Liquid metal batteries are proposed for low-cost grid scale energy storage. During their operation, solid intermetallic phases often form in the cathode and are known to limit the capacity of the cell. Fluid flow in the liquid electrodes can enhance mass transfer and reduce the formation of localized intermetallics, and fluid flow can be promoted by careful choice of the locations and topology of a battery's electrical connections. In this context we study four phenomena that drive flow: Rayleigh-B\'enard convection, internally heated convection, electro-vortex flow, and swirl flow, in both experiment and simulation. In experiments, we use ultrasound Doppler velocimetry (UDV) to measure the flow in a eutectic PbBi electrode at 160{\deg}C and subject to all four phenomena. In numerical simulations, we isolate the phenomena and simulate each separately using OpenFOAM. Comparing simulated velocities to experiments via a UDV beam model, we find that all four phenomena can enhance mass transfer in LMBs. We explain the flow direction, describe how the phenomena interact, and propose dimensionless numbers for estimating their mutual relevance. A brief discussion of electrical connections summarizes the engineering implications of our work

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Abstract. Liquid metal batteries (LMBs) consist of two liquid metal electrodes and a molten salt ionic conductor sandwiched between them. The density ratios allow for a stable stratification of the three layers. LMBs were already considered as part of energy conversion systems in the 1960s and have recently received renewed interest for economical large-scale energy storage. In this paper, we concentrate on the magnetohydrodynamic aspects of this cell type with special focus on electro-vortex flows and possible effects of the Tayler instability.

A numerical model for simulating electro-vortical flows in OpenFOAM is developed. Electric potential and current are solved in coupled solid-liquid conductors by a parent-child mesh technique. The magnetic field is computed using a combination of Biot-Savart's law and induction equation. Further, a PCG solver with special regularisation for the electric potential is derived and implemented.Finally, a performance analysis is presented and the solver is validated against several test cases. gradient and therefore drives a flow. For an illustrative example, see Shercliff [17].Numerical simulation of electro-vortex flow is easy when modelling only the fluid, or a non-conducting obstacle inside a fluid. However, in most realistic cases, electric current passes from solid to liquid conductors and vice versa.The electric potential in these regions must therefore be solved in a coupled way. The classical, segregated approach means solving an equation in each region, and coupling the potential only at the interfaces by suitable boundary conditions [11]. While that is easy to implement, convergence is rather poor.An implicit coupling of the different regions by block matrices is a sophisticated alternative for increasing convergence [18]. However, it is memory-intensive and by no means easy to implement.In this article we will present an alternative effective option for region coupling in OpenFOAM. We solve global variables (electric potential, current density) on a global mesh with a variable electric conductivity according to the underlying material. We then map the current density to the fluid regions and compute the electromagnetic induced flow there. This parent-child mesh technique was already used for the similar problem of thermal conduction [19,20] and just recently for the solution of eddy-current problems with the finite volume method [21]. Mathematical and numerical model OverviewThe presented multi-region approach is based on a single phase incompressible magnetohydrodynamic (MHD) model [11,22]. The flow in the fluid is described by the Navier-Stokes equation (NSE)with u denoting the velocity, t the time, p the modified pressure, ν the kinematic viscosity and ρ the density. The fluid flow is modelled as laminar only; adding for the constant and induced magnetic field in the quasi-static limit [32].

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