MATHEMATICAL MODELING OF MELTING AND FREEZING PROCESSES Vasilios Alexiades The University of Tennessee and Oak Ridge National Laboratory Alan D. Solomon Consultant Formerly at Oak Ridge National Laboratory OHEMISPHERE PUBLISHING CORPORATION A member of the Taylor & Francis Group Washington Philadelphia London
CONTENTS Preface xi 1. PROBLEM FORMULATION 1 1.1 AN OVERVIEW OF THE PHENOMENA INVOLVED IN A PHASE CHANGE 2 PROBLEMS 5 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 X 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 X 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 v
2.3. THE EFFECT OF DENSITY CHANGE 59 2.3.A Physical effects 59 2.3.B Expansion with bulk movement due to p L < p s 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 change 87 2.4.E Steady-state of a finite slab 87 2.4.F Phase equilibrium and the Gibbs-Thomson effect 88 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 2.5.1 Rapid freezing of a finite slab with stining 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 source 118 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 llux 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
Vlll CONTENTS 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 4.1.1 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 4.3.1 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
CONTENTS ix 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