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Perturbation theory for metal pad roll instability in cylindrical reduction cells

Published online by Cambridge University Press:  18 September 2019

W. Herreman Open the ORCID record for W. Herreman [Opens in a new window] ,
C. Nore Open the ORCID record for C. Nore [Opens in a new window] ,
J.-L. Guermond ,
L. Cappanera ,
N. Weber  and
G. M. Horstmann Open the ORCID record for G. M. Horstmann [Opens in a new window]
W. Herreman*
Affiliation:
LIMSI, CNRS, Université Paris-Sud, Université Paris-Saclay, Orsay, F-91405, France
C. Nore
Affiliation:
LIMSI, CNRS, Université Paris-Sud, Université Paris-Saclay, Orsay, F-91405, France
J.-L. Guermond
Affiliation:
TAMU, Texas A&M, College Station, TX 77843, USA
L. Cappanera
Affiliation:
CAAM, Rice University, 6100 Main St., Houston, TX 77005-1827, USA
N. Weber
Affiliation:
Helmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstr. 400, 01328 Dresden, Germany
G. M. Horstmann
Affiliation:
Helmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstr. 400, 01328 Dresden, Germany
*
†Email address for correspondence: [email protected]
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Abstract

We propose a new theoretical model for metal pad roll instability in idealized cylindrical reduction cells. In addition to the usual destabilizing effects, we model viscous and Joule dissipation and some capillary effects. The resulting explicit formulas are used as theoretical benchmarks for two multiphase magnetohydrodynamic solvers, OpenFOAM and SFEMaNS. Our explicit formula for the viscous damping rate of gravity waves in cylinders with two fluid layers compares excellently to experimental measurements. We use our model to locate the viscously controlled instability threshold in cylindrical shallow reduction cells but also in Mg–Sb liquid metal batteries with decoupled interfaces.

JFM classification

Multiphase and Particle-laden Flows: Multiphase flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 878 , 10 November 2019 , pp. 598 - 646
Copyright
© 2019 Cambridge University Press 

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References

Antille, J. P., Descloux, J., Flueck, M. & Romerio, M. V. 1999 Eigenmodes and interface description in a Hall–Heroult cell. In Light Metals, pp. 333–338. TMS.Google Scholar
Ashour, R., Kelley, D. H., Salas, A., Starace, M., Weber, N. & Weier, T. 2018 Competing forces in liquid metal electrodes and batteries. J. Power Sources 378, 301–310.Google Scholar
Banerjee, S. K. & Evans, J. W. 1990 Measurements of magnetic fields and electromagnetically driven melt flow in a physical model of a Hall–Héroult cell. Metall. Trans. B 21 (1), 59–69.Google Scholar
Bojarevics, V. & Pericleous, K. 2006 Comparison of MHD models for aluminium reduction cells. In Light Metals, pp. 347–352. TMS.Google Scholar
Bojarevics, V. & Pericleous, K. 2008 Shallow water model for aluminium electrolysis cells with variable top and bottom. In Light Metals, pp. 403–408. TMS.Google Scholar
Bojarevics, V. & Romerio, M. V. 1994 Long waves instability of liquid metal–electrolyte interface in aluminium electrolysis cells: a generalization of Sele’s criterion. Eur. J. Mech. (B/Fluids) 13, 33–56.Google Scholar
Bojarevics, V. & Tucs, A. 2017 MHD of large scale liquid metal batteries. In Light Metals 2017, pp. 687–692. Springer.Google Scholar
Bradwell, D. J., Kim, H., Sirk, A. H. C. & Sadoway, D. R. 2012 Magnesium–antimony liquid metal battery for stationary energy storage. J. Am. Chem. Soc. 134 (4), 1895–1897.Google Scholar
Cappanera, L., Guermond, J.-L., Herreman, W. & Nore, C. 2018 Momentum-based approximation of incompressible multiphase fluid flows. Intl J. Numer. Meth. Fluids 86 (8), 541–563.Google Scholar
Case, K. M. & Parkinson, W. C. 1957 Damping of surface waves in an incompressible liquid. J. Fluid Mech. 2 (02), 172–184.Google Scholar
Davidson, P. A. & Boivin, R. F. 1992 Hydrodynamics of aluminium reduction cells. In Light Metals, pp. 1199–1204. TMS.Google Scholar
Davidson, P. A. & Lindsay, R. I. 1998 Stability of interfacial waves in aluminium reduction cells. J. Fluid Mech. 362, 273–295.Google Scholar
Descloux, J., Flueck, M. & Romerio, M. V. 1991 Modelling for instabilities in Hall–Heroult cells: mathematical and numerical aspects. In Magnetohydrodynamics in Process Metallurgy, pp. 107–110. TMS.Google Scholar
Descloux, J., Flueck, M. & Romerio, M. V. 1994 Stability in aluminium reduction cells: a spectral problem solved by an iterative procedure. In Light Metals, pp. 275–281. TMS.Google Scholar
Descloux, J. & Romerio, M. V. 1989 On the analysis by perturbation methods of the anodic current fluctuations in an electrolytic cell for aluminium. In Light Metals, pp. 237–243. TMS.Google Scholar
Flueck, M., Hofer, T., Picasso, M., Rappaz, J. & Steiner, G. 2009 Scientific computing for aluminium production. Intl J. Numer. Anal. Model. 6 (3), 489–504.Google Scholar
Flueck, M., Janka, A., Laurent, C., Picasso, M., Rappaz, J. & Steiner, G. 2010 Some mathematical and numerical aspects in aluminum production. J. Sci. Comput. 43 (3), 313–325.Google Scholar
Gerbeau, J.-F., Le Bris, C. & Lelièvre, T. 2006 Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Clarendon Press.Google Scholar
Gerbeau, J.-F., Lelièvre, T. & Le Bris, C. 2003 Simulations of MHD flows with moving interfaces. J. Comput. Phys. 184 (1), 163–191.Google Scholar
Gerbeau, J.-F., Lelièvre, T. & Le Bris, C. 2004 Modeling and simulation of the industrial production of aluminium: the nonlinear approach. Comput. Fluids 33 (5), 801–814.Google Scholar
Guermond, J.-L., Laguerre, R., Léorat, J. & Nore, C. 2007 An interior penalty Galerkin method for the MHD equations in heterogeneous domains. J. Comput. Phys. 221 (1), 349–369.Google Scholar
Guermond, J.-L., Laguerre, R., Léorat, J. & Nore, C. 2009 Nonlinear magnetohydrodynamics in axisymmetric heterogeneous domains using a Fourier/finite element technique and an interior penalty method. J. Comput. Phys. 228 (8), 2739–2757.Google Scholar
Herreman, W., Nore, C., Cappanera, L. & Guermond, J.-L. 2015 Tayler instability in liquid metal columns and liquid metal batteries. J. Fluid Mech. 771, 79–114.Google Scholar
Horstmann, G. M., Weber, N. & Weier, T. 2018 Coupling and stability of interfacial waves in liquid metal batteries. J. Fluid Mech. 845, 1–35.Google Scholar
Horstmann, G. M., Wylega, M. & Weier, T. 2019 Measurement of interfacial wave dynamics in orbitally shaken cylindrical containers using ultrasonic pulse-echo techniques. Exps. Fluids 60 (4), 56.Google Scholar
Ibrahim, R. A. 2005 Liquid Sloshing Dynamics: Theory and Applications. Cambridge University Press.Google Scholar
Kim, H., Boysen, D. A., Ouchi, T. & Sadoway, D. R. 2013 Calcium–bismuth electrodes for large-scale energy storage (liquid metal batteries). J. Power Sources 241, 239–248.Google Scholar
Lamb, H. 1945 Hydrodynamics, vol. 43. Cambridge University Press.Google Scholar
Lukyanov, A., El, G. & Molokov, S. 2001 Instability of MHD-modified interfacial gravity waves revisited. Phys. Lett. A 290 (3), 165–172.Google Scholar
Molokov, S. 2018 The nature of interfacial instabilities in liquid metal batteries in a vertical magnetic field. Eur. Phys. Lett. 121 (4), 44001.Google Scholar
Molokov, S., El, G. & Lukyanov, A. 2011 Classification of instability modes in a model of aluminium reduction cells with a uniform magnetic field. Theor. Comput. Fluid Dyn. 25 (5), 261–279.Google Scholar
Moreau, R. J. & Ziegler, D. 1986 Stability of aluminum cells – a new approach. In Light Metals, pp. 359–364. TMS.Google Scholar
Munger, D. & Vincent, A. 2006a Direct simulations of MHD instabilities in aluminium reduction cells. Magnetohydrodynamics 42 (4), 417–425.Google Scholar
Munger, D. & Vincent, A. 2006b Electric boundary conditions at the anodes in aluminum reduction cells. Metall. Mater. Trans. B 37 (6), 1025–1035.Google Scholar
Munger, D. & Vincent, A. 2006c A level set approach to simulate magnetohydrodynamic instabilities in aluminum reduction cells. J. Comput. Phys. 217 (2), 295–311.Google Scholar
Nore, C., Quiroz, D. C., Cappanera, L. & Guermond, J.-L. 2016 Direct numerical simulation of the axial dipolar dynamo in the Von Kármán sodium experiment. Europhys. Lett. 114 (6), 65002.Google Scholar
Pedchenko, A., Molokov, S. & Bardet, B. 2017 The effect of ‘wave breakers’ on the magnetohydrodynamic instability in aluminum reduction cells. Metall. Mater. Trans. B 48 (1), 6–10.Google Scholar
Pedchenko, A., Molokov, S., Priede, J., Lukyanov, A. & Thomas, P. J. 2009 Experimental model of the interfacial instability in aluminium reduction cells. Eur. Phys. Lett. 88 (2), 24001.Google Scholar
Potocnik, V. 1988 Modeling of metal-bath interface waves in Hall–Heroult cells using ester/phoenics. In Light Metals, pp. 227–235. TMS.Google Scholar
Potocnik, V. & Laroche, F. 2001 Comparison of measured and calculated metal pad velocities for different prebake cell designs. In Light Metals, pp. 419–425. TMS.Google Scholar
Reclari, M., Dreyer, M., Tissot, S., Obreschkow, D., Wurm, F. M. & Farhat, M. 2014 Surface wave dynamics in orbital shaken cylindrical containers. Phys. Fluids 26 (5), 052104.Google Scholar
Renaudier, S., Bardet, B., Steiner, G., Pedcenko, A., Rappaz, J., Molokov, S. & Masserey, A. 2016 Unsteady MHD Modeling Applied to Cell Stability, pp. 579–584. Springer International Publishing.Google Scholar
Romerio, M. V. & Antille, J. 2000 The numerical approach to analyzing flow stability in the aluminum reduction cell. Aluminium 76 (12), 1031–1037.Google Scholar
Sele, T. 1977 Instabilities of the metal surface in electrolytic alumina reduction cells. Metall. Mater. Trans. B 8 (4), 613–618.Google Scholar
Severo, D., Gusberti, V., Schneider, A.-F., Pinto, E. C. V. & Potocnik, V. 2008 Comparison of various methods for modeling the metal-bath interface. In Light Metals, p. 413. TMS.Google Scholar
Severo, D. S., Schneider, A.-F., Pinto, E. C. V., Gusberti, V. & Potocnik, V. 2005 Modeling magnetohydrodynamics of aluminum electrolysis cells with ANSYS and CFX. In Light Metals, pp. 475–480. TMS.Google Scholar
Sneyd, A. D. 1985 Stability of fluid layers carrying a normal electric current. J. Fluid Mech. 156, 223–236.Google Scholar
Sneyd, A. D. & Wang, A. 1994 Interfacial instability due to MHD mode coupling in aluminium reduction cells. J. Fluid Mech. 263, 343–360.Google Scholar
Sreenivasan, B., Davidson, P. A. & Etay, J. 2005 On the control of surface waves by a vertical magnetic field. Phys. Fluids 17 (11), 117101.Google Scholar
Steiner, G.2009 Simulation numérique de phénomènes MHD: application à l’électrolyse de l’aluminium. PhD thesis, École Polytechnique Fédérale de Lausanne.Google Scholar
Sun, H., Zikanov, O. & Ziegler, D. P. 2004 Non-linear two-dimensional model of melt flows and interface instability in aluminum reduction cells. Fluid Dyn. Res. 35 (4), 255–274.Google Scholar
Urata, N., Mori, K. & Ikeuchi, H. 1976 Behavior of bath and molten metal in aluminum electrolytic cell. WAA Translation from: J. Japan Inst. Light Met. 26 (11), 30.Google Scholar
Viola, F. & Gallaire, F. 2018 Theoretical framework to analyze the combined effect of surface tension and viscosity on the damping rate of sloshing waves. Phys. Rev. Fluids 3, 094801.Google Scholar
Wang, K., Jiang, K., Chung, B., Ouchi, T., Burke, P. J., Boysen, D. A., Bradwell, D. J., Kim, H., Muecke, U. & Sadoway, D. R. 2014 Lithium-antimony-lead liquid metal battery for grid-level energy storage. Nature 514 (7522), 348–350.Google Scholar
Watson, G. N. 1995 A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press.Google Scholar
Weber, N., Beckstein, P., Galindo, V., Herreman, W., Nore, C., Stefani, F. & Weier, T. 2017a Metal pad roll instability in liquid metal batteries. Magnetohydrodynamics 53 (1), 129–140.Google Scholar
Weber, N., Beckstein, P., Galindo, V., Starace, M. & Weier, T. 2018 Electro-vortex flow simulation using coupled meshes. Comput. Fluids 168, 101–109.Google Scholar
Weber, N., Beckstein, P., Herreman, W., Horstmann, G. M., Nore, C., Stefani, F. & Weier, T. 2017b Sloshing instability and electrolyte layer rupture in liquid metal batteries. Phys. Fluids 29 (5), 054101.Google Scholar
Weber, N., Galindo, V., Stefani, F. & Weier, T. 2014 Current-driven flow instabilities in large-scale liquid metal batteries, and how to tame them. J. Power Sources 265 (0), 166–173.Google Scholar
Weber, N., Galindo, V., Stefani, F., Weier, T. & Wondrak, T. 2013 Numerical simulation of the Tayler instability in liquid metals. New J. Phys. 15 (4), 043034.Google Scholar
Ziegler, D. P. 1993 Stability of metal/electrolyte interface in Hall–Héroult cells: effect of the steady velocity. Metall. Mater. Trans. B 24 (5), 899–906.Google Scholar
Zikanov, O. 2015 Metal pad instabilities in liquid metal batteries. Phys. Rev. E 92 (6), 063021.Google Scholar
Zikanov, O. 2018 Shallow water modeling of rolling pad instability in liquid metal batteries. Theor. Comput. Fluid Dyn. 32 (3), 325–347.Google Scholar
Zikanov, O., Sun, H. & Ziegler, D. P. 2004 Shallow water model of flows in Hall–Héroult cells. In Light Metals, pp. 445–452. TMS.Google Scholar
Zikanov, O., Thess, A., Davidson, P. A. & Ziegler, D. P. 2000 A new approach to numerical simulation of melt flows and interface instability in Hall–Heroult cells. Metall. Mater. Trans. B 31 (6), 1541–1550.Google Scholar
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