On an extended physical and mathematical model of high-temperature silicification of porous carbon products

Abstract

We have developed an extended physical and mathematical model of the gaseous silicon transfer from a melt mirror to a porous carbon product at high temperatures under medium vacuum conditions. In this model, an assumption of the non-compressibility of the gas mixture is used that makes it possible to simplify the problem significantly and at the same time to obtain a reasonable result, confirmed in experiment. The model is simplified to the greatest extent possible, and in the future it can be used for numerical implementation of more complex physical formulations, including non-stationary three-dimensional silicification of arbitrary-shaped porous carbon products under the condition of a complex location of the silicon sources in the retort of the furnace. The model is described by a system of three differential equations making it possible to calculate the average mass flow rate of the gas mixture and the diffusive-convective transfer of silicon concentration to the product. In this paper, the unsteady flow regime of silicon vapor and remaining gas is considered. The approach presents a two-dimensional formulation of the problem in the case of a rectangular cavity with the conditions of gas slip and adherence to the upper and lower walls of the computational domain. For each formulation of the problem, we obtained a numerical solution of the system of differential equations that predicts the filling of the entire working space of the furnace with gaseous silicon in a very short time, which explains the abnormally large flow of silicon vapor deep into the porous carbon material that was observed in experiments. The numerical solution was obtained using the finite difference method; an explicit difference scheme was used for convenience and ease of implementation. The obtained results are expressed in terms of the known verified values of the material parameters of the gas mixture and confirm that the vapor-liquid-phase process of carbon materials silicification is possible in a reasonable time.