between the cells, etc.
experience in the current simulation programs: NEWBUS [1],
reports from Reynolds Metals Company, and Russian program
TOK from the VAMI institute. Current distribution in a busbar
network can be described to a reasonable approximation accuracy
suitable for engineering purposes by linear resistance elements. The
electric currents and voltages in such a circuit are governed by the
Kirchhoff laws:
taken around a loop (or 'mesh') of a circuit is zero;
of a circuit is zero.
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
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).
previously referenced programs and involves a tedious job of
locating all possible meshes of the circuit that 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.
found to be more convenient. According to this method the
potentials at the nodes are the independent unknowns. The use of
node potentials rather than mesh currents makes for greater
efficiency in the analysis of circuits that are predominantly parallel
in character, since the number of simultaneous equations that have
to be solved is significantly smaller. The following equation set
arises for M nodes each of which has N directly connected neighbor
resistances:
reference nodes in the liquid metal of previous cell and '-I' current
leaves the nodes at the liquid metal of the downstream cell. For all
other nodes the external current - right side of the equation - is
zero. This is just another statement of the current law, and the
voltage law is satisfied because the sum of the potential differences,
with
connected in series, and this property is essential in order to
generate automatically the set of equations to solve.
between two neighbor nodes multiplied by the connecting
resistance gives the current in each resistance. The main task of the
present program is to find the current distribution in the bar
network, yet a further improvement in accuracy can be achieved if
computing Joule heating:
temperature of a bar. For this purpose we integrate the temperature
T
dependence (4). This set can also be solved by linear algebra
solvers, e.g., from the package LAPACK.
equation set solved again to iterate the whole procedure while the
convergence is achieved. The convergence is easily established for
bars with reasonable cross sections and sufficiently effective heat
transfer to the ambient air and to the neighbor bars. We incorporate
in the program also the heating at the ends of first and last bars
connected to the anodes and the cathode carbon by assigning the
user defined temperatures at these ends.
current distribution in the busbars. This is calculated by coupling
the electric current in the fluid zone to the resistance network
representing the elements division of individual anodes and
cathode collector bars as well as the whole busbar circuit between
two adjacent cells. The electric current in the fluid zones must be
computed from the continuous media equations governing the DC
current (which can change in time with the waves and anode
burnout process):
are accounted for. The electric potential in the fluid is governed by
the equation:
given current distribution j
in turn is coupled to the computed potential distribution from (24)).
At the interface between the liquid metal and the electrolyte the
continuity of the potential and the electric current normal
component must be satisfied.
MHD-Valdis uses the versatile 1D line element busbar network
generator called BUSNET to create the 1D busbar mesh reading a
very compact user input file. Then in a few seconds, it solves the
non-linear problem and generates a TECPLOT [13] compatible
output file and a standard printout ASCII output file called
BARSOUT.
reproduces exactly the same busbar geometry as the second 1D