Modelling microsilica particle formation and growth due to the combustion reaction of silicon monoxide with oxygen

Microsilica particles arise as a byproduct of silicon furnace operation, created inside high temperature flames due to the combustion reaction of silicon monoxide with oxygen. These nanoparticles, which grow as silicon dioxide vapour condenses on the surface of existing particles, are used in a variety of composite materials. The size and quality of the particles affect the performance of the material used for such applications, and hence control of these quantities is of importance to manufacturers. Motivated by this, we derive a mathematical model that connects local thermal and chemical concentrations conditions to the formation and growth of microsilica particles. We consider two distinct reductions of our general model: the case of initially well-mixed or spatially homogeneous chemical species (modelling the region within the flame or reaction zone), and the case of initially spatially separated chemical species, in which diffusion will play a dominant role in providing material to a combustion front (modelling a larger cross section, which contains a reaction zone with limiting quantities of fuel which must diffuse into the reaction zone). In both cases, we provide asymptotic solutions for the temperature, chemical concentrations, and number density function of microsilica particles in the oxygen rich limit, and compare them to numerical simulations. Motivated by realistic furnace control mechanisms, we treat the relative quantity of oxygen to other fuel components and the saturation concentration of silicon dioxide as a control parameters, discussing how each may be used to modify the properties (such as size and abundance) of microsilica particles formed. One physically interesting finding is the theoretical description of a bimodal distribution for microsilica particle size which was previously observed in experiments.