A Theoretical Heat Transfer Model for Unidirectional Solidification of Pure Metals on a Coated Sinusoidal Mold with Constant Boundary Temperature

A theoretical model is presented in this study for investigating the heat transfer problem during solidification of pure metals on a coated sinusoidal mold. The previous works are extended by considering effects of both the sinusoidal coating layer's properties and prescribed temperature boundary condition at the lower surface of the mold on the solidification process. The thermal diffusivities of the shell, coating and mold materials are assumed to be infinitely large, and it helps us to solve two-dimensional heat conduction problem analytically. The effects of the ratios between the thermal conductivities of coating, shell and mold materials, coating thickness and the amplitude ratios between the wavelengths of coating and mold surfaces on the solidification process are investigated in detail. Furthermore, the inverse design problem for the directional solidification process is briefly discussed. The results show that the thickness of the coating causes the decrease in the amplitude of the undulation at the solidification front for all cases considered. When the ratio between the amplitudes of both surface wavelengths of the mold is negative, the decrease in the ratio between thermal conductivities of the shell and coating materials causes more uniform growth in the shell regardless of the amplitudes of wavelength ratio between the surfaces of the coating layer. A bandwidth for thermal conductivity ratio between the shell and coating materials is determined depending on the process parameters for more uniform growth.