Analytical and Numerical Solution for a Solidifying Liquid Alloy Slab

Numerical and analytical solutions are presented for the temperature and concentration distributions during the solidification of a binary liquid alloy slab. The slab is taken to be of a finite depth but infinite in the horizontal direction. The solidification process is started by withdrawing a fixed amount of heat from the lower surface of the slab. The upper surface of the slab is subjected to both radiation and convective conditions. The solution gives the concentration and temperature profiles and the interface position as a function of time. Due to the smallness of the mass diffusion coefficient in the solid, the numerical solution method breaks down whenever the ratio of the diffusivities in the solid and the liquid falls below a certain value. An analytical method is developed which gives accurate solution for any value of the diffusivity ratio.