Mathematical Modeling Of Melting And Freezing Processes

V. Alexiades

• Publisher: Routledge
• Year: 2018
• ISBN:
  9781351433273
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Industrial & Technical Chemistry

This reference book presents mathematical models of melting and solidification processes that are the key to the effective performance of latent heat thermal energy storage systems (LHTES), utilized in a wide range of heat transfer and industrial applications. This topic has spurred a growth in research into LHTES applications in energy conservation and utilization, space station power systems, and thermal protection of electronic equipment in hostile environments. Further, interest in mathematical modeling has increased with the speread of high powered computers used in most industrial and academic settings. In two sections, the book first describes modeling of phase change processes and then describes applications for LHTES. It is aimed at graduate students, researchers, and practicing engineers in heat transfer, materials processing, multiphase systems, energy conservation, metallurgy, microelectronics, and cryosurgery.

CHAPTER 1

PROBLEM FORMULATION

The purpose of simulation is to gain understanding of the process being simulated. Building understanding is complex, involving iterative use of experiment and observation on the “physical plane” and model building and analysis on the “conceptual plane.” The act of formulating a mathematical model tests our understanding of the physical process. Do we know what is important, and what may be ignored? Do we believe that we know the underlying relationships between the entities making up our process well enough to formulate them in mathematical terms?

This book is concerned with simulating the processes of melting and freezing on a macroscopic scale. Very few processes are more familiar to us than these. Yet our interest in them, motivated by processes of increasing complexity, has grown steadily in recent years, while our intuition is increasingly tested. On the one hand we know that it will take half a day to defrost a piece of meat. On the other hand, we have not the slightest intuition about the performance of a material used to store solar energy as the latent heat of melting as it cycles through repeated freeze/thaw cycles. The growing use of Silicon brings us into intimate contact with processes involving freezing of supercooled liquid; alloy processing in space leads us to surface-tension driven convection; laser induced melting brings us to the edge (and perhaps beyond it) of credibility of the heat equation, our fundamental tool for heat transfer modeling on a macroscopic scale. Time scales have broadened. When Stefan formulated the classical phase change model, the credible time scale for processes of interest could be measured in days to several years. In recent years the authors have dealt with problems whose time scales are measured in pico-seconds, and with problems whose time scales are literally 20,000,000 years. Thus our need to know has expanded, and with it, our intuition about even simple processes has waned. To build our intuition we need experiment and observation on the one hand, and better simulation tools on the other.

In this chapter we formulate the classical Stefan-type model of melting and freezing. Historically this problem has been regarded as lying on the border between tractable and intractable problems. It is nonlinear, and its principal difficulty lies in the fact that one of its unknowns is the region in which it is to be solved. For this reason it is called a “moving boundary problem.” Our intuition tells us that for most reasonable phase change processes energy conservation prevails. This is manifested in two ways:...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1. PROBLEM FORMULATION
• 2. PROBLEMS WITH EXPLICIT SOLUTIONS
• 3. ANALYTICAL APPROXIMATIONS
• 4. NUMERICAL METHODS – THE ENTHALPY FORMULATION
• 5. APPLYING THE TECHNIQUES OF MODELING
• BIBLIOGRAPHY
• INDEX