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Homogenised model for the electrical current distribution within a submerged arc furnace for silicon production

Part of: Optics, electromagnetic theory - Basic methods Thermodynamics and heat transfer

Published online by Cambridge University Press:  13 August 2021

ELLEN K. LUCKINS ,
JAMES M. OLIVER ,
COLIN P. PLEASE ,
BENJAMIN M. SLOMAN  and
ROBERT A. VAN GORDER Open the ORCID record for ROBERT A. VAN GORDER [Opens in a new window]
ELLEN K. LUCKINS
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK emails: Ellen.Luckins@maths.ox.ac.uk, oliver@maths.ox.ac.uk, Colin.Please@maths.ox.ac.uk
JAMES M. OLIVER
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK emails: Ellen.Luckins@maths.ox.ac.uk, oliver@maths.ox.ac.uk, Colin.Please@maths.ox.ac.uk
COLIN P. PLEASE
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK emails: Ellen.Luckins@maths.ox.ac.uk, oliver@maths.ox.ac.uk, Colin.Please@maths.ox.ac.uk
BENJAMIN M. SLOMAN
Affiliation:
Elkem ASA, Technology, Fiskaaveien 100, Kristiansand 4621, Norway email: ben.sloman@elkem.no
ROBERT A. VAN GORDER
Affiliation:
Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand email: rvangorder@maths.otago.ac.nz
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Abstract

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.

Keywords

Homogenisation theorysubmerged arc furnaceheat and mass transferendothermic chemical reactionfree boundary problem

MSC classification

Primary: 78M40: Homogenization
Secondary: 80A20: Heat and mass transfer, heat flow
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 5 , October 2022 , pp. 828 - 863
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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