Liquid metal batteries (LMBs) are discussed today as a cheap grid scale energy storage, as required for the deployment of fluctuating renewable energies. Built as a stable density stratification of two liquid metals separated by a thin molten salt layer, LMBs are susceptible to short-circuit by fluid flows. Using direct numerical simulation, we study a sloshing long wave interface instability in cylindrical cells, which is already known from aluminium reduction cells. After characterising the instability mechanism, we investigate the influence of cell current, layer thickness, density, viscosity, conductivity and magnetic background field. Finally we study the shape of the interface and give a dimensionless parameter for the onset of sloshing as well as for the short-circuit.

The increasing deployment of renewable energies requires three fundamental changes to the electric grid: more transmission lines, a flexibilisation of the demand and grid scale energy storage. Liquid metal batteries (LMBs) are considered these days as a promising means of stationary energy storage. Built as a stable density stratification of two liquid metals separated by a liquid salt, LMBs have three main advantages: a low price, a long life-time and extremely high current densities. In order to be cheap, LMBs have to be built large. However, battery currents in the order of kilo-amperes may lead to magnetohydrodynamic (MHD) instabilities, which -in the worst case -may short-circuit the thin electrolyte layer. The metal pad roll instability, as known from aluminium reduction cells, is considered as one of the most dangerous phenomena for LMBs. We develop a numerical model, combining fluid-and electrodynamics with the volume-of-fluid method, to simulate this instability in cylindrical LMBs. We explain the instability mechanism similar to that in aluminium reduction cells and give some first results, including growth rates and oscillation periods of the instability 1 . MotivationAccording to prognoses of the International Energy Agency, the worldwide energy demand will grow from the year 2011 to 2035 by two thirds. In the same period, the share of renewable energies is predicted to rise from 20 to 31 % [2]. These renewable energies are highly fluctuating; in order to stabilise voltage and frequency in the electric grid, new transmission lines must be built 1 This article is an extended version of a conference paper of the proceedings of the 10th PAMIR conference [1]. electrolyte alloy metal (a) short circuit ! (b) Figure 1: Scheme of a liquid metal battery with typical inventory (a), and short circuit due to a strong fluid flow in the upper metal compartment (b). arXiv:1612.03656v1 [physics.flu-dyn]

Liquid Metal Batteries (LMBs) are a promising concept for cheap electrical energy storage at grid level. These are built as a stable density stratification of three liquid layers, with two liquid metals separated by a molten salt. In order to ensure a safe and efficient operation, the understanding of transport phenomena in LMBs is essential. With this motivation we study thermal convection induced by internal heat generation. We consider the electrochemical nature of the cell in order to define the heat balance and the operating parameters. Moreover we develop a simple 1D heat conduction model as well as a fully 3D thermo-fluid dynamics model. The latter is implemented in the CFD library OpenFOAM, extending the volume of fluid solver, and validated against a pseudo-spectral code. Both models are used to study a rectangular 10×10 cm Li||Bi LMB cell at three different states of charge.

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].

Eddy-current problems occur in a wide range of industrial and metallurgical applications where conducting material is processed inductively. Motivated by realising coupled multi-physics simulations, we present a new method for the solution of such problems in the finite volume framework of foam-extend, an extended version of the very popular OpenFOAM software. The numerical procedure involves a semi-coupled multi-mesh approach to solve Maxwell's equations for non-magnetic materials by means of the Coulomb gauged magnetic vector potential A and the electric scalar potential φ. The concept is further extended on the basis of the impressed and reduced magnetic vector potential and its usage in accordance with BiotSavart's law to achieve a very efficient overall modelling even for complex three-dimensional geometries. Moreover, we present a special discretisation scheme to account for possible discontinuities in the electrical conductivity. To complement our numerical method, an extensive validation is completing the paper, which provides insight into the behaviour and the potential of our approach.