Anisotropy and natural convection during solidification and melting

Abstract

The effects of anisotropy of thermal conductivity and natural convection on solidification have been studied numerically. A fixed grid enthalpy-based formulation was developed to model convection and anisotropic conduction during solidification of pure materials and alloys in a rectangular cavity. The time dependent governing equations, describing the conservation of mass, momentum, energy and concentration were solved using a vorticity-stream function formulation. A finite difference-finite volume method was employed, incorporating an improved discretization method and a modified Samarskii-Andreyev ADI scheme with internal iterations. The interface was tracked with the use of an interfacial energy equation. A monotonic second-order upwind scheme (MSOU) was used for convective fluxes with central differences for the diffusion terms of concentration. Comparisons between the present calculations, analytical solutions, existing experimental results and other numerical methods are very good. The improved discretisation method is shown to have an excellent performance as it can solve the discontinuity of temperature, velocity, vorticity and stream function across the solid-liquid interface. Effects of anisotropic conduction on the temperature distribution through a gallium crystal are examined. The results show that anisotropy distorts the isotherms, especially at the adiabatic boundaries, and also decreases the overall heat transfer at the isothermal walls. Effects of aspect ratio, Stefan number, liquid superheat and boundary conditions and anisotropy during solidification are investigated. A study of solidification from either the side wall or the top wall of a cavity containing pure gallium show that natural convection has a significant effect on rate of solidification and the shape of the solid-liquid interface. The results, covering a range of values of Rayleigh number, aspect ratio and anisotropy characteristics, show how anisotropy affects the growth morphology and the flow structure. The effects of liquid aspect ratio on oscillatory convective flow during solidification are studied and compared with those for pure natural convection. Solidification from the side wall of a cavity containing a gallium-0.5% wt indium alloy was considered. The results show that anisotropy distorts the interface shape, and hence the interface shape has an effect on solute redistribution and flow patterns. The code was also used for natural convection driven melting problems of pure gallium where the interface shape is more irregular than in solidification problems. A correlation of the melting rate is given in terms of non-dimensional time, Rayleigh number, Stefan number and aspect ratio.