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

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

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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References

Andresen, B. (1995) Process Model for Carbothermic Production of Silicon Metal. Master’s thesis, NTH, Norway.Google Scholar
Baer, M. R. & Nunziato, J. W. (1986) A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flow 12(6), 861–889.CrossRefGoogle Scholar
Bakken, J. A., Gu, L., Larsen, H. L. & Sevastyanenko, V. G. (1997) Numerical modeling of electric arcs. J. Eng. Phys. Thermophys. 70(4), 530–543.CrossRefGoogle Scholar
Bermúdez, A., Muñiz, M. C., Pena, F. & Bullón, J. (1999) Numerical computation of the electromagnetic field in the electrodes of a three-phase arc furnace. Int. J. Numerical Methods Eng. 46(5), 649–658.3.0.CO;2-C>CrossRefGoogle Scholar
Bowman, B. & Krüger, K. (2009) Arc Furnace Physics, Verlag Stahleisen Düsseldorf.Google Scholar
Boyer, F., Guazzelli, é. & Pouliquen, O. (2011) Unifying suspension and granular rheology. Phys. Rev. Lett. 107(18), 188301.CrossRefGoogle ScholarPubMed
Brennen, C. E. (2005) Fundamentals of Multiphase Flow, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Budd, C., Chapman, S. J., Lacey, A. A. & Ockendon, J. R. (1990) Electric arc problem. Study group report. http://www.maths-in-industry.org/miis/651/1/p2.pdf.Google Scholar
Budd, C., Jones, A., Biesenbach, H. & Halvorsen, S. Elkem arc problem. Study group report, personal communication from Elkem.Google Scholar
Byrne, H. & Norbury, J. (1994) Stable solutions for a catalytic converter. SIAM J. Appl. Math. 54(3), 789–813.Google Scholar
Cowley, M. D. (1974) Integral methods of analysing electric arcs: I. Formulation. J. Phys. D Appl. Phys. 7(16), 2218.CrossRefGoogle Scholar
Davidson, P. A. (2001) Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Dhainaut, M. (2004) Simulation of the electric field in a submerged arc furnace. In: Proceedings of the Tenth International Ferroalloy Congress, pp. 605–613.Google Scholar
Eidem, P. A. (2008) Electrical Resistivity of Coke Beds. PhD thesis, NTNU, Norway.Google Scholar
Forterre, Y. & Pouliquen, O. (2008) Flows of dense granular media. Ann. Rev. Fluid Mech. 40, 1–24.CrossRefGoogle Scholar
González-Fariña, R., Münch, A., Oliver, J. M. & Van Gorder, R. A. (2020) Modeling microsilica particle formation and growth due to the combustion reaction of silicon monoxide with oxygen. SIAM J. Appl. Math. 80(2), 1003–1033.CrossRefGoogle Scholar
Gustavsson, N. (2004) Evaluation and Simulation of Black-Box Arc Models for High-Voltage Circuit-Breakers. Master’s thesis, Institutionen för systemteknik, Norway.Google Scholar
Halvorsen, S. A., Schei, A. & Downing, J. H. (1992) A unidimensional dynamic model for the (ferro-) silicon process. In: Electrical Furnace Conference Proceedings, Vol. 50, pp. 45–59.Google Scholar
Hannesson, T. H. (2016) The Si Process Process, Elkem Iceland.Google Scholar
Herland, E. V., Sparta, M. & Halvorsen, S. A. (2019) Skin and proximity effects in electrodes and furnace shells. Metall. Mater. Trans. B 50(6), 2884–2897.Google Scholar
Hinch, E. J. (1991) Perturbation Methods, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Howell, J. R. & Siegel, R. (2002) Thermal Radiation Heat Transfer, 4th ed., Taylor and Francis, Boca Raton, FL.Google Scholar
Hoyaux, M. F. (1968) Arc Physics, Springer, New York, NY.CrossRefGoogle Scholar
Internal Elkem Data, personal communication.Google Scholar
Jones, G. R. & Fang, M. T. C. (1980) The physics of high-power arcs. Rep. Prog. Phys. 43(12), 1415–1465.CrossRefGoogle Scholar
Kadkhodabeigi, M. (2011) Modeling of Tapping Processes in Submerged Arc Furnaces. PhD thesis, NTNU, Norway.Google Scholar
Kadkhodabeigi, M., Tveit, H. & Berget, K. H. (2010) Silicon process – new hood design for tapping gas collection. In: Twelfth International Ferroalloys Congress, pp. 109–119.Google Scholar
Kierzenka, J. & Shampine, L. F. (2001) A BVP solver based on residual control and the MATLAB PSE. ACM Trans. Math. Software 27(3), 299–316.CrossRefGoogle Scholar
Kiradjiev, K. B., Halvorsen, S. A., Van Gorder, R. A. & Howison, S. D. (2019) Maxwell-type models for the effective thermal conductivity of a porous material with radiative transfer in the voids. Int. J. Thermal Sci. 145, 106009.CrossRefGoogle Scholar
Lago, F., Gonzalez, J. J., Freton, P. & Gleizes, A. (2004) A numerical modelling of an electric arc and its interaction with the anode: Part I. The two-dimensional model. J. Phys. D Appl. Phys. 37(6), 883–897.CrossRefGoogle Scholar
Lowke, J. J. (1979) Simple theory of free-burning arcs. J. Phys. D Appl. Phys. 12(11), 1873–1886.CrossRefGoogle Scholar
Modest, M. F. (2013) Radiative Heat Transfer, Academic Press, Amsterdam, The Netherlands.CrossRefGoogle Scholar
Mrozowski, S. (1952) Semiconductivity and diamagnetism of polycrystalline graphite and condensed ring systems. Phys. Rev. 85(4), 609.CrossRefGoogle Scholar
Ni, J. & Beckermann, C. (1991) A volume-averaged two-phase model for transport phenomena during solidification. Metall. Trans. B 22(3), 349–361.CrossRefGoogle Scholar
Pfender, E. (1978) Electric arcs and arc gas heaters. Gaseous Electron. 1(5), 291–298.Google Scholar
Ramakrishnan, S., Stokes, A. D. & Lowke, J. J. (1978) An approximate model for high-current free-burning arcs. J. Phys. D Appl. Phys. 11(16), 2267–2280.CrossRefGoogle Scholar
Rivière, P. & Soufiani, A. (2012) Updated band model parameters for $\textrm{H}_2\textrm{O}$ , $\textrm{CO}_2$ , $\textrm{CH}_4$ and CO radiation at high temperature. Int. J. Heat Mass Transfer 55(13–14), 3349–3358.CrossRefGoogle Scholar
Rooney, C. M., Please, C. P. & Howison, S. D. (2020) Homogenisation applied to thermal radiation in porous media. Eur. J. Appl. Math., 1–22. https://doi.org/10.1017/S0956792520000388.Google Scholar
Sævarsdóttir, G. A. (2002) High Current AC Arcs in Silicon and Ferrosilicon Furnaces. PhD thesis, NTNU, Norway.Google Scholar
Sævarsdóttir, G. A. & Bakken, J. A. (2010) Current distribution in submerged arc furnaces for silicon metal/ferrosilicon production. In: The 12th International Ferroalloys Congress, pp. 717–728.Google Scholar
Scheepers, E., Adema, A. T., Yang, Y. & Reuter, M. A. (2006) The development of a CFD model of a submerged arc furnace for phosphorus production. Minerals Eng. 19(10), 1115–1125.Google Scholar
Schei, A., Tuset, J. K., Tveit, H., et al. (1998) Production of High Silicon Alloys, Tapir Trondheim, Norway.Google Scholar
Schlitz, L. Z., Garimella, S. V. & Chan, S. H. (1999) Gas dynamics and electromagnetic processes in high-current arc plasmas. Part I. Model formulation and steady-state solutions. J. Appl. Phys. 85(5), 2540–2546.Google Scholar
Shampine, L. F. & Reichelt, M. W. (1997) The MATLAB ODE suite. SIAM J. Sci. Comput. 18(1), 1–22.CrossRefGoogle Scholar
Sloman, B. M. (2018) Mathematical Modelling of Silicon Furnaces. PhD thesis, University of Oxford, UK.Google Scholar
Sloman, B. M., Please, C. P. & Van Gorder, R. A. (2018) Asymptotic analysis of a silicon furnace model. SIAM J. Appl. Math. 78(2), 1174–1205.CrossRefGoogle Scholar
Sloman, B. M., Please, C. P. & Van Gorder, R. A. (2019) Homogenization of a shrinking core model for gas–solid reactions in granular particles. SIAM J. Appl. Math. 79(1), 177–206.CrossRefGoogle Scholar
Sloman, B. M., Please, C. P. & Van Gorder, R. A. (2020) Melting and dripping of a heated material with temperature-dependent viscosity in a thin vertical tube. J. Fluid Mech. 905, A16.CrossRefGoogle Scholar
Sloman, B. M., Please, C. P., Van Gorder, R. A., Valderhaug, A. M., Birkeland, R. G. & Wegge, H. (2017) A heat and mass transfer model of a silicon pilot furnace. Metall. Mater. Trans. B 48(5), 2664–2676.CrossRefGoogle Scholar
Tesfahunegn, Y. A., Magnusson, T., Tangstad, M. & Sævarsdóttir, G. A. (2018) Effect of electrode shape on the current distribution in submerged arc furnaces for silicon production - a modelling approach. J. South. Af. Inst. Min. Metall. 118(6), 595–600.Google Scholar
Thompson, A. B., Richardson, G., Dellar, P., McGuinness, M. & Budd, C. (2010) Arc phenomena in low-voltage current limiting circuit breakers. Study group report. Phenomena in low-voltage current limiting circuit breakers.Google Scholar
Valderhaug, A. M. (1992) Modelling and Control of Submerged-arc Ferrosilicon Furnaces. PhD thesis, NTNU, Norway.Google Scholar
Wang, X., Liu, J., Gong, Y., Li, G. & Ma, T. (1997) An electrostatic magnetohydrodynamics theory for resistive-viscous helical instabilities of arc discharges. Phys. Plasmas 4(8), 2791–2797.CrossRefGoogle Scholar
Westermoen, A. (2007) Modelling of Dynamic Arc Behaviour in a Plasma Reactor. PhD thesis, NTNU, Norway.Google Scholar