Skip to main content Accessibility help
×
Home

Coupling and stability of interfacial waves in liquid metal batteries

  • G. M. Horstmann (a1), N. Weber (a1) and T. Weier (a1)

Abstract

We investigate the coupling dynamics of interfacial waves in liquid metal batteries and its effects on the battery’s operation safety. Similar to aluminium reduction cells, liquid metal batteries can be highly susceptible to magnetohydrodynamically exited interfacial instabilities. The resulting waves are capable of provoking short-circuits. Owing to the presence of two metal-electrolyte interfaces that may step into resonance, the wave dynamics in liquid metal batteries is particularly complex. In the first part of this paper, we present a potential flow analysis of coupled gravity–capillary interfacial waves. While we are focusing here on liquid metal batteries with circular cross-section, the theory is applicable to arbitrary stably stratified three-layer systems. Analytical expressions for the amplitude ratio and the wave frequencies are derived. It is shown that the wave coupling can be completely described by two independent dimensionless parameters. We further provide a decoupling criterion that suggests that wave coupling will be present in most future liquid metal batteries. In the second part, the theory is validated by comparing it with multiphase direct numerical simulations. An accompanying parameter study is conducted to analyse the system stability for interfaces coupled to varying degrees. Three different coupling regimes are identified involving characteristic coupling dynamics. For strongly coupled interfaces we observe novel instabilities that may have beneficial effects on the operational safety.

Copyright

Corresponding author

Email address for correspondence: g.horstmann@hzdr.de

References

Hide All
Abramowitz, M. & Stegun, I. A.(Eds) 1972 Handbook of Mathematical Functions, 10th edn. (Applied Mathematics Series) , vol. 55. National Bureau of Standards.
Agruss, B., Karas, H. R. & Decker, V. L.1962 Design and development of a liquid metal fuel cell. Tech. Rep. ASD-TDR-62-1045.
Aqra, F. & Ayyad, A. 2011 Theoretical estimation of temperature-dependent surface tension of liquid antimony, boron, and sulfur. Metall. Mater. Trans. B 42 (3), 437440.
Beljajew, A. I., Rapoport, M. B. & Firsanowa, L. A. 1957 Metallurgie des Aluminiums, vol. 2. VEB Verlag Technik.
Bojarevics, V. & Pericleous, K. 2006 Comparison of MHD models for aluminium reduction cells. In Proc. TMS Light Met., pp. 347352.
Bojarevics, V. & Romerio, M. V. 1994 Long waves instability of liquid metal–electrolyte interface in aluminium electrolysis cell: a generalization of Sele’s criterion. Eur. J. Mech. (B/Fluids) 13, 3356.
Bojarevics, V. & Tucs, A. 2017 MHD of large scale liquid metal batteries. In Light Metals (ed. Ratvik, A.). The Minerals, Metals & Materials Series. Springer.
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100, 335354.
Bradwell, D. J.2011 Liquid metal batteries: ambipolar electrolysis and alkaline earth electroalloying cells. PhD thesis, Massachusetts Institute of Technology.
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), 18951897.
Cairns, E. J., Crouthamel, C. E., Fischer, A. K., Foster, M. S., Hesson, J. C., Johnson, C. E., Shimotake, H. & Tevebaugh, A. D. 1967 Galvanic Cells with Fused-Salt Electrolytes. Argonne National Laboratory.
Cairns, E. J. & Shimotake, H. 1969a High-temperature batteries. Science 164, 13471355.
Cairns, E. J. & Shimotake, H. 1969b Recent advances in fuel cells and their application to new hybrid systems. Adv. Chem. 90, 321350.
Cappanera, L., Guermond, J.-L., Herreman, W. & Nore, C. 2017 Momentum-based approximation of incompressible multiphase fluid flows: momentum-based approximation of incompressible multiphase fluid flows. Int. J. Numer. Meth. Fluids 86 (8), 541563.
Chum, H. L. & Osteryoung, R. A. 1980 Review of Thermally Regenerative Electrochemical Systems. Solar Energy Research Institute.
Chum, H. L. & Osteryoung, R. A. 1981 Review of Thermally Regenerative Electrochemical Cells. Solar Energy Research Institute.
Davidson, P. A. 2001 An Introduction to Magnetohydrodynamics, Cambridge Texts in Applied Mathematics. Cambridge University Press.
Davidson, P. A. & Lindsay, R. I. 1998 Stability of interfacial waves in aluminium reduction cells. J. Fluid Mech. 362, 273295.
Ferziger, J. H. & Perić, M. 1996 Computational Methods for Fluid Dynamics. Springer.
Gerbeau, J.-F., Le Bris, C. & Lelievre, T.2001, Simulations of MHD flows with moving interfaces. Tech. Rep. RR-4277. INRIA.
Gerbeau, J.-F., Le Bris, C. & Lelièvre, T. 2006 Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Numerical Mathematics and Scientific Computation. Oxford University Press.
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), 27392757.
Herédy, L. A., Iverson, M. L., Ulrich, G. D. & Recht, H. L. 1967 Development of a thermally regenerative sodium–mercury galvanic system part I. Electrochemical and chemical behavior of sodium–mercury galvanic cells. In Regenerative EMF Cells, pp. 3042.
Herreman, W., Nore, C., Cappanera, L. & Guermond, J.-L. 2015 Tayler instability in liquid metal columns and liquid metal batteries. J. Fluid Mech. 771, 79114.
Ibrahim, R. A. 2005 Liquid Sloshing Dynamics Theory and Applications. Cambridge University Press.
International Atomic Energy Agency 2008 Thermophysical Properties of Materials for Nuclear Engineering: A Tutorial and Collection of Data. International Atomic Energy Agency.
Issenmann, B., Laroche, C. & Falcon, E. 2016 Wave turbulence in a two-layer fluid: coupling between free surface and interface waves. Eur. Phys. Lett. 116, 64005.
Janz, G. J., Allen, C. B., Bansal, N. P., Murphy, R. M. & Tomkins, R. P. T. 1979 Physical Properties Data Compilations Relevant to Energy Storage. II. Molten Salts: Data on Single and Multi-Component Salt Systems, U.S. Department of Commerce.
Janz, G. J., Tomkins, R. P. T., Allen, C. B., Downey, J. R., Gardner, G. L., Krebs, U. & Singer, S. K. 1975 Molten salts: clorides and mixtures. J. Phys. Chem. Ref. Data 4 (4), 8711178.
Karas, H. R. & Mangus, J. D.1963 First quarterly technical progress report on research and development of an advanced laboratory liquid metal regenerative fuel cell. Tech. Rep. EDR 3344. Allison Division of General Motors Corporation.
Kelley, D. H. & Sadoway, D. R. 2014 Mixing in a liquid metal electrode. Phys. Fluids 26 (5), 057102.
Kim, H., Boysen, D. A., Newhouse, J. M., Spatocco, B. L., Chung, B., Burke, P. J., Bradwell, D. J., Jiang, K., Tomaszowska, A. A., Wang, K., Wei, W., Ortiz, L. A., Barriga, S. A., Poizeau, S. M. & Sadoway, D. R. 2013 Liquid metal batteries: past, present, and future. Chem. Rev. 113 (3), 20752099.
Köllner, T., Boeck, T. & Schumacher, J. 2017 Thermal Rayleigh–Marangoni convection in a three-layer liquid-metal-battery model. Phys. Rev. E 95, 053114.
Lukyanov, A., El, G. & Molokov, S. 2001 Instability of MHD-modified interfacial gravity waves revisited. Phys. Lett. A 290, 165172.
Lyon, R. N.(Ed.) 1954 Liquid-Metals Handbook, Oak Ridge National Laboratory.
Mohapatra, S. C., Karmakar, D. & Sahoo, T. 2011 On cappillary gravity-wave motion in two-layer fluids. J. Engng Maths 71, 253277.
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), 261279.
Munger, D. & Vincent, A. 2006 Electric boundary conditions at the anodes in aluminum reduction cells. Metall. Mater. Trans. B 37B, 10251035.
Munger, D. & Vincent, A. 2008 A cylindrical model for rotational MHD instabilities in aluminum reduction cells. Theor. Comput. Fluid Dyn. 22, 363382.
Pearson, T. G. & Phillips, H. W. L. 1957 The production and properties of super-purity aluminium. Metall. Rev. 2 (8), 305360.
Platzer, B. & Noll, G. 1983 Möglichkeiten zur analytischen Beschreibung der turbulenten Strömung in unbewehrten und teilbewehrten Rührkesseln mit radialfördernden Rührern. Chem. Techn. 35 (5), 235238.
Rusche, H.2002 Computational fluid dynamics of dispersed two-phase flows at high phase fractions. PhD thesis, Imperial College London.
Santalo, L. A. 1993 Vectores Y Tensores Con Sus Aplicaciones. Buenos Aires.
Sele, T. 1977 Instabilities of the metal surface in electrolytic alumina reduction cells. Metall. Trans. B 8 (4), 613618.
Shen, Y. & Zikanov, O. 2016 Thermal convection in a liquid metal battery. Theor. Comput. Fluid Dyn. 30 (4), 275294.
Shimotake, H., Rogers, G. L. & Cairns, E. J. 1969 Secondary cells with lithium anodes and immobilized fused-salt electrolytes. Ind. Engng Chem. Process Des. Dev. 8 (1), 5156.
Smithells, C. J., Gale, W. F. & Totemeier, T. C. 2004 Smithells Metals Reference Book, 8th edn. Elsevier Butterworth-Heinemann.
Sneyd, A. D. & Wang, A. 1994 Interfacial instability due to MHD mode coupling in aluminium reduction cells. J. Fluid Mech. 263, 343359.
Sobolev, V. 2007 Thermophysical properties of lead and lead–bismuth eutectic. J. Nucl. Mater. 362 (2-3), 235247.
Sobolev, V. 2010 Database of Thermophysical Properties of Liquid Metal Coolants for GEN-IV. SCK CEN.
Spatocco, B. L., Burke, P. J. & Sadoway, D. R. 2014 Low temperature liquid metal batteries for grid-scaled storage. US Patent Nr. 0099522 A1.
Stefani, F., Galindo, V., Kasprzyk, C., Landgraf, S., Seilmayer, M., Starace, M., Weber, N. & Weier, T. 2016 Magnetohydrodynamic effects in liquid metal batteries. IOP Conf. Ser. Mater. Sci. Engng 143, 012024.
Stefani, F., Weier, T., Gundrum, T. & Gerbeth, G. 2011 How to circumvent the size limitation of liquid metal batteries due to the Tayler instability. Energy Convers. Manage. 52, 29822986.
Steiner, G.2009 Simulation numérique de phénomènes MHD: Application à l’électrolyse de l’aluminium. PhD thesis, École polytechnique fédérale de Lausanne.
Swinkels, D. A. J. 1971 Molten salt batteries and fuel cells. In Advances in Molten Salt Chemistry (ed. Braunstein, J., Mamantov, G. & Smith, G. P.), vol. 1, pp. 165223. Plenum Press.
Ubbink, O.1997 Numerical prediction of two fluid systems with sharp interfaces. PhD thesis, University of London.
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), 348350.
Weaver, R. D., Smith, S. W. & Willmann, N. L. 1962 The sodium–tin liquid–metal cell. J. Electrochem. Soc. 109 (8), 653657.
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), 129140.
Weber, N., Beckstein, P., Galindo, V., Starace, M. & Weier, T.2018 Electro-vortex flow simulation using coupled meshes. Comput. Fluids (in press, DOI: https://doi.org/10.1016/j.compfluid.2018.03.047).
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, 054101.
Weber, N., Galindo, V., Priede, J., Stefani, F. & Weier, T. 2015a The influence of current collectors on Tayler instability and electro vortex flows in liquid metal batteries. Phys. Fluids 27, 014103.
Weber, N., Galindo, V., Stefani, F. & Weier, T. 2015b The Tayler instability at low magnetic Prandtl numbers: between chiral symmetry breaking and helicity oscillations. New J. Phys. 17 (11), 113013.
Weber, N., Galindo, V., Stefani, F., Weier, T. & Wondrak, T. 2013 Numerical simulation of the Tayler instability in liquid metals. New J. Phys. 15, 043034.
Weier, T., Bund, A., El-Mofid, W., Horstmann, G. M., Lalau, C.-C., Landgraf, S., Nimtz, M., Starace, M., Stefani, F. & Weber, N. 2017 Liquid metal batteries – materials selection and fluid dynamics. IOP Conf. Ser.: Mater. Sci. Engng 228, 012013.
Weller, H. G., Tabor, G., Jasak, H. & Fureby, C. 1998 A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput. Phys. 12 (6), 620631.
Woolfenden, H. C. & Parau, E. I. 2011 Numerical computation of solitary waves in a two-layer fluid. J. Fluid Mech. 688, 528550.
Zhang, S. & Zhao, X. 2004 General formulations for Rhie–Chow interpolation. In ASME 2004 Heat Transfer/Fluids Engineering Summer Conference, ASME.
Zikanov, O. 2015 Metal pad instabilities in liquid metal batteries. Phys. Rev. E 92, 063021.
Zikanov, O.2017 Shallow water modeling of rolling pad instability in liquid metal batteries. arXiv:1706:08589v1.
Zinkle, S. J. 1998 Summary of physical properties for lithium, Pb-17Li, and (LiF)n⋅BeF2 Coolants. In APEX Study Meeting, Sandia National Laboratories.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Coupling and stability of interfacial waves in liquid metal batteries

  • G. M. Horstmann (a1), N. Weber (a1) and T. Weier (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed