One-dimensional solidification of pure materials with a time periodically oscillating temperature boundary condition

A finite difference method is used to solve a one-dimensional solidification problem with a periodic boundary condition prescribed at the bottom of the mold of finite thickness. The temperature distributions in the solidified shell and mold, the position of the moving freezing front, and its velocity are evaluated. Analytical results are obtained for the limiting cases and then compared with the numerical predictions to establish the validity of the model and the numerical approach. Interactive effects of the process parameters such as Stefan number of the solidified shell material, the mold thickness, the thermal conductivity and thermal diffusivity between the shell and mold materials on the evolution of the freezing front and its velocity are investigated in detail. The results show that the solidified materials with larger Stefan number grow slower than those with relatively smaller Stefan number. The impact of oscillating mold temperature boundary on the growth of shell thickness is particularly significant at earlier stages of the process and more pronounced for smaller Stefan numbers. Increasing mold thickness or thermal conductivity ratio between the shell and mold materials slows down the evolution of the shell thickness.