Solutions of some rasially symmetric problems of heat conduction with phase change

Abstract

Some analytical and numerical solutions pertaining to radially symmetric problems of heat conduction with phase change have been presented in this thesis. The emphasis is on finding the growth rate of the moving boundary. The present method of solution holds good for melting ablation as well as solidification to a spherical mould. It is also applicable to some problems concerning two-dimensional radially symmetric solidification in an infinite cylindrical mould, and this has been demonstrated with the help of an example. The analytical solution is essentially valid for short time, but in some particular cases such as the liquid at the fusion temperature initially, the solidified volume is considerable. The time for which the short-time analytical solution for the growth rate of the moving boundary is valid has been cross-checked with the numerical schemes. The contents of the thesis are as follows: Chapter I is an introductory part which consists of the literature survey pertaining to analytical and numerical solutions of these phase-change problems. This chapter also includes a brief introduction to the method of solution employed in the present work. In brief, the conditions at the moving boundary have been satisfied by introducing some fictitious initial temperatures in some fictitious extensions of the original region. Chapter II: A one-dimensional radially symmetric problem of the melting ablation of a solid sphere has been studied with time-dependent flux. An infinite series solution has been assumed for the moving boundary. The first few terms of the series solution for the moving boundary have been obtained successfully. The method of solution is simple and gives meaningful results. In principle, many more terms in the series solution can be obtained, but the algebra becomes very tedious after seven or eight terms. There does not seem to be any rigorous method to determine the order of convergence of this series, and therefore analytical results have been checked with numerical solutions obtained by the Landis scheme (the grid points change with the growth of the moving boundary in this scheme). The comparisons of the analytical results and numerical results have been made with the help of graphs and tables. Several material parameters and physical parameters have been considered. The series has been found to be very reliable when the coefficients in the series are decreasing, and in this case, the extent of time for which the solution is valid can be easily found by looking at the order of the last term. However, the series gives a short-time solution even if the coefficients are increasing, but in this case, the series is valid only for a short time. Several interesting particular cases can be derived from the present solution. Chapter III deals with the problem of radially symmetric solidification of a superheated liquid in a spherical mould. The boundary conditions could be flux-prescribed or temperature-prescribed and have been taken as time-dependent. Once again, the short-time analytical solution for the growth of the moving boundary has been compared with numerical solution. A more recent numerical scheme known as the Enthalpy Scheme has been employed for numerical computations. When the melt is at the melting temperature, the analytical results are valid for considerable length of time. In this particular case, the temperature in the solid (there are no temperature gradients in the liquid) can be easily determined analytically. Chapter IV: The problem of radially symmetric solidification in an infinite mould with a spherical cavity has been investigated. The melt could be superheated and time-dependent flux boundary conditions have been considered. The instantaneous starting of solidification of a superheated liquid has been considered. Results are also given for the melting ablation problem. Numerical results are given for the case when the melt is at the fusion temperature. Chapter V deals very briefly with the two-dimensional radially symmetric solidification in an infinite cylindrical mould with temperature-prescribed boundary conditions.