Classical Stefan Problem (eBook)

Basic Concepts, Modelling and Analysis

S.C. Gupta (Autor)

eBook Download: PDF

2003 | 1. Auflage
404 Seiten
Elsevier Science (Verlag)
978-0-08-052916-5 (ISBN)

Infos

This volume emphasises studies related to
classical Stefan problems. The term Stefan problem is
generally used for heat transfer problems with
phase-changes such
as from the liquid to the solid. Stefan problems have some
characteristics that are typical of them, but certain problems
arising in fields such as mathematical physics and engineering
also exhibit characteristics similar to them. The term
``classical distinguishes the formulation of these problems from
their weak formulation, in which the solution need not possess
classical derivatives. Under suitable assumptions, a weak solution
could be as good as a classical solution. In hyperbolic Stefan
problems, the characteristic features of Stefan problems are
present but unlike in Stefan problems, discontinuous solutions are
allowed because of the hyperbolic nature of the heat equation. The
numerical solutions of inverse Stefan problems, and the analysis of
direct Stefan problems are so integrated that it is difficult to
discuss one without referring to the other. So no strict line of
demarcation can be identified between a classical Stefan problem
and other similar problems. On the other hand, including every
related problem in the domain of classical Stefan problem would
require several volumes for their description. A suitable
compromise has to be made.
The basic concepts, modelling, and analysis of the classical
Stefan problems have been extensively investigated and there seems
to be a need to report the results at one place. This book
attempts to answer that need. Within the framework of the
classical Stefan problem with the emphasis on the basic concepts,
modelling and analysis, it tries to include some weak
solutions and analytical and numerical solutions also. The main
considerations behind this are the continuity and the clarity of
exposition. For example, the description of some phase-field
models in Chapter 4 arose out of this need for a smooth transition
between topics. In the mathematical formulation of Stefan
problems, the curvature effects and the kinetic condition are
incorporated with the help of the modified Gibbs-Thomson relation.
On the basis of some thermodynamical and metallurgical
considerations, the modified Gibbs-Thomson relation can be
derived, as has been done in the text, but the rigorous
mathematical justification comes from the fact that this relation
can be obtained by taking appropriate limits of phase-field
models. Because of the unacceptability of some phase-field models
due their so-called thermodynamical inconsistency, some consistent
models have also been described. This completes the discussion of
phase-field models in the present context.
Making this volume self-contained would require reporting and
deriving several results from tensor analysis, differential
geometry, non-equilibrium thermodynamics, physics and functional
analysis. The text is enriched with appropriate
references so as not to enlarge the scope of the book. The proofs
of propositions and theorems are often lengthy and different from
one another. Presenting them in a condensed way may not be of much
help to the reader. Therefore only the main features of proofs
and a few results have been presented to suggest the essential
flavour of the theme of investigation. However at each place,
appropriate references have been cited so that inquisitive
readers can follow them on their own.
Each chapter begins with basic concepts, objectives and the
directions in which the subject matter has grown. This is followed
by reviews - in some cases quite detailed - of published works. In a
work of this type, the author has to make a suitable compromise
between length restrictions and understandability.

---

This volume emphasises studies related to classical Stefan problems. The term "e;Stefan problem"e; is generally used for heat transfer problems with phase-changes such as from the liquid to the solid. Stefan problems have some characteristics that are typical of them, but certain problems arising in fields such as mathematical physics and engineering also exhibit characteristics similar to them. The term ``classical"e; distinguishes the formulation of these problems from their weak formulation, in which the solution need not possess classical derivatives. Under suitable assumptions, a weak solution could be as good as a classical solution. In hyperbolic Stefan problems, the characteristic features of Stefan problems are present but unlike in Stefan problems, discontinuous solutions are allowed because of the hyperbolic nature of the heat equation. The numerical solutions of inverse Stefan problems, and the analysis of direct Stefan problems are so integrated that it is difficult to discuss one without referring to the other. So no strict line of demarcation can be identified between a classical Stefan problem and other similar problems. On the other hand, including every related problem in the domain of classical Stefan problem would require several volumes for their description. A suitable compromise has to be made. The basic concepts, modelling, and analysis of the classical Stefan problems have been extensively investigated and there seems to be a need to report the results at one place. This book attempts to answer that need.

Inhalt

Cover 1
Contents 7
Chapter 1. The Stefan Problem and its Classical Formulation 19
1.1 Some Stefan and Stefan-like Problems 19
1.2 Free Boundary Problems with Free Boundaries of Codimension-two 36
1.3 The Classical Stefan Problem in One-dimension and the Neumann Solution 37
1.4 Classical Formulation of Multi-dimensional Stefan Problems 41
Chapter 2. Thermodynamical and Metallurgical Aspects of Stefan Problems 57
2.1 Thermodynamical Aspects 57
2.2 Some Metallurgical Aspects of Stefan Problems 65
2.3 Morphological Instability of the Solid--Liquid Interface 70
2.4 Non-material Singular Surface: Generalized Stefan Condition 73
Chapter 3. Extended Classical Formulations of n-phase Stefan Problems with n > 1
3.1 One-phase Problems 79
3.2 Extended Classical Formulations of Two-phase Stefan Problems 84
3.3 Stefan problems with Implicit Free Boundary Conditions 94
Chapter 4. Stefan Problem with Supercooling: Classical Formulation and Analysis 103
4.1 Introduction 103
4.2 A Phase-field Model for Solidification using Landau Ginzburg Free Energy Functional 104
4.3 Some Thermodynamically Consistent Phase-field and Phase Relaxation Models of Solidification 112
4.4 Solidification of Supercooled Liquid Without Curvature Effect and Kinetic Undercooling: Analysis of the Solution 122
4.5 Analysis of Supercooled Stefan Problems with the Modified Gibbs Thomson Relation 130
Chapter 5. Superheating due to Volumetric Heat Sources: The Formulation and Analysis 147
5.1 The Classical Enthalpy Formulation of a One-dimensional Problem 147
5.2 The Weak Solution 151
5.3 Blow-up and Regularization 157
Chapter 6. Steady-State and Degenerate Classical Stefan Problems 160
6.1 Some Steady-state Stefan Problems 160
6.2 Degenerate Stefan Problems 161
Chapter 7. Elliptic and Parabolic Variational Inequalities 166
7.1 Introduction 166
7.2 The Elliptic Variational Inequality 167
7.3 The Parabolic Variational Inequality 188
7.4 Some Variational Inequality Formulations of Classical Stefan Problems 192
Chapter 8. The Hyperbolic Stefan Problem 214
8.1 Introduction 214
8.2 Model I: Hyperbolic Stefan Problem with Temperature Continuity at the Interface 217
8.3 Model II: Formulation with Temperature Discontinuity at the Interface 225
8.4 Model III: Delay in the Response of Energy to Latent and Sensible Heats 234
Chapter 9. Inverse Stefan Problems 242
9.1 Introduction 242
9.2 Well-posedness of the solution 243
9.3 Regularization 249
9.4 Determination of Unknown Parameters in Inverse Stefan Problems 262
9.5 Regularization of Inverse Heat Conduction Problems by Imposing Suitable Restrictions on the solution 267
9.6 Regularization of Inverse Stefan Problems Formulated as Equations in the form of Convolution Integrals 270
9.7 Inverse Stefan Problems Formulated as Defect Minimization Problems 275
Chapter 10. Analysis of the Classical Solutions of Stefan Problems 289
10.1 One-dimensional One-phase Stefan Problems 290
10.2 One-dimensional Two-phase Stefan Problems 318
10.3 Analysis of the Classical Solutions of Multi-dimensional Stefan Problems 333
Chapter 11. Regularity of the Weak Solutions of Some Stefan Problems 340
11.1 Regularity of the Weak solutions of One-dimensional Stefan Problems 340
11.2 Regularity of the Weak solutions of Multi-dimensional Stefan Problems 347
Appendix A. Preliminaries 356
Appendix B. Some Function Spaces and Norms 363
Appendix C. Fixed Point Theorems and Maximum Principles 367
Appendix D. Sobolev Spaces 369
Bibliography 373
Captions for Figures 399
Subject Index 401

Textauszug

Chapter 2

Thermodynamical and Metallurgical Aspects of Stefan Problems

2.1 Thermodynamical Aspects

Several generalizations and extensions of the two-phase problem (1.4.4)–(1.4.8) are possible but before dealing with them, we give some definitions. Many terms like equilibrium temperature, thermal conductivity, latent heat, enthalpy, etc., have been used earlier without defining or explaining them for the reason that greater concern was shown for the mathematical formulation. Some thermodynamical aspects of Stefan problems will be discussed in this section.

2.1.1 Microscopic and macroscopic models

A system is called microscopic if it is of ‘small dimensions’, roughly of the size of an atom or a molecule, i.e., of the size of 10−10 meters. In a microscopic model, thermal, mechanical or chemical phenomenon is studied at an atomic or molecular level. In a macroscopic model, the system is ‘large enough’ of the order of one micron, be visible with a microscope using ordinary light. A macroscopic system consists of large numbers of atoms or molecules. There is an important difference between a microscopic model and a macroscopic model. In the microscopic model, the description of an individual particle motion/action, even if available would not disclose the gross behaviour/properties of the system. For example, it is a striking fact, and one which is difficult to understand in microscopic detail that simple atoms forming a gas can condense abruptly to form a liquid with very different properties.

It is well known that a transport phenomenon whether it is electrical, heat or mass transfer, occurs due to changes in the energy levels of atoms or molecules. Imagine applying Newtonian laws of motion to 1020 molecules and obtaining information for each molecule in a transport phenomenon such as heat transfer. It is an awesome task. All the mathematical formulations discussed earlier are based on macroscopic modelling. Does it meant that microscopic modelling is not required? However complicated, microscopic models cannot be totally discarded. Kinetic theory of matter (cf. [35, 36]) applies the laws of mechanics to individual molecules of a system and enables one to calculate, for example, the numerical values of heat capacities, heats of transformation, viscosity. These parameters can be explicitly defined in a macroscopic model, but their numerical values can be obtained only on the basis of a molecular model. As stated earlier some of the equations of mathematical physics, for example, the heat equation can be easily derived on the basis of a microscopic model.

The advantage of the macroscopic model or a continuum hypothesis, and in which we are interested is that the gross behavior of the system can be studied and laws of Newtonian mechanics can be applied to the bulk matter. An approach more general than macroscopic modelling is of statistical thermodynamics which, ignores the detailed consideration of molecules as individuals, and applies statistical methods to find the distribution of very large number of molecules that make up a macroscopic piece of matter over energy states of the matter. The equations of conservation of mass, momentum and energy arising in continuum mechanics can be obtained as particular cases of the Boltzmann equation [36].

Both kinetic theory and statistical thermodynamics were first developed on the assumption that the laws of mechanics deduced from the behavior of matter in bulk, could be applied without change to particles like molecules and electrons. As science progressed, it became evident that at least in some respects this assumption was not correct, that the conclusions drawn from it by logical methods did not agree with experimental facts. For example, experiments suggest that the specific heat of many solids at constant volume approach the Dulong-Petit [35] value of 3R (R is universal gas constant) at high temperatures, but decreases to zero at very low temperatures. This behavior of solids can be explained with the help of a quantum mechanics approach. The failure of small scale systems to obey the same laws as large scale systems led to the development of quantum theory and quantum mechanics. Statistical thermodynamics is best treated today from the view point of quantum mechanics. On a microscopic scale classical mechanics does not apply and must be replaced by quantum mechanics. For further details of quantum theory, the reader is referred to [35, 36, 37].

2.1.2 Laws of classical thermodynamics

We shall be dealing here mostly with equilibrium thermodynamics. Thermodynamics is the study of energy and its transformation. There are many different types of energy but most studies of thermodynamics are primarily concerned with two forms of energy: heat and work. Thermodynamics deals with the macroscopic properties of matter and is an empirical science. It is developed on a small number of principles which are generalizations made from experience.

Thermodynamic equilibrium:

When a system is in thermal, mechanical and chemical equilibrium, it is said to be in thermodynamic equilibrium. In thermal equilibrium, the temperature will be the same at all points of the system. In mechanical equilibrium, all motions, expansions or contractions of the system are absent. Note that atoms are still in motion. When all the chemical reactions stop then the system is in chemical equilibrium.

Reversible and quasi-static process:

A process is called reversible if the initial state of the system can be restored with no observable effects in the system and its surroundings. If a process is not reversible, it is called irreversible. If a process is carried out in such a way that at any time the system departs only infinitesimally from the equilibrium state it is called a quasi-static process.

First law of thermodynamics:

Internal energy (internal heat energy) of a system is the sum of all the individual kinetic energies of motion and energies of interaction (potential energies) of the particles in the system. Internal energy can be transformed to do work and produce heat. One form of the first law of equilibrium thermodynamics is

(2.1.1)

Here, dU is the change in the internal energy of the system from equilibrium state a to another equilibrium state b, d′Q is the heat flow into the system during the change of state and d′W is the work done by the system when the system changes its equilibrium state from a to b. The dash indicates that the quantities are not exact differentials. Internal energy is a state property, i.e., internal energy in state b does not depend on the process by which the system has been brought from state a to state b. dU is an exact differential but not d′Q and d′W. It can be easily shown that work is path-dependent and so is heat flow [35]. Equation (2.1.1) holds for both reversible and irreversible processes.

Second law of thermodynamics: Entropy

Some changes in a system can take place only in one direction. Consider an isolated system in which a body at temperature T1 is in contact with a heat reservoir at temperature T2 > T1. Heat will flow from the reservoir to the body and raise its temperature to T2. Is it possible for the body to cool down to temperature T1 by releasing heat to the reservoir? The change in this direction is not possible. It may be noted that the total energy of the system consisting of the body and the reservoir is conserved even if a reverse change takes place. Therefore if we are looking for some property of the system whose change can tell us the direction in which the reverse change is possible, then it cannot be energy. This property of the system is called entropy, denoted by S and defined as

(2.1.2)

Here, 1 and 2 are the two equilibrium states, of a system. Entropy is defined only for reversible processes. The second law of thermodynamics, states that processes in which the entropy of an isolated system (i.e., dQ = 0) would decrease do not exist. Specific entropy (entropy per unit mass) will be denoted by .

If hot water is mixed with cold water, then the entropy of the cold water will increase more than the decrease in the entropy of the hot water which can be checked from (2.1.2). So there will be an increase in the entropy of the system consisting of both cold and hot water. This tells us that heat cannot flow from cold water to hot water as decrease in the entropy of an isolated system is not possible. Unlike energy or momentum, entropy is not conserved. The first law of thermodynamics states that energy can neither be created nor destroyed and the second law states that entropy can not be destroyed but it can be created. The process is called isentropic if the entropy of the system does not change during the process.

2.1.3 Some thermodynamic variables and thermal parameters

Specific heat capacity or specific heat:

If no...

Produkt-Details

• Erscheint lt. Verlag: 22.10.2003
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• ISBN-10: 0-08-052916-X / 008052916X
• ISBN-13: 978-0-08-052916-5 / 9780080529165