Mathematical modelling of electrical, thermal, and chemical processes occurring within a silicon furnace

Thesis

Silicon is produced in submerged arc furnaces (SAFs), with the heat needed for the endothermic chemical reactions provided by an electric current. The interacting physical processes occurring in SAFs are tightly coupled, and vary over a range of timescales. As a result, SAF modelling efforts have generally neglected to include many of the chemical, thermal, and electrical effects. In this thesis, we develop a mathematical framework to understand the mechanisms by which these processes interact.

We propose a new, simplified model for electric arcs in a SAF, with heat radiation as the dominant heat loss mechanism, based on dimensional analysis of a magnetohydrodynamic model. To understand the alternating current effects, we incorporate this arc model into an equivalent-circuit model for the SAF electrical system. We investigate the effects of hot gases flowing through the porous bed of raw materials in a SAF, by studying the interaction of a counter-current flow with an endothermic, temperature-dependent chemical reaction in the asymptotic limit of large Péclet number. Having better understood these mechanisms, we then develop a tractable model for the coupled processes taking place within a SAF, combining our arc model with simplified models for the raw material bed. In order to efficiently model processes on both the fast timescale of the alternating current, and the slower timescale of heat transfer and flow of the raw materials, we perform a multiple-timescale homogenisation.

Our modelling efforts give insight into the interaction of a counter-current flow with an endothermic chemical reaction across a range of parameter values for the heat transfer coefficient between the phases. Our arc model is derived from first principles, and compares favourably with empirical models currently used in the metallurgy literature and industry. Our homogenised furnace model provides a simple framework to explore interacting furnace processes of industrial relevance; solutions provide insight into the distribution of current in the furnace, the evolution of the internal structure, and how stoking may improve furnace efficiency.

Authors: Luckins, EK

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: heat radiation; reacting flow; asymptotic analysis; magnetohydrodynamics; electromagnetism; heat and mass transfer; silicon furnace; mathematical modelling; electric arc
• Subjects: Applied mathematics