Silicon is produced in submerged arc furnaces which are heated by electric currents passing through the furnace. It is important to understand the distribution of heating within the furnace in order to accurately model the silicon production process, yet many existing studies neglect aspects of this current flow. In the present paper, we formulate a model that couples the electrical current to thermal, material flow and chemical processes in the furnace. We then exploit disparate timescales to homogenise the model over the timescale of the alternating current, deriving averaged equations for the slow evolution of the system. Our numerical simulations predict a minimum applied current that is required in order to obtain steady-state solutions of the homogenised model and show that for high enough applied currents, two spatially heterogeneous steady-state solutions exist, with distinct crater sizes. We show that the system evolves to the steady state with a larger crater radius and explain this behaviour in terms of the overall power balance typically found within a furnace. We find that the industrial practice of stoking furnaces increases the overall rate of material consumption in the furnace, thereby improving the efficiency of silicon production.

The most common technological route for metallurgical silicon production is to feed quartz and a carbon source (e.g., coal, coke, or charcoal) into submerged-arc furnaces, which use electrodes as electrical conductors. We develop a mathematical model of a silicon furnace. A continuum approach is taken, and we derive from first principles the equations governing the time evolution of chemical concentrations, gas partial pressures, velocity, and temperature within a one-dimensional vertical section of a furnace. Numerical simulations are obtained for this model and are shown to compare favorably with experimental results obtained using silicon pilot furnaces. A rising interface is shown to exist at the base of the charge, with motion caused by the heating of the pilot furnace. We find that more reactive carbon reduces the silicon monoxide losses, while reducing the carbon content in the raw material mixture causes greater solid and liquid material to build-up in the charge region, indicative of crust formation (which can be detrimental to the silicon production process). We also comment on how the various findings could be relevant for industrial operations.

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\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . Reactions between gases and solid particles are commonly modeled using a shrinking core framework, where a sharp interface between an inner unreacted core and an outer product shell moves inward until the reaction is complete. However, for some physical systems this sharp divide is not present, and so a better model is needed to capture a transition region for the reaction. We are interested in large particles made of many small grains where there are strong interactions between microscale granular and macroscale particulate effects, and where a shrinking core model represents behavior at the microscale level. We obtain homogenized equations for macroscale behavior by exploiting the small ratio of granular to particle lengthscales. These macroscale equations allow for a diffuse reaction front, as well as a sharp interface between reacted and unreacted solid material. We analyze the resulting model asymptotically in the limits where the reaction time is rate-limited by chemical kinetics, and separately by diffusion, determining the thickness of the reaction front. Numerical simulations support the law of additive reaction times, which states that the total reaction time is given as the sum of the conversion times under these limits. We further show how the model results can be extended to incorporate the transport of the product gas out of the solid particle, using a binary Fickian diffusion model. In metallurgical production the particle size and porosity are known to influence reaction times. Our results help quantify these effects and may be an aid in raw material selection.\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . homogenization, gas-solid reactions, porous media, shrinking core model, diffusion, asymptotics \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs . 35B27, 35C20, 80A30, 92E20 \bfD \bfO \bfI .

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