caa a22 4500 475839870 CHVBK 20180406123858.0 cr unu---uuuuu 170329e20000801xx s 000 0 eng 10.1023/A:1004762404996 doi (NATIONALLICENCE)springer-10.1023/A:1004762404996 Mathematical modelling of heat transfer during electron-beam autocrucible melting by means of the steady-state Stefan problem [Elektronische Daten] [Yu.A. Mitropolsky, A.A. Berezovsky, Yu.V. Zhernovyi] 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. Kluwer Academic Publishers, 2000 melting nationallicence electron beam nationallicence circular scanning nationallicence steady-state Stefan problem nationallicence auxiliary Green-function method nationallicence Mitropolsky Yu.A. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska Str., 252601, Kyiv-4, Ukraine aut Berezovsky A.A. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska Str., 252601, Kyiv-4, Ukraine aut Zhernovyi Yu.V. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska Str., 252601, Kyiv-4, Ukraine aut Journal of Engineering Mathematics Kluwer Academic Publishers 38/2(2000-08-01), 173-190 0022-0833 38:2<173 2000 38 10665 https://doi.org/10.1023/A:1004762404996 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1004762404996 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Mitropolsky Yu.A. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska Str., 252601, Kyiv-4, Ukraine aut NATIONALLICENCE 700 1- Berezovsky A.A. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska Str., 252601, Kyiv-4, Ukraine aut NATIONALLICENCE 700 1- Zhernovyi Yu.V. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska Str., 252601, Kyiv-4, Ukraine aut NATIONALLICENCE 773 0- Journal of Engineering Mathematics Kluwer Academic Publishers 38/2(2000-08-01), 173-190 0022-0833 38:2<173 2000 38 10665 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer