Penetration of molten silicon into a bed of fines

During the casting process of silicon the mould must be buffered from the high temperatures of the molten silicon to prevent the mould melting and this is typically done by the operators laying down a layer of crushed silicon particles (fines) prior to pouring of the molten silicon. It is useful for operators to know how deep they should make the layer the fines so as to adequately separate the molten silicon from the mould. In this paper, we consider a model for the penetration of molten silicon into the pre-laid layer of silicon fines, which provides a predictive tool for estimating the necessary depth of fines in order to prevent the molten silicon touching the mould. The mathematical model developed here considers the flow of molten silicon as a Darcy flow and solidification due to heat flow as a one-phase Stefan problem. We are able to find a numerical solutions to this model, and from this we are able to extract data regarding the penetration depth of the molten silicon into the fines before solidification occurs. Our model and numerical solution can been seen as a first step toward understanding this important part of the casting process for silicon.