New approach to simulation of radiative-conductive heat transfer in semi-transparent crystals being pulled from a melt

A method is suggested for the solution of multi-dimensional radiative heat transfer problems arising in growing crystals. The basic idea of the approach lies in the construction of a special division of the unit sphere into a set of solid angles (cells) and in the approximation of the radiation intensity in each solid angle by the P₁ approximation. The radiant transport equation is satisfied in the mean over each elementary cell and the system of partial differential equations of the second order relative to the local zeroth-moments of radiation intensity are obtained. It is shown that the solid angle subdivision can be carried out in different ways with respect to specific features of the heat transfer problem under consideration. As a result even a very rough partition permits satisfactory accuracy of the numerical solutions. One of the main advantages of the method consists in using solid angle subdivisions which can be varied from point to point of the spatial domain. The latter gave possibility to simulate the radiative heat transfer in a circular cylinder of finite length with specular side surface. On this basis the calculation of the temperature field in a cylindrical sapphire crystal being pulled from the melt has been carried out without any restriction on the size of the crystal