Analytical and numerical solutions of some moving boundary problems. vol.1

Abstract

Title of the Thesis: Analytical and Numerical Solutions of Some Moving Boundary Problems This thesis presents analytical and numerical solutions of some Moving Boundary Problems (MBP’s), especially those pertaining to solidification/melting. Since solidification problems are mathematically analogous to melting problems, it is sufficient to study either of them. So only solidification problems will be addressed here except when reference to melting problems is specifically called for. In mathematical literature, solidification problems are generally referred to as Stefan problems and are studied from many perspectives. A typical feature of Stefan problems is that apart from the fixed boundary of the region under consideration, there exists a boundary which is not known a priori and on which some boundary conditions, often very complicated, are prescribed. For example, in the case of solidification, the unknown boundary is the solid-liquid interface, and thermodynamic equilibrium conditions and dynamic compatibility conditions could be prescribed on this boundary. Such problems are highly nonlinear. Analytical and numerical solutions are difficult to obtain in the case of problems with conditions that are to be satisfied on unknown boundaries. Such challenges make the search for solutions to these problems more intellectually exciting. Problems involving unknown boundaries are encountered in physics, chemistry, biology and several fields of engineering. The unknown boundary is variously designated in different disciplines as freezing/melting front, phase-change boundary, free boundary etc. An unknown boundary which is time-dependent is called a moving boundary, and problems in which moving boundaries occur are called ‘moving boundary problems’. The problems categorised as Stefan problems in mathematical literature represent a subclass of MBP’s. Since this thesis deals also with some problems that are beyond the purview of Stefan problems, its title uses the more general term ‘moving boundary problems’. Coming within the scope of the thesis are: (i) Analytical and numerical solutions to solidification problems with heat transfer alone and with both heat transfer and mass transfer, (ii) Numerical solutions of multi-dimensional solidification problems with extended freezing temperature range, and (iii) Numerical solutions of composite slabs with boundaries that deform and move with time. The third point signifies the author’s attempt at coupling solid mechanics, which formed the focus of his doctoral research (1969), with his later interest in MBP’s. Solidification problems can be divided into three main classes. Class I problems are those in which solidification starts simultaneously at all the points of the boundary of the region occupied by the melt. The freezing front can be assumed to be smooth. These problems can be multi-dimensional in space coordinates. In some special situations, they can be modelled mathematically as one-dimensional problems as well. In Class II problems, solidification starts over a portion, while in Class III it starts at a point of the boundary of the region under consideration. These problems have to be necessarily multi-dimensional. In these cases, the freezing front, which is the union of the spread of solidification along the surface and the growth towards the interior, is a non-smooth boundary. Although for very simple situations some exact analytical solutions are available, exact analytical solutions to Stefan problems for any given geometry, and initial and boundary conditions are extremely difficult and can even be considered inconceivable to obtain. The author’s contribution consists in the development of a method for obtaining short-time analytical solutions to all the three classes of solidification problems and a numerical method for solving multi-dimensional solidification problems with extended freezing temperature range. Most of the analytical and numerical solutions presented here are concerned with mathematical formulations which are known as classical formulations in the literature. In these formulations, energy equations are written separately for each of the regions considered. It is assumed that there is a sharp interface between any two regions and that the energy balance and mass balance across it are suitably taken into account. Method for Analytical Solutions The solid and liquid regions existing at any time are first embedded in the original region occupied by the melt. This facilitates the construction of the solution of Fourier heat equations for well-defined geometries. The initial temperature of the melt is known. For the solid region, however, initial temperature is fictitious and unknown as initially no solid is present. In order to satisfy the boundary conditions, this original region is extended suitably and fictitious initial temperatures in these extensions are assumed. A major advantage of this approach is that by employing Green’s function for unbounded medium, the integral representations are made simple and amenable to repeated differentiations with respect to time variable. The initial temperatures help satisfy the boundary conditions at the fixed boundaries and at the freezing front. Suitable series expansions are assumed for the known and unknown quantities. Unknown coefficients occurring in these expansions can be determined with the help of boundary conditions if (i) they are differentiated repeatedly with respect to ?t, which is the square root of time and (ii) the limits t ? 0+ are taken for each differentiation. The method usually yields an analytical solution valid for small values of time, but in some cases, long-time analytical solutions can also be obtained systematically. Solutions have been constructed to problems concerning semi-infinite regions and finite slabs, radially symmetric problems pertaining to the inside and outside regions of spheres and cylinders, and axisymmetric problems of inside and outside regions of long cylinders.The amount of heat extracted from the system at the fixed boundaries should be equal to the amount of heat given out by the system in cooling at all intermediate times till the solidification is complete. It is possible to extend this numerical scheme to three-dimensional problems. Organization of the Thesis The thesis is divided into ten chapters. Chapter I, introductory in nature, examines only techniques which are widely applicable to analytical and numerical solutions of Stefan problems. In Chapter II, a short-time analytical solution of a radially symmetric two-phase (solid and liquid) solidification problem in a cylindrical mold has been obtained by using Boley’s embedding technique described in Chapter I. Later on, the author developed a new embedding technique for obtaining short-time analytical solutions. The analytical solutions reported in Chapters III to VIII have been obtained with the help of the author’s new embedding technique. Chapter III comprises Class I, two-phase, multi-dimensional solidification problems and their solutions under different physical situations, geometries and boundary conditions. Short-time solutions of freezing front and temperatures in both solid and liquid regions have been obtained with the help of the new technique. Chapter IV gives analytical and numerical solutions of one-dimensional, two-phase solidification problems. The finite difference numerical scheme proposed here uses the notion of ‘moving grid points’. The numerical scheme runs smoothly till total solidification is achieved. Analytical and numerical solutions have been checked against each other for short-time validity. A criterion for ascertaining the period for which a short-time solution is valid was evolved and tested. If the successive coefficients in the series for moving boundary are decreasing in absolute value, then, by calculating the last term of the truncated series, the error can be estimated. The methods of analytical and numerical solutions developed by the author were extended to binary alloy solidification problems. Mathematical formulations with sharp freezing front model were considered. Although not very realistic, as invariably mushy region develops in alloy solidification, this model was pursued because the primary concern of the author was to develop an effective numerical scheme which works for two-phase alloy solidification problems. The checking of integral mass balance at all intermediate times till the solidification is complete indicated that the discontinuities in concentration and concentration gradients were accurately calculated. Linear and quadratic phase diagrams were considered. The jump in the densities of the two phases at the freezing front was taken into account and advective terms arising out of it were incorporated in the numerical scheme. Chapter V describes the results of these studies. Chapter VI deals with Class II problems. Flux and temperature-prescribed boundary conditions have been taken to start solidification on a limited portion of the surface. On the unsolidified boundary, the boundary conditions could be of a different type. The temperature boundary condition seems to be interesting as some of the calculated quantities have removable singularities whose limits are well defined. The solution corroborates the physics of the problem. The spread along the surface can have growth rates not commonly found. Class III problems have been studied in Chapter VII. The solidification starts at a point on the surfaces of cylindrical and semi-infinite mold. The initial temperature of the melt is assumed to be equal to the melting temperature at one point. The spread of solidification along the surface for small time can have unusual asymptotic behaviours such as the fourth root of time. Classical enthalpy formulation forms the focus of Chapter VIII. Moving boundaries due to distributed heat sources in a one-dimensional slab have been studied. Depending on the initial temperature, the solid-mush boundary can have several types of growth rates. The numerical solution of liquid-mush boundary is interesting and has been obtained in a novel way. The oxygen-diffusion problem is treated as a particular case of this general problem, and the analytical and numerical solutions proposed by the author are compared with those in the literature. In Chapter IX, the numerical scheme proposed in the previous chapters has been adapted for the two-dimensional solidification problems with extended freezing temperature range. In this case the numerical scheme has to accurately determine two moving boundaries, viz., solid-mush boundary and liquid-mush boundary which are coupled. Classical formulations for solid, mush, and liquid regions with appropriate boundary conditions at the moving boundaries have been considered. The freezing front can have curvature. A check on the accuracy of the results was provided by the integral heat balance which held at all intermediate times till total solidification was achieved. The extension of the numerical scheme to two-dimensional solidification problems required a lot more innovative ideas and numerical experimentation. The method of lines in space has been used in conjunction with the notion of moving grid points. This method can be extended to three-dimensional problems. In Chapter X, the method of fractional time steps and splitting of operators has been used to obtain a numerical solution to the problem of transient heat transfer in a three-dimensional, anisotropic, composite rectangular slab. Heat transfer is coupled with thermal expansion, the latter changing the shape and size of the composite slab. Numerical results have been presented for moving boundaries and transient temperatures.