MATHEMATICAL
MODELING
OF MELTING
AND FREEZING
PROCESSES
Vasilios Alexiades
HEMISPHERE PUBLISHING CORPORATION
A member of the Taylor & Francis Group
Washington Philadelphia London
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MATHEMATICAL MODELING OF MELTING AND FREEZING PROCESSES
Copyright © 1993 by Hemisphere Publishing Corporation. All rights reserved. Printed in the
United States of America. Except as permitted under the United States Copyright Act of 1976,
no part of this publication may be reproduced or distributed in any form or by any means, or
stored in a database or retrieval system, without the prior written permission of the publisher.
1 2 3 4 5 6 7 8 9 0 B R B R 9 8 7 6 5 4 3 2Cover design by Michelle Fleitz.
Preface............................................. x
1. PROBLEM FORMULATION 1
1.1 AN OVERVIEW OF THE PHENOMENA INVOLVED
IN A PHASE CHANGE 2
1.2. FORMULATION OF THE STEFAN PROBLEM 6
1.2.A Introduction 6
1.2.B Assumptions 6
1.2.C Heat conduction 8
1.2.D Boundary conditions 17
1.2.E Interface conditions 19
1.2.F The Stefan Problem 22
PROBLEMS 26
1.3. GENERAL MELTING AND SOLIDIFICATION PROCESSES 29
PROBLEMS 32
2. PROBLEMS WITH EXPLICIT SOLUTIONS 33
2.1. THE ONE-PHASE STEFAN PROBLEM 34
2.1.A Introduction 34
2.1.B The Neumann Solution 36
2.1.C Dimensionless form 37
2.1.D The root @lambda@ vs the Stefan Number 38
2.1.E Example: Melting a slab of ice 40
2.1.F The case of small Stefan Number 43
PROBLEMS 44
2.2. THE TWO-PHASE PROBLEM ON A SEMI-INFINITE SLAB 46
2.2.A Problem statement and solution 46
2.2.B Dimensionless form 48
2.2.C Approximations to the root @lambda@ 49
2.2.D Approximating the finite slab case 50
2.2.E Energy content and Stefan Numbers 51
2.2.F Shape of melting and cooling curves 52
2.2.G An example 55
PROBLEMS 56
2.3. THE EFFECT OF DENSITY CHANGE 59
2.3.A Physical effects 59
2.3.B Expansion with bulk movement due to rhoL<=rhoS 61
2.3.C Application to cryosurgery 64
2.3.D Void formation 68
2.3.E Conservation laws and interface conditions 72
PROBLEMS 77
2.4. SOLIDIFICATION OF A SUPERCOOLED MELT 79
2.4.A How supercooling arises 79
2.4.B One-phase supercooled solidification 82
2.4.C Two-phase supercooled solidification 84
2.4.D Supercooled solidification with density change87
2.4.E Steady-state of a finite slab 87
2.4.F Phase equilibrium and the Gibbs-Thomson effect88
2.4.G Mullins-Sekerka morphological stability
analysis 94
PROBLEMS 96
2.5. CHANGE OF PHASE IN A BINARY ALLOY 98
2.5.A Introduction 98
2.5.B The phase diagram 99
2.5.C Interdiffusion 102
2.5.D A simple binary alloy solidification model 104
2.5.E The Rubinstein similarity solution 106
2.5.F Shortcomings of the simple model 107
2.5.G Uncoupled models of alloy solidification 109
2.5.H The Tien-Geiger model for freezing over an
extended range 110
2.5.I Rapid freezing of a finite slab with
stirring of melt 112
PROBLEMS 116
2.6. SIMILARITY SOLUTIONS IN CYLINDRICAL
AND SPHERICAL GEOMETRIES 117
2.6.A Similarity solutions 117
2.6.B Axially-symetric melting due to a line source118
2.6.C Freezing of supercooled liquid 119
PROBLEMS 120
2.7. BENCHMARK SOLUTIONS OF MULTIDIMENSIONAL
PROBLEMS FOR SIMULATION VERIFICATION 121
2.7.A Necessity of benchmark solutions 121
2.7.B Phase-change in a box 121
2.7.C Summary of the method 123
PROBLEMS 123
3. ANALYTICAL APPROXIMATIONS 125
3.1. THE QUASISTATIONARY APPROXIMATION 126
3.1.A Introduction 126
3.1.B One-phase Stefan Problem with imposed
temperature 127
3.1.C One-phase Stefan Problem with imposed flux 134
3.1.D The case of convective boundary condition 137
3.1.E Volumetric heating 140
PROBLEMS 141
3.2. QUASISTATIONARY APPROXIMATION OF AXIALLY
OR RADIALLY SYMMETRIC PROCESSES 144
3.2.A Introduction 144
3.2.B Outward melting of a hollow cylinder 144
3.2.C Inward melting of a cylinder 147
3.2.D Inward melting of a sphere 148
3.2.E Simple approximations of a heat storage
process 149
PROBLEMS 152
3.3. PERTURBATION METHODS FOR ONE-PHASE PROBLEMS 154
3.3.A Introduction 154
3.3.B Landau transformation 155
3.3.C Front location as independent variable 157
3.3.D Rapid freezing of dilute alloys 160
PROBLEMS 161
3.4. THE MEGERLIN AND HEAT-BALANCE-INTEGRAL
METHODS 162
3.4.A Introduction 162
3.4.B The Megerlin method 162
3.4.C The Heat-Balance-Integral method 165
PROBLEMS 167
3.5. SOME MELTING TIME RELATIONS 168
3.5.A Introduction 168
3.5.B Melt-time for a simple PCM body with
imposed temperature 168
3.5.C Melt-time for a simple PCM body with
convective boundary condition 171
3.5.D Melt-time for a rectangular body under
imposed temperature 173
3.5.E Freezing of a PCM cylinder array 174
PROBLEMS 178
4. NUMERICAL METHODS - THE ENTHALPY FORMULATION 180
4.1. NUMERICAL HEAT TRANSFER 181
4.1.A Introduction 181
4.1.B Control volume discretization of the
conservation law 184
4.1.C Discretization of boundary conditions 191
4.1.D The discrete problem 192
4.1.E Explicit time updating 194
4.1.F Implicit time updating 197
4.1.G Heat conduction in 2 or 3 dimensions 199
4.1.H Internal heat source 202
4.1.I Some programming suggestions 203
PROBLEMS 205
4.2. BRIEF OVERVIEW OF NUMERICAL METHODS
FOR PHASE CHANGE PROBLEMS 210
4.3. THE ENTHALPY METHOD IN ONE SPACE DIMENSION 212
4.3.A Introduction 212
4.3.B The enthalpy method 212
4.3.C A time-explicit scheme 217
4.3.D Performance of the explicit scheme on a
one-phase problem 218
4.3.E Implicit schemes 224
4.3.F Implicit scheme by Newton iteration 229
4.3.G Performance of the schemes on the two-phase
Stefan problem 231
4.3.H Performance on problems with unequal
properties 236
4.3.I Implicit versus explicit schemes 238
4.3.J Use of the enthalpy method for a
multi-layered slab 239
PROBLEMS 241
4.4. MATHEMATICAL FRAMEWORK
OF THE ENTHALPY FORMULATION 242
4.4.A Introduction 242
4.4.B Weak derivatives 244
4.4.C Examples of weak formulation of problems 248
4.4.D Classical formulation of Stefan Problems
in 3-dimensions 250
4.4.E Weak formulation of the Stefan Problem 253
PROBLEMS 260
4.5. CONVERGENCE OF THE ENTHALPY SCHEME
AND EXISTENCE OF THE WEAK SOLUTION 260
4.5.A Introduction 260
4.5.B Structure of the proof 261
4.5.C Proof of CLAIM 1 264
4.5.D Proof of CLAIM 2 269
4.5.E Proof of CLAIM 3 272
4.5.F Note on error estimates 273
PROBLEMS 273
5. APPLYING THE TECHNIQUES OF MODELING 274
5.1. LATENT HEAT STORAGE
AND PHASE CHANGE MATERIALS 276
5.1.A Introduction 276
5.1.B A simple heat storage example 277
5.1.C Trombe wall 279
PROBLEMS 283
5.2. NUMERICAL SIMULATION
OF A LATENT HEAT TROMBE WALL 284
PROBLEMS 290
5.3. DEVELOPMENT OF A SIMULATION CODE
FOR A LHTES SYSTEM IN A SPACE STATION 290
5.3.A Introduction 290
5.3.B The system of interest 291
5.3.C Rough sizing of the storage system via
the quasistationary approximation 296
5.3.D Numerical simulation 298
5.3.E Some simulation results 299
PROBLEMS 304
BIBLIOGRAPHY 305
INDEX 321