SHALLOW WATER MODEL FOR ALUMINIUM ELECTROLYSIS CELLS WITH VARIABLE TOP AND BOTTOM Valdis Bojarevics and Koulis Pericleous University of Greenwich, School of Computing and Mathematical Sciences, 30 Park Row, London, SE109LS, UK V.Bojarevics@gre.ac.uk Keywords: aluminium electrolysis cells, MHD stability, interface waves, shallow layer, dynamic multiphysics modelling Abstract The MHD wave instability in commercial cells for electrolytic aluminium production is often described using ‘shallow water’ models. The model [1] is extended for a variable height cathode bottom and anode top to account for realistic cell features. The variable depth of the two fluid layers affects the horizontal current density, the wave development and the stability threshold. Instructive examples for the 500 kA cell are presented. Introduction The MHD stability problem for aluminium electrolysis cells is of increasing importance due to significant electrical energy costs, disruptions in the technology and control of environmental pollution rate. The electric current with the associated magnetic field, are intricately involved in the oscillation process and a possible instability of the interface between liquid aluminium and electrolyte. This interaction results in the wave frequency shift from the purely hydrodynamic ones (see [2] and references therein). Moreau and Evans [3] introduced the linear friction model for the wave motion and the horizontal circulation, and attempted to introduce models for the electrolyte side channel effects on the circulation. Actually, the linear friction and the variable bottom effects are used widely in the sea wave theoretical studies [4]. The linear friction is a simplification of the more general nonlinear bottom friction term appearing in the shallow water models, see for example [5]. The systematic perturbation theory for the shallow layer fluid dynamics and, similarly, for the electric current problems, permitting to reduce the three-dimensional problem of the aluminium cell to a two-dimensional shallow layer problem was developed in [6]. This work mathematically proved the wave oscillation frequency shift due to the magnetic interaction and the possibility of a resonant growth. The wave model has been extended to the weakly nonlinear case using the Boussinesq formulation including the linear dispersion terms [7]. The intense turbulence generated by the horizontal circulation velocity is a critical feature to determine the level of damping friction level. A correct damping level permits just a small amplitude self- sustained oscillations observed in real cells, known as ‘MHD noise’. The fully coupled real cell problem requires time dependent, extended electromagnetic field simulation including the fluid layers, the whole bus bar circuit and the ferromagnetic effects [1]. The present paper extends the ‘shallow layer’ theory and the complete dynamic MHD model to the cases of variable bottom of aluminium pad and the variable thickness of the electrolyte due to the anode nonuniform burn-out process and the presence of the side channels. Mathematical model for waves at the interface between two liquid shallow layers The electric current to the individual cell is supplied from above via massive anode bus bars, from which anode rods connect to the carbon anodes. The liquid electrolyte layer beneath the anode blocks is relatively poor electrical conductor of a small depth (4-5 cm) if compared to its horizontal extension (4-5 m in width and 15-20 m in length). The electrolyte density (ρ 2 = 2.1e3 kg/m 3 ) is just slightly lower than the liquid aluminium (ρ 1 = 2.3e3 kg/m 3 )in the bottom layer of typical depth 15–30 cm. The “shallow layer” approximation assumes that the horizontal dimensions L x and L y are much larger than the typical depth H for each of the layers, and, in addition to this, the interface wave amplitude A is assumed to be small relative to the depth H. In the present extension of the theory for a variable layer depth we will assume that the layer deformation is similarly small. Thus the two small parameters of the problem are the nondimensional depth δ = H/L y and the amplitude ε = A/H. The resulting fluid dynamic equations become two-dimensional after the depth averaging procedure is applied to the horizontal momentum equations. The equations for the combined horizontal velocity (horizontal circulation u 0 , plus ε-order ε u ˆ wave motion) are: 1 0 ˆ ˆ ( ) i o H H dz ε ε − = − = + ∫ u u u u (1) 0 1 0 2 ˆ ˆ ˆ ˆ ( ) ( ) ˆ ˆ ˆ , t j k k j j j j -1 k e k j j i j z u u u pH u Re u E f EH f ρ∂ ∂ ∂ ε∂ ς μ ∂ν∂ δ ∂ + =− − − + + + − (2) where the continuity of the pressure at the interface is satisfied by introducing the pressure 0 ( ) pH at the common interface 0 0 /( ) (, ,) H H L xyt δ ες = = . (3) The summation convention is assumed over the repeating indexes k (1 or 2, respectively for x, y coordinates). The nondimensional variables are introduced using the following typical scales: gH u = 0 for the wave velocity; gH L / for time t, 2 0 1 u ρ for 403 Light Metals 2008 Edited by: David H. DeYoung TMS (The Minerals, Metals & Materials Society), 2008