Mathematical modelling of silicon furnaces

Silicon is a widely utilised material, with use in the production of semiconductors, textiles, and metallic alloys. It is produced by feeding quartz rock and carbon into huge furnaces, which are heated by electrodes. In this thesis, we develop mathematical models to better understand furnace behaviour. Of particular interest is crust formation, a phenomenon whereby a blockage emerges, preventing the raw materials from falling naturally under gravity to the lower, hotter part of the furnace where the necessary chemical reactions occur. We begin by developing a continuum model for the evolution of chemical concentrations, temperature, and gas flow in a vertical cross-section of the furnace. This model is analysed in the context of 'pilot furnace' experiments, which are smaller, cylindrical furnaces heated in laboratory conditions. Numerical simulations of our model are shown to compare well with these experiments. We further perform parameter sweeps of our model, to give recommendations for operating strategies. Next, we undertake an asymptotic analysis of this model, to identify the dominant chemical and thermal behaviour within the pilot furnace. We determine chemical balances in different spatial and temporal regions by imposing a temperature profile. We also obtain a simplified thermo-chemical furnace model, which captures the main behaviour of our original model, but comprises only two partial differential equations. We then present a model of crust formation, in which furnace crust is treated as a single fluid of thermally-dependent viscosity, located between two moving boundaries. Numerical simulations show that solutions of this model can either tend towards a steady state, indicative of crust formation in practice, or the raw materials can fall indefinitely. The latter case is desirable in real furnaces, although it may not always be feasible. We discuss bifurcations between these two behaviours in our model, and the significance of such bifurcations to avoiding the build-up of crust. Finally, we focus on the reaction between solid carbon particles and silicon monoxide gas. This motivates us to develop a homogenised model for reactions between solid particles and gas, which has many possible applications. We comment on simplifications of this model for both diffusion-limited and chemically-limited reactions.