MATHEMATICAL MODELING OF MELTING AND FREEZING PROCESSES

Transcription

1 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

2 CONTENTS Preface xi 1. PROBLEM FORMULATION AN OVERVIEW OF THE PHENOMENA INVOLVED IN A PHASE CHANGE 2 PROBLEMS FORMULATION OF THE STEFAN PROBLEM A Introduction B Assumptions C Heat conduction D Boundary conditions E Interface conditions F The Stefan Problem 22 PROBLEMS GENERAL MELTING AND SOLIDIFICATION PROCESSES 29 PROBLEMS PROBLEMS WITH EXPLICIT SOLUTIONS THE ONE-PHASE STEFAN PROBLEM A Introduction B The Neumann Solution C Dimensionless form D The root X vs the Stefan Number E Example: Melting a slab of ice F The case of small Stefan Number 43 PROBLEMS THE TWO-PHASE PROBLEM ON A SEMI-INFINITE SLAB A Problem statement and solution B Dimensionless form C Approximations to the root X D Approximating the finite slab case E Energy content and Stefan Numbers F Shape of melting and cooling curves G An example 55 PROBLEMS 56 v

3 2.3. THE EFFECT OF DENSITY CHANGE A Physical effects B Expansion with bulk movement due to p L < p s C Application to cryosurgery D Void formation E Conservation laws and interface conditions 72 PROBLEMS SOLIDIFICATION OF A SUPERCOOLED MELT A How supercooling arises B One-phase supercooled solidification : C Two-phase supercooled solidification D Supercooled solidification with density change E Steady-state of a finite slab F Phase equilibrium and the Gibbs-Thomson effect G Mullins-Sekerka morphological stability analysis 94 PROBLEMS CHANGE OF PHASE IN A BINARY ALLOY A Introduction B The phase diagram C Interdiffusion D A simple binary alloy solidification model E The Rubinstein similarity solution F Shortcomings of the simple model G Uncoupled models of alloy solidification H The Tien-Geiger model for freezing over an extended range Rapid freezing of a finite slab with stining of melt 112 PROBLEMS SIMILARITY SOLUTIONS IN CYLINDRICAL AND SPHERICAL GEOMETRIES A Similarity solutions B Axially-symetric melting due to a line source C Freezing of supercooled liquid 119 PROBLEMS BENCHMARK SOLUTIONS OF MULTIDIMENSIONAL PROBLEMS FOR SIMULATION VERIFICATION A Necessity of benchmark solutions B Phase-change in a box C Summary of the method 123 PROBLEMS 123

4 3. ANALYTICAL APPROXIMATIONS THE QUASISTATIONARY APPROXIMATION A Introduction B One-phase Stefan Problem with imposed temperature C One-phase Stefan Problem with imposed llux D The case of convective boundary condition E Volumetric heating 140 PROBLEMS QUASISTATIONARY APPROXIMATION OF AXIALLY OR RADIALLY SYMMETRIC PROCESSES A Introduction B Outward melting of a hollow cylinder C Inward melting of a cylinder D Inward melting of a sphere E Simple approximations of a heat storage process 149 PROBLEMS PERTURBATION METHODS FOR ONE-PHASE PROBLEMS A Introduction B Landau transformation C Front location as independent variable D Rapid freezing of dilute alloys 160 PROBLEMS THE MEGERLIN AND HEAT-BALANCE-INTEGRAL METHODS A Introduction B The Megerlin method C The Heat-Balance-Integral method 165 PROBLEMS SOME MELTING TIME RELATIONS A Introduction B Melt-time for a simple PCM body with imposed temperature C Melt-time for a simple PCM body with convective boundary condition D Melt-time for a rectangular body under imposed temperature E Freezing of a PCM cylinder array 174 PROBLEMS NUMERICAL METHODS - THE ENTHALPY FORMULATION NUMERICAL HEAT TRANSFER 181

5 Vlll CONTENTS 4.1.A Introduction B Control volume discretization of the conservation law C Discretization of boundary conditions D The discrete problem E Explicit time updating F Implicit time updating G Heat conduction in 2 or 3 dimensions H Internal heat source Some programming suggestions 203 PROBLEMS BRIEF OVERVIEW OF NUMERICAL METHODS FOR PHASE CHANGE PROBLEMS THE ENTHALPY METHOD IN ONE SPACE DIMENSION A Introduction B The enthalpy method C A time-explicit scheme D Performance of the explicit scheme on a one-phase problem E Implicit schemes F Implicit scheme by Newton iteration G Performance of the schemes on the two-phase Stefan problem H Performance on problems with unequal properties Implicit versus explicit schemes J Use of the enthalpy method for a multi-layered slab 239 PROBLEMS MATHEMATICAL FRAMEWORK OF THE ENTHALPY FORMULATION A Introduction B Weak derivatives C Examples of weak formulation of problems D Classical formulation of Stefan Problems in 3-dimensions E Weak formulation of the Stefan Problem 253 PROBLEMS CONVERGENCE OF THE ENTHALPY SCHEME AND EXISTENCE OF THE WEAK SOLUTION A Introduction B Structure of the proof C Proof of CLAIM D Proof of CLAIM E Proof of CLAIM F Note on error estimates 273 PROBLEMS 273

6 CONTENTS ix 5. APPLYING THE TECHNIQUES OF MODELING LATENT HEAT STORAGE AND PHASE CHANGE MATERIALS A Introduction B A simple heat storage example C Trombe wall 279 PROBLEMS NUMERICAL SIMULATION OF A LATENT HEAT TROMBE WALL 284 PROBLEMS DEVELOPMENT OF A SIMULATION CODE FOR A LHTES SYSTEM IN A SPACE STATION A Introduction B The system of interest C Rough sizing of the storage system via the quasistationary approximation D Numerical simulation E Some simulation results 299 PROBLEMS 304 BIBLIOGRAPHY 305 INDEX 321