Contact :

Marie Charlotte Renoult, associate professor

Context

Liquid Metal Batteries (LMBs) have recently been introduced to solve a major problem concerning the storage of energy from the electricity grid. Since the grid has almost no storage capacity, the electricity produced must then be immediately consumed, favoring storable energy sources such as fossil fuels to the detriment of intermittent energy sources such as renewable. As the energy transition is underway, a multitude of new technologies concerning accumulators, the most familiar means of energy storage, are being studied, whether for stationary applications as mentioned above or for mobility applications such as in transport or the Internet of Things. In addition to LMBs, we can notably cite all-solid-state batteries or supercapacitors. Regarding stationary applications, LMBs seem to be ideal candidates due to their low overall cost of ownership and their high cyclability.

LMBs are composed of two electrodes (an anode and a cathode) separated by an electrolyte. Their particularity is that these three components are liquid. It is therefore necessary to study their flows and in particular the conditions of their stability inside these BMLs to avoid a short circuit that would occur in the event of contact between the two electrodes. For this, among many hydrodynamic instabilities that can even coexist within a liquid metal battery, only the interfacial instabilities due to the redistribution of the electric current and not to the thermal one are studied here. These instabilities can have the behavior of a progressive wave for infinite systems or by considering disturbances of small wavelengths; they can however have the behavior of a standing wave, rotating along the walls for bounded systems or by considering disturbances of large wavelengths.

Liquid Metal Battery model.

Effect of the coupling parameter (H) for a system with a gravitationally stable interface (top)
and a gravitationally unstable interface (bottom).

Theory without magnetic field (progressive wave)

We have conducted a linear stability analysis to study the effect of viscosity and surface tension in an infinite three-layer fluid system subjected to the gravity field alone. We have characterized the coupling between the interfaces in order to better understand previous experiments. The results show that the behavior of the interfaces depends on a coupling parameter: the thickness of the medium layer adimensionalized by the wavenumber of the perturbation. Direct numerical simulations performed with the Archer code have validated the theory until the nonlinear effects are significant and have identified three regimes according to the increasing value of the coupling parameter for a system where the upper interface is gravitationally stable and the lower interface is gravitationally unstable: the succession of two coupled regimes, one dominated by the behavior of a gravity wave, the other dominated by the behavior of a Rayleigh-Taylor instability and a decoupled regime.

The external magnetic field can destabilize a system
where both interfaces are gravitationally stables.

Theory with magnetic field (progressive wave)

We then studied the effect of the magnetic field on a three-layer system representative of BMLs. The eigenvalue problem was solved by varying an operating parameter of the battery: a dimensionless magnetic field, b, as well as the coupling parameter H, in order to determine the time coefficients of the perturbation. This allows to obtain a map indicating the stability of the system for a given couple (H,b). A critical magnetic field bC due to the electric current imposed on the system and from which the latter can potentially destabilize is then determined. This increases and therefore favors stability when the dimensionless density of the lower layer increases and when the dimensionless density of the upper layer decreases. Naturally, when the dimensionless surface tensions increase, the stability domain of the system widens. Viscosity has no effect on the stability domain; it only slows down the evolution of the perturbation.

Direct numerical simulation of interfacial instability
in a bounded system performed with ARCHER.

Side effects (stationary wave)

In order to briefly study the stationary nature of the interfacial instability in liquid metal batteries, We used an energetic method to study the edge effects as well as the effects of the magnetic field on a system composed of two fluid layers. This notably allowed to find the stability criterion of the aluminum reduction cells proposed by Sele, thus allowing to lay the foundations for the calculation of a generalized Sele criterion taking into account more difficulties.

Finally, new direct numerical simulations were carried out with the in-house Archer code. These simulations first allowed to find the flow of a wave rotating along the walls and to show that the system was stable only when the stability criterion determined by Sele in the 70s was verified. However, for certain values ​​of the physical parameters, the behavior of the interface whose amplitude reaches a threshold value and then whose wave continues to rotate along the walls was not predictable by this theory.