studied ways of improving and easing the mathematical
i th
analysis. The mathematical manipulations required for the heat balance
I - The region x> 0 is initially liquid at a constant temperature To, with the
method, for anything other than relatively simple problems, can be very lengthy
s
surface x = 0 maintained at zero temperature for t > 0 (Figure 2.1).
and complicated. Furthermore, the choice of a satisfactory approximation to
re
the temperature distribution is a major difficulty with the method. For
so
example, the use of a high order polynomial for the temperature profile makes
x M(t)
(2.31)
,
00
x=O
= Aerf[ 2(a:t)1J2 ]
Tl(x,t)
=To -
Berfc[
2(al~)112 ]
(2.34) (2.35)
(2.36)
(2.37)
where A and B are constants which satisfy (2.30)&(2.35) and (2.31)&(2.34)
bo
2 l a la-T-l aT_0 2
T.(x,t)
e th
•
2
,
-1
The solution of problem I can be written [16J as
e
aT. _ a (iT.
when x
To
T.(x,t)=O
nc re
techniques, since it is possible to obtain an analytical solution. In general, the
4
fe re
The method is illustrated for a number of simple heat transfer cases [16J.
at
The equations to be solved are (2.30), (2.31), (2.32), (2.33) with the following additional boundary conditions
e pl
including that of the melting boundary, at successive times.
at
l, fu
than subdividing the space variable. The solution of a system of differential
respectively. Equation (2.33) requires that
ok
16
17
Chapter 2 Numerical methods for MBPs
Chapter 2 Numerical methods for MBPs
If yo
For the special case in which the liquid is initially at its melting point, i.e.
Aerf[ M(t)112 ] = To - Berfc[ M(t)112 ] = T b 2(as t) 2(a/ t)
To
(2.38)
= T b , (2.41) reduces to
u
d fin
i th
M(t) = 2w(as t)1I2
wexp(aherf(w) =
..Jt, that is
To satisfy (2.38) for any value of t, M must be proportional to
s
where w is a real constant to be determined from the condition (2.32). Making use of(2.36), (2.37) and (2.39) wmust be the solution of
Assume
re
r
••pC-m ) -
)112] -
KsTb{a; erfc w( :;
csT b
(2.41)
x = 0 ~:: .... ;::::~i:;l: :~:~j~~:: :;;:~:::~y~: .':-:;:~;;
.,
x> 0
!C"";'
I. j
LlqUld
_.,.__ .,.' T b, as shown in
solid (Figure 2.4).
s
:::~TK~'t,
"
Solid
T-
-0
'fffffr'
ti::;:::::::::~:~ ;: : :;.:.:.:.:.: ~::;.:.:.:.:.:.:.:.:.:.:.:r::::::::~::::;:::::::::;:::;:;::=;:::;::;:::.
ce
x>O
ur
so
::oJr~i!.T=T'>Tb
l, fu
t> 0
(2.51)
KsTb{a; exp( -alm / as)
mL...{;
Kl{a;(To -Tb)erfc[m(al / a s )1I2] = cl(To -Tb )
(2.52)
2(as~t)1/2])
T"l(X,t)=B+cerf[ 2(a:t)1/2]
fe re
2
T S1 (x,t) = A[ 1 + erf[
e as
where mis the root of exp(-m 2) erf(m)
The solution of the problem can be written as
e pl
M(t) = 2m(al t)1/2
nc re
Tl(x,t) = To -nerfc[
The temperature distributions in both regions are given by
(2.53)
(2.56)
(2.57)
The temperature continuity
relations at x=O require that
A=B
, Ks1A.,Ja;; = K"lCF::
bo
- T b) erf[ 2(alxt) 112] Tl(x,t) = To - (To erf(m)
0 as x ~ - 0 0 ,
e th
Tb erfC[ x ] erfc[ meal / a s )1I2] 2(as t)1I2
~
2(al~)IJ2 ]
(2.55)
e
Equation (2.55) makes T S1 (x,t)
Ts(x,t) =
)~'~~;i~~,i!~][~~:i~~]i~i!!liii!ii~~~!!j~l::Ch
Figure 2.4: Case IV
e us
Similarly to the previous cases, the interface is written as
x - M(t
t=O
11
Figure 2.3: Case III
x < O ~KS1,cS1 x=O~n~~~~ _ ~E2!i
X 0 initially at T b2 and x =0 at zero temperature, (2.41)
so
different regions can be written as
T~
(2.60), bearing in mind that Cl, in (2.41) and (2.60), is the value given by (2.65).
After determination of w from (2.60), the temperature distributions for the
T (x t)-
(2.65)
T~ s T s Tb,.
l;.. ~,~,~,~,~,~,~ ....,....,,..,.....".,....." ... '~ ... ~,~ ... :'~,~ ':'~ ':,.
:'~/~":":/>:' Phase 2
>.:':':':':':':':':
1
e
r. O. It is assumed that the material has two
the latent heat L and the specific heat Cl' of the homogenous liquid
transformation temperatures T bl and T~, V>
ok
bo
temperature T bl and T~. Assuming that the latent heat in the mushy region is liberated linearly with temperature, this effect can be taken into
T~
>Tbt >0, at which the latent
Chapter 2 Numerical methods for MBPs
Chapter 2 Numerical methods for MBPs
22
23
If ~
VII - The effect of density change
yo
heats L l and
are liberated respectively. The subscripts 1,2 and 3 are used to
u
denote the regions 1 (where the temperature varies from 0 to T bl ), 2 (T b1 , Tb-z)
When the change in density of the material, due to the phase changes, is
interfaces: x
= Ml(t)
d fin
and 3 (Tb.z' V), respectively (Figure 2.5). Consequently there will be two moving separating phases 1 and 2, and x
= M2(t)
taken into account, in a similar manner to (2.36), the temperature in the solid region can be written as
which separates
phases 2 and 3. Following the same procedure as for previous cases, the
i th
moving boundaries can be written as
s
X T.(x,t) = Aerf[ 2(a t) 112 ] (2.68)
Since the liquid and solid have different densities, then, when the interface moves a distance dM, the liquid moves with a velocity ur given by
so
re
Ml(t) = 2wl(aIt)1I2
where Wl and W2 are the solutions of the following system of equations
ur K 2..j(i;(Tb-z -TbJexp(-wr a l/a 2}
-
Cl
-
C2
ax
different regions can be written as
PIal
dt
(2.71)
ax
al at
x + w(Ps - Pl){a;] 2(al t)1I2 Pl{a;
(2.77) requires that Tl(x,t) ~ To as x ~
nc re
er~b~l) (erf[ 2(a;t)1I2 ]J
fe re
Tl(x,t) = To - Berfc[
T I(X,t) =
_l)dM dt
(2.75)
(2.76)
If M = 2w(as t)1I2, (2.76) is satisfied by
e as
Having determined iteratively wI and W2' the temperature distribution in the
=_(Ps PI
a2~1 + (P. - PI) dM arl _ -.!. arl = 0
e pl
K 2.,J'a; erf[ W2{ a2/(3)1I2]
erf(Wl) - erf[WI( al/(2)l/2]
r
The heat equation in the moving liquid is then written as
l, fu
K3~(To-Tb2}exp[-w~(a2/a3)]_ w2L2~ (2.70)
(Tb-z - T bl }exp(-w~)
U
(2.69)
e us
{erf(W2)-erfc[Wl(al/a2)1I2]}Kl~
_ wlLl~
ce
T bl exp(-w~) erf(w l )
(2.74)
s
M 2(t) = 2w2(a2 0 112
00.
(2.77)
The temperature continuity at the
moving boundary x=M(t), requires that
= To -
(2.73)
(2.78)
Following the same procedure as case I, and making use of (2.78), co is determined from the solution of
ok
(To-Tb-z) erfc[--X--=-=-] erfc[ W2(a2 / (3)1I2] 2(a3 0 1l2
Aerf(w) = To - Berfc[wPS{a;] = T b PI {a;
bo
T 3(x,t)
(2.72)
e th
(Tb-z - T bl )erf[ x/2(a2 0 112 ] + T bl erf(W2) - Tb-z erf[ WI( ad(2)1I2] [112] erf(w2)-erf wl(ada2)
e
T 2 (x,t) =
Chapter 2 Numerical methods for MBPs
Chapter 2 Numerical methods for MBPs
24
25
If yo
The variation of the position of the moving boundary is given by (2.79)
u
exp(-w 2 ) erf(w)
d fin i th
2.4. Fixed grid finite difference methods
Pj+1
){ Pj Pj + 1 } = Pj - ( .t t.t L1x2 Pj + 1 Tib-l,j --p;-Tib,j
(2.82)
Various finite difference schemes have been proposed for approximating
s re
both the boundary conditions and the partial differential equation at the neighbouring grid points. For instance, Murray and Landis [28] used a fixed
Using finite difference techniques, the heat flow equation is approximated by
grid network. In particular, in the sub-space containing the fusion front at any
temperature Ti,j' at xi = iL1x and time t j = jt.t on a fixed grid in the (x,t) plane. At any time t j = jt.t, the phase change boundary will usually be located
time, they introduce two fictitious temperatures, one obtained by quadratic extrapolation from temperatures in the solid region and the other from
between two neighbouring grid points; for example: between ibL1x and
temperatures in the liquid region. The fusion temperature and the current
ur
so
finite difference replacements for the derivatives in order to compute values of
ce
+ 1)L1x, as shown in Figure 2.6.
(2.81). For the motion of the fusion front, an expression analogous to (2.82) is used based on the Taylor extrapolation formula. Furzeland [29] suggested using an approximation for (iJT / be eliminated by use of(2.80).
ax)
on the MB with a fictitious temperature to
e as
ib +2
ib + 1
compute temperatures near the interface instead of using special formulae like
e pl
9~H ib
are then substituted into the standard approximation, such as (2.80), to
l, fu
x=M(t)
ib -1
position of the interface are incorporated in the fictitious temperatures, which
e us
(ib
The advantages of implicit finite difference equations have been investigated
Figure 2.6: Location of the moving boundary in a fixed grid
by Ehrlich [30], Koh et al. [31], Saitoh [32], and Bonnerot and Jamet [33]. For
fe re
example Meyer [34] obtained a second order approximation for the movement of The numerical solution of the single-phase problem, defined by equations (2.1)
the MB by using a three-point backward formula. Ciment and Guenther [35]
to (2.6), provides a simple example and proceeds as follows: the temperature
employed a method of spatial mesh refinement on both sides of the MB,
nc re
T i,j+1' as a function of temperatures of the previous step, is given by
previously analysed by Ciment and Sweet [36]. They also incorporated the idea used earlier by Douglas and Gallie [37], of adjusting the time step so that the
i=0,ib -1
(2.80)
MB coincides with one of the refined mesh points. Huber [38] used an implicit
e
T·I,J+ . 1 = T·I,J. +(~){T' L1x 2 1- l.J' -2T·I,J. +T,1+ 1,J,}
J
Lazaridis [40 I used explicit finite difference approximations on a fixed grid to solve two-phase solidification problems in both two and three space dimensions.
(2.81)
ok
'-~T'.} P' "b,J
2t.t){ T-"b,J'+1 = T ib, J' + ( -L1x 2 -11 T,"b- 1,J Pj+
simplified by Rubinstein [4J and Fasano and Primicerio [39].
bo
By using three-point Lagrange interpolation [27], instead of (2.80), the temperature at x = ibL1x is calculated using
e th
scheme for approximating the heat flow equation in each time step, but he tracked the MB explicitly using the Stefan condition; this approach was later
He wrote the heat balance condition on the MB in Patel's expressions and
26
Chapter 2 Numerical methods for MBPs
Numerical methods for MBPs
Chapter 2
If
iII') = (at
yo
developed numerical schemes based on an auxiliary set of differential equations, which express the fact that the ME is an isotherm. These auxiliary
i
Xi
M(t)
u
(iII') a dt at + ax 2
dM
T
27
(2.85)
2
forms are compatible with Patel's expressions in enabling the boundary
d fin
movement to be computed in the direction of the coordinate axes, rather than
The position of the moving boundary M(t) is updated at each time step by
along the normal to itself. Standard finite difference approximations to the heat
using, for example, a suitable finite difference form of the boundary condition
-iII' /ax) on
(dM / dt
Westwater [41J to convection problems and by Tien and Churchill [42] to
s
i th
flow equation were used at grid points far enough from the ME. Near the boundary, formulae for unequal intervals were incorporated into the auxiliary equations. To avoid loss of accuracy associated with singularities, which can
=
X
= M(t).
The method was extended by Heitz and
cylindrical coordinates. Crank and Gupta [43J avoided the complications due to the unequal grid size
profiles were used. The mathematical manipulations are very lengthy and
near the MB by moving the whole uniform grid system with the velocity of the
complex indeed.
ME. They described two methods of obtaining the interpolated values of
ur
so
re
arise when the MB is too near a grid point, localised quadratic temperature
temperature at the new grid points, to be used for the next time step, based on
ce
2.5. Variable space grid finite difference methods
cubic splines or polynomials. Later Gupta [44,45] avoided the interpolations by
e us
using a Taylor expansion of space and time variables and derived an equation
Several techniques of modifying the grid have been proposed, all with the
which can be seen as a particular case of (2.85).
aim of avoiding the increased complication and the loss of accuracy associated
interval, L1x = M(t) / q, is different for each time step. The ME is always on the Xi
grid line. They differentiated partially with respect to x. Thus, for the line = iL1x the heat equation becomes
decided to determine,
as part of the
solution, a variable time step such that the MB always coincides with a grid
line in space. They used the fully implicit finite difference equation; this leads
(2.83)
fe re
to a system oflinear equations to be solved at each time step.
(~} =(~)J:}+(~)x
Gupta and Kumar [46] formulated the same set of finite difference equations as Douglas and GaBie but they used the interface condition to update the time
Xi
moved according to
the expression
nc re
Murray and Landis assumed that a general grid point at
Douglas and GaBie [37] instead of using a fixed time step and searching for
the position of the moving boundary,
e as
qth
2.6. Variable time step grid finite difference methods
e pl
Landis [28] kept the number of space intervals between a fixed and moving boundary constant, for example equal to q, for all time. Consequently the space
l, fu
with unequal intervals near the MB in the fixed grid network. Murray and
step .1t. They thus avoided the instability which develops as the depth of the ME increases; the instability is generated as the time step enters an infinite loop
e
because it becomes very sensitive to rounding errors. Goodling and Khader [47J
dt
M(t) dt
(2.84)
e th
dx i =~dM
incorporated the finite difference form of the MB condition into the system of equations to be solved. The system is solved for an arbitrary value of the
The one-dimensional heat equation becomes
bo
temperature of the node neighbouring the MB, which is then updated from the boundary condition. Gupta and Kumar [48] found this latter method fails to
ok
converge as the computation progresses in time, because it is too sensitive to a small change in temperature. Gupta and Kumar's results are in good
28
Chapter 2 Numerical methods for MBPs
29
Chapter 2 Numerical methods for MBPs
If yo
agreement with those obtained by using the the integral method of Goodman
Suppose the moving boundary is moving from the nodal point (ib,j) to
and other variable time step methods. Gupta and Kumar [49] also adapted the
u
(ib + 1,j + 1), then the problem is to compute: the time step L1t j necessary for the
method of Douglas and Gallie to solve the oxygen diffusion problem (implicit
d fin
ME to move a single spatial step, and the temperature distribution at the time
moving boundary problems). Their results compare favourably with those
tj+l
obtained by Hansen and Hougaard [50] and by Dahmardah and Mayers [51].
(i.e. Ti,j+l' i
= O,N) from
(see Figure 2.7).
i th
To illustrate the variable time-step methods, consider the following problem,
= a aT
,
iJx2
0:5 x :5 M(t)
ax
t~O
M(t):5x:51,
,
x
= M(t)
t>
(2.89)
a
=a
(2.90)
ib -1
central finite difference, gives
(2.92)
Rearrangement of (2.92) gives
= Ti,j
,
i=1,i b
(2.93)
e
i=1,2,
N
(2.91)
where rJ = a L1t j / Llx 2 is the Fourier number.
e th
,
m=O
In order to determine L1tj , both sides of(2.86) are integrated with respect to x, from x = a to x = M(t), making use of(2.87). This gives M(t)
:t
]
ok
to x = (m+ 1)Llx.
bo
where L1tm denotes the time interval in which the boundary moves from
= mLlx
2Ti,j+l + Ti+l,j+l] Llx 2
nc re
j-l
x
= a(Ti-1,j+l -
fe re
Tj,j+l - Ti,j L1t.)
(1+2rj)Ti ,j+l-rj(Ti _l,j+l+Ti+l,j+l)
t j = LL1t m
ib +2
ib + 1
ib
2.6.1. The Extended Douglas and Gallie's (EDG) method
Llx during the time interval. Any point (Xi,t j ) in the space-time domain is
given by the coordinates
,
I
~-----
_________ I
Figure 2.7: Movement of the MB x = M(t) in a variable lime step grid network
e as
at each time stepj, is chosen such that the boundary moves a distance
xi=iLlx
v
,
Let Ti,j be the temperature at the coordinates xi,t j . Replacing the left side of (2.86) by a backward finite difference at the point (Xi,t j ) and the right side by a
The region 0:5 x:5 1 is divided into N sections of length Llx = 1/ N. The time .t1t j'
I
)
(2.88)
where h is a convective heat transfer coefficient and IJI is a real constant.
step
I
e pl
M(O)
,
I I I
Phase 2
.t1x
(2.87)
a
~---------r-----
l, fu
aT
dM dt
,
t >
'
.t1t j
(2.86)
e us
=1
=a ,
a
ce
T(x,t)
, x
t>
ur
aT ;;; = hT + IJI
,
I'
Phase I
so
at
1
)+1
I
re
2
aT
x=M(t)
s
in dimensionless variables,
a known temperature distribution at the time tj
IT(X,t)dx-(a+1)M(t) =-a(hT(O,t)-IJI) [
(2.94)
30
Chapter 2 Numerical methods for MBPs
31
Chapter 2 Numerical methods for MBPs
If yo
Further integration of (2.94) with respect to t , and making use of (2.90), gives
u
Similarly, the (k+1)th iteration for calculating ,1tj can be written from (2.97)
a'lfl + ah T(O,t)dt
= (a + l)M -
o
f
d fin
f
as
M
t
T(x,t)dx
(2.95)
k 1
= t j + 1 , gives
re b
+1) T i +l,j+1
a( h + vfl'O,j+1)
e as
matches with the one obtained by satisfying the boundary conditions (2.87) and (2.89). The special case of ib = 1 is discussed in section 2.6.3.
Goodling and Khader [47] used the same set of simultaneous equations, (2.98) and (2.99), to determine the temperatures at time t j + 1 at the k th iteration. The boundary condition (2.89) is replaced by the following finite difference equation
e pl
It should be noted that in (2.96), the finite difference replacement of the integrals have been made such that the value of ,1tj , obtained from (2.97),
2.6.2. Goodling and Khader's (GK) method
l, fu
(2.97)
k k _&1 (T 'b+l,j+1 _T'b,J+l . ) = &-
,1t~
accuracy in ,1tj is achieved. The k th iteration for solving the system of (ib + 1)
,1t k J
linear equations is
(2.101)
J
nc re
giving
fe re
By choosing a suitable estimate of ,1tj , the temperature at any grid point is computed from (2.93) and the estimated value of ,1tj is subsequently reevaluated from (2.97). This iterative process is repeated until the desired
(2.100)
a new temperature distribution T i: j + 1 , i = 0, ib . This process is repeated until the desired accuracy in ,1tj is achieved.
e us
,1t j
i
.=1
k)
the new estimate of ,1tJ from (2.100). ,1tJ is substituted in (2.98) and (2.99) to give
ce
Mter manipulation, (2.96) becomes
t;
(
+1)
Choosing ,1tJ equal to ,1tj _1 which is already known, T~j+1' i = O,ib are determined from (2.98) and (2.99). These values are in turn used to determine
(2.96)
ur
so
a'lflj+1 + ah L(To,p ,1tp_1) = (a + 1)(j + l),1x - & LTi+l,j+1 p~l i=l
j
i
~ Ti.1,j+1
a h + vfl'O,j+1
J
ib+1
(a + 1)(j + 1)& - alpt j + ah ~ (To,p ,1tp_1) - &
p=l
,1t· + =
s
j+1
b
j
(a+ l)(j + 1)& - aljltj + ahL(To,p ,1tp_1) - &
i th
The finite difference replacement of (2.95), when t
f
j
0
&2 (2.102)
k
1-Tib ,j+1
e
rj
(Ti~l,j+1 + Tt'tl,j+1) = Ti,j
= ljI&
= a ,1tj / &2 and T i:+l,j+1 = 1 from (2.88) for all k.
(2.98)
(2.99)
The method suggests that after choosing a value of Ttj+1' ,1tJ is recalculated from (2.102). The value of ,1tJ is used to calculate T~j+1' i = O,ib Using (2.98). Equation (2.99) is then used to test the accuracy of the selected value of Ttj+1' If (2.99) is not satisfied, T ib ,j+1 is estimated again and a new ,1tj is obtained from (2.102), which in turn is used to calculate new values of Ti.J+l , i
= O,ib .
ok
where rj
= 1,ib
bo
Ttj+1 - (1 + h&)Ti,j+1
i
e th
(1 + 2rj)T!~j+1 -
This process is repeated until (2.99) is satisfied within a
33
Chapter 2 Numerical methods for MBPs
Chapter 2 Numerical methods for MBPs
32
If yo
T ib ,j+1
The first time step
was selected is not explained
is calculated by combining both the boundary
11to
condition (2.87) and the interface condition (2.89). The finite difference of (2.87)
u
specified error. However, the way in which in detail by the authors.
Gupta and Kumar [48], in a comparative study of numerical methods for
d fin
gives
moving boundary problems, tackle this point. They tried taking T?b. )'+1 equal to Tib-l,j as a suitable estimate; then kept increasing it by small increments until
1 L1x'
- ( T11 -To
s
i th
a change of sign in the error in (2.99) was detected; then T ib ,j+1 was interpolated either by the method of chords or the bisection method. They found
(2.105)
•1) = hTo•1+ lfI
and the finite difference of (2.89) gives
re
that this procedure does not always work. In the problem they solved [48J the procedure fails after a number of time-steps; the method of solving (2.98)
so
L1x
11tJ
from the following
+ (11tj-1 -
11tj-2) + e
process is repeated until a change of sign in the error occurs; then the value of
L1x( 1 + hL1x )
(2.107)
(h + lfI)
In sections 2.6.1, 2.6.2 and 2.6.3 the computational procedure, of passing from one time step to another, for each method is explained. At the beginning of computation where the moving boundary is moving from x
is interpolated by the method of bisection.
e as
11tj
= 1, the following is obtained
e pl
is incremented by approximately 0.2% and the
11tj
Tl,l
11to =
l, fu
where eis small. Ttj+1 is calculated from (2.100); values of T~j+1' i = O,ib are then determined from (2.98); (2.99) is checked to ensure that it is within the specified error; if not,
node (1,1) and therefore
(2.103)
e us
11tJ = 11tj-1
Combining (2.105), (2.106) and remembering that the moving boundary is at the
ce
they estimated
T ib ,j+1'
ur
Therefore, they adopted the following approach: instead of estimating
(2.106)
--.!..-(TO,l -Tl,l) = 11t L1x o
becomes unstable and the error in (2.99) behaves in an irregular manner.
any of the EDG, GK and MVTS methods, the time step
=0 to x = L1x,
11to
using
is calculated from
(2.107) without iteration and the temperature at x = 0 is calculated using
2.6.3. The Modified. Variable Time Step (MVTS) method
fe re
either (2.105) or (2.106). In this method [48) the same set of simultaneous equations, (2.98) and (2.99), written as
2.7. The enthalpy fonnulation teclmiques
nc re
are to be solved as in the EDG and GK methods. The interface condition (2.89) is
These techniques are based on the relationships which exist between the (2.104)
e
different thermodynamic properties of the material such as enthalpy, heat capacity and latent heat of fusion. These methods are strictly limited to melting
e th
11tY1=~ I-T~ .
'b.)+l
(freezing) problems, where the enthalpy formulation automatically satisfies the
= O,ib
equal to
11tj _1
initially, the temperature distribution
is obtained using (2.98) and (2.99). Using the value of
T?b· )'+1
in
(2.104) the new estimate 11t} is obtained. The process is repeated until the specified error in 11tj is reached.
jump in heat flux across an isothermal moving boundary. For instance, these methods cannot be used for implicit moving boundary problems, where the moving boundary is present due to physical phenomena other than a phase change [52J.
ok
T,O)'+l ,i •
11tJ
bo
Choosing
34
Chapter 2
35
Chapter 2 Numerical methods for MBPs
Numerical methods for MBPs
If yo
,~ ~{:."
u
2.7.1. Post-iterative (Isothermal)
Solid region Mushy region Liquid region
TT 2
(2.109)
d fin
This is probably the simplest method r53,54]. For each node the usual finite where
C + (L/LlT);
change is occurring the temperature is set back to the temperature of transition
T 1 and T 2 are the temperatures at the lower and the higher ends of the mushy region, respectively. The apparent capacities are
and the equivalent energy is added to an enthalpy budget for that node. Once the
calculated using the temperatures at the nodes, and the resulting finite
difference or finite element equations are used. For the nodes where the phase
s
i th
budget is equal to the latent heat for the volume associated with that node, the
Cmu
=
difference or finite element equations are based on these apparent capacities.
re
temperature is allowed to vary according to the Fourier conduction equation.
so
2.7.4. Enthalpy method
ur
2.7.2. Post-iterative (Mushy)
In the enthalpy method [60 - 64], the heat conduction equation is written as
ce
This is similar to the isothermal case except that a mushy range (see appendix A) is also taken into consideration [55,56]. The latent heat is released
e us
at
over a temperature range, and is assumed to be a function of temperature
(generally linear) in that range. The procedure is as in 2.7.1 except that the
l, fu
temperature will be set to
= T 2 -( ~)LlT
(2.108)
are: heat in enthalpy budget, latent heat of fusion and difference of temperature as finite element formulations,can be used for the partial differential equations.
H(T)= cT + L(T-T 1 }
JcT
T 2 -T 1
Tl~T~T2
T > T2
Solid region Mushy region
(2.111)
Liquid region
fe re
of the mushy range. Again both implicit and explicit finite difference, as well
T0
(3.82)
Assuming the following thermal dependencies
without loss of accuracy or stability. It consists of automatic generation of Virtual Sub-time-Steps (VSSs), which are progressively eliminated as the
O:S x :S 1 ,
,
e th
The VSIET can also be used for non-linear problems [26]; it permits the use of large time steps for any of the explicit equations (equation (3.68) or (3.74»
temperature distribution Ti,j' i
ax
nc re
2Ki _l,jKi ,j K i- 1J2 ,j = K i- 1,j + Ki,j
()T =-a(K(T)()T)
C(T)-
fe re
separating the grid points. K i - 1J2 ,j and K i +1J2 ,j can also be evaluated using the
e as
In equation (3.75), a linear interpolation is used for the thermal conductivity
(3.83)
70
Chapter 3
Computational performance of FDMs
Chapter 3
71
Computational performance of FDMs
If
and choosing the following boundary conditions
yo
similar accuracy; however, (3.69) requires fewer operations.
u
ry(x) ::: aexp( -c x)
times more accurate than the classical ones. Equations (3.69) and (3.71) achieve
!P(t) ::: aexp(bt)
,
rp(t) ::: a exp(bt - c)
(3.84)
d fin
t he following is an exact solution for equations (3.79) to (3.82)
Table 3.7: Comparison of percentage deviations (X10 2) at the centre of the slab
i th
T(x,t) ::: aexp(bt - c x)
time
(3.85)
s so
re
where a, b, c and d are real constants.
The problem defined by (3.79) to (3.82) is solved using the classical equations
ur
(i.e. (3.74)&(3.75)), the equation of Patankar (i.e. (3.74)&(3.76)) and the finite
ce
difference equations due to Zerroukat and Chatwin (i.e. (3.68)&(3.69) and (3.68)&(3.71)). The VSIET is incorporated with all the equations. For accuracy
e us
assessment, the numerical results are compared with the analytical solution and expressed in terms of percentage deviations defined as PD. . ::: 100 x (Ti •j ) Nu -
".J
(Ti •j ) An () T· . '.J An
(3.86)
that both the classical finite difference equations under-estimate the solution; whereas, (3.68)&(3.71) give an over-estimated solution; however, the numerical
-D.l788 -D.2261 -D.2520 -D.2588 -D.2647 -D.2697 -D.25 17 -D.24 19 -D.2255 -D.2101
Analytic
(3.68)&(3.69) PDN /2,j
T(~,tJ)
+0.03765 +0.03617 +0.01738 -D.00835 -D.00802 -D.03082 +0.00000 +0.00711 +0.02050 +0.02627
16.2117 16.8733 17.5619 18.2786 19.0246 19.8010 20.6091 21.4501 22.3255 23.2367
0.05647 0.05426 0.01738 0.00000 0.00802 0.00000 0.00740 0.00711 0.02050 0.02627
Table 3.8: Comparison of maximum percentage deviation (XT02) time
Classical
Patankar
SLCp
(3.74)&(3.75) MPD j 0.1694 0.2080 0.2346 0.2457 0.2614 0.2620 0.2369 0.2276 0.2227 0.2071
(3.74)&(3.76) MPD j 0.1788 0.2261 0.2520 0.2633 0.2698 0.2697 0.2517 0.2468 0.2299 0.2140
j
1 2 3 4 5 6 7 8 9 10
ZerroukaL and ChaLwin
(3.68)&(3.71) MPD j 0.07297 0.07683 0.04801 0.02667 0.01864 0.00852 0.01557 0.00786 0.02601 0.03123
(3.68)&(3.69) MPD j 0.07021 0.05844 0.05581 0.02728 0.02071 0.03240 0.00818 0.00869 0.02050 0.02761
e th
greater than those of the equations of Zerroukat and Chatwin. It also shows
-D. 1694 -D.2080 -D.2346 -D.242 I -D.2567 -D.2620 -D.2369 -D.2276 -D.2119 -D.2036
ZerroukaL and ChaLwin
(3.68)&{3.71) PDN /2.j
e
Table 3.7 shows that the relative errors using the classical equations are
0.01 0.02 0.D3 0.04 0.05 0.06 0.07 0.08 0.09 0.10
nc re
units of variables, when a:::20, b=4, c=1I2, d=2.0, &=0.1, L\t=O.Ol and rf ::: 0.35, are shown in Tables 3.7 and 3.8.
Patankar
fe re
Numerical results of the problem defined by (3.79) to (3.82), omitting the
1 2 3 4 5 6 7 8 9 10
e as
where "An and Nu" refer to temperature calculated Analytically and Numerically respectively.
tj
e pl
J'li : : 0, N) •
j
l, fu
MPDJ. ::: max(IPDi
Classical
Q.741&{3.75) (3.74)&{3.76) PDN /2.j PDN /2,j
SLep
3.5.2. Three-dimensional problems
bo
In this section, a new finite difference equation for non-linear three-
shows the maximum percentage deviation at each time step. Overall the new
dimensional heat transfer problems, which is a generalisation of the ID case,
finite difference equations, due to Zerroukat and Chatwin, are approximately 10
is presented and tested. To avoid the stability constraints of using small time
ok
results due to (3.68)&(3.69) oscillate around the analytical solution. Table 3.8
steps and therefore increasir..g the memory size requirements, the virtual time
Chapter 3
72
Chapter 3
Computational performance of FDMs
73
Computational performance of FDMs
If As
yo
step elimination technique is again incorporated in the overall scheme.
Using a 3-point Lagrange formula gives
u
seen for 1D problems, the VSIET uses two vectors, let's say A and B, and transposes them as the computation moves one virtual step. For higher
d fin
2 uJ +b (au) a -a 2 ( ax 1=0 1=0
dimensions, the same procedure is used; the only difference is that A and B are
ax
2D or 3D arrays for 2D and 3D problems, respectively. The numerical scheme
a {u(x-L1x,y,z,0)-2u(x,y,z,0)+u(x+L1x,y,z,O)} ==-2 L1x
(3.92)
b + -{u(x + L1x,y,z,O)- u(x - L1x,y,z,O)} 2L1x
i th
is used to solve a normalised problem which has an analytical solution and results are compared to both the analytical solution and that of other classical
s
finite difference equations.
Consider the following partial differential equation
so
re
2 c -a 2 uJ + d (au) ( iJy 1=0 iJy 1=0
(3.87)
e us
2 e -a 2 uJ +((au) ( az 1=0 az 1=0
where u=u(x,Y,z,t) and a, b, c, d, e and (are real constants. Let ¢(x,Y,z) and e(t) be functions so that the solution of(3.87) can be written as
= t/J(x,Y,z)e(t)
(3.88)
Using the notation UiJ.k to denote the variable u at the grid point (Xi,yj,Zk,t m )
e pl
= iL1x, Yj = j.1y,
= k&,
Zk
tm
= m.1t.
The finite difference form of
equation (3.87) is obtained by rewriting (3.91) for the time t m + 1 =tm +.1t,
= ¢(x,Y,z)
;
i.e.
e(o)
=1
(3.89)
e as
U(X,Y,z,O)
assuming that to = t m and making use of (3.92)-(3.94). This gives
fe re
Substituting (3.88) into (3.87) and dividing by ¢e we obtain
m+l u"",l, "k
(3.90)
where
m "kexP = u"t,J,
e
',J.
1
m
'.J,
m
',J-
(r
z
(&)
(3.91)
and
J
(3.95)
m '+ l"k ,J, m
'.J+ •
m k = - 2 u·m . k + ( 1- - - u"m . k 1 + ( { .1z) m. k 1 Z '.J. .. 1+ -2e - U"'.J. ',J, 2e ',J. +
ok
1
J
bo
= u(X,y,z,O)exp
ax
y
m u'_l .J,
e th
m
ax
J
2 (1 -bL1x) (bL1x) U" 2-a ' "k+ 1+-2a Y )U 2 (1 ---U"l,k+ d.1Y ) (d.1 1 +-Y ""k=-U""k+ '"lk 2c 2c
integrate (3.90) with respect of t from t=O to t, giving after using (3.89) 2 2 au au a2u2 b -+c-+ a u d -+e-+ a u2 (auJ a-+ ( 2 iJy az az iJy 1-0 u(x,Y,z,O)
m
x -m -I X exp [r -m- ym ""k exp -m- Zm ""k . ,""k ), !,j, ",), Ui,j,k Ui,j.k Ui,j.k
m m 'k+ X ',J. ""k =- u"'.J,
whence, as the right hand side of (3.90) is independent of time, we may
t
(r
nc re
a2 a2 _I_de =!.(a;P¢ +bat/J +c ¢ +d a¢ +e ¢ +(at/JJ 2 e(t) dt t/J dx dx di dy dz 2 dz
U(X,Y,z,t)
(3.94)
&
+-{u(x,y,z + .1z,O) - u(x,y,z - .1z,O)} 2&
where xi
with the initial condition
==-2 e {u(x,y,z-&,0)-2u(x,y,z,0)+u(x,y,z+.1z,0)}
{
l, fu
U(X,Y,z,t)
(3.93) + -{u(x,y + .1y,z,O) - u(x,y - .1y,z,O)} 2.1y
ce
ax
- .1y,z,O) - 2u(x,y,z,0) + u(x,y + .1y,z,O)}
.1y d
ur
au a 2u au a 2u au a 2u au -=a-+b-+c-+d-+e-+{at ax 2 iJy2 iJy az 2 az
== - c 2 {u(X,y
(3.96)
74
Chapter 3 Compwational performance of FDMs
Chapter 3
Computational performance of FDMs
75
If -
.1x 2
ry
c,1t = ,1y2
d fin
:
0'
: ti":-.::::: _._ ~ij,k+l:
__ ... I _
1
+ r x xm i,i,k + ry ym i,i,k + rz zm) i,i,k
(lJ-l,k)
,
1
(3.98)
- - -
I
tiy z
1
1
so
I.
~. 1
A_
UA
Me
~'1
ur
Figure 3.7: Spatial position of grid points with cqual intervals in each dircction
By replacing Utj,k = 0 in (3.98), the finite difference equation, for the case of
ce
uri,k
'
1
liz
re
u"':J'~""'O (m ~~-~) ui,i,k
(i+l
1
s
i,i,k
- -.
-lj,k)
i th
u m+1
(ij+ l,k)
(3.97)
When utj,k ~ 0, the exponential in (3.91) can be approximated by the first two terms of the Taylor series giving
(ij,k-l)
__ _ _ _ _ _ _
e,1t rz = ,1z2
u
x -
yo
r _ a,1t
= 0, is obtained
',J"
1+-2e
(3.99)
1
For the pointP(xi'Yi,Zk), as shown in Figure 3.7, using the notation
l, fu
dt"y) u·m. 'k } +r {( 1 - - ( 1+-2c ',J+" z 2e
2c
e us
{(1 -d,1y) - - u·m'-'k+ f&)u·',J,m'k-l+ (f &)U"k m } ',J, +
m+l =r {( 1 -b.1x) m (b.1x) m, 'k } +r u"k - -au·_, ',J, x 2 ' "J,'k+ 1 + 2a- u·I+"J, y
( ~aK]
1 aK) _ Ki.:.l,i,k -Ktr:'1,i,k _ - Yx (-- C ax p 2.1xCtj,k m K i,i,k+l -K,mJ' , ,k-l = yz J:. aK) ( C ()z p 2.1xCi~i,k
e pl
C ()y
the non-linear heat conduction equation given by
= Yy
2,1yC[J,k
p
(3.102)
K) =K0,k=a ( C p Ci,i,k
e as
The heat flow in materials with variable thermal properties is governed by
m -Krn., k K i,i+l,k ',J-"
ar)
ar]
ar)
(3.100)
(-ar) at
2
a T =(a-+y 2
ax
arax
ar
p
ar)
2 2 T r -+a aT+r- + aa- + • ()z x ()y2 y ()y az 2
nc re
ar
a ( K(T)- +a ( K(T)- +a ( K(T)C(T)-=at ax ax ()y ay az az
fe re
equation (3.101) becomes
(3.103) p
e
It can be seen that (3.103) has the form of equation (3.87); therefore, in a similar
Equation (3.100) can also be written as
J
J
y m ex (-r z x "m+1 = Tm .. ex ( .. -.. exp(-r .. T ',J,k -rX ',J,k ',J,k P Trn. p Trn. ym ',J,k Trn.- Zm ',J,k
(3.101)
l.,j,k
L,j,k
ok
where
bo
carat = aKax arax +Ka2T + aK ar +Ka2T + aK ar +Ka2T ax 2 ()y ay ()y2 az az az 2
e th
manner to (3.87), (3.103) has the following finite difference solution
",j,k
J
(3.104)
76
Chapter 3
Chapter 3
Computational performance of FDMs
77
Computational performance of FDMs
If yo
(1 +fJ!"· t,l, k)Trr:.l, Jt. k - (1- fJ!"· ",), k)T!"l, . k y!". 1k ",l, k = 2T!"· L,l, k - (1 + T/!'" k)T.m·_l,k - (1- T/!'" k )T!". I.,J+, t
l,J
(3.105)
l,),
d fin
t,j,
Unequal spacings
I+},
u
X!"· t,J, k = 2T!"· I,), k -
_________
r !"'k)T!"'k l-(l- r !"'k)T m' k 1 Z~n'k t,l, =2T!"'k-(1+ t,j, "",), ',l, "",j, ",j, +
-
t>z
i th
where
s re
m fJ £,j, !"· k = (K!"l, . k - K!"l, 1.- ), 1+ J,. k) / 4KI,},.k m 4K·L,l, k
t,j, -
I.,),
(3.106)
= a',J,k L1x 2
r y =r x
(:r
rz
= rx
(:r
Tt'
~x
Y.m. = [2I,J,k -c.m. I,J,k
m m . . k)]T m . _ [2(1+ fJ-.I,J,. k)]T!". _ [2(1- 1JI,J, I,J,k 1+ ~x l-l,J,k ~x(l+ ~x) I+l,J,k
T/i~J,k(l-~Y)]T!" _[2(1+ T/[J'k)]T!". _[2(1- T/[J'k)]T!". ~y I,J,k 1+ ~y I,J-l,k ~y(l+ ~y) I,J+l,k
= [2- (iJ'k(l-~z)]T!". _[2(1+ ~J'k)]T!'" ~z
I,J,k
ok
•
m
= [2- fJ-.',J,. k(1-,I:':ox )] T!".
bo
•
(3.110)
(3.111)
e th
While for the 1-D problem, the same equation as that reported in [22] is obtained, namely,
_ X!". ',J,k
J
e
(3.109)
J (
nc re
where
ttj
J
with respect to the grid point (ij,k), as shown in Figure 3.8, then the
fe re
eliminating the terms relating to the z direction. Then
T!,,+l = T m exp(-r x X!"
~X t>x
m+1 = Tm m ex -rx X-.. -ry y-m) -r z Z-m .. .. exP( .. ex P( .. T ',J,k ',J,k T!".- ',J,k P T!".- ',J,k Tm.- ',J,k ',J,k I,J,k I,J,k
The finite difference form for 2-D problems is obtained by omitting and
J
'
If for instance the six neighbouring grid points (i-1j,k), (i+1j,k), (ij-1,k),
e as
(1- TI'ij,k)TtJ+l,k} + rz{( 1+ (['J,k)Tti,k-1 + (1- (iJ,k)T[J,k+1}
t,J
..:
temperature at the point (ij,k) at the time tm + 1 is calculated using
(3.108)
(-r y
'
I
(ij+1,k), (ij,k·1), (ij,k+1) are situated respectively at L1x, ~xL1x, Lly, ~yLly, LIz,
e pl
T'ij:l = rx{( 1+ f3tJ,k )T[~l,J,k + (1- f3tJ,k )Tt:'l,J,k} + ry{( 1+ TI'ij,k)T['J-1,k +
J
... -
-
Figure 3.8: Schematic representation of grid points with unequal spacing
~zLlz
(-r
.....
Y
(3.107)
As with (3.98), when Tt'J,k = 0, instead of(3.104) the following equation is used
m m+1 = Tm .. exp -T!".x X',J .. exp .. T .. ym ',J ',J T!". ',J
-
z
t>
/ ' '
I
14-t>X
l, fu
rx
LIt_
- - - - - - -:- - -
I
e us
m
..' j(l,j,k+1)' I I
: Ci,j-1,k)
y
,7
I
I I
fY,
---z
ur
t,j,
-
• (i-1,j,k)
so
= (K!". ",J- 1,kl- ,K!"· j + l,k) / m. k 1) / 4K!"· k (!". k = (K!". k 1- K L,l, + !". k T/ I.,),
-
l+~z
I,J,k-1
_[2(1-
~J'k)]T!".
~z(l+~.)
I,J,k+1
(3.112)
Chapter 3
78
Chapter 3 Computational performance of FDMs
COmputational performance of FDMs
79
If yo
Tm . )(1+1) ( -I,),k
u
and
m .k} / 2K!"· k {J-.m. k = {(.!..=..k)K!'" k + k - ( ~x(l + ~x) )K'+ 1,). ',). ~x ',), 1 + ~x )K!"l,' ,-). '.).
l~y~Y ]KtJ'k +( l+l~y ]KtJ-l,k -( ~y(l~~) ]KtJ+l,k}/ 2Kt'j.k (3.113)
(_1
l+~z JK!". ',), k-
(
1-(~z(l+~z) 1 JK',). k I} /2K!"· k
[(
m
Ll,j,k
)(1)]
(3.117)
m
+ Kl,j,k
T-l,j,k
)(1=0)
=
m dx { } TO,j,k + 2K m F x(tm+l) + Fx(tm) O,j,k
(3.118)
rn
re
= {(~JK!". ~z ',), k +
m
s
r.m. k 0,,'.).
= {(
K
where (l) is an iteration index, and the initial value can be taken as
i th
Tit:}.k
1,),k
1
d fin
(_1
= T m. + 2dx{Fx(tm+l) + Fx(tm)}
+
'.),
so
Validation
ur
Equations(3.112) and (3.113) are obtained in a similar manner to the equally In order to validate the new finite difference equation, consider the following
the grid points are unequally spaced. When ~x
ce
spaced case; the only difference is that in the three-point Lagrange polynomial,
problem
= ~y = ~z = 1, i.e. equal nodal
e us
spacing in each direction, equations (3.112) and (3.113) are identical to (3.105) and (3.106) respectively.
CdT
l, fu
at
Boundary conditions
, =~(KdT)+~(KdT)+~(KdT) ax ax if)' ay az az
O$;x$;l O$;y$;l
, t> 0
(3.119)
O$;z$;l
ax
= Fx(t) x=O
(3.114)
T(O,y,z,t)
e as
K(aT)
e pl
For convenience, the following boundary conditions are imposed
Suppose that the boundary x=O is subjected to a time-dependent heat flux
The finite difference form of(3.114) can be written as
(3.115)
,
T(x,O,z,t)
= aexp(bt -
ex - ez)
= aexp(bt-e-dy-ez)
,
T(x,l,z,t)
= aexp(bt-ex-d-ez)
T(x,y,l,t) = aexp(bt-ex-dy-e)
nc re
1
K1j.k + KI:j,k T1j.k - TI:),k = ..!{F (tm+l) + Fx(tm)} ( 2 2dx 2 x
ez)
T(x,y,O,t) = aexp(bt - ex - dy)
fe re
T(l,y,z,t)
= aexp(bt -dy -
(3.120)
(3.121)
and the initial condition
e
T(x,y,z,O) = aexp(-ex - dy- ez)
(3.122)
e th
where
If C and K vary with T following the functions
2dx{Fx (tm+l) + Fx(tm)}
K(T) = {loge(T)
2 d2 2 C(T)={e + , +e {l+log c (T)}
ok
Equation (3.116) can be solved iteratively as
(3.116)
bo
m
T~L.k = Tl,j.k + K(T~L.k)+KI:j,k
(3.123)
80
Chapter 3
Chapter 3
CompLUational performance of FDMs
Computational performance of FDMs
81
If where a, b, c, d, e and fare constants. Hence, the problem governed by equations (3.119) to (3.122) has the exact solution
yo
2Ki~l,j,kK;''j,k
u
2K m
d fin
Ki':.1I2,j,k
= Kf", k + Kt:'l,j,k ',J,
K i,j+1I2,k
= Ki'j,k + K i,j+l,k m
m K i J' k+1I2
2K J,kKtj,k+l = Kf". m k + K J',k+l
Km'k
i,j-l,k ',J, m K ,j-1I2,k = Kt'j-l,k + K i,j,k
(3.124)
T(x,y,z,t) = aexp(bt - cx - dy - ez)
2Kf"· t,), kKt':.l,j,k
Ki~1I2,j,k = Ki~l,j,k + K;''j,k im
m
m
(3.128)
m
i th
2Ki,j,k-lKf"'k ',J, Km m i,j,k-1I2 = Kf", k-l + K i,j,k ',J,
'
I
I.,
I
",l"
,
i
s
This normalised problem is solved here by both the new finite difference solution and the classical one, which is given by
2Ki' j,k K t'j+l,k
re
Table 3.9: Comparison of different finite difference solutions for a 3D problem a=20, b=10, c=0.6, d=1.5, e=l,.f=4.5, &=1/11, ,1y=l/15, ,1z=l/13 and ,1r-O.OOI
so
T m .+ 1 _ Tf";l'l epin epout epin _ epout epin _ epout ef", ',J,k ',J,k = r - r + Y Y + Z z ',J,k ,1t L1x ,1y L1z
(3.125)
ur
time step
ce
where
ep;n
= Kt'j-1/2,k(Tt'j-l,k - Tt'j,k) / ,1y , ep~ut = K;''j+1/2,k(Ttj,k - Tt'j+l,k) / ,1y (3.126)
ep~n = KimJ, k-l/2 (TimJ,k-1 - TtJ,k) / ,1z , ep~ut = KtJ, k+ 1/2 (TtJ,k - TtJ,k+ 1) / L1z "
I
I
I
I
,
I
,
,
,
l, fu
e us
ep~n = Kt':.1/2,j,k(Tt\j,k - Tt'j,k) / L1x , ep~ut = Kt':.1/2,j,k(Ttj,k - Tt':.l,j,k) / L1x
I
e pl
Classic equation
P:llankar
Zerroukat&Chatwin
(3.125)&(3.127)
(3.125)&(3.128)
(3.104)
Analytic
nt
m PD5,7,6
m PD5,7,6
m PD5,7,6
m T5,7,6
I 2 3 4 5 6 7 8 9
-0.00374 -0.00766 -0.01153 -0.01552 -0.01923 -0,02340 -0.02732 -0.03256 -0.03790
-0.00394 -0.00808 -0.01212 -0.01632 -0.02024 -0.02464 -0.02876 -0.03401 -0.03981
0.00118 0.00222 0.00320 0.00416 0.00532 0.00628 0.00689 0.00709 0.00542
4.28746 4.33038 4.37373 4.41752 4.46174 4.50640 4.55151 4.59702 4.64296
Note- PDt- k is the percentage deviation from the analytical solution at the grid point situated at the 'ioordinates (x=iL1x ,y=jL1y , z=k!Jz , l=mL1l) in the space-time domain (x,y,z,t)
The problem governed by equations (3.119) to (3.122) is solved by the classical
fe re
y, or z is calculated as the average of that of the two neighbouring grid points in the same direction
e as
The thermal conductivity at the midpoint of two grid points for any direction x,
equation (3.125) and that of Zerroukat and Chatwin (3.104). Table 3.9 shows that, as with I-D problems, the new finite difference equation (3.104) m = (K-~l ",l,'k + K ",l,,k) / 2
Kt'j,k+ 1/2
(3.127)
=(Kt'j ,k + Kt'j ,k+ 1) / 2
J.M. Bettencourt,
a.c.
Eng. 17,931-938 (1981).
a.c.
Zienkiewicz and C.T. Parekh, Int. J. Numer. Meth. Eng. 2,
66-71 (1970). 3.
ok
2.
Zienkiewicz and G. Cantin, Int. J. Numer. Meth.
bo
the inverse of thermal resistances at the same positions as suggested by Patankar [25J; this gives
1.
e th
The conductivities at the midpoints given in (3.127) can also be determined as
3.6. References
e
Kt'j+1/2,k = (KiJ,k +Ktj+l,k)/2
m K ",l,. k-l/2
= (Kf". ",l, k-l + Kf", t,l, k) / 2
outperforms the classical ones by a factor of order ten.
Kt':-1/2,j,k = (Kt'j,k + K;'':.1,J,k) / 2
Kf"'_1/2 " , l , k = (Kf"'_1 t,J, k + Kf", l,l, k) / 2
nc re
K!".:.1/2 r. ,l,'k
D.W. Nicolson, Acta Mechanica 38, 191-198 (1981).
82
Chapter 3
Chapter 3 Computational performance of FDMs
Computational performance of FDMs
If Problems, Pineridge Press, Swansea (1985). Z. Zlatev and G.P. Thomson, Int. J. Numer. Meth. Eng. 14, 1051-1061 (1979). H.C. Carslaw and J.C. Jaeger (eds.), Conduction of Heat in Solids, Clarendon press, Oxford (1959).
24.
Y. Jaluria and KE. Torrance (eds.), Computational Heat Transfer,
25.
Hemisphere P.C., Washington D. C. (1986). S. V. Patankar (eds.), Numerical Heat Transfer and Fluid Flow,
26.
Hemisphere P.C., Washington D. C. (1980). M. Zerroukat, Numerical Computation of Moving Boundary
i th
7.
M.C. Bhattacharya, Commu. Appl. Numer. Meth. 6, 173-184 (1990).
d fin
6.
23.
u
5.
RW. Lewis and K Morgan (eds.), Numerical Methods in Thermal
yo
4.
so
RM. Furzeland, J.Inst. Maths. Appl. 26, 411-429 (1980).
ur
10.
re
D. Poirier and M. Salcudean, Trans. ASME J. Heat Transfer 110, 562-570 (1988).
9.
A
s
1503-1520 (1992). 8.
Phenomena Ph.D. Thesis, University of Glasgow, (1993).
M. Zerroukat and C.R. Chatwin, Int. J. Numer. Meth. Eng. 35,
C. Haberland and A. Lahrmann, Int. J. Numer. Meth. Eng. 25,
ce
593-609 (1988).
12.
M.C. Bhattacharya, Int. J. Numer. Meth. Eng. 21, 239-265 (1985).
13.
M.C. Bhattacharya, Appl. Math. Modelling 10,68-70 (1986).
1317-1331 (1987).
14.
C. Haberland and A. Lahrmann, in Numerical Methods in Thermal
J. Crank and P. Nicolson, Proc. Camb. Phil. Soci. 43,50-67 (1947).
16.
G.D. Smith (ed.), Numerical Solution of Partial Differential Equations,
17.
M. Zerroukat and C.R Chatwin, Technical Report TRJGLU/MZ920830, Dept. Mech. Eng., University of Glasgow, (1992). O.C. Zienkiewicz (ed.), The Finite Element Method in Engineering
Science (2nd ed.), McGraw-Hill, London (1971). G. Comini, S. Del Guidice, RW. Lewis and O.C. Zienkiewicz,
Int. J. Numer. Meth. Eng. 8,613-624 (1974). 21.
G. Aguirre-Ramirez and J.T. Oden, Int. J. Numer. Meth. Eng. 7,
Problems Vol. VIlI, RW. Lewis (ed.), Pineridge press, Swansea, (1993).
ok
M. Zerroukat and C.R Chatwin, in Numerical Methods in Thermal
bo
345-357 (1973). 22.
e th
20.
E. L. Wilson and R.E. Nickell, Nuclear Eng. Design 4, 276-286 (1966).
e
19.
nc re
18.
fe re
Clarendon press, Oxford (1978).
e as
15.
e pl
Problems Vol. V, RW. Lewis and K Morgan (eds.), Pineridge press, Swansea, (1987).
l, fu
M.C. Bhattacharya and M.G. Davies, Int. J. Numer. Meth. Eng. 24,
e us
11.
83
If u
yo d fin s
i th so
re ur
4
ce
Explicit Computation of Moving Boundary Problems
l, fu
e us Summary
e pl
This chapter presents and validates an improved explicit variable time-step
e as
method, which combines the explicit exponential finite difference equation, a
variable time-step grid network and virtual unequal space-increments around
fe re
the moving boundary. It also outlines some techniques to improve the methods presented in chapter 2, by incorporating the more accurate finite difference equations developed in chapter 3.
e
nc re e th ok
bo
Chapter 4
86
Explicit computation of MBPs
Chapter 4
Explicit computation of MBPs
87
If yo
Nomenclature
F
Heat flux input
h H
Heat transfer coefficient Enthalpy
K
Thermal conductivity
M
Liquid/solid interface position
N
Total number of space
equation is used the mesh size must be carefully chosen to avoid problems with
x
Space variable
convergence and..stability; this method requires considerable computer memory
a
Diffusivity
size if computations are performed for extended periods of time. Although the
d fin Em in
).
implicit finite difference equations have less demanding stability requirements,
iteration
fixed grid methods using either the explicit or the implicit difference equations,
Latent heat of fusion/unit
break down when the moving boundary jumps a distance larger than a single
volume
space increment during a single time step [31; in other words, these methods
Subscripts
are inadequate for heat transfer processes with phase changes where the
s
Time-step error allowed to stop
i th
Slab thickness
u
a
re
elements
i,j
Space/time indices
PDt
Percentage deviation for time
ib
The node index at the moving
PDST
Percentage deviation for Corrected Fourier number in
m
Virtual sub-time step index in
solid regions are at any instant in time divided into a fixed number of equally
the liquid region
spaced intervals. However, the size of the intervals will increase or decrease as
Time variable q
Number of virtual sub-time steps in the solid region Solid Superscript
Iteration index
suggest in their methods that the thickness is to be divided into N equally spaced intervals of length .1x; the time step .1t, which is variable, is the time taken for the moving boundary to move a distance.1x. Due to stability problems
only the implicit difference equations have been used so far with these methods, where at each time step a set of
(N + 1)
simultaneous linear equations must be
solved. When the time step is relatively large, the Euler explicit and the implicit
fe re
k
is very small (for example laser surface melting). Douglas and Gallie [5)
e as
s
Temperature of the fluid in contact with the slab
in commencing the computation, especially when the liquid or solidified depth
Number of virtual sub-time steps in the liquid region
Temperature at the moving boundary
the proportion of solidlliquid alters; this kind of subdivision presents difficulties
the solid region
e pl
Temperature
Virtual sub-time step index in
l, fu
p
e us
n
and Landis method [4] the time increment is kept constant and the liquid and
ce
Fixed Fourier number (::;; 0.5)
To
ur
Liquid
the solid region
T Tb
divided into equal sized increments, the other being variable. In the Murray
l
Corrected Fourier number in
rr
Using variable grid methods only one dimension, either time or space, is
surface temperature the liquid region res
so
ret
boundary
moving boundary may have a high instantaneous velocity during the process.
equations suffer from either significant oscillations or loss of accuracy [6,7); in
relatively slow velocity. It is clear that the choice of the numerical method for a specific problem depends not only on the nature of the heat transfer process itself, where an a
e th
problems (2) may be classified into two categories:
accurate to solve heat transfer problems where the liquid/solid interface has a
e
According to Gupta 11J the numerical methods for solving moving boundary
nc re
other words, the implicit variable time step method may not be sufficiently
4.1. Introduction
fixed grid methods,
priori view is necessary, but also on the available computational platform.
(ii)
variable grid methods.
This chapter presents some numerical methods, where the stability is
bo
(i)
automatically maintained irrespective of the mesh size and the velocity of the
lengths of equal magnitude. For these methods the Euler explicit and implicit
moving boundary. An additional benefit of these methods is that they do not
finite difference equations can be utilised. When the explicit finite difference
require a large amount of computer memory. This is due to the incorporation
ok
In the first category the space and time dimensions are divided into step
88
Chapter 4
89
Explicit computation of MBPs
Chapter 4
Explicit computation of MBPs
If = aLIt / L\x2 is the Fourier number for a given time step Lit and a mesh
where r
facilitate maximal exploitation of vectorisation and parallel computation, due to
element L\x. Equation (4.3) is an efficient approach when used for relatively slow heat transfer processes. However, when the cooling rate is relatively high, which
u
yo
of the virtual sub-interval time step elimination technique; the methods also
d fin
their explicit computation procedures.
results in steep temperature gradients, I(Ti-l,j + T i +1,j )/(2Ti,j)1 may exceed unity. Numerical experiments show that a scheme based on (4.3) has a higher
4.2. The Explicit Variable Time Step (EVI'S) method
i th
probability of breakdown than that based on (4.2), if used for heat transfer
The EVTS method [8] is described here for a general heat transfer problem
s
processes with higher cooling rates such as subsequent cooling after laser
where a solid material melts under the effect of a variable heat flux prescribed
re
surface heating. Therefore, the EVTS method is based on the more stable
at one boundary and loses heat by convection through the other. It is convenient
so
linear exponential equation, given by (4.2).
to test the method by adapting this general case to solve two specific problems
heat through its surface by convection [5,9],
Consider the one-dimensional general melting process where a piece of solid
a solid material at its melting temperature which melts due to a variable heat flux input at one boundary [10,11].
e us
B)
4.2.1. Mathematical formulation of two-phase Stefan problems
a liquid maintained at its fusion temperature which solidifies by losing
ce
A)
ur
which have been dealt with by other authors
One-dimensional heat transfer in a homogeneous material of diffusivity a is
()x2
,
05 x5 a
, 0 5 t 5 -r
(4.1)
thickness of the material, respectively.
: yy)')'}??
'LLLL
1777
''Y'y S~lid ,y
t), the Following the same procedure used in chapter 3, section 3.3, the finite
finite difference solution of (4.61) can be written as
Hi,j+1
= Hi,jexpL Llx~~
(4.59)
~,J
where When Hi,j
= 0, the exponential in (4.59) is approximated by the first two terms
)
(4.62)
I,j
III . . 'Y ',j
= 0, namely,
=2T·',j.-(1+f3.. ',j )T,- 1,.-(1-f3.. j ',j )T.t+ I,j'
ok
and
bo
of the Taylor series expansion; eliminating Hi,j from the resulting equation gives the finite difference equation for the special case of Hi,j
I,j r··
H . "'i,j
e th
1
. . (2Ui,j -Ui-l,j -ui+l,j)}
= Hi,jexp ( -
e
Hi,j+l
nc re
difference equation of (4.54) can be written as
(4.63)
Chapter 4
110
Explicit computation of MBPs
Chapter 4
Explicit computation of MBPs
111
If K i-l,j - K i+l,j A 17..
Lit ri,j = Ki,j L1x 2
Following the same procedure as that of section 3.4, and expressing the
(4.64)
enthalpy Ht'j~l at the grid point (Xi,Yj,Zk,t m+1), in the space-time domain given
u
O
(5.1)
ax
, X = 8(..) ,
u = .B....(X - 80 )2 2D
80
,
0~ X
~ 80
= 8(0) = (2D Uo R
(5.2)
or2:0
or> 0
or=O
r 2
(5.3)
(5.4)
(5.5)
where the concentration of oxygen free to diffuse at depth X is U(X, or), the time
bo
is or and D, R are the diffusion coefficient and the rate of oxygen consumption per unit volume of the medium, respectively. Equations (5.4) and (5.5) are the solution of the steady state rll. where U o is the concentration at which the
ok
without increasing the minimum array size requirement, by using virtual time steps which are progressively eliminated as the computation moves forward.
8( or)
,
e th
value; this avoids the break down that would occur for a fixed grid network
u = -au = 0
e
moving relatively fast and the time step is automatically adjusted to a smaller
~
X=O
ax
nc re
with a relatively slow velocity; this results in a high computational efficiency
, 0~X
au =0
fe re
advantages of the variable time step grid and explicit equations, whilst
ax
with the following boundary conditions
e as
method [11] to solve the oxygen diffusion as an example of implicit moving
aor
e pl
step is a major difficulty, as it is crucial for the convergence and the stability of the scheme.
l, fu
concentration. Furthermore, the first estimate of the time step length at each
118
Chapter 5
Implicit moving boundary problems
Chapter 5
Implicit moving boundary problems
119
If approximate analytical solution. When the velocity of the moving boundary
depth of oxygen penetration. To simplify the problem the variables are made dimensionless as follows
increases, the problem is computed using a numerical scheme. Their solution
u
yo
surface is maintained until a steady state is reached, and So is the maximum
D t=-2't" So
d fin
x
x=So
D
1
M=.§.... So
thus:
(5.6)
i th
u=--2U =--U RSo 2Uo
proceeds as follow~. The solution of (5.7) subjected to the boundary condition (5.8) and the initial condition (5.10) can be found by using Laplace transforms,
u(x,t) = %(1-x)2
s
The system of equations (5.1) to (5.5) is thus reduced to the following nondimensional formulation
, O$x$l , t=0
,0 $ x $ 1 , t
~
0
t=O
The computation of (5.12) shows that the convergence of the infinite series is very rapid; therefore, the terms corresponding to n=O give sufficient accuracy
(5.9)
(5.10) (5.11)
series can be neglected to obtain an accuracy less than 10- 5 . Hence, the concentration u(x,t) for the interval oftime 0$ t $0.02 can be approximated by
U(x,t)
= %(1- x)2 -2~exp[-(2~ r]+x erfc(2~)
(5.13)
0$ x $ 1 , 0 $ t $ 0.02
fe re
The absence of (dM / dt) from (5.9) makes this an implicit moving boundary problem.
over a small interval of time. Furthermore, for 0$ t $0.02, the second and third
e as
=1
, t> 0
,
(5.12)
(5.8)
t~O
e pl
M(t)
x = M(t)
2]) - ~(_l)n ~ {(2n +2-x)erfc ( 2n+2-x 2..Jt )
- (2n + x)erfc ( 2n+x)} 2..Jt
l, fu
U=t(1-X)2
,
,
r]
(5.7)
e us
au u=- =0 ax
, x=O
[
ce
ax
t> 0
2n+x -exp -( 2..Jt)
ur
au = 0
,
so
re
2 au = a u _ 1 , 0 $ x $ M(t) at ax 2
+2~~(_1)n{exp[_(2n;~-x
For times t$ 0.2, the moving boundary is assumed to be stationary at x=l;
nc re
52.Numeri~compu~tion~hem~
once it starts to move Crank and Gupta [1] resort to a numerical scheme. The concentrations at the intermediate grid points, in a fixed grid network, are calculated using a simple explicit finite difference formula. Near the moving
problem defined in section 5.2 are reviewed. compared in the following section
boundary a Lagrange type formula is used.
Their numerical results are
e
In this section, some of the available numerical methods for solving the
e th
Suppose that the concentration distribution at the time t j is already known,
bo
5.3.1. Fixed-grid finite difference methods
and the moving boundary is located between ib and it, -1, as shown in Figure
The problem was first solved by Crank and Gupta [11; at the beginning of the process, when the velocity of moving boundary is negligible, they used an
ok
5.1; then, the problem is to compute the concentration: distribution at the time
t j + 1• from that of t j , and the new position of the moving boundary.
Chapter 5
120
Implicit moving boundary problems
Chapter 5
121
Implicit moving boundary problems
If yo
J+l1
••R:J [ t 1
which satisfies
,
X ~
0 , x'~ 0
, for all t
G(x,x' ,t') =
(5.25)
,
forallt
o
0
(5.31)
a 2G aG a 2u au G(---J-u(-+-J]dxdt 2 ax at ax 2 at
t'M(t)
U(x"t')=~(l-x')2- f
(5.27)
fG(x,x',t'-t)dxdt
o
0
(5.32)
.. 1
+ 2:1
l, fu
f f
~~ (x,x' ,t'-O]dt
properties of the delta function, gives
e us
where 8 is the Dirac delta function. Consider the following integral t'M(t)[
8(x - x')8(t - t')+
(5.26)
ce
x'>O
j[
Substitution of G(x,x',t') from (5.31) into (5.30), and making use of the
ur
,
(5.30)
0
o
so
and G(O,x',t'-O=O
G(X,x',t'-t)dxdt+~f G(x,x',t')(1-x)2dx
0
re
at
1
Combining (5.25) and (5.24), the following can be deduced
s
ac = -8(x _ x')8(t' -t)
o
i th
2 a G+ ax 2
U(X',t')=-f f
(5.24)
,t > t'
d fin
1o
t'M(t)
ff riG ax2 (x,x' ,t' -t)(l- x)2 dxdt o0
e pl
Integrating the second integral of (5.32) by parts twice with respect to x, and
Making use of(5.7) and (5.25), the integral (5.27) is in fact equal to
e as
making use of (5.26) and (5.24), (5.32) becomes
t'M(t) f G(x,x',t'-t)dxdt+u(x',t')
o
0
(5.28)
u(x',t') =
t'
1
o
0
M(t)
f G(x,x',t'-t)dxdt
(5.33)
nc re
Integrating by parts and using (5.24), (5.27) can also be written as
t'
%0-X')2 - f G(O,x',t'-t) dt+ f
fe re
f
After some manipulation, (5.33) can be written in the form t' a
X=M(t)
where t
= M-1(x)
f
I
[Gut: (X)dx
(5.29)
0
is the inverse function of x=M(t).
Making use of the
ok
equal, the following is obtained
where
%0- x')2 - 2~(~) exp( - ~~~ J+ x'erfc(2~ ) + R(x' ,t')
bo
conditions (5.8), (5.9), (5.10), (5.26) and bearing in mind that (5.28) and (5.29) are
u(x' ,t') =
e th
o
dt-
e
aG] f[ G-.!!:..-uax ax .>:=0
1
(5.34)
Chapter 5
124
Implicit moving boundary problems
Chapter 5
Implicit moving boundary problems
125
If 2
yo
:~]-erfc[ 2(t'1--t)X~2]
1 2 -2 [t x ] u(x,t)=2'(I-X) ;exp(X -'4t) +xerfc( 2.,[t
V
u
R(x' ,t') = .!ft' {erfc[ M(t) 2 2(t' -t) o
d fin
(5.35)
+ erfc[ M(t) + x' ] + erfc[_I_+_X-:;-'~]}dt 2(t' _t)1I2 2(t' _t)1I2
+4
i th
s
re
Hansen and Hougaard calculated x=M(t) and u(x,t) directly from (5.38) and
so
(5.39) for small times 0::; t ::;0.02, but for t>0.02 they used a numerical scheme.
ur
- +ex [(X_X')2] -ex [(X+X')2]}d t (t' _t)1I2 P 4(t' -t) P 4(t' -t)
When t
~
0, (5.36) has the following asymptotic
solution
where
(5.37)
~.[
V
V
nc re
e
e th
(5.38)
B(fj,t)
Using (5.38), the asymptotic solution of u(x,t), for small times, can be found to
(5.43)
ok
bo
k) f k+l . - D(f·J' t·J J
be
(5.42)
Equation (5.40) can be solved using the following iterative procedure
2
M(t) =
rf
f') - erfc [2-fj) D(fj,tj ) = 1 + erf [ 2Jt; 2.jt}
or
1-{;r exp(- 41J
rf
fe re
= 1-2erfc(21]
fj f an 1I2{-I+exp[ (f j -fn)2]_exp [_ (2- - n )2]} {iC n=l (t j -tn) 4(t j -tn) 4(tj -tn)
f [ 2]) (5.41) +2erfc I-f,) -2erfc [2-f.) +2 ...:L. l+exp [-(2- J.)2] -exp-!..L [ 2 tj 2 tj 1! 4t j 4t j
and M(t)
t
B(fj,tj)=~
e as
For convenience and simplicity, in (5.36) the variables x' and x denote M(t') and M(t), respectively.
(5.40)
(5.36)
e pl
o
= I-M(jtJt) = B(fj,t j ) D(fj,t j )
l, fu
{iC
fj
e us
- - ft'
+ exp[ _(1;/)2]}
For t>0.02, (5.36) is written in the form
ce
l-exp[
(5.39)
identical to equation (5.13) of Crank and Gupta.
the position of the moving boundary, an integral equation is obtained by differentiating (5.34) with respect to x and using the boundary condition (5.9). This equation, whose detailed derivation can be found in [3], can be written as
_2m{ (1~:,')2] 1 1 {I
1 (
---exp(2-X)2) - - - - +---exp -(2+X)2)} --(2-x)2 4t (2+X)2 4t
Except for the additional term R(x,t) in (5.34), it can be seen that (5.39) is
When the position of the interface x=M(t) is determined, the concentration u(x,t) anywhere in the. space-time can be calculated from (5.34). To compute
I-X ' ] ' )erfc (I+X'] X' ] +(I-x')erf (2-R x'= 1-2erfc( 2-R +(I+x 2-R
f;{I -3 1!
where fJ can be obtained by extrapolation from the three previous values fj-3, f j - 2 and fj-l as a suitable initial guess. The iterative process is stopped when
126
Chapter 5
Chapter 5
Implicit moving boundary problems
127
Implicit moving boundary problems
If yo
the difference between two successive values of fj is less than a specified
M jtl
u
minimum error.
tit!
d fin
f ",(x) dx f o tj
5.3.3. AdaptivErmesh finite element methods
Mj
~ dt+ ",(x)
dt (5.47)
tj
M(t)
i th
To solve the oxygen diffusion problem, Miller et aZ. [6] used an adaptive-mesh
~Io
- [Z ~ M(t)
M(t)]
dx - ["'(X) dx dt
s
finite element method. In this method, the approximate solution of the problem, for the discrete times t j , is written as
f ~~
f ",(x) dx M itl
tjtl1
[
t=M-l(x)
so
re
Making use of (5.8) and (5.9), (5.47) becomes
ib
M itl
(5.44)
Mj
f u(x,t j+1)",(x) dx o
ur
U)·(x) ='" do. ·(x) £...i Q.',). '1",) i=l
f
u(x,tj)",(x) dx
ce
(5.48)
0
tit!
where Qi,j are time-dependent coefficients to be determined so that Uj(x) is the
e us
= -
best approximation of u(x,t j ) in the space-time domain, in which the basis functions ¢i,j(X) are defined. Let Dj be the change in U during the time-
= t j +1 - t j , obtained by differentiation of (5.7). Then Qi,j+1 can be
determined using the Galerkin equations
l, fu
interval Lit
f 0
dx + f ",(x) dx 0
quadrature rule, the right hand of (5.45) can be written as
Mitl
l
e as
(5.45)
f dt ti
M(t)]
Replacing u and", in (5.48) by U and ¢i,j+1 respectively, and applying the ()-
e pl
f[Uj+1(X)-Uj(X)]¢i,j+1(X)dx = f D j (X)¢i,j+1(X)dx
Z~~
[M(t)
-8 Lit
[d;;l
d¢~+l dx +
l
Mj
duo d .. dx dx
M)}
(1- 8)L1t f - ) ¢,,)+1 dx+ f"··
o
nc re
and t, gives
[¢i,j+1 dx (5.49)
fe re
Introducing a test function ",(x) into (5.7) and integrating with respect to both x
Mjt!} '1",)+1 dx
0
The substitution of (5.44) into (5.45) and making use of (5.49), produces the .!
M(t)
o
0
~~ ",(x) dx =
M(t)
f dt f
0
J
(~~ - 1 ",(x) dx
following system of equations, written in matrix form, (5.46)
e th
f dt f
tit!
e
tj
[A+ 8(t j +1-t j )B]Qj+1 = R-(tj+1- t )S
0
ok
Interchanging the order of integration for the left hand side of (5.46) and integrating by parts the right hand side, gives
bo
where
(5.50)
128
Chapter 5
129
From the right hand side of the condition (5.9), it can be deduced that the solution near the boundary is proportional to (M - X)2. If fix) represents the
yo
fIi;: t :>: t m ; Figure 6.1a)
increases with no change of phase. The governing equations for this stage are
'
M(t) :>: x :>: a
= 0 V(t)
M(t) x
=a
(c)
Figure 6.1: Three phase Stefan problem with different boundary conditions
(6.2)
e th
x=O, O:>:t:>:tm
(6.1)
x
e
-K.(~.)=F(t),
, 0:>: t :>: te
Heat flux input~ F(t)
nc re
2
a T. c· at - • ax 2
x=a
(b)
fe re
Due to the positive heat flux input, the temperature throughout the solid
M(t)
tv :>: t :>: te
e pl
then, depending on the process duration, the following stages will occur.
dT. _ K
(a)
l, fu
Assuming that the heat transfer is one-dimensional and the material thermal
x= 0
x=a
x=O
6.2.2. Melting stage ( t m :>: t:>: tv ; Figure 6.1b)
, x=a , O:>:t:>:te
(6.3)
bo
-K.(~.)=h(T.(x,t)-TO)
ok
When the surface temperature T(O,t) reaches the melting point of the material (T m) a MB, x=M(t), appears which separates the liquid from the solid. It is assumed that the liquid and the solid have the same density and
Chapter 6
146
Chapter 6
Multiple moving boundary problems
Multiple moving boundary problems
147
If yo
consequently there is no displacement of the liquid surface. The liquid occupies the region 05 x 5 M(t) and the solid occupies the region M(t) 5 x 5 a; the liquid/solid interface M(t) moves in the x direction and temperature increases
A dV u dt
u
= Kl(aTl)-F(t) ax
,
x=V(t)
,
t u S,t5te
(6.7)
d fin
where Au is the latent heat of vaporisation per unit volume.
throughout the solid and liquid regions. In addition to (6.1) and (6.3), the following equations are necessary to describe this stage:
aaxT2l
'
-
ax
V(t)
,
tm 5 t 5 t u
,
x=M(t)
,
t m 5t5te
=0
, t 5 tu , t 5 tm
M(t) = 0
=T m T(x,t) = T u
, x
, t ~ tm
,
= M(t) x = V(t)
,
T(x,t) = To
,
05x5a, t=O
T(x,t)
(6.5)
t
(6.8)
~ tu
ce
Sax
=0
(6.4)
ur
A dM =K.(aT")_Kl(aTl ) mdt
x
t m 5 t 5 te
so
,
,
re
~ ) = F(t)
V(t) < x 5 M(t)
s
- K{
l
The following conditions apply for all three stages
i th
aTl - K
Cl-at-
2
(6.6)
e us
In the following sections, two different methods, capable of solving the multi-
where Am' V(t) and t u are the latent heat of fusion per unit volume, the respectively. The subscripts land s distinguish liquid and solid phases respectively.
vapour. Assuming that the vapour is removed as soon as it appears (i.e. only
equation
2
caT -- K
at
a T2 =0
ax
'
(x,t)
E
R
(6.9)
nc re
two phases remain), the liquid occupies the region V(t) 5 x 5 M(t) and the solid
Let T(x,t) be a function which satisfies the following partial differential
fe re
the material, a second MB, x=V(t), appears which separates the liquid from the
6.3. The conservative finite element method ofBonnerot and Jamet
e as
When the surface temperature T(O,t) reaches the vaporisation point (Tu ) of
based on finite elements [3] and the second on finite differences [5].
e pl
6.2.3. Vaporisation stage ( tu 5 t 5 te ; Figure 6.1c)
phase problem defined in section 2, are illustrated in detail. The first method is
l, fu
vapour/liquid interface position and the time at which vaporisation starts,
occupies M(t) 5 x 5 a (the ablation case, where the whole liquid region is
and G an arbitrary sub-domain of R and aG its boundary. Multiplying (6.9) by
instantaneously evaporated and the problem becomes a single moving boundary
an arbitrary function q> and integrating by parts in the domain G, gives
e
problem, is not considered herein). The two MBs continue to propagate in the x regions until the time (te ) of the end of the heat transfer process. The final time
G
aq> cT-dxdt+
at
If
aT aq> K--dxdt
G
ax ax I~ q>(CTdx+K~ dt)=O
-
ok
the liquid/solid interface x=M(t) has reached x=a). This stage is governed by (6.1), (6.3), (6.4), (6.6) and the following condition at the second MB, x=V(t):
If
bo
for the test problems in section 6.5 is the time for complete melting (i.e. when
-
e th
direction and the temperature increases throughout the solid and liquid
For the particular case of q>=1, (6.10) is reduced to
(6.10)
148
Chapter 6
Chapter 6 Multiple moving boundary problems
149
Multiple moving boundary problems
If yo
f~(CTdx+K~ dt) = 0
(6.11)
u
-
l'
r KI(aTaxl ) %=v dt=f F (t)dt+A Jrv
d fin
i th
Cs{fr:dx-fT.dx+fTSdx-
s
n.
re
\
r..
r..
\ Tv
rM
ur
G.
Figure 6.2
n/
r T I dx} + K s1 r ax
n/
t'
(6.16)
o
t
E s + E l + Em + Ell + Ed
=Ep
(6.17)
where Es ' EI , Em and Ell are the energies used to: heat the solid, heat the liquid, melt the solid and vaporise the liquid, respectively. Ed and E p are the energy
e pl
can be written
Tsdx}+CI{fr:ldx - f Tldx+
rM
in the form
l, fu
Applying (6.11) to each domain Gs and Gl , shown in Figure 6.2, the following
1
aTs dt - AmL1M - AII L1V = f F(t)dt
Jr v
e us
x
ro
Equation (6.16) represents the conservation of energy, which can also be written
ce
Ds
n.
+1 T I dx -
so
To
(6.15)
Combining (6.12), (6.13), (6.14) and (6.15), the following is obtained
To, Ds , D~, T M and T M , Dl> D;, Tv, as shown in Figure 6.2.
t'~
L1V
t
Consider two adjacent domains G. and GI , with their respective boundaries
t
II
lost from the boundary x=a and the energy provided at the surface x=O,
n.
csTsdx+
n.
i
c.T.dx-
ro
r J
rM
K. aT. dt _
ax
ro
,clTldx-
n/
rM
r clT1dx+ Jrr clTldx- Jrr clTldx In/ v
M
i
rM
aTl KI-dt-
ax
I
rv
K. ~s dt = 0
K I -aTl d t=O
(6.12)
E s =Cs{fr:sdx-
(6.13)
ax
n.
f f.
Tsdx+Tm L1M}
n.
EI = CI{fr:1 dx - T l dx + T m L1M - Til L1V} n/ n/ Em =-Am L1M , Ell = -All L1V Ed = Ksf aTs dt r o ax
e
+
r J
nc re
f.
respectively. These energies are given by
c.T.dx
fe re
+
i
e as
i ~sTsdx- i
t'
E p = f F(t)dt
e th
The integration of(6.6) and (6.7) with respect to t, from t=t to t=t', gives
(6.18)
t
Ks(aTS )
ax
%=M
dt-
r
JrM
KI(aTI )
ax
%=M
dt=A mL1M
(6.14)
the following problem
ok
r
JrM
bo
To illustrate the general features of Bonnerot and Jamet's method [3], consider
Chapter 6
150
Chapter 6
Multiple moving boundary problems
Multiple moving boundary problems
151
If =
f
yo
2
,
M(t) < x < Vet)
, t> 0
(6.19)
and
u
e dT -K d '; at dX
(6.20)
x=u(t), u=M,Y
(6.21)
, M(O) < x < V(O) on
Dj(T,f,rp) =
, u=M,Y
!2 +
),
-;;e~:.....--.::..~.:. .......
.
, J - Jm
q=Jm
-k
+ w Xj+f)
, X=M,V
and V j the value of V at the time t j (6.40)
ur
Xj+f) = (1- w)Xj+f)
• k+l
so
re
for the next iteration -k+l
ofT
tjm ~ t
the (x,t) domain given by the coordinates
i th
J
be the time step indices so that
ce
6.4.1. Heating of solid stage
where w is a relaxation factor 0 is' throughout the liquid are carried out in a similar manner as melting only.
ok
,1tJ
(6.64)
Chapter 6
162
Chapter 6
Multiple moving boundary problems
Multiple moving boundary problems
163
If yo
where
u
with a convective heat transfer coefficient h=O. Bonnerot and Jamet [3] demonstrated the solution of the problem with two different sets of data. (6.65)
d fin
~j ={C'm~2JrK1(Tt-l,J+I-Tm)-Ks(Tm-Tt+l,J+l)]r
1
Problem la: a=l, T(x,O) = 27 , F=2500, cl=c.=4.944, K 1 = K. = 0.259 , Am = 2160 , Au = 37200, T m = 1454 , T u = 3000
i th
If equation (6.63) is not satisfied, a relaxation procedure is applied to determine
c,k)
_J_
(6.67)
This data set permits all three stages (i.e. heating of solid, melting and vaporisation) to occur. For the present method a choice of Lix = 0.02 was used so as to have approximately the same total number of grid points as the BJ
ce
= .1tJ- 1 .
(6.66)
ur
where w is a relaxation factor and .1tJ
so
[
.1tYl=.1tJ 1-100w
re
k
s
the time step for the (k+l)th iteration
to as the ZC method, a relaxation factor w=3.0, C,min =0.05% and rr =0.30 have been used. A comparison of results using the BJ method and the ZC method are shown in Table 6.1, and Figures 6.6, 6.7, and 6.8; these are
that suggested by Gupta and Kumar [14] (i.e. .1tj+l = ~j ). For the case of two
for others. In these circumstances such oscillatory behaviour can be avoided by
6.5. Test problems
Problem lb:
a = 1 , T(x,O) = 27 , F = 2500 ,
Cl = Cs = 104lp, K 1 = 1. 73 , K. = 0.865 , Am = 400p , A. u = 10700p, T m = 638 , T u = 2480 , P = 2.77
(6.68)
e
nc re
evaluating the time step for the (k+1)th iteration using: .1tj+l = i( .1tj + ~j) [15].
section 6.4.1.
fe re
Remark 2: Experiments show that a given value of w may give satisfactory convergence for some time steps and may exhibit non-convergent oscillations
Remark 3: Since the time step during the melting and vaporisation stages for the ZC method is variable,the M quoted in Tables 6.1 and 6.2 is the .1to defined in
e as
not in others; however, equation (6.66) is general and has proved satisfactory for all the computations performed.
discussed in section 6.6.
e pl
MBs such as problem 2, the procedure may converge in certain cases and may
l, fu
e us
Remark 1: When the energy balance at x = M(t) takes into account the temperature gradient in one phase only, as in test problem 2 of section 6.5, the time step at the next iteration .1tj+l can be determined by a similar procedure to
method, where .1t / Lix = 1/ 16. For the present method, which will be referred
where p is the material density.
e th
Problem 1: Collapse of a solid wall
With this data set, the melting interface reaches the adiabatic boundary
problem is similar to solving the problem previously described in section 6.2 but
problem. The same parameters ( ZC (Lix, rr, w, C,min), BJ {.1t / Lix, Emin)) have been used as in problem 1a. The comparison of results of the ZC and BJ
ok
other extremity x=a is thermally insulated (i.e. (dI' / dx).x=CI = 0) [3]. Solving this
before vaporisation occurs; the process is therefore reduced to a single MB
bo
The methods described in sections6.3 and 6.4 have been used to solve the collapse of a solid wall due to a constant heat flux input F at x=O, whilst the
164
Chapter 6
Multiple moving boundary problems
Chapter 6 Multiple moving boundary problems
165
If yo
methods are shown in Table 6.2 and Figures 6.9, 6.10 and 6.11; these are discussed in section 6.6.
u
the virtual distorted grid network is omitted in order to reduce the cpu-time; therefore, the temperature near the vapourlliquid interface is computed in a similar manner to that near the liquid/solid interface. Numerical results when
d fin
Problem 2
T v =1.5Tm , w=2.0, r(=0.35 and Em in=O.Ol% are shown in Table 6.3 and Figures 6.12, 6.13 and 6.14. For accuracy assessment, the Maximum Percentage Deviation for Temperature (MPDT) is defined as
s
i th
In order to further validate the accuracy of the ZC method, the following normalised two MB problem, whose exact analytical solution is known, was
re
considered [6]. A solid material melts due to a variable heat flux input applied at the liquid surface. The temperature throughout the solid region is always assumed to remain at the melting point. The governing equations are 2
ax
iJt
Vet) s x s M(t) ,Os t s te
(
~=-exp al t)
X = M(t)
,Os t S t.
,M(t)5;x5;l
(6.73)
for OSxS1
(6.74)
The exact analytical solution is given by
j
= 100x (tj)NU -(tj)An
(6.79)
(t j )An
and
V(t) = al (t-tv ) 1
te = al
(6.75)
= 100x (Vj)Nu -(Vj)An
J
(6.80)
(Vj)An
ok
Since in this particular problem the condition at the second moving boundary (vapourlliquid) is not coupled to the temperature gradient in the liquid region,
PDV.
bo
al
PDt
e th
M(t) = al t
1 =-loge(T v)
(6.78)
e
tv
,
O,}.}
The Percentage Deviation for the Vaporisation (PDV) interface position is given by
Tl(x,t) = exp(al t-x)
(6.77)
where
,t~O
, x = Vet) , t ~ tv
, TI (x,O)=l
MPDt = max{(IPDt j !),} =
nc re
M(O)=O
=1
m
(6.72)
fe re
{TTv
tv S t S t.
100x TNu(Xiotj)-TAn(Xi,tj) TAn(Xi,tj )
=
The Maximum Percentage Deviation for time (MPDt) is given by
e as
Tl(x,t) =
(6.71)
e pl
Tv dV = exp(al tv) , al dt
PDTi,j
(6.70)
l, fu
1 dM __ ()Tl a;dt"ax '
, x=O , OStStv
(6.76)
where PDTi•j is the percentage deviation from the analytical temperature calculated at the nodal point (Xi,t j ) which is given by
(6.69)
e us
()Tl
'
ce
()Tl - l iJ T2l -a
ur
so
MPDT = max{((!PDTi,jl),i = O,N),} = O,}e}
where the subscripts An and Nu refer to the value calculated Analytically and Numerically, respectively.
Chapter 6
Chapter 6 Multiple moving boundary problems
166
167
Multiple moving boundary problems
If yo
1.2
u
6.6. Numerical results and discussion
x=M(t), BJ method x=M(t), ZC method x=V(tl, BJ method x=V(tl, ZC method
1.0
d fin
3
Numerical results show that for problem la, which corresponds to a low
0.8
I:
conductivity material compared to that used in problem lb, the vaporisation
0 ~
'0;
i th
occurs before the liquid/solid interface reaches the adiabatic boundary. On the
0
0.6
c.
contrary, in problem lb, the wall collapses before the appearance of a
QI U
s
~
vapour/liquid interface and therefore the problem is one of melting only.
0.4
....I: QI
re
-
Table 6.1, Figures 6.6, 6.7 and 6.8 show that for problem la, both the BJ and
so
ZC solutions are in good agreement; a maximum relative error of only 2%
0.2
occurs between the two solutions. For problem la, the BJ method has a cpu-
ur
0.0
4
2
0
time of 40 seconds, whereas the ZC solution takes only 9 seconds (non-vectorised
time (t)
6
8
10
ce
program) on an IBM3090/l50VF (Computer Centre, University of Glasgow). Figure 6.6 (problem la): Position of the liquid/solid and vapour/liquid interfaces versus time
e us
The ZC scheme is more than 4 times faster than the BJ method. Using
vectorisation, the speed up factor due to vectorisation varies from 2.5 to 3.5 for the BJ method, whereas it varies from 5 to 6 for the ZC method. This however,
cycle of variable dependencies).
e pl
variables are determined with recurrent do-loops (i.e. do-loops that form a
l, fu
is due to the fact that explicit methods are more suitable for vectorisation than
implicit finite difference and finite element methods [16] where the unknown
4000
e as ~
~
~
3000
x=aJ2, BJ method x=aJ2, ZC method x=O, BJ method x=O, ZC method
Eo-
ce
tm L1t BJ method 0.23485 1/4 0.23401 1/8 W6 0.23400 Z method 0.23352 1/4 0.23398 1/8 0.23390 1/16
Figure 6.8 (Problem la): Velocity of the moving boundaries versus time
1.92536 1.92383 1.92381
2349.18 2347.98 2347.96
1.89063 1.88747 1.88719
2341.45 2341.40 2341.40
1.50
e as
x=M(t), BJ method x=M(t), ZC method
~
~
~II
1.25
c::
1.00
fe re
~ 0
',.3
.Iii 0
0.75
0Q) C)
.
..;':!
...,
0.50
Q)
c::
e
0.25 0.00 0.0
e th
-
nc re 0.5
. () 1.0 time t
bo
1.5
2.0
Figure 6.9 (problem 1b): The liquid/solid interface position versus time
ok
greater thermal-resistance of the liquid. When the V/L interface appears, the US interface slows sharply due the energy loss from vaporisation. The V/L interface accelerates sharply in the initial instance, then continues to accelerate at a lower rate throughout the process. The US interface velocity reaches a minimum then starts to increase again; this is due to the fact that the heat flux entering the solid region decreases sharply when approaching the adiabatic extremity (i.e. there is less energy absorbed as the whole solid approaches the melting temperature). The heat flux at the US interface decreases with a smaller gradient than the heat flux entering the solid region resulting in a greater proportion of the energy being available to melt the solid material; meanwhile, the V/L interface converges towards a constant acceleration due to the constant heat flux input. Before the total melting or the collapse of the wall, the US interface velocity starts to decrease due to the decreasing heat flux through the liquid and the absence of heat flux in the solid region.
e pl
after its appearance; its propagation speed then starts to decrease due the
l, fu
e us
Figure 6.8 shows the velocity history of the two MBs for problem la, where
the US interface starts with zero velocity and reaches a maximum speed just
TCO,t e )
te
Chapter 6
170
Chapter 6
Multiple moving boundary problems
171
Multiple moving boundary problems
If 2500
E=
ClJ
Q
ClJ
...
c..
1.6
0.8 time (t) 1.2
2.0
ce
Figure 6.10 (problem lb): Temperature versus time at different depths
'-'
l, fu
't ClJ
0.6
0.4
.... 0
0.2
100
For problem 2, Figure 6.12 shows that MPDT, MPDt , PDt j., PDt j. and PDV j. all decrease as N increases for problem 2. In other words, the accuracy of the ZC method increases by increasing the total number of space elements; this
fe re
'C:; 0
a;
:>
40 60 80 Number of space elements (N)
e as
dMldt, BJ method dMldt, ZC method
0'
;.3
$
20
results in a greater cpu-time, as shown in Figure 6.13.
en
'S
2
e pl
~
'0 ~
J•
Figure 6.12 (problem 2, a=O.I): Variation of percentage deviations from the analytical solution with the total number of space elements N
e us
0.8 ClJ
... .S
•
4
o
ur
0.4
so
0.0
1\
ClJ
'-'
re
0
•
•II
1ll: ClJ
500
I
MPDT MPDt PDtj • PDt· PDV·.
bII
s
1000
Eo
extended times, explicit methods do have instability constraints which make
re
6
9
12
15
18
Velocity of the moving boundaries
The ZC method combines both the VSIET and explicit computations. The VSIET ensures that the stability of the scheme is automatically maintained without increasing the storage requirements. This results in highly efficient use of the computational platform. The explicit procedures maximise the gain
21
ur
3
higher demands on computer memory requirements due to the large amount of data that must be stored.
so
O. 0
parallel computations. However, if computations are to be performed for
s
.5c
cpu-time of large scale computation. Explicit methods, where the unknown variables are determined with no recurrence (i.e. a group of statements that form a cycle of variable dependencies), are highly suitable for vectorisation and
i th
'S: .,
1.2-
d fin
c0
J
u
~
Figure 6.14 where the percentage deviation for the different variables in the solution remains constant with the variation of moving boundary velocity. Vectorisation is an essential tool to increase performance and reduce the
yo
1.
ce e us
Figure 6.14 (problem 2, L1x=0.02): Variation of percentage deviations with the velocity of moving boundaries
moving boundary, the accuracy is very satisfactory. Maximum errors for the different variables of the solution are always generated at the appearance of a
l, fu
Figure 6.12 also shows that despite the singularities that appear with each
due to vectorisation or parallelism of the code. Other benefits of the method are simplicity and ease of programming; in the ZC method only simple linear equations must be solved compared to many numerical integrations and the solution of systems of linear equations for the BJ method.
e pl
ME. These errors decrease with time as shown in Table 6.3.
6.7. References
V Nu
0.094535 0.194535 0.294535 0.394535 0.494535 0.594535
3.
R. Bonnerot and P. Jamet, J. Comput. Phys. 41, 357-388 (1981).
4.
R. Bonnerot and P. Jamet, J. Comput. Phys. 32, 145-167 (1979).
5.
M. Zerroukat and C.R. Chatwin, J. Comput. Phys., (in press)
6.
M. Zerroukat and C.R. Chatwin, Technical Report TRJGLUIMZ920910,
PDV
-2.677884 -1.561637 -1.163964 -0.945447 -0.801985 -0.697940
unaffected by the nature of the heat transfer problem. This can be seen from
8. 9.
ok
Due to the incorporation of the VSIET, the accuracy of the present scheme is
7.
Dept. Mech. Eng., University of Glasgow, (1992). M. Zerroukat and C.R. Chatwin, Int. J. Numer. Meth. Eng. 35, 1503-1520 (1992). J.R. Cannon and M. Primicerio, SIAM J. Math. Anal. 4, 141-148 (1973). J.R. Cannon, J. Douglas and C.D. Hill, J. Math. Meeh. 17, 21-33 (1967).
bo
0.092003 0.191497 0.291106 0.390805 0.490569 0.590385
VAn
e th
PDt -1.426458 -1.249921 -1.136717 -1.047945 -0.969506 -0.892385 -0.820732 -0.755906 -0.698196 -0.646725
2.
Heat Flow and Diffusion, Clarendon Press, Oxford (1979). J. Crank (ed.), Free and moving boundary problems, Clarendon Press, Oxford (1984).
e
tAn 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000 3.5000 4.0000 4.5000 5.0000
J.R. Ockendon and W.R. Hodgkins (eds.), Moving Boundary Problems in
nc re
tNu 0.492868 0.987501 1.482948 1.979041 2.475761 2.973228 3.471273 3.969764 4.468581 4.967664
1.
fe re
M(t) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
e as
Table 6.3 (Problem 2): A comparison of computed history of the moving boundaries M(t) and V(t), using the ZC method, with the analytical solution Percentage deviations are also tabulated
174
Chapter 6
Multiple moving boundary problems
If yo
B. Sherman, J. Math. Anal. Appl. 33,449-466 (1971). W.L. Heitz and J.W. Westwater, Int. J. Heat Mass Transfer 13,
12.
1371-1375 (1970). S.C. Hsu, S. Chakravorty and R. Mehrabian, Metall. Trans. 9 B,
13.
221-229 (1978). M.C. Bhattacharya, Communications Appl. Numer. meth. 6,
14.
173-184 (1990). RS. Gupta and D. Kumar, Int. J. Heat Transfer 24, 251-259 (1981).
15.
M. Zerroukat and C.R. Chatwin, Technical Report TRJGLUIMZ920820,
16.
Dept. Mech. Eng., University of Glasgow, (1992). M. Zerroukat and C.R. Chatwin, Technical Report TRJGLUIMZ920830,
u
10. 11.
d fin
s
i th
so
re
7
ur ce
Dept. Mech. Eng., University of Glasgow, (1992).
Heat Flow During Laser Heat Treatment of Materials
l, fu
e us Summary
e pl
The finite difference schemes developed throughout the previous chapters are
e as
here utilised to simulate a practical problem. It consists of the melting and re-
crystallisation of silicon and cast-iron, when subjected to radiation by a
fe re
scanning cw-laser beam. The model takes into account the temperature dependence of material properties, the temperature-dependent surface reflectivity and the variation of the incident power density with time and
nc re
surface-temperature. The problem is to compute the temperature distribution, the melt depth, and the effect of different combinations of laser parameters
e
such as the incident power, beam diameter and the scanning speed.
The
agreement.
e th
numerical results are compared with experimental results and show good
ok
bo
176
Chapter 7 Heat flow during laser heat treatment of materials
Chapter 7 Heat flow during laser heat treatment of materials
177
If 7.1. Introduction
yo
Nomenclature
Heat capacity per unit volume of material
cp
K
Heat capacity of air Laser power density Acceleration due to gravity Heat transfer coefficient Thermal conductivity of material
M
Liquid/solid interface position
N
1l:x,t)
Total number of space elements Time at which melting starts Time at which the process ends Temperature at depth x, time t
Tm
Melting temperature
To r
Temperature of air
R
Reflectivity of the surface material
substrate providing self-quenching. To achieve a desired effect on a specific material it is necessary to make the optimum choice of process parameters such as power, scan velocity and beam diameter. To avoid a costly time-consuming experimental trial-and-error
ur
so
re
te
beam is utilised to rapidly heat a thin surface region, with the underlying bulk
s
tm
i th
g h
Laser processing of surfaces, which involves rapid localised heating of a material to modify its physical properties by altering the surface structure or redistributing impurities, has found many applications in manufacturing and particularly in semi-conductor technology [1-3]. Typically a high-power laser
d fin
F
u
c
ce
mathematical model describing the laser-material interaction process.
e us
Fourier number
search to obtain the optimal combination, and in order to assess the influence of the laser parameters on the process, it is necessary to construct a Laser materials processing technology allows beam-workpiece interaction
l, fu
times ranging from a few picoseconds duration to several seconds; the concomitant incident energy densities, which induce observably reproducible
rr
Fixed Fourier number (::; 0.5)
p
Total incident power of the laser beam
X
Thickness of the work-piece
v /(
Scanning speed Thermal conductivity of air
a
Stefan-Boltzman constant
J1.
Dynamic viscosity of air
these extreme conditions, the linearity between the heat flux and temperature
P
Density of air
gradient breaks down; for these circumstances the energy transport into the
a
DifIusivity
material can be described using the kinetic theory of heat transfer rather than
L1x, At
Space element, time step length
the Fourier conduction theory (4). However, for interaction times greater than
.t
Latent heat of fusion/unit volume
some picoseconds, the heat conduction process is correctly described by the
w
Laser beam radius
Fourier conduction equation [5].
becomes comparable both to the radiation absorption length, which is of the
e as
order of 20 nm in most metals, and to the mean free path l of the heat carriers
(i.e. the conduction electrons for which l =20 nm at room temperature). Under
e
nc re
fe re
ok
Virtual time step index Last virtual time step index for a given real time step
chapters, are utilised to simulate the melting and re-crystallisation of silicon when subjected to radiation by a scanning cw-Iaser beam [6]. The process is physically described as follows. A laser beam moving with a uniform velocity scans a silicon substrate of virtually infinite length and width but finite thickness. The incident power is partly reflected and partly absorbed,
bo
Liquid / Solid
e th
n
e pl
Space / time indices
Superscript k
with high power and picoseconds interaction times, the heat diffusion length
In this chapter, the finite difference schemes, developed in previous
Subscripts
i,J l, s
effects, range from several mJ/cm 2 to some thousands of J/cm 2 • For processes
Chapter 7 Heat flow during laser heat treatment of materials
Chapter 7 Heatflow during laser heat treatment of materials
178
179
If melting starts and time at which solidification ends, respectively. The subscripts s and l refer to solid and liquid regions respectively.
yo
depending on the reflectivity of the surface. Some of the absorbed energy is lost by re-radiation and convection from both the upper and lower surfaces into the
u
Equations (7.1) to (7.3) may also be written with a heat source term, due to
surrounding environment, whilst the rest is conducted throughout the
d fin
volume absorption of the radiation. However, calculations show that energy
quenched by the solid substrate and the heat lost from the surfaces. The
absorption is predominantly a surface phenomenon [8]; hence, the heat source
numerical scheme is also applied to the laser heating and melting of nodular
term is neglected in (7.1) to (7.3). The boundary conditions are written as
i th
substrate. When the beam has completely passed, the melted zone is self
cast iron; the results are compared with published experimental results.
-Kl(Tl{'~/)=F(T/,t)-h(Tl,TO)(Tl-TO)
s Theory of1aser-beam interaction with materials
,
x=o
,
tm~t~te
(7.4)
so
re
7.2.
-Ks(Ts {
As the laser-beam diameter is very small compared with the lateral
a:xs ) = h(Ts,To)(Ts - To)
,
x
=X
, 0~t
~ te
(7.5)
ur
dimensions of the material, and the power distribution is symmetric with
ce
respect to the beam centre (i.e. Gaussian), the approximation of the heat flow by
where h is a heat transfer coefficient and F(T,t) is the power density absorbed
a one-dimensional model is not expected to introduce significant errors [7]; the
e us
at the centre of the beam by the surface material, which is at temperature Tat
maximum heat effect occurs when the surface is irradiated by the central part
time t. Assuming that the laser beam has a Gaussian distribution of radius w
of the beam. The process can be adequately modelled as a I-D moving boundary the heat conduction into a material of thickness X with change of phase, are
at
ax
ax
,
,
,
M(t)~x~X
t m ~t~te
,
t>O
(7.1)
(7.2)
t 1
-
n
R(T)I.~, '+(" ~ w
O~t ~
..
(7.6)
t> ..
where .. = 2w / v and R are the beam-workpiece interaction time and the
fe re
Cs(Ts)a~S = ~(Ks(Ts)~')
O~x~M(t )
F(T,,) "
e as
aT/ a ()aT/) c/(T/)-=K/(T/ -
at the lie point, if the laser beam has a total incident power P and is moving with a constant velocity v, then F(T,t) is given by
e pl
written as
l, fu
problem with variable thermal properties. The governing equations, describing
reflectivity of the surface, respectively. From both surfaces (x =0 and x =X) the
,td:
=Ks(Tm{~')-K/(Tm{a:x/)l '
,
tm~t~te
heat transfer coefficient, due to both convection and radiation, of a plate of (7.3)
= T/(x,t) = T m
zK(Tr ) cp (Tr )2gp Z(Tr )(T 1 -Tz )X X [ K(Tr ) (T 1 + T z )J1.(Tr )
ok
melting/solidification per unit volume, melting temperature, time at which
h(T1,Tz ) = 0[1- R(T1)](T~ + Ti)(T 1+ T z )+
bo
capacity per unit volume and thermal conductivity respectively. M, ,t, T m' tm and te refer to the liquid/solid interface position, latent heat of
the function h(Tl' T z) is given by
e th
where T(x,t), c(T) and K(T) denote temperature at the coordinates (x,t), heat
thickness X at temperature T 1 submerged within a fluid at temperature T z ,
e
Ts(x,t)
x=M(t)
nc re
heat is lost by a combination of free-convection and radiation. Expressing the
3JY
(7.7)
Chapter 7 Heat flow during laser heat treatment of materials
Chapter 7 Heat flow during laser heat treatment of materials
180
181
If yo
8.257 x 1O-3 (T - 273) + 12.0
where T r = i(T l + T 2 ), a and g are temperature of the air film, Stefan-
3
j
K(T) = -4.450 x 10- (T - 973) + 17.78 O. 360(T - 1573) + 15.11 33.11
u
Boltzman constant and gravity respectively. Also cp ' Jl, P, and /( denote heat capacity, dynamic viscosity, density and thermal conductivity of air respectively; these variables are temperature dependent. The correlation factors
273~T ~
973
973 ~ T
1573
~
(7.10)
d fin
1573 $ T ~ 1623 T> 1623
z and yare equal to: z =0.14 and y =113, for the surface facing upwards; z =0.58
4.670 x 10-3 (T - 273)+ 194
i th
and y =1/5 for the surface facing downwards [9].
j
1573 $ T
c(T) = 0.123(T -1573) + 8. 01 -0. 123(T - 1623) + 14.17 8.01
s
7.2.1. Application to s1licon
273$T$1573 ~
1623
(7.11)
so
re
1623 $ T $ 1673 T> 1673
The variation of the thermal conductivity and the heat capacity/unit volume The units in equation (7.10) and (7.11) are: T[OK], c[106 J/m 3 ] and K [W /mOK]. The melting temperature of nodular cast-iron is T m = 1573 OK
ur
of silicon with temperature is given by [10]
ce
K(T)=299X10 2 (_1_) , C(T)=2.336X10 6 (T-159) T-99 T-99
and the latent heat of fusion per unit volume A. = 791134 X 10 6 J / m 3 • The
(7.8)
e us
reflectivity as a function of temperature is approximated by a linear function [14], which is given by
The units in (7.8) are T [OK], K[W /mOK] and c[J / m 3 0K]. The melting
volume A. = 3282 X 106 J / m 3 . For liquid silicon the thermal conductivity evidence [11,12] shows that the reflectivity of silicon varies linearly with accurate function, for a large range of temperatures including melting, is
R(T) = 0.372 + 2.693 x 1O-5 T+2.691x 1O-15 T4,
T ~ 3130 OK
(7.9)
(7.12)
The functions describing the variation of thermal properties of air with
temperature are obtained by approximating the experimental data in [15] by a polynomial of degree 5.
fe re
given by [10]
T[OK]
,
e as
temperature for temperatures less than the melting point. However, a more
R(T) = 0.72 - 25 X 10-5 (T - 293)
e pl
increases to 0.64 [W/mK], while c(T) is taken to be constant [7]. Experimental
l, fu
temperature of silicon is T m = 1690 OK and the latent heat of fusion per unit
7.3. Numerical computation scheme
nc re
Prior to melting (T(O,t) < T m) when only one phase is present, the
7.2.2. Application to nodular cast-iron
numerical computation proceeds as follows.
Consider that the temperature
e
distribution Ti,j' i = O,N is known and we wish to compute the temperature volume of nodular cast-iron with temperature are approximated to fit the experimental data in [13] by
distribution at the time t j+ 1 = t j + L1t. The temperature distribution Ti~j, i = O,N at any Virtual Sub-time Step (VSS) of index n (n = 1,k), situated at
e th
The dependence of thermal conductivity and the heat capacity per unit
= tj = (t j + ~L1t). is calculated from
that of previous VSS using
ok
bo
t
n-l n-l -ri,j n-l n-l n-l n-l n-l } n T·'.J . = T·'.J . exp [ 2 T.'.J . -(1+{3.. T~-:-l '.J )T.,- 1,'J -(1-{3'',J. )T.,+ 1,.J J {
',J
(7.13)
183
Chapter 7 Heat flow during laser heat treatment of materials
Chapter 7 Heat flow during laser heat treatment of materials
182
If yo
where
where
u n
ai,j
n
K~·
aiJ'=~ , n
(n. )r r max ai,j,L=O,N
n n Kt:l, J. - K i+l,j f3i,j = n A v i.
d fin
n _ ri,j -
Ci,j
1
(7.14)
Qr-: =
t}
,J
i th
and rr $; 0.5 is a real constant fixed throughout the computation process. At some intermediate VSS n =k, so that tj ~ tj+l> the temperature distribution at t = t j+l is calculated from that of t = tj-l using equation (7.13) (i.e. k. 0 N) Wl'th ri,j k-l II-l( 2 T i,j+l= T i,j,L=, =ai,j tj+l-t jk-l)/ L1x.
l l 1+ .t: + e: - l)f3!1-:l]T~-:l - (.t: + '='.t:f3!1-: - (1- f3!1-: I.,J )Tr:-l,l. .. "d )Tm' i = ; • 1[[2 _~(1- f3~jl)lTi~jl - [(1- ~)( 1- f3~jl )]Tt~.-l,lj - (1 + f3i~jl)Tm' i =i + 1(7.17) m ~
~
&,)
~
I.,j
J
"71l
s
also
re
so
f3!1-: l = ',J
H(I- ~)Ki~jl+ ~Ki':-(j -K(Tm)]1 4KCj l
, i = im
H-~Cjl+(~-1)Ki~-(j+K(Tm)]/4Ki~jl,
(7.18)
i=im +1
ur
ce
When the temperature of the surface, exposed to laser radiation, exceeds the melting point, the heat in the superheated region 0 $; x $; Xl (i.e. T(x,tm ) > Tm) is converted to heat of fusion, giving an initial melt depth X2 at a uniform melting
X2
=----{- J
(7.15)
(T(x,tm)-Tm)dx
o
',J
',J
J
i
= im,im + 1
(7.16)
x, from X =0 to X =M(t), and (7.2) from x =M(t) to x =X and using (7.3) gives
M
X
CI dTl dx+JC s dT. dx=-). dM + Ql(t)-Q2(t) iJt iJt dt J o M
ok
T~-:l
xi, is the variation of the melting front during each virtual time step Otj = tj - tr l . The xi are obtained as follows. Integrating (7.1) with respect to
where
bo
n-l
T·n . =T·n-l exp - ri,j - - Q n-l ..
(7.19)
n=l
e th
and im + 1. For grid lines with 0 $; i $; im- l and im+2 $; i $; N equation (7.13) is used. For the grid lines i m and im + 1, instead of(7.13) the following is used
= M j + LXi
e
to that for the heating of solid stage, except for the nodes near the moving interface. Consider that the moving boundary is located between the lines im
M j +l
nc re
M j + l from those at the previous time t j' The computation procedure is similar
II
fe re
temperature distribution Ti,j+l' i = O,N as well as the melting front position
and im+2 $; i $; N and (7.16) for i = im , im + 1. The position of the melting front is given by
e as
During the melting and solidification stages, the problem is to compute the
(
VSS (i.e. Ti,j+l = Ti~j' i = O,N). Equation (7.13) is used for grid lines with
e pl
< L1x); therefore, T(x,t m) in (7.15) is approximated by a linear function of X; hence, x2 is directly obtained without recourse to numerical integration.
(Xl
',J
The temperature distribution at time t j + l is calculated from that of the last
o $; i $; im-l
Generally the superheated region extends less than a space increment
',J
boundary.
l, fu
x,
c(T )
e us
temperature [16]. This can be expressed mathematically as
and ~ = mOd(Mrl,L1x). Equations (7.17) and (7.18) are obtained using the approximations used in [17] and in chapter 4, for the nodes around the moving
(7.20)
184
Chapter 7 Heat flow during laser heat treatment of materials
185
Chapter 7 Heat flow during laser heat treatment of materials
If yo
where Ql(t) and Q2(t) are the heat flux absorbed by the top surface (i.e. right side of (7.4)) and the heat flux lost by the bottom surface (i.e. right side of (7.5)), respectively. Further integration of (7.20) with respect to t, from t = tTl to t = t~ = t~-l + Ot~J ' ... aives J J
7.4. Numerical results and discussion
u
d fin
i th
ti M
ti X
tj I M
ti
ti
l::
(7.21)
(,)
§
I
so
-5 0.. 0
(A.1)
e as
e pl
with the boundary condition T(x,t)
= To
, x
=0
, t
=0
(A.2)
,
=0
(A.3)
and initial conditions
=0
,
x> 0
nc re
fe re
T(x,t)
M(t)=O
t
(A.4)
, t=O
e
Two further conditions are needed on the moving boundary; these are
e th
ax
dt
1
,
x = M(t)
,
t> 0
ok
bo
=0 -K ()T = pL dM
T(x,t)
(A.5)
192
If
AS. General formulation of a moving boundary problem
u
yo
where p, c, K and L are density, heat capacity, thermal conductivity and latent heat of fusion respectively.
The general melting or solidification process of a material is described as follows. At time t = to, the two regions R l and ~ which are bounded by the fixed boundary surfaces aRl and a~ respectively and the moving boundary
d fin
A2. Two-phase ice melting problem
193
Appendices
Appendices
M
= M(X, to)
with X the space coordinate vector, as shown in Figure A.l.
i th
A more general formulat.ion can be expressed by the fact the heat flow occurs in both phases. The heat parameters may all be functions of T, x and t , i.e. K = K(T,x,t) and c = c(T,x,t). There may also be a heat source Q = Q(Tm,x,t)
s
M(X,to)
re
(or sink) at the moving boundary. The heat conduction equation (A. 1) becomes
x
E
R;
, i = 1,2
are the regions defined by
= {(x,t) : 0 < x < M(t) , t > o} = {(x,t) : M(t) < x < a, t > o}
(A.7)
l, fu
R2
e us
Rl
Figure A.l: Two different regions separated by a moving boundary M(X,t)
e pl
and the condition at the moving boundary (A.5) becomes
At later time t the moving boundary M moves into a new position M(X,t), in the domain R = R l U R 2 • The problem is to locate M in space at any time and
(A.B)
e as
K 2 (Tm,x,t)aT2 -Kl(Tm,x,t)aTl =pL dM -Q(Tm,x,t) ax ax dt
aRl
(A.6)
ce
~
,
ur
where R l and
= ~ (K;(T;,x,t) a:x;)
so
p;c;(T;,x,t) ~;
subsequently the temperature distributions Tl(X,t) and T 2 (X,t) in the regions
respectively.
temperature, the heat generation terms E and Q within the material and at the
fe re
where a, T m are the thickness of the slab and the melting temperature,
R l and ~ respectively. Taking into consideration the dependence of the thermal properties on the
moving boundary respectively, equation (A.6) becomes An alternative formulation may be used to
sheets of metal as an example of mushy region formulation. Several authors [3physical significance of such formulations.
(A.9)
with the boundary conditions
ar: - h;T; = g;
aT·
bo
7] have presented formulations for different problems and discussed the
,i=1,2
e th
range of temperature "Mushy region". Atthey [2] studied the welding of two
P;C;~;='i1(K;'i1T;)+E, xER;
e
describe the change of phase; this assumes that the phase change occurs over a
nc re
It can also be postulated that the change of phase is not well represented by a
sharp isothermal boundary.
, x EaR;
= 1,2
ok
and on the moving boundary M(X,t)
, i
(A. 10)
194
If yo
u
a a a -(K T) ]2 = -(K 2T 2)--(K1T 1 ) [ an Ian an
= T 1(X,t) =T m
where U(x,t) denotes the concentration of the oxygen free to diffuse at distance x from the outer surface of the medium at time t, R is the rate of consumption of oxygen per unit volume of the medium and D is the diffusion coefficient. The oxygen diffusion problem is formulated in two stages, first as a steady
= pLVn +Q (All)
d fin
T1(X,t)
state solution (details can be found in [11]) and secondly as a moving boundary problem. After the surface x=O has been sealed, the oxygen which is already in
i th
,
T 2(X,0)
= T 2(X)
,
denoted by M(t). The problem is then expressed by the following equations
M(X,O) = Mo(X)
(A12)
2 au = D a u _ R at ax 2
ur
so
= T 1(X)
concentration moves towards x=O. Let the position of zero concentration be
re
T1(X,0)
the region continues to be consumed; consequently, the point of zero
s
with the initial conditions
195
Appendices
Appendices
where n is the outward normal to the moving boundary and V n is the velocity of
ce
this boundary along the normal. The functions E, Q and g can be also functions ofXandt.
au
e us
ax
=
0 < x < M(t)
,
° , x=
0
,
t >0
(A.14)
(A15)
, t>0
Equation (All) can also be written in Patel's expressions for each space
au
direction x, y and z, which are more practical forms. Details of Patel's
U=a;=O
l, fu
expressions can be found in reference [8] and [9].
R
2
2D
, t~O
,O~x~Mo
, t=O
(A16)
(A.17)
where M o = M(O) is the deepest penetration of oxygen into the medium, and is given by the solution of the steady state [11]. Also, t=O is the moment when the
e as
In the problems formulated in AI, A2 and A3, the moving boundary condition connects the variable T or its derivatives with the velocity of the
x=M(t)
U(x,t)=-(x-Mo )
e pl
M. Implicit moving boundary problems (oxygen diffusion problem)
,
surface is sealed. It should be noticed that the velocity of the moving boundary is
explicit relationship does not exist on the moving boundary; these are called
absent in the boundary condition given by equation (A16).
implicit moving boundary problems [10]. In other words they correspond to the
The problem, arising from the diffusion of oxygen in a medium which
Appendix B: Calculation oftemperatures around the moving boundary (Chapter 4)
e
simultaneously consumes the oxygen [11,12], is an example of implicit moving
nc re
special case of L=O in (A.B).
fe re
moving boundary (e.g. equation (A8». Some problems exist in which such an
To calculate the temperature T,:,},m,p the following approximations are
the moving boundary. The diffusion process is described by the partial
made: (i) The moving boundary between t = t} and t = tj+l is approximated by a
differential equation
e th
boundary problems, due to the nature of the boundary condition prevailing at
bo
step wise function of pj steps.
ax 2
-R
(A13)
ok
au =D at
a2u
196
Appendices
Appendices
197
If t = tj
yo
(ii) The temperature profile between
t = tj+1 is approximated by
and
= at &2 8~ = at 8tt/(pj /m)2 = k (mL1x/ pj rct,j
t
r2
u
a step wise function of pj steps (i.e. the temperature remains constant within a virtual sub-time step).
d fin
I
0) m=m+1
£j
. exp(-x 2 ) lerfc(x) = "0 - x erfc(x)
erfc(x) = 1-erf(x)
e pl
end do
erf(x) =
11:~2
f x
exp(-y2) dy
(D.5)
o
fe re
D1. Computation of temperatures at the first time step
(D.4)
and
e as
Appendix D: Numerical scheme ofmultiple moving boundary problems (Chapter 6)
(D.3)
11:
l, fu
t j = tj- 1 + {(m -l)rr + rj }L1x 2 ui,j = F(Ai_I,Ai,Ai+l,i,ib,rj) ; i = O,ib
(D.2)
where
e us
G( .4;b- 1,Aib ,rj) = 0
ce
Bi = F(Ai_l,Ai,Ai+l>i,ib,rr) end do end while
r[
iL1x 1/2 J+T0 ' O emin) calculate pj and qj and the values of r~,j and r~,j
d fin
where
= J" -
do n = J.qJ
so
VO:CN + I-i".) = VO:(VI:CN + I-im ») VI:CN + 1- i".) = VO:CN + 1- i".)
ur
For simplification we use the notation VOCd) to represent a Vector Output of
end do
ce
the temperatures of d discrete points, at any instant in time, calculated from
to distinguish liquid and solid respectively; the definition of all other variables
calculate eJ and the time step L1tr 1 for the next iteration k = k + 1 end while
are identical to the main text. Problems such as the one described in section 6.2
assign VOICi".) and VOsCN + I-im ) as temperature distribution
the Vector Input VICd) from the previous time step. Subscripts l and s are used
the following type sta~
end while Vaporisation staie
calculate tu and the initially removed material
do }
.
calculate Po and rc
..
l.", =J-J m
+1
I>
0 ,L1tj =L1tj_l
} = }+I
do n = I,Po make the transposition VIsCN + 1) = VOscN + 1) for the next VSS Cn + 1)
calculate
nc re
calculate VOscN + 1) = VOs(VIsCN + 1»)
fe re
do while (Iej emin ) calculate pj and qj ; the values of r:l,j and r:s,j do n= l,pj
do while CTCO,t) < T m)
Xj,n-l,p and locate the surface node
is
votCi", -is )= VO;(VI7Cim -is») VItCim - is) = votCi", - is)
end do do n= 1,qj
e
end do
end while
calculate end while
ej and the time step L1tj+l for the next iteration
ok
Meltin{l stQ{le
end do
bo
VO:CN + 1-im )= VO:(VI:CN + 1-i",») VI:CN + 1-im )= VO:CN + 1-im )
distribution at the time t j + 1 correct tm and recalculate the vector VOsCN + 1) for the corrected time tm
e th
assign the final VOsCN + 1) as the temperature
do while (TCO,t) < T u )
X2
= }u,}e
e as
calculate temperature distribution at}=1 Canalytical approximation)
in the liquid and solid, respectively, at the time t j + 1
e pl
Heatini of solid
l, fu
e us
can be computed using the method described in section 6.4 by an algorithm of
k = k+1
204
Appendices
AUTHOR INDEX
205
If yo
assign VO/(im) and VOs(N + l-im) as temperature distribution
u
in the liquid and solid, respectively, at the time end do stop
tj+l
d fin
rod
i th
References
Aguirre-Ramirez, G. 82
43, 44, 45, 51, 52, 83, 112, 114,
Albrecht, J. 40
115, 116, 118, 119, 125, 129, 131,
Atthey, D.R 112, 192, 204
132,136,137,139,173,204 Crowley, A.B. 42
s
'.
re
2.
D.R Atthey, J. Inst. Math. Appl. 13, 353-366 (1974).
3.
L.L Rubinstein, Bull. Math. Biophys. 36,365-377 (1974).
4.
L.L Rubinstein, J. Inst. Math. Appl. 24, 259-277 (1979).
5.
L.L Rubinstein, IMA J. Appl. Math. 28, 287-299 (1979).
6.
E. Magenes (ed.), Free boundary Problems vols. I and II, Instituto
ur
so
J. Stefan, Sber. Akad. Wiss. Wien. 98,473-484 (1889).
ce
boundary problems, Research Notes in Mathematics 59, Pitman, London (1982). A. Lazaridis, Int. J. Heat Mass Transfer 13, 1459-1477 (1970).
10.
G.G. Sackett, SIAM J. Numer. Anal. 8, 80-96 (1971).
11.
J. Crank and RS. Gupta, J. Inst. Math. Appl. 10, 19-33 (1972).
12.
J. Crank and RS. Gupta, J. Inst. Math. Appl. 10, 296-304 (1972).
13. 14.
Del Guidice, S. 42, 82
Berger,A.E. 115,139
Doherty, P.C. 41
Bettencourt, J.M. 81
Douglas, J. 25, 27,28,29,41,87, 99,112,173 Draper, C.W. 189
Blanchard,D.38,42
Duley, W.W. 189
Boggs, P.T. 40
Dusinberre, G.M. 41
Boley, B.A. 39, 40 Bonnerot, R 5,6,25,41,143,147, 149,163,173
Ehrlich, L.W. 25, 41 Elliot, C. 204
Burke, H.H. 189
Estenssoro, L. 40 Fasano, A. 25, 40, 41
Cantin, G. 81
Ferris,D.H. 115,139
M.C. Bhattacharya, Int. J. Numer. Meth. Eng. 21, 239-265 (1985).
Carslaw, H.S. 40, 82
Fox, L. 39, 40, 42
M.N. Ozisk (ed.), Boundary value problems of heat conduction,
Chakravorty, S. 174,189
Fremond, D. 38,42
Int. textbook company, Scranton, Pennsylvania (1968).
Chatwin, C.R 5, 6, 66, 70, 71, 81,
Frey, J. 189
82, 112, 116, 132, 135, 136, 137,
Fujii, J. 189
e
M. Abramowitz and LA. Stegun (eds.), Handbook of mathematical functions, Dover Publications INC, New York (1965).
nc re
Cannon, J.R 173
fe re
9.
Bell, G.E. 14,40, 42
e as
P.D. Patel, Am. Inst. Aeronautics Astro. J. 6,2454-2464 (1968).
Davies, M.G. 82
e pl
8.
Dahmardah, H.O. 28,41, 140
BankofI, S.G. 39
174,198,204
l, fu
C. Elliott and J.R Ockendon, Weak and Variational methods for moving
Baines, M.J. 136,137,139
Bhattacharya, M.C. 46, 82, 83,
e us
Nazionale di Alta Matematica Francescon Severi, Rome, (1980).
15.
Comini, G. 42, 82 Crank,J. 4,6,27,38,39,40,41,
Alwan, A.A. 189
1.
7.
Abdullah, Z. 42 Abramowitz, M. 204
138,139,140,143,154,173,174, Churchill, S.W. 27,41 Clapeyron, B.P.E. 1, 6 Colony, R 41
GaBie, J.M. 25,27,28,29,41,87, 99,112
ok
Collatz, L. 40
82,112,139
bo
Ciment, M. 25,41,139
Furzeland, RM. 6, 25, 39, 40, 41,
e th
189
Gay, P. 187, 189 Gladwell, I. 40
206
Author index
Author index
207
If yo
Goodling, J.S. 27, 31,41,99, 112
Murray, W.D. 25,26,40,87,112
Sherman, B. 174
Khader, M.S. 27,31,41,99,112 Koh, J.C.Y. 25, 41 Kubota, K 189
Nicolson, D.W. 81
Smith, G.D. 82
Nicolson, P. 4, 39, 43, 44, 45, 52, 82,116,131
Sparrow, E.M. 42
Nickell, RE. 82 Noble, B. 14,40
Stefan, J. 1, 6, 191, 204
Sliepcevich, C.M. 42
u
Goodman, T.R 8, 9, 12, 13, 14, 28,40,99,112 Guenther, RB. 25,41
Jellison Jr., G.E. 189
d fin
Gupta, RS. 27, 28, 32, 41, 86, 99,
Kumar, D. 27, 28, 32, 41, 99, 112,
129, 130, 132, 134, 135, 136, 137,
116,130,134,135,136,137,139, 140,162,174
s
Ockendon,J.R. 40,42,173,204
Kuttler, J.R. 35,42
so
Haberland, C. 82, 112
Hashemi, H.T. 42
Lame, M.M. 1, 6
Hendel, RH. 189
Lardner, T.J. 40 Lazaridis, A. 25,41,204
Hill, S. 115, 139
Lewis, RW. 42,82,189
Hodgkins, W.R 40, 42, 173 Hougaard, P. 28, 41, 115, 121,
Mashaie, A. 41
125,136,137,138,139
Mayers, D.F. 28, 41, 140
Tarr, N.G. 189
Parekh, C.T. 81
Thomson, G.P. 82
Patankar, S.V. 68, 70, 71, 80, 81,
Tien, L.C. 27, 41 Torrance, KE. 83
Patel, P.D. 25, 26, 194, 204 Pham, Q.T. 35,42
Waechter, D. 189
Pohle, F.V. 40
Wait, R. 40
Poirier, D. 39,42,82
Westwater, J.W. 27,41,157,174
Poots, G. 40
Wilson, D.G. 40
Price, J.F. 41
Wilson, E.L. 82
Primicerio, M. 25, 40, 41, 173
Wood, A.S. 42
e as
Magenes,E.6,40,204
Ozisk, M.N. 204
e pl
Hoffman, KH. 39, 40, 112
Tacke, KH. 37,42
83
l, fu
Hill, C.D. 173
e us
Heitz, W.L. 27,41,174,157
Landis, F. 25,26,40,87, 112 Langford, D. 40
Oden, J.T. 82
Thomas, RE. 189
ce
136,137,138,139
Lahrmann, A. 82, 112
ur
Hansen, E.B. 28,41, 115, 121, 125,
Stegun, LA. 204 Sweet, RA. 25, 40
re
139,140,162,174,204
i th
112, 114, 115, 116, 118, 119, 125,
Sunderland, J.E. 39
Mayhew, Y.R 189
Huang, P.N.S. 40
Mazzoldi, P. 189
Huber, A. 25,41
Mehrabian, R 174, 189
Rogers, G.F.C. 189
Zerroukat, M. 5, 6, 66, 70, 71, 81,
Hunt, C.E. 189
Meyer, G.H. 25,41,42
Rogers, J.C.W. 139
82, 83, 112, 116, 132, 135, 136,
Hunter, LW. 35,42
Miaoulis, LN. 189
Rose,L.G.D.D.189
137,138,139,140,143,154,173,
Mikic, B.B. 189
Rubinstein, L.L 25, 39, 204
174,189
115, 126, 128, 130, Sackett, G.G. 204
e th
136,137,139 Moody, J.E. 189
Salcudean, M. 39, 41, 42, 82
J al uria, Y. 83
Morgan, K 82
Schvan, P. 189
Morton, KW. 136,137,139
Shamsunder, N. 42
Muehlbauer, J.C. 39
Shea, J.J. 12, 13, 14, 40
149,163,173
ok
Jaeger,J.C.40,82
Zlatev, Z. 82
bo
Modine, FA 189
Saitoh, T. 25, 41
Jamet, P. 5, 6, 25, 41, 143, 147,
Zienkiewicz, O.C. 42, 81, 82
e
Imura, H. 189
Miller, J,V.
nc re
Imber, M. 40
fe re
Hsu, S.C. 174,157,189
208
SUBJECT INDEX
Subject index
209
If adiabatic boundary 162, 163 algorithms 135, 161, 199,200,202,203
complex geometries 2, 65, 77 computational performance factor 45,53-64
d fin
adaptive mesh 126-130
u
ablation 146
yo
A
90,99,145
s
re
so
coordinate transformation 35, 115,
45,52,116,131 beam diameter 178-180
curved trapezoidal element 143
bisection interpolation 32
curved triangular element 143 cylindrical coordinates 27
D decision theory 2
non-linear problems 65-81
density 12, 44, 163
one-dimensional problems 43-70
density changes 23
heat balance method single phase ice melting 9-11 two phase problems 11-14
enthalpy technique 106-112
heat capacity 12,44,86,142,176
exponential FD equation 46-49
homograph approximation 38
extrapolation 25, 113, 116, 139
I
F
ice-melting 9-14, 191, 192
fictitious temperature 25, 53, 200
implicit finite difference methods 25,53,64,87
finite difference methods
incident power 177, 178, 179, 185, 186
finite element methods
three-dimensional problems 71-81
diffusion coefficient 114, 117
multiphase problems 147-154
Fourier number 29,44,86,114,176
distorted grid 143, 159, 160, 161, 165
freezing index 38 44,45,52,129
E
comparative performance 51-64
effective heat capacity 36, 37
iteration 3(}'33, 97,125,130,154,162
ok
effect of density change 23
197
iron see cast-iron
bo
fully implicit equation 4,27,39,43,
collapse of wall 162-164
interpolation 24,27,32,95,160, 161,
e th
68-71, 80-81
e
118
Dirac delta function 122, 123
classical finite difference methods
integral method 9, 99 integration by parts 122, 123, 126, 147
fixed grid FD methods 2, 24, 86, 110,
diffusivity 12, 44, 142, 176
cast-iron 175,178,180,187,188
instability 27,116
nc re
diffusion see Oxygen diffusion
adaptive mesh 126-130
c
see Oxygen diffusion problem
comparison of methods 51-64
test problems 54-58
diffusion length 177
implicit moving boundary problems
boundary value problems 46-81
fe re
comparison 54-58
121,122,123,124,125,126
e as
boundary conditions 3, 53, 78, 109
Hansen and Hougaard's method
e pl
cubic splines 27, 115
experimental results 178, 180, 181,
variable time step 88-106
l, fu
B
H
explicit computations
e us
Crank-Nicolson scheme 4, 39, 43, 44,
exact solutions see analytical
explicit FD methods 85-112
ce
asymptotic solution 115, 124
Green's function 113, 121
187,188
ur
cpu-time 54,59, 166
gravity 176, 180
87,115
convergence 27,99,100,102,105
cracks in solids 2
boundary value problems 54-83
Goodman integral method 9-14
evaporation 146, 157, 163
convective boundary condition 28,
159,198
Gauss elimination technique 53, 154
three dimensions 110-112 Euler FD equation 4, 39, 43, 44, 45, 52,
array size 44, 54
Blanchard and Fremond method 38
Gaussian distribution 178, 179
one dimension 33-38,106-110
147-154
approximate solutions 99, 119 arbitrary meshes 77
gas phase see vapour phase
i th
141,154
enthalpy 8,33,35,37,106-110 enthalpy techniques
conservative finite element method
apparent heat capacity 34, 35, 36, 37 appearing / disappearing phases
Galerkin equations 126, 129
conservation of energy 149, 151
analytical solutions 8-23, 54-58, 70, 80,100,156,200
energy balance 149, 151
K
G
kinetic theory 177
210
Subject index
Subject index
211
If yo
supercooled liquid 17
Kirchhoff transformation 35, 107
u
Lagrange formula 24, 66, 73, 115, 119 Laplace transforms 119
quadratic extrapolation 25, 113, 116,
Neumann's method 14-24
53,105,130
T
quadrature formulae 152, 153
s
results 185-189
system of linear equations 27,30,45,
160,161,197
non-linear PDEs 65,66, 74, 109, 110, 178
superheated solid 182
quadratic interpolation 24,26,27,
nodal volume 36, 37
computation 181-184
superheated liquid 157, 158 139
i th
laser heat treatment of material
negative concentration 116, 132
d fin
L
N
Q
Tacke's method 37
re
non-linear problems
theory 178-181
R
boundary value problems 65-81
Taylor series 27,48,58, 64, 108,
so
121, 198
moving boundary problems 107,
radiation 175, 179, 185, 188
latent heat of vaporisation 142, 147
178
rate of oxygen consumption 114, 117
test problems 54-58, 99-105, 162-165
reflectivity 177, 180, 181, 188
thermal conductivity 12,44, 86, 142,
46-64
ce
linear boundary-value problems
ur
latent heat of fusion 12, 86, 142, 176
o
one phase Stefan problems 28-33, 100
one·dimensional problems 9-39
oscillation of numerical schemes 70
three-dimensional problems 110
oxygen concentration 114, 117
e us
linear moving boundary problems
melt depth 185-188
comparison 135-139 fixed grid FD 118-121 formulation 117-118
melting range see mushy region
integral method 121-125 variable grid FD 130-135
multi-dimensional problems
moving boundary problems 110
p penetration of radiation 179
three phase Stefan problem see
semi-conductors 2, 177 shock waves 2
multi-phase Stefan problems transient problems 43-64
short time analytical solution 119, 124,132,201
trapezoidal element 143
silicon 175, 180, 185, 186, 188
trapezoidal rule 131, 184
simulation 2, 177, 188
triangular finite element 143
simple problems 14, 191
truncation method 115
single-phase Stefan problems 9,
two-dimensional problems 25, 76
28-33,100
nc re
boundary value problems 71-81
moving boundary problems 110
fe re
moving grid system 27, 115
boundary value problems 71-81
scanning speed 175, 176, 185, 186
e as
melting of ice see ice-melting melting of finite slab 11-14
s
e pl
adaptive mesh FE 126-130
176 thermal resistance 68, 80, 168 three dimensional problems
l, fu
oxygen diffusion problem
M
relaxation 130, 154, 162 Runge-Kutta method 14
two moving boundary problems see multi-phase Stefan problems
percentage deviation 54, 70, 101, 165
singularities 129, 172
Pham's method 35
smoothed results 115
comparison 165-173
plasma ppysics 2
solidification 17, 19,99, 178
FD methods 154-162
polynomial interpolation 27, 115,
spatial mesh refinement 14, 39
discretisation 90-94
specific energy 38
formulation 89-90
multi-phase Stefan problems
porous media 2
stability 5,27,45,87,89,95, 105
post-iterative enthalpy 34
steady state 114, 117, 118
power of the laser beam 177-186
step-wise functions 94, 95, 96, 195
u
ok
formulation 144-147
bo
160,161,197
mushy region 20,34,36,37, 192
computation 94-98
e th
FE methods 147-154
Murray and Landis method 25, 26, 87
two-phase Stefan problems
e
analytical solution 21-22
unsteady state see transient problems
212
Subject index
If u
yo
unequal spacings 27,77
V
variable thermal properties see non-
welding 192
l, fu
e us
weighted time step method 50-51
ce
W
ur
vectorisation 59, 61, 88, 139, 166, 173 viscosity 176, 180
so
variable space grid methods 26-27
re
variable time step methods 27-33
s
linear problems
i th
variable change 35, 115, 159, 198
d fin
validation 69, 79, 164 vaporisation 143,146,157
z 67,98,100
e as
zero temperature 9, 15, 18, 19, 48, 54,
e pl
zero concentration 114, 132, 133
e
nc re
fe re e th ok
bo
If
u yo
d fin
is th
ur so re
ce
l, fu e us
e as e pl
en er f re
ce
e
th
ok
bo