XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2685
connections between the radius of Plateau borders in he foam, the radius of the bubbles and the volume
fraction of the liquid in the foam see Bürger et al. 2022 for all details.
The nonlinear function D(ϕB) models the capillarity presents when bubbles are in contact, given by:
,
D^z d 1h^n 2h
0
1,
for
for B
c
n n
B c
B 1
S
term cap
S S b
##
#z
~~
z z
z z =
-++
-
+-^1
^z
^n
^zBh h
h2
h
Z0
[
\
]
]y (0.6)
where ω(ϕB) :=(1= –ϕB)nS+1((n
S +1)ϕB +1) and dcap is a capillarity-to-gravity constant (parameter) present
in the froth.
Finally, we define the total convective flux for the solids appearing in the governing system by:
,:=
,,z,t
F
if
if
1
0
0 1,
1.
S B
S
B B
B
#
zB,z,th z
z
z z
z
-
=
u^z d n
Z
[
\
]F
](0.7)
MATERIALS AND METHODS
The experiments were carried out at the Metallurgical
Engineering Department (Universidad de Concepción) by
using a laboratory-scale flotation column. This column is
made of acrylic to visualize the internal phenomena that
occur both in the collection and cleaning areas. The col-
umn has a volume of 54.7 liters, an interior diameter of
6~inches and 2.8m height. The air is injected from the
lower central part of the column through a sparger located
at 0.07 m from the bottom whose pores are 1mm in diam-
eter. The locations of the inlets and outlets are detailed in
Figure 1. The column is instrumented as shown in Figure 2
and Table 1.
The use of solids for the validation experiences was
not considered. In this contribution, as a frother reagent,
a mix (1:1) of MIBC (methyl isobutyl carbinol, an organic
chemical compound used primarily as a frother in mineral
flotation) and polyglycol with a dosage of 100 gr/liter of
water was utilized.
Five steady-state experiments were performed see
Table 2 for details on the conditions. 36 data points in the
interval [0, 1.630] of the underflow velocity qU =QU/A
controlled by a pump were used.
Experimental tests were performed in duplicate and
randomly, and we report only the average values. The stan-
dard deviation of the tests was less than 5%.
RESULTS AND DISCUSSION
Experimental Determination of Stability Regions
The resulting upper and lower limits of qF, within which
a pulp–froth interface is present in zone 3, are showed
in Figures 3 and 4. The enclosed region between the two
curves can be considered as a stability region since, for
values of (qU, qF) therein, a stable pulp–froth interface in
zone 3 was observed experimentally. On the other hand,
for choices of (q qU
F ,),outside that region, unstable opera-
tion was observed, which means that either no froth layer
was produced at all or that the froth layer reached into zone
2 and possibly that bubbles left through the underflow. A
stable pulp– froth interface in zone 2 can exists as a valid
stationary solution (corresponding to a mode of operation
in which the pulp feed acts as a submerged feed source),
but such steady states are very difficult to control, and we,
therefore, address them as “unstable operation”. This pro-
cedure is consistent with the theoretical steady-state analy-
sis developed by Bürger et al. 2020, and in particular the
construction of operating charts in which any theoretical
situation in which the froth level cannot be accommo-
dated within zone 3 is deemed “unstable,” independently
of whether the parameters permit a froth layer within zone
2 or not (i.e., bubbles leave through the underflow).
Comparison Between the Model and Experimental
Stationary Data
The proposed model contains several parameters. Table 3
shows the parameters used for the numerical simulation,
and for the justification of the selected values, we refer to
Bürger et al. 2022.
All the numerical results were obtained with a spatial
discretization of N =800 computational cells, which means
a spatial step size of Δz =3.50 × 10–3 m .In this contribu-
tion we just consider the steady state version of the model
PDE equations. We refer to Betancourt et al. 2023, for
dynamic simulations.
Comparisons between the stationary model and the five
steady-state experiments can be seen in Figure 5. The white
region in the (qU, qF)-plane correspond to steady state in
which the frost level lays in zone 3 (see Figure 1). We notice
connections between the radius of Plateau borders in he foam, the radius of the bubbles and the volume
fraction of the liquid in the foam see Bürger et al. 2022 for all details.
The nonlinear function D(ϕB) models the capillarity presents when bubbles are in contact, given by:
,
D^z d 1h^n 2h
0
1,
for
for B
c
n n
B c
B 1
S
term cap
S S b
##
#z
~~
z z
z z =
-++
-
+-^1
^z
^n
^zBh h
h2
h
Z0
[
\
]
]y (0.6)
where ω(ϕB) :=(1= –ϕB)nS+1((n
S +1)ϕB +1) and dcap is a capillarity-to-gravity constant (parameter) present
in the froth.
Finally, we define the total convective flux for the solids appearing in the governing system by:
,:=
,,z,t
F
if
if
1
0
0 1,
1.
S B
S
B B
B
#
zB,z,th z
z
z z
z
-
=
u^z d n
Z
[
\
]F
](0.7)
MATERIALS AND METHODS
The experiments were carried out at the Metallurgical
Engineering Department (Universidad de Concepción) by
using a laboratory-scale flotation column. This column is
made of acrylic to visualize the internal phenomena that
occur both in the collection and cleaning areas. The col-
umn has a volume of 54.7 liters, an interior diameter of
6~inches and 2.8m height. The air is injected from the
lower central part of the column through a sparger located
at 0.07 m from the bottom whose pores are 1mm in diam-
eter. The locations of the inlets and outlets are detailed in
Figure 1. The column is instrumented as shown in Figure 2
and Table 1.
The use of solids for the validation experiences was
not considered. In this contribution, as a frother reagent,
a mix (1:1) of MIBC (methyl isobutyl carbinol, an organic
chemical compound used primarily as a frother in mineral
flotation) and polyglycol with a dosage of 100 gr/liter of
water was utilized.
Five steady-state experiments were performed see
Table 2 for details on the conditions. 36 data points in the
interval [0, 1.630] of the underflow velocity qU =QU/A
controlled by a pump were used.
Experimental tests were performed in duplicate and
randomly, and we report only the average values. The stan-
dard deviation of the tests was less than 5%.
RESULTS AND DISCUSSION
Experimental Determination of Stability Regions
The resulting upper and lower limits of qF, within which
a pulp–froth interface is present in zone 3, are showed
in Figures 3 and 4. The enclosed region between the two
curves can be considered as a stability region since, for
values of (qU, qF) therein, a stable pulp–froth interface in
zone 3 was observed experimentally. On the other hand,
for choices of (q qU
F ,),outside that region, unstable opera-
tion was observed, which means that either no froth layer
was produced at all or that the froth layer reached into zone
2 and possibly that bubbles left through the underflow. A
stable pulp– froth interface in zone 2 can exists as a valid
stationary solution (corresponding to a mode of operation
in which the pulp feed acts as a submerged feed source),
but such steady states are very difficult to control, and we,
therefore, address them as “unstable operation”. This pro-
cedure is consistent with the theoretical steady-state analy-
sis developed by Bürger et al. 2020, and in particular the
construction of operating charts in which any theoretical
situation in which the froth level cannot be accommo-
dated within zone 3 is deemed “unstable,” independently
of whether the parameters permit a froth layer within zone
2 or not (i.e., bubbles leave through the underflow).
Comparison Between the Model and Experimental
Stationary Data
The proposed model contains several parameters. Table 3
shows the parameters used for the numerical simulation,
and for the justification of the selected values, we refer to
Bürger et al. 2022.
All the numerical results were obtained with a spatial
discretization of N =800 computational cells, which means
a spatial step size of Δz =3.50 × 10–3 m .In this contribu-
tion we just consider the steady state version of the model
PDE equations. We refer to Betancourt et al. 2023, for
dynamic simulations.
Comparisons between the stationary model and the five
steady-state experiments can be seen in Figure 5. The white
region in the (qU, qF)-plane correspond to steady state in
which the frost level lays in zone 3 (see Figure 1). We notice