A cylindrical model for rotational MHD instabilities in aluminum reduction cells
Tóm tắt
Large-scale horizontal vortices associated with deformations of the aluminum-electrolyte interface have been observed in operating aluminum reduction cells as well as in physical and numerical models. To expose their importance, we analyze a particular class of magnetohydrodynamic (MHD) interfacial instabilities which are induced by rotation. As we focus on a single vortex, a cylindrical geometry is preferred. Two analytical models are proposed. In a first model based on the MHD shallow-water approximation, we consider a vortex that has a solid rotation profile to obtain a wave equation and a dispersion relation. A more realistic second model includes a viscous rotation profile and the treatment of the base-state interface deformation. Energetics of the flow gives further insight on how an initial perturbation evolves as an oscillatory or a non-oscillatory instability, depending on the direction of rotation. We find that the mechanism at the very origin of these instabilities is neither due to a shear between the two layers—and are therefore not Kelvin–Helmholtz instabilities—nor simply due to magnetic force alone, but rather to the indirect action of the centripetal pressure due to the rotation induced by magnetic force.
Tài liệu tham khảo
Banerjee S.K., Evans J.W.: Measurements of magnetic fields and electromagnetically driven melt flow in a physical model of a Hall–Héroult cell. Metall. Trans. B. 21B(1), 59–69 (1990)
Bojarevičs V., Pericleous K.: Nonlinear MHD stability of aluminium reduction cells. In: Cebers, A. (eds) Proceedings of the joint 15th Riga and 6th PAMIR international conference, vol. II, pp. 87–90. Institute of Physics of University of Latvia, Latvia (2005)
Bojarevičs, V., Pericleous, K.: Comparison of MHD models for aluminium reduction cells. In: Light Metals 2006, pp. 347–352. TMS (2006)
Bojarevičs V., Romerio M.: Long wave instability of liquid metal–electrolyte interface in aluminium electrolysis cells: a generalization of Sele’s criterion. Eur. J. Mech. B 13(1), 33–56 (1994)
Chandrasekhar S.: Hydrodynamic and hydromagnetic stability, Chap. XI, pp. 508. Dover, Mineola (1961)
Davidson P.A.: An introduction to magnetohydrodynamics. Cambridge University Press, Cambridge (2001)
Davidson P.A., Lindsay R.I.: Stability of interfacial waves in aluminium reduction cells. J. Fluid Mech 362, 273–295 (1998)
Dupuis, M., Bojarevičs, V.: Impact of using selective collector bar rodding on the MHD stability of a 500-kA aluminium electrolysis cell. In: CIM 2005 Light Metals, pp. 19–33. Met. Soc. CIM (2005)
Evans J.W., Zundelevich Y., Sharma D.: A mathematical model for prediction of currents magnetic fields, melt velocities, melt topography and current efficiency in Hall–Héroult cells. Metall. Trans. B 12, 353–360 (1981)
Kundu P.: Fluid mechanics, 2nd edn. Academic Press, London (2002)
Kurenkov A., Thess A., Zikanov O., Segatz M., Droste C., Vogelsang D.: Stability of aluminum reduction cells with mean flow. Magnetohydrodynamics 40(2), 3–13 (2004)
Lukyanov A., El G., Molokov S.: Instability of MHD-modified interfacial gravity waves revisited. Phys. Lett. A 290(3), 165–172 (2001)
Morris S.J.S., Davidson P.A.: Hydromagnetic edge waves and instability in reduction cells. J. Fluid Mech. 493, 121–130 (2003)
Munger D.: Simulation numérique des instabilités magnétohydrodynamiques dans les cuves de production de l’aluminium. Master’s thesis. Université de Montréal, Montréal (2004)
Munger D., Vincent A.: Direct simulations of MHD instabilities in aluminium reduction cells. Magnetohydrodynamics 42(4), 417–425 (2006)
Sele T.: Instabilities of the metal surface in electrolytic alumina reduction cells. Metall. Trans. B 8, 613–618 (1977)
Shin, D., Sneyd, A.D.: Metal pad instabilities in aluminium reduction cells. In: Peterson, R.D. (ed.) Light Metals 2000, pp. 279–283. The Minerals, Metals & Materials Society (2000)
Sneyd A.D., Wang A.: Interfacial instability due to MHD mode coupling in aluminium reduction cells. J. Fluid Mech. 263, 343–359 (1994)
Sun H., Zikanov O., Finlayson B.: Effects of background melt flow and interface distortion on the stability of Hall-Héroult cells. Magnetohydrodynamics 41(3), 273–287 (2005)
Urata N., Mori K., Ikeuchi H.: Behavior of bath and molten metal in aluminum electrolytic cell. J. Japan Inst. Light Metals 26(11), 573–583 (1976)
Ziegler D.P.: Stability of metal/electrolyte interface in Hall–Héroult cells: effect of the steady velocity. Metall. Trans. B 24B, 899–906 (1993)
Zikanov O., Thess A., Davidson P.A., Ziegler D.P.: A new approach to numerical simulation of melt flows and interface instability in Hall–Héroult cells. Metall. Mater. Trans. B 31(6), 1541–1550 (2000)