Conference publications

Abstracts

XIII conference

Mathematical modelling of crystallization process of liquid metal on the basis of nonlinear equation of thermal conductivity with account for impact of external pressure

Kabanov P.G.

Moscow state university STANKIN, The Department of Applied mathematics, Russia, 127994, Moscow, Vadkovskiy per. 1, Phone: 972-95-20, E-mail: vadius-st@yandex.ru

The problem of mathematical modelling of liquid metal crystallization under pressure is considered in the present work. This theme is of particularly great importance for modern foundry industry, for example, during description of casting process with crystallization under pressure.

This task is solved on the basis of a boundary problem for heat conductivity equation:

with boundary conditions of the third kind,

T(x,r,0)=T0 , cэф = , where y(t ) – is a part of rigid phase at the temperature T, L – latent heat of alloy crystallization. The pressure p plays the role of a control parameter.

In such statement, values ρ, λ, α continuously depend on the temperature and pressure. We divide the whole area (p,T) by several subareas where values ρ, λ are approximated by some average constant values ρ*, λ*. On each of these subareas the initial nonlinear equation of heat conductivity becomes linear, in which ρ(T,p)= ρ*, λ(T,p)= λ*. Having determined temperature field, we can numerically find crystallization front as a time function with the help of condition T(x*,r*,t)=Tcryst [2].

As an example, temperature field in the cylinder at the height of 11 cm and the radius of 5 cm, which is a typical size of casting used in the industry, has been modelled in the work. Also for such cylinder the disposition of the crystallization front has been received. Results are well coordinated with data of work [2]. Calculations were controlled by checking the law of energy conservation on the surface of phase change.

Literature.

1. Batyshev A.I. Liquid metal stamping. –M: publishing house «Engineering», 1979

2. Romanov M.N. The solution of crystallization task of axisymmetric casting under con-stant pressure by finite-element method // Foundry production. 1987. Number 8. Pp. 85-88