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SPECIAL
ALUMINIUM SMELTING INDUSTRY
Theory
The theory is described in a more detailed
manner in reference [4]. Let us briefly mention
that the alumina distribution in the bath
is determined by the following partial differential
equation:
c: Alumina concentration in the bath
D: Alumina diffusion coefficient
u: Bath velocity field generated by MHD
Lorentz force and by the release of bubbles
If we consider fluctuations around the stationary
state:
(2)
The velocity field is generated by Lorentz
force field and by the bubbles. When the
number of bubbles produced, per m 2 and per
second, is too large, we cannot use a numerical
approach describing the motion of each
bubble separately.
There are essentially two standard ways to
overcome this difficulty. The first consists of
performing some kind of averaging over the
equations and over the corresponding fields.
The second bypasses the averaging and directly
postulates the flow equations for each
phase.
One of the main difficulties encountered
when performing an averaging process is related
to the possible jumps that fields can suffer
at the boundaries between the two phases.
One way to overcome this problem consists
in extending the domain of definition of each
motion equation to the domain occupied by
the two phases. This is achieved by multiplying
each equation by the characteristic function
corresponding to its domain of definition.
Derivatives are then performed in the sense
of distributions which allows us to keep track
of these discontinuities in the averaging process.
Whatever choice we make, the resulting
equations will contain terms which reflect
the interaction between the two phases. The
exact shapes of these terms are not known;
they have to be defined through constitutive
equations.
(3)
Taking into account the first law of Fick, eq.
3 becomes: in Fig. 3. From the two figures, the alumina
concentration field appears as only slightly
modified by the velocity field. However, when
An estimation shows that D turb is of the order
of 0.2 m 2 /s.
Industrial cell
(4)
The problem has been solved for a 180 kA cell
using two point feeders. On the feeders, the
alumina concentration is set to 5% of the bath
weight. When presenting a stationary solution,
this assumes continuous feeding. But we can
easily analyse the
impact of dump
feeding. Fig. 1 corresponds
to the
stationary alumina
distribution,
when the velocity
is neglected. The
concentration is
shown under the
anodes. The two
feeder locations
appear clearly in
the figure. The
asymmetry of the
diffusion pattern
reflects the larger
channel width at
the feeders. A difference
of close
to 2.5% alumina
concentration can
be observed at
the surface of the
anodes. The vertical
variation of
alumina is 0.5%
under the feeders,
but it is negligible
away from
the feeders.
Fig. 2 shows
the velocity field
generated by
the bubbles and
Lorentz force in
this particular
cell.
The impact of
the velocity field
on the stationary
solution of the
alumina concentration
is shown
considering the concentration evolution, the
time needed for reaching the stationary state
is reduced by a factor 2 when the velocity field
is acting. Therefore the velocity field plays an
important role in the feeding process (alumina
dumps).
To highlight the role of the velocity field,
Fig. 4 shows the difference between the alumina
concentration field due only to the diffusion
compared with that in presence of the
velocity field. The greatest differences are observed
at the ends of the cell, due essentially
to the MHD effects. High negative values re-
Fig. 1: Alumina concentration at the anode surface assuming no velocity field
Fig. 2: Velocity field in the bath of the cell
Fig. 3: Alumina concentration at the anode surface in presence of the velocity field
Fig. 4: Alumina concentration variation due to the velocity field
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