Fixed Cartesian grid based numerical model for solidification process of semi-transparent materials II: Reflection and refraction or transmission of the thermal radiation at the solid-liquid interface

Mathematical and numerical models of solidification process in materials which were semi-transparent both in the solid and liquid phases were developed in this paper. These models took into account different optical and thermophysical properties in phases, herein for the first time different refractive index in the solid and the liquid phase. Also optical phenomena like reflection and refraction or transmission of the thermal radiation at transparent as well as either specular, partially specular and partially diffusive or diffusive solid-liquid interface were also considered. Conditions for the radiation intensity at the transparent solid-liquid interface were formulated according to the specular reflection and Snell's laws. The numerical model was based on the Fixed Grid Front Tracking Method combined with the Immersed Boundary Technique for phase change process and on the Pixelation Technique for optical phenomena at the solid-liquid interface. Subsequently, comparisons of obtained results with results presented in the literature for one-dimensional two layers slab proved the correctness and accuracy of the proposed approach. Also the effect of different refractive index in the solid and the liquid phase on solidification process in an idealized square cavity were studied.