Sinusoidal perturbation solution for solidification of pure materials on a planar mold of finite thickness

A linear perturbation method is used to solve two-dimensional heat conduction problem in which a liquid becomes solidified by heat transfer to a plane mold of finite thickness. Heat flux drawn from the lower surface of the mold is approximately uniform, but contains a small sinusoidal perturbation in one space dimension. Numerical results are obtained for the consequent sinusoidal perturbation in the solid/melt boundary as a function of time. Analytical results are obtained for the limiting case in which diffusivities of the solidified shell and the mold materials are infinitely large. These results are then compared with the numerical predictions to establish the validity of the model and the numerical approach. The results document the effects of both solidified shell and the mold material diffusivities on the growth of a perturbation in a nominally plane solidification front. It is demonstrated that the magnitude of this perturbation increases with the diffusivity of the mold, and this effect of mold diffusivity is overweighed by the diffusivity of the casting material. The influence of other process parameters such as the mold thickness, thermal contact resistance at the mold–shell interface, and thermal conductivity ratio on the growth of solidified shell thickness is also investigated in detail.