Modelling the reduction of quartz in a quartz–carbon pellet

Traditional refining of silicon generates carbon dioxide emissions that are released into the atmosphere. An alternative process under experimental exploration uses quartz particles coated in a layer of porous carbon as the raw material, instead of lumps of quartz and carbon. The quartz–carbon pellets are processed at a lower temperature than in an industrial furnace, and hence, different chemical reactions are dominant, reducing the greenhouse gas emissions. The quartz core shrinks as it is consumed, and the carbon is converted to silicon carbide, which can subsequently be processed into silicon. We develop a model for chemical and transfer processes within a single quartz–carbon pellet. We derive governing equations for the concentration of silicon monoxide, carbon monoxide, and carbon dioxide, and conservation equations on the moving quartz interface. Furthermore, we then focus on a reduced model in an industrially relevant distinguished limit, and solve numerically the resulting leading-order system. We show examples of reaction-limited behaviour as well as diffusion-limited behaviour. Both regimes are physically admissible due to the large potential range of the parameters. Finally, we sweep through the parameter space, and characterize the dynamics based on the utilization of the carbon and the silicon yield. We find that the diffusion-limited regime is best for carbon utilization and silicon yield, as the silicon monoxide reacts with the carbon before it is transported out of the pellet.