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solution, which in turn is coupled to the computed potential distribution from the equation
(2) ). At the interface between the liquid metal and the electrolyte the continuity of the
potential and the electric current normal component must be satisfied. Since the depths of the
liquid layers are extremely small if compared to their horizontal extension, the shallow layer
approximation is very efficient to solve this 3-dimensional problem. It can be shown (7) that
the solution, for instance in the aluminium layer, can be obtained from the following
equation:

(3)

Where the aluminium pad interface H

is variable in time and horizontal coordinates, and

the current distribution at the top and the bottom depend on the iterative solution from the
finite element network of the bus bars, anodes, pins, collector bars and the collector bar
rodding procedure (see e.g. Figure 1).

As the beginning of the computer simulation the MHD package generates

automatically a very large set of Kirchhoff equations from the relatively simple unified data
input. The current distribution in the bus bar network can be described to reasonable
approximation accuracy suitable for engineering purposes by linear resistance elements. The
electric currents and voltages in such a complex, multiply connected circuit are governed by
the Kirchhoff laws. The two Kirchhoff laws are:

1) the voltage law: The algebraic sum of the potential differences taken around a

loop (or `mesh') of a circuit is zero;

2) the current law: The algebraic sum of the currents into a node of a circuit is zero.
If directly applied, these laws contain unknown potential differences for each

resistance and unknown currents for each mesh of a given circuit. There are two methods to
reduce the number of independent unknowns and the equations respectively. These are based
on combining the two laws and are referred to as either mesh analysis or nodal analysis.

The mesh analysis is based on currents as unknowns. A set of mesh currents I

are chosen to traverse all complete loops of the circuit. Since each mesh current flows right
through any junction (node) in its path, the current law is automatically satisfied. Then it is
left to apply the voltage law for each of the meshes by replacing the potential differences
with the algebraic sums of currents through the particular resistance multiplied by this
resistance (Ohms law). For the reduction cell situation this procedure was used in the past
and involves a tedious job of locating all possible meshes of the circuit, what is rather hard
to automate in a computer program. Therefore this approach involves an intelligent input
from the program user and results in long labor days with a possibility of potential errors.
Principal changes in the circuit, such as anode or cathode element disconnection or new
branching, require significant reconsideration of the equation set.