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Stability theory for metal pad roll in cylindrical liquid metal batteries

Published online by Cambridge University Press:  26 April 2023

W. Herreman Open the ORCID record for W. Herreman [Opens in a new window] ,
L. Wierzchalek ,
G.M. Horstmann Open the ORCID record for G.M. Horstmann [Opens in a new window] ,
L. Cappanera Open the ORCID record for L. Cappanera [Opens in a new window]  and
C. Nore Open the ORCID record for C. Nore [Opens in a new window]
W. Herreman*
Affiliation:
Université Paris-Saclay, CNRS, Laboratoire FAST, F-91405 Orsay, France
L. Wierzchalek
Affiliation:
Université Paris-Saclay, CNRS, Laboratoire FAST, F-91405 Orsay, France
G.M. Horstmann
Affiliation:
Helmholtz-Zentrum Dresden – Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany
L. Cappanera
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA
C. Nore
Affiliation:
Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, F-91405 Orsay, France
*
†Email address for correspondence: wietze.herreman@universite-paris-saclay.fr
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Abstract

When liquid metal batteries are charged or discharged, strong electrical currents are passing through the three liquid layers that we find in their interior. This may result in the metal pad roll instability that drives gravity waves on the interfaces between the layers. In this paper, we investigate theoretically metal pad roll instability in idealised cylindrical liquid metal batteries that were simulated previously by Weber et al. (Phys. Fluids, vol. 29, no. 5, 2017b, 054101) and Horstmann et al. (J. Fluid Mech., vol. 845, 2018, pp. 1–35). Near the instability threshold, we expect weakly destabilised gravity waves, and in this parameter regime, we can use perturbation methods to find explicit formulas for the growth rate of all possible waves. This perturbative approach also allows us to include dissipative effects, hence we can locate the instability threshold with good precision. We show that our theoretical growth rates are in quantitative agreement with previous and new direct numerical simulations. We explain how our theory can be used to estimate a lower bound on cell size beneath which metal pad roll instability is unlikely.

JFM classification

MHD and Electrohydrodynamics: MHD and Electrohydrodynamics Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 962 , 10 May 2023 , A6
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Bojarevics, V. 1998 Nonlinear waves with electromagnetic interaction in aluminum electrolysis cells. Prog. Astronaut. Aeronaut. 182, 833–848.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, 33–33.Google Scholar
Bojarevics, V. & Tucs, A. 2017 MHD of large scale liquid metal batteries. In Light Metals 2017, pp. 687–692. Springer.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Case, K.M. & Parkinson, W.C. 1957 Damping of surface waves in an incompressible liquid. J. Fluid Mech. 2 (02), 172–184.CrossRefGoogle Scholar
Davidson, P.A. & Lindsay, R.I. 1998 Stability of interfacial waves in aluminium reduction cells. J. Fluid Mech. 362, 273–295.CrossRefGoogle Scholar
Eltishchev, V., Losev, G., Kolesnichenko, I. & Frick, P. 2022 Circular surface wave in a cylindrical MHD cell. Exp. Fluids 63 (8), 127.CrossRefGoogle Scholar
Gerbeau, J.-F., Le Bris, C. & Lelièvre, T. 2006 Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Clarendon.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Herreman, W., Nore, C., Guermond, J.-L., Cappanera, L., Weber, N. & Horstmann, G.M. 2019 Perturbation theory for metal pad roll instability in cylindrical reduction cells. J. Fluid Mech. 878, 598–546.CrossRefGoogle 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.CrossRefGoogle Scholar
Kelley, D.H. & Weier, T. 2018 Fluid mechanics of liquid metal batteries. Appl. Mech. Rev. 70 (2), 020801.CrossRefGoogle Scholar
Landau, L.D. & Lifshitz, E.M. 1987 Fluid Mechanics. Pergamon.Google Scholar
Lukyanov, A., El, G. & Molokov, S. 2001 Instability of MHD-modified interfacial gravity waves revisited. Phys. Lett. A 290 (3), 165–172.CrossRefGoogle Scholar
Molokov, S. 2018 The nature of interfacial instabilities in liquid metal batteries in a vertical magnetic field. Europhys. Lett. 121 (4), 44001.CrossRefGoogle Scholar
Nore, C., Cappanera, L., Guermond, J.-L., Weier, T. & Herreman, W. 2021 Feasibility of metal pad roll instability experiments at room temperature. Phys. Rev. Lett. 126 (18), 184501.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Ouchi, T., Kim, H., Ning, X. & Sadoway, D.R. 2014 Calcium–antimony alloys as electrodes for liquid metal batteries. J. Electrochem. Soc. 161 (12), A1898.CrossRefGoogle Scholar
Pedchenko, A., Molokov, S. & Bardet, B. 2016 The effect of ‘wave breakers’ on the magnetohydrodynamic instability in aluminum reduction cells. Metall. Mater. Trans. B 1–5.Google Scholar
Pedchenko, A., Molokov, S., Priede, J., Lukyanov, A. & Thomas, P.J. 2009 Experimental model of the interfacial instability in aluminium reduction cells. Europhys. Lett. 88 (2), 24001.CrossRefGoogle Scholar
Sele, T. 1977 Instabilities of the metal surface in electrolytic alumina reduction cells. Metall. Mater. Trans. B 8 (4), 613–618.CrossRefGoogle Scholar
Sneyd, A.D. 1985 Stability of fluid layers carrying a normal electric current. J. Fluid Mech. 156, 223–236.CrossRefGoogle Scholar
Sneyd, A.D. & Wang, A. 1994 Interfacial instability due to MHD mode coupling in aluminium reduction cells. J. Fluid Mech. 263, 343–360.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Tucs, A., Bojarevics, V. & Pericleous, K. 2018 a Magneto-hydrodynamic stability of a liquid metal battery in discharge. Europhys. Lett. 124, 24001.CrossRefGoogle Scholar
Tucs, A., Bojarevics, V. & Pericleous, K. 2018 b Magnetohydrodynamic stability of large scale liquid metal batteries. J. Fluid Mech. 852, 453–483.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Weber, N., Beckstein, P., Galindo, V., Herreman, W., Nore, C., Stefani, F. & Weier, T. 2017 a Metal pad roll instability in liquid metal batteries. Magnetohydrodynamics 53 (1), 129–140.CrossRefGoogle Scholar
Weber, N., Beckstein, P., Herreman, W., Horstmann, G.M., Nore, C., Stefani, F. & Weier, T. 2017 b Sloshing instability and electrolyte layer rupture in liquid metal batteries. Phys. Fluids 29 (5), 054101.CrossRefGoogle Scholar
Xiang, L. & Zikanov, O. 2019 Numerical simulation of rolling pad instability in cuboid liquid metal batteries. Phys. Fluids 31 (12), 124104.Google Scholar
Zikanov, O. 2015 Metal pad instabilities in liquid metal batteries. Phys. Rev. E 92 (6), 063021.CrossRefGoogle ScholarPubMed
Zikanov, O. 2018 Shallow water modeling of rolling pad instability in liquid metal batteries. Theor. Comput. Fluid Dyn. 32 (3), 325–347.CrossRefGoogle 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.CrossRefGoogle Scholar
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