Perturbation theory for metal pad roll instability in cylindrical reduction cells

Abstract

We propose a new theoretical model for metal pad roll instability in idealized cylindrical reduction cells. In addition to the usual destabilizing effects, we model viscous and Joule dissipation and some capillary effects. The resulting explicit formulas are used as theoretical benchmarks for two multiphase magnetohydrodynamic solvers, OpenFOAM and SFEMaNS. Our explicit formula for the viscous damping rate of gravity waves in cylinders with two fluid layers compares excellently to experimental measurements. We use our model to locate the viscously controlled instability threshold in cylindrical shallow reduction cells but also in Mg–Sb liquid metal batteries with decoupled interfaces.

Efficient mixing by swirling electrovortex flows in liquid metal batteries

“…(2018) and used in the LMB context in Herreman et al. (2015, 2019 a , b , 2020). All the fields are decomposed on a Fourier basis in the azimuthal direction and finite element bases in the meridian plane.…”

Fractality of metal pad instability threshold in rectangular cells

“…This approximation, which is used already by Sneyd & Wang (1994), is applicable to aluminium reduction cells because The two-layer system is subject to a downward gravity force with the free fall acceleration g and external magnetic field B 0 = B 0 e z which is assumed to be vertical and uniform. Horizontal components of the magnetic field, which are known to have a stabilizing effect as long as they are predominantly generated by the current passing through the system (Sneyd 1985;Herreman et al 2019), are as usual assumed to be negligible for the metal pad instability. This is because the effect of the horizontal magnetic field being determined by its horizontal gradient (Sneyd 1985) is ∼ H/L = ε 1 relative to that of the vertical magnetic field which is determined by the magnitude of this field (Sneyd & Wang 1994).…”

“…Horizontal components of the magnetic field, which are known to have a stabilizing effect as long as they are predominantly generated by the current passing through the system (Sneyd 1985; Herreman et al. 2019), are as usual assumed to be negligible for the metal pad instability. This is because the effect of the horizontal magnetic field being determined by its horizontal gradient (Sneyd 1985) is relative to that of the vertical magnetic field which is determined by the magnitude of this field (Sneyd & Wang 1994).…”

“…The low stability threshold of the square cell is attributed to the occurrence of several pairs of gravity wave modes with the same frequency (Urata, Mori & Ikeuchi 1976). The same applies also to cylindrical cells, which are considered as an option for the liquid metal batteries (Herreman et al 2019;Horstmann, Wylega & Weier 2019). According to the electromechanical model of Davidson & Lindsay (1998), such cells are expected to be inherently unstable.…”

“…The same applies also to cylindrical cells, which are considered as an option for the liquid metal batteries (Herreman et al. 2019; Horstmann, Wylega & Weier 2019). According to the electromechanical model of Davidson & Lindsay (1998), such cells are expected to be inherently unstable.…”