Asymptotic analysis of a silicon furnace model - Sloman, B; Please, C; Van Gorder, R (Society for Industrial and Applied Mathematics, 2018)
Silicon is produced from quartz rock in electrode-heated furnaces by using carbon as a reduction agent. We perform an asymptotic analysis of a heat and mass transfer model of an experimental pilot furnace in order to determine the dominant chemical and thermal behaviour of the system. First, by prescribing a steady state temperature profile in the furnace we explore the leading order reactions in different spatial regions, as well as early time behaviour. We are able to obtain asymptotic solutions which compare well with numerical simulations. Utilising the dominant balances found when the temperature is prescribed, we next reduce the full model to two coupled partial differential equations for the time-variable temperature profile within the furnace and the concentration of solid quartz. These equations account for diffusion, an endothermic reaction, and the external heating input to the system. A moving boundary is found and the behaviour on either side of this boundary explored in the asymptotic limit of small diffusion. We note how the simplifications derived may be useful for future models and industrial furnace operation, and comment on insights from the model about furnace crust formation.