Critical modes of convection induced by Joule heating in a liquid metal battery*

Liquid metal batteries (LMB)  consists of three super-imposed liquids in  stable stratificationof density  where two liquid metals (electrodes) sandwich a molten salt that serves as an electrolyte [1]. An electric current of density J₀ is applied through the liquid system. As the electrical conductivity of the electrolyte is much lower than the electrical conductivities of liquid metals, the electric current induces an internal Joule heating with the density q  through the electrolyte. In conduction regime, the temperature profile is linear in both the liquid metals and it has a parabolic profile in the electrolyte. The upper part of the electrolyte and the upper electrode are thermally unstable while the lower part of the electrolyte and the lower electrode are thermally stable.

Many hydrodynamic phenomena can develop in the battery: natural convection, interfacial instabilities, electro-magneto-hydrodynamic instabilities [1]. In the present study, we have neglected the magnetic effects induced by the applied current or any external magnetic fields. We have formulated the natural convection model in LMB with the appropriate boundary conditions.  We have performed linear stability analysis and determined the critical parameters of natural convection in the battery for imposed temperatures of LMB boundaries. We found that,  above a critical value of the electric current J_0c, natural convection modes develop in the electrolyte and they invade the upper electrode while the lower electrode remains stable.  A parametric study of the critical modes is performed with respect to the relative size of the elecrodes with respect to the electrolyte.  The dependence of the critical value of Rayleigh number with the Marangoni number is analysed for three types of batteries [2, 3,4].

References

[1] D. H. Kelley and T. Weier, Appl. Mech. Rev. 70 (2), 020801 (2018).

[2] Y. Shen and O.  Zikanov, Theor. Comput. Fluid Dyn. 30, 275-294 (2016).

[3] T. Köllner, T. Boeck and J. Schumacher, Phys. Rev. E 95, 053114 (2017).

[4] D. J. Bradwell, H. Kim, A. H. Sirk and D. R. Sadoway, J. Am. Chem. Soc. 134(4), 1895-1897 (2012).