Modelling alternating current effects in a submerged arc furnace

Modelling the production of silicon in a submerged arc furnace (SAF) requires accounting for the wide range of timescales of the different physical and chemical processes: the electric current which is used to heat the furnace varies over a timescale of around $10^{-2}\,$ s, whereas the flow and chemical consumption of the raw materials in the furnace occurs over several hours. Models for the silicon furnace generally either include only the fast-timescale, or only the slow-timescale processes. In a prior work, we developed a model incorporating effects on both the fast and slow timescales, and used a multiple-timescales analysis to homogenise the fast variations, deriving an averaged model for the slow evolution of the raw materials. For simplicity, in the previous work we focussed on the electrical behaviour around the base of a single electrode, and prescribed the current in this electrode to be sinusoidal, with given amplitude. In this paper, we extend our previous analysis to include the full electrical system, modelled using an equivalent circuit system. In this way, we demonstrate how the two furnace-modelling approaches (on the fast and slow timescales) may be combined in a computationally efficient way. Our previously derived model for the arc resistance is based on the assumption that the dominant heat loss from the arc is by radiation (we will refer to this as the radiation model). Alternative arc models include the empirical Cassie and Mayr models, which are commonly used in the SAF literature. We compare these various arc models, explore the dependence of the solution of our model on the model parameters and compare our solutions with measurements from an operational silicon furnace. In particular, we show that only the radiation arc model has a rising current-voltage characteristic at high currents. Simulations of the model show that there is an upper limit on the length of the furnace arc, above which all the current bypasses the arc and flows through the surrounding material.