Locating the baking isotherm in a Søderberg electrode: analysis of a moving thermistor model. (English) Zbl 1475.80004
Summary: Søderberg electrodes feature prominently in the operation of metallurgical electrical furnaces. The electrode material must bake before entering the furnace; failure to bake will lower the efficiency of the process and may cause physical harm to the furnace itself through a soft breakage. As such, ensuring that the baking isotherm remains within the region of the electrode outside of the furnace is essential. We propose a mathematical model for a Søderberg electrode taking into account the heat, mass, and current transfer mechanisms at play, along with realistic boundary conditions on the outside of the electrode that are strongly heterogeneous in height. The resulting model describes a strongly heterogeneous cylindrical “thermistor” which moves slowly downward and is acted on by current clamps which provide Joule heating. Although it is often ignored in the literature on thermistor problems, we find that the Péclet number resulting from the downward motion strongly influences the position of the baking isotherm. Aside from some specific reductions leading to analytical solutions, the general form of the model is complicated enough to require numerical simulations. Still, our modeling approach provides us with a qualitative understanding of many aspects of the Søderberg electrode baking process and permits us to identify three parameters of key importance to the positioning of the baking isotherm. In particular, our results suggest desired ranges for the lowering rate of the electrode (in terms of a Péclet number), the radius of the electrode, and the strength of the Joule heating due to an applied current, which are the three aspects which may be controlled (to varying degrees) in industrial applications.
MSC:
| 80A19 | Diffusive and convective heat and mass transfer, heat flow |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 35Q79 | PDEs in connection with classical thermodynamics and heat transfer |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
References:
| [1] | S. Antontsev and M. Chipot, The thermistor problem: Existence, smoothness uniqueness, blowup, SIAM J. Math. Anal., 25 (1994), pp. 1128-1156. · Zbl 0808.35059 |
| [2] | G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967. · Zbl 0152.44402 |
| [3] | T. Bergstrøm, S. Cowley, A. C. Fowler, and P. E. Seward, Segregation of carbon paste in a smelting electrode, IMA J. Appl. Math., 43 (1989), pp. 83-99. · Zbl 0679.76105 |
| [4] | A. Bermúdez, J. Bullón, and F. Pena, A finite element method for the thermoelectrical modelling of electrodes, Commun. Numer. Methods Eng., 14 (1998), pp. 581-593. · Zbl 0922.65089 |
| [5] | J. Beukes, H. Roos, L. Shoko, P. Van Zyl, H. Neomagus, C. Strydom, and N. Dawson, The use of thermomechanical analysis to characterise Söderberg electrode paste raw materials, Miner. Eng., 46 (2013), pp. 167-176. |
| [6] | J. J. Bezuidenhout, Computational Fluid Dynamic Modelling of an Electric Smelting Furnace in the Platinum Recovery Process, Ph.D. thesis, Stellenbosch University, 2008. |
| [7] | C. Burstedde, O. Ghattas, G. Stadler, T. Tu, and L. C. Wilcox, Parallel scalable adjoint-based adaptive solution of variable-viscosity Stokes flow problems, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 1691-1700. · Zbl 1227.76027 |
| [8] | X. Chen and A. Friedman, The thermistor problem for conductivity which vanishes at large temperature, Quart. Appl. Math., 51 (1993), pp. 101-115. · Zbl 0803.35051 |
| [9] | G. Cimatti, A bound for the temperature in the thermistor problem, IMA J. Appl. Math., 40 (1988), pp. 15-22. · Zbl 0694.35139 |
| [10] | G. Cimatti, Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions, Quart. Appl. Math., 47 (1989), pp. 117-121. · Zbl 0694.35137 |
| [11] | A. Fitt and J. Aitchison, Determining the effective viscosity of a carbon paste used for continuous electrode smelting, Fluid Dyn. Res., 11 (1993), pp. 37-59. |
| [12] | A. Fitt and P. Howell, The manufacture of continuous smelting electrodes from carbon-paste briquettes, J. Engrg. Math., 33 (1998), pp. 353-376. · Zbl 0922.76278 |
| [13] | A. Fowler, I. Frigaard, and S. Howison, Temperature surges in current-limiting circuit devices, SIAM J. Appl. Math., 52 (1992), pp. 998-1011. · Zbl 0800.80001 |
| [14] | M. Gockenbach and K. Schmidtke, Newton’s law of heating and the heat equation, Involve, 2 (2009), pp. 419-437. · Zbl 1185.35117 |
| [15] | T. Hannesson, The Si Process Drawings, Elkem, Oslo, Norway, 2016. |
| [16] | S. Hejazi and J. Azaiez, Stability of reactive interfaces in saturated porous media under gravity in the presence of transverse flows, J. Fluid Mech., 695 (2012), pp. 439-466. · Zbl 1250.76085 |
| [17] | S. Hejazi, P. Trevelyan, J. Azaiez, and A. De Wit, Viscous fingering of a miscible reactive a + b \(\rightarrow\) c interface: A linear stability analysis, J. Fluid Mech., 652 (2010), pp. 501-528. · Zbl 1193.76058 |
| [18] | S. Howison, A note on the thermistor problem in two space dimensions, Quart. Appl. Math., 47 (1989), pp. 509-512. · Zbl 0692.35094 |
| [19] | S. Howison, J. Rodrigues, and M. Shillor, Stationary solutions to the thermistor problem, J. Math. Anal. Appl., 174 (1993), pp. 573-588. · Zbl 0787.35033 |
| [20] | R. Innvaer, The Søderberg electrode system. Recent research and development. New challenges, in International Ferro-Alloys Congress, New Orleans, LA, 1989, pp. 216-226. |
| [21] | R. Innv\aer, K. Fidje, and T. Sira, 3-dimensional calculations on smelting electrodes, Model. Identification Control, 8 (1987), pp. 103-115. |
| [22] | R. Innvaer, K. Fidje, and R. Ugland, Effect of current variations on material properties and thermal stresses in Söderberg electrodes, Proc. INFACON IV, Rio de Janeiro, Brazil, 1986, pp. 321-330. |
| [23] | R. Innv\aer, A. Vatland, and L. Olsen, Operational parameters for Soderberg electrodes from calculations, measurements, and plant experience, in Mintek 50, Johannesburg 1984, 1985. |
| [24] | K. Karalis, N. Karkalos, G. Antipas, and A. Xenidis, Pragmatic analysis of the electric submerged arc furnace continuum, R. Soc. Open Sci., 4 (2017), 170313. |
| [25] | A. Lacey, Thermal runaway in a non-local problem modelling ohmic heating: Part i: Model derivation and some special cases, Eur. J. Appl. Math., 6 (1995), pp. 127-144. · Zbl 0843.35008 |
| [26] | A. Lacey, Thermal runaway in a non-local problem modelling ohmic heating. Part ii: General proof of blow-up and asymptotics of runaway, Eur. J. Appl. Math., 6 (1995), pp. 201-224. · Zbl 0849.35058 |
| [27] | A. Lacey, Thermo-electrical stability in an electrode, Math. Indust., (1998). |
| [28] | L. Landau and E. Lifshitz, Course of Theoretical Physics. Vol. 6: Fluid Mechanics, Pergamon Press, New York, 1959. |
| [29] | B. Larsen, J. P. M. Amaro, S. Z. Nascimento, K. Fidje, and H. Gran, Melting and densification of electrode paste briquettes in Søderberg electrodes, Elkem, (1998), pp. 1-10. |
| [30] | B. Larsen, H. Feldborg, and S. Halvorsen, Minimizing thermal stress during shutdown of Soderberg electrodes, in Thirteenth International Ferroalloys Congress Efficient Technologies in Ferroalloy Industry, Almaty, Kazakhstan, 2013, pp. 453-467. |
| [31] | H. Larsen, Current distribution in the electrodes of industrial three-phase electric smelting furnaces, in Proceedings of the 2006 Nordic COMSOL Conference, 2006. |
| [32] | B. Li, B. Wang, and F. Tsukihashi, Modeling of electromagnetic field and liquid metal pool shape in an electroslag remelting process with two series-connected electrodes, Metall. Mater. Trans. B, 45 (2014), pp. 1122-1132. |
| [33] | O. Manickam and G. Homsy, Simulation of viscous fingering in miscible displacements with nonmonotonic viscosity profiles, Phys. Fluids, 6 (1994), pp. 95-107. · Zbl 0922.76168 |
| [34] | I. Mc Dougall, C. Smith, B. Olmstead, and W. Gericke, A finite element model of a søderberg electrode with an application in casing design, Proceedings of INFACON X, Cape Town, South Africa, 2004, pp. 575-583. |
| [35] | R. Meyjes, J. Venter, and U. Van Rooyen, Advanced modelling and baking of electrodes, in Twelfth International Ferroalloys Congress: Sustainable Future, Helsinki, 2010, pp. 779-788. |
| [36] | L. R. Nelson and F. X. Prins, Insights into the influences of paste additions and levels on Søderberg electrode management, in Tenth International Ferroalloys Congress, Vol. 1, Cape Town, South Africa, 2004, pp. 418-431. |
| [37] | H. Plsson and M. R. Jnsson, Finite Element Analysis of Proximity Effects in Søderberg Electrodes, Technical report, University of Ireland, 2000. |
| [38] | C. R. Ray, P. K. Sahoo, and S. S. Rao, Electrode management-Investigation into soft breaks at 48 MVa FeCr closed furnace, in Proceedings of the Eleventh Internatonal Ferroalloys Congress (INFACON), New Delhi, India, 2007, pp. 18-21. |
| [39] | L. Roberts, E. Nordg\aard-Hansen, Ø. Mikkelsen, S. A. Halvorsen, and R. A. Van Gorder, A heat and mass transfer study of carbon paste baking, Int. Commun. Heat Mass. Transf., 88 (2017), pp. 9-19. |
| [40] | Y. Sheng, G. Irons, and D. Tisdale, Transport phenomena in electric smelting of nickel matte: Part ii. Mathematical modeling, Metall. Mater. Trans. B, 29 (1998), pp. 85-94. |
| [41] | L. Shoko, J. Beukes, and C. Strydom, Determining the baking isotherm temperature of söderberg electrodes and associated structural changes, Miner. Eng., 49 (2013), pp. 33-39. |
| [42] | A. Skjeldestad, M. Tangstad, L. Lindstad, and B. Larsen, Temperature profiles in Søderberg electrodes, in Production Technologies and Operation, Kiev, 2015. The Fourteenth International Ferroalloys Congress, 2015, pp. 327-338. |
| [43] | B. M. Sloman, C. P. Please, and R. A. Van Gorder, Asymptotic analysis of a silicon furnace model, SIAM J. Appl. Math., 78 (2018), pp. 1174-1205. · Zbl 1394.80009 |
| [44] | B. M. Sloman, C. P. Please, and R. A. Van Gorder, Melting and dripping of a heated material with temperature-dependent viscosity in a thin vertical tube, J. Fluid Mech., 905 (2020), A16. · Zbl 1460.80014 |
| [45] | B. M. Sloman, C. P. Please, R. A. Van Gorder, A. M. Valderhaug, R. G. Birkeland, and H. Wegge, A heat and mass transfer model of a silicon pilot furnace, Metall. Mater. Trans. B, 48 (2017), pp. 2664-2676. |
| [46] | C. W. Söderberg, Electrode for Electric Furnaces and Process for Manufacturing the Same, https://patentimages.storage.googleapis.com/87/af/9c/f05dace9cc857a/US1440724.pdf, 1923. |
| [47] | C. Tan and G. Homsy, Stability of miscible displacements in porous media: Rectilinear flow, Phys. Fluids, 29 (1986), pp. 3549-3556. · Zbl 0608.76087 |
| [48] | K. Torklep, Viscometry in Paste Production, Light Metals 1988, 1988, pp. 237-244. |
| [49] | G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1995. · Zbl 0849.33001 |
| [50] | J. Xia and T. Ahokainen, Numerical modelling of slag flows in an electric furnace, Scand. J. Metall., 33 (2004), pp. 220-228. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
