The shallow water approximation applied

The industrial production of aluminium via the electrolysis process involves two superposed horizontal liquid layers subjected to a vertical electric current supplied by carbon electrodes. The lower layer, consisting of molten aluminium, rests on the cathode, while the upper layer serves as the electrolyte, covered by the anode. Oscillations or waves at the interface between these layers, analogous to surface waves in seas or lakes, are often perturbed due to the influence of electric currents and the resulting magnetic fields, which generate electromagnetic (Lorentz) forces capable of amplifying these oscillations and negatively affecting the process. To accurately model the interface motion, the shallow water equations—depth-averaged Navier-Stokes equations often used in oceanography—are applied based on the aspect ratio of the electrolytic bath. This study explores various approximation orders of these equations to account for essential nonlinear and dispersion terms typically overlooked in existing literature concerning interface instabilities in aluminium reduction cells, where predominantly linear theory is utilized. A finite volume resolution incorporating electromagnetic sources has been developed for integration into a three-dimensional computational fluid dynamics code, which also addresses heat transfer within the cell.