Mathematical modelling of heat transfer during electron-beam autocrucible melting by means of the steady-state Stefan problem

Abstract

A two-dimensional axisymmetric mathematical model of electron-beam autocrucible melting is developed and examined. Here, the hypothesis is used that forced convective heat transfer in the melt may be modelled with the help of the coefficient of effective thermal conductivity, λE. A simplified approach is used in which λE is assumed to be known. In another approach the value of λE depends on a prescribed value of a mean melt-stirring velocity and a mean liquid-pool radius which is determined in the course of solving of the problem. With the help of the Kirchoff transformation and a Green function we may reduce the problem to a nonlinear Hammerstein integral equation. Here, a dependence of the thermal-conductivity coefficient on the temperature, λS(T), at the cooled surfaces is disregarded and constant (mean) values of λS are utilized. In order to solve the problem in the case where this dependence λS(T) is taken into account, an axuiliary Green-function method is proposed which also permits to take into account a change of the heat-exchange coefficients on the autocrucible. This reduces the problem to a system of three integral Hammerstein equations. Numerical solutions of the nonlinear integral equations are obtained with the help of a variational (projective-net) method for the case of circular scanning of an electron beam over the heated surface. The computational results are well consistent with experimental data.