2688 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
the four dimensional space of (qU, qG, qF, qW )-values that
a froth layer exists. Both the model and the experiments
verify that, once a steady state has been found with a froth
level, a small change in any of the bulk velocities will either
make the froth layer be flushed out upwards or the entire
column filled with bubbles, which also leave through the
underflow.
The theoretically derived PDE model automatically
captures several different phenomena (hindered bubble
rise, hindered settling of particles and the formation of
foam) without any boundary conditions. The model pre-
dicts that a desired steady-state solution with a froth layer
above the feed level is only possible in a thin region in
(qU, qG, qF, qW )-space. The model involves nonlinearities
in both the convective and diffusive parts and is strongly
degenerate in the diffusive part and, therefore, gives rise
to discontinuities in the concentration profiles, which was
confirmed experimentally.
The qualitative agreement between our model and the
experiments is interesting and shows the possibilities for
further investigations and model calibration. A reason for
the discrepancies between the model output and the experi-
mentally determined regions in the operating charts, when
a stationary froth level in zone 3 is possible, is the following.
In the upper strip of the white region, the model predicts a
froth level zfr (the yellow surface) close to the effluent level
zE, which means a very thin layer of froth. It may well be
that such thin froth layers could not registered as valid in
the experimental observations. It is therefore natural that
the upper experimental red line lies some distance below
the upper line of the white region. The fact that the lower
red line, at least when wash water is present, lies further
down in the grey region, means that a froth level is observed
Figure 5. Experiments 1–3 (no wash water): Comparisons between the model with stationary conditions. Here, the white
region shows the theoretical conditions for a pulp–froth level above the feed level. The red lines in that plane (see also Figures
3 and 4) show the experimental lower and upper values of q
F for each given q
U ,in between which, a pulp–froth level was
observed. The yellow surface is the graph of the estimated height of the pulp–froth interface by the model. Experiments 4–5
were performed by adding wash water
the four dimensional space of (qU, qG, qF, qW )-values that
a froth layer exists. Both the model and the experiments
verify that, once a steady state has been found with a froth
level, a small change in any of the bulk velocities will either
make the froth layer be flushed out upwards or the entire
column filled with bubbles, which also leave through the
underflow.
The theoretically derived PDE model automatically
captures several different phenomena (hindered bubble
rise, hindered settling of particles and the formation of
foam) without any boundary conditions. The model pre-
dicts that a desired steady-state solution with a froth layer
above the feed level is only possible in a thin region in
(qU, qG, qF, qW )-space. The model involves nonlinearities
in both the convective and diffusive parts and is strongly
degenerate in the diffusive part and, therefore, gives rise
to discontinuities in the concentration profiles, which was
confirmed experimentally.
The qualitative agreement between our model and the
experiments is interesting and shows the possibilities for
further investigations and model calibration. A reason for
the discrepancies between the model output and the experi-
mentally determined regions in the operating charts, when
a stationary froth level in zone 3 is possible, is the following.
In the upper strip of the white region, the model predicts a
froth level zfr (the yellow surface) close to the effluent level
zE, which means a very thin layer of froth. It may well be
that such thin froth layers could not registered as valid in
the experimental observations. It is therefore natural that
the upper experimental red line lies some distance below
the upper line of the white region. The fact that the lower
red line, at least when wash water is present, lies further
down in the grey region, means that a froth level is observed
Figure 5. Experiments 1–3 (no wash water): Comparisons between the model with stationary conditions. Here, the white
region shows the theoretical conditions for a pulp–froth level above the feed level. The red lines in that plane (see also Figures
3 and 4) show the experimental lower and upper values of q
F for each given q
U ,in between which, a pulp–froth level was
observed. The yellow surface is the graph of the estimated height of the pulp–froth interface by the model. Experiments 4–5
were performed by adding wash water