XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2689
close to, but above, the feed level zF, i.e., almost the entire
zone 3, is filled with froth. The theoretical model does not
fully capture this behavior and predicts that, for points (qU,
qF) below the white region, the entire zone 3 is filled with
foam, and there are possibly bubbles dragged down to the
underflow. This discrepancy between the model and experi-
ments near the location of the pulp–froth interface when
wash water is applied should be further investigated.
In the model development in Bürger et al 2022, sev-
eral reasonable assumptions (partially verified by reported
experiments) for the drainage in the froth were assumed
to hold for volume fractions close to but above the criti-
cal concentration ϕC in order to obtain a unified model. It
appears that further modelling is needed for the behavior
near the pulp–froth interface. That said, we suggest that
that the foam model and the description of that interface
by a critical concentration is consistent with the approach
by Neethling and Cilliers in 2002 and 2003 (which is fur-
ther elaborated, e.g., in Neethling et al. 2018).
CONCLUSIONS
From the results showed in previous sections it is possible
to say that the model presented in section 2, plus consti-
tutive equations and specifications of control functions, is
based on several existing theories (the drift–flux and sol-
ids–flux models as well as the model of foam drainage, see
for instance Stevenson et al. 2008 and Vandenberghe et al.
2005) and sets the ground for a complete simulator of a
flotation column in one space dimension without the need
to impose boundary conditions or track, for instance, the
pulp–froth interface. We once again refer to Bürger et al.
2022 for an exposition of all technical details
The comparison with experimental results conducted
in this work are the first results that indicate that the model
is consistent with experimental observations. That said,
further experiments and comparisons with simulations
should be conducted with a focus on transient behavior and
involving solids.
With respect to the potential use of the model in real
flotation practice (for instance, to optimize the flotation
performance) we mention that the model presented is an
advancement of the phenomenological models currently
reported in relation to the description of the foam level
within the column as well as the gas hold-up. Although
it does not yet consider the attachment and detachment
mechanism of particles to bubbles, the model is reasonably
accurate in determining the stable operating zones, which
would allow its use as a complement to current control sys-
tems [Tian et al. 2018, Azhin et al. 2021, Quintanilla et al.
2021a]. Examples of control systems based on the involved
phenomenological models (coupled PDEs) have been
reported and used in other unit operations [Diehl 2001,
Diehl 2008, Diehl &Farås 2013, Betancourt et al., 2013,
2014, Torfs et al. 2015].
The present approach captures the multiphase hydro-
dynamics of aggregates (bubbles) and gangue particles in
the column but does not model the aggregation process
itself (that is, the attachment of hydrophobic (valuable)
particles to gas bubbles). That process usually takes place
in the collection zone (zone 2 in Figure 1). To add realism
and to explore the interdependence of velocities and reac-
tion kinetics, the flotation model (3) should be extended
to include the process of attachment of hydrophobic (valu-
able) particles. One option consists of considering the
valuable and gangue particles as two independent disperse
solid phases (while the present approach only includes the
gangue) and adding another field variable that describes the
local state of aggregation. This procedure leads to two addi-
tional PDEs for the two new variables and likely involves
spatial variants of known kinetic models for the adhesion
of particles (as reviewed, for instance, in Wang et al. 2018).
REFERENCES
[1] Azhin, M. Popli, K. Prasad, V. Modelling and
boundary optimal control design of hybrid column
flotation. Can. J. Chem. Eng. 2021, 99 (Suppl. 1),
S369–S388.
[2] Bascur, O.A. A unified solid/liquid separation frame-
work. Fluid/Part. Sep. J. 1991, 4, 117–122.
[3] Bergh, L.G. Yianatos, J.B. Experimental studies on
flotation column dynamics. Miner. Eng. 1994, 7,
345–355
[4] Bergh, L.G. Yianatos, J.B. Flotation column auto-
mation: State of the art. Control Eng. Pract. 2003,
11, 67–72.
[5] Bergh, L.G. Yianatos, J.B. The long way to multivari-
ate predictive control of flotation processes. J. Process
Control 2022, 21, 226–234.
[6] Betancourt, F., Bürger, R., Ruiz-Baier, R., Torres,
H., &Vega, C. A. On numerical methods for hyper-
bolic conservation laws and related equations mod-
elling sedimentation of solid-liquid suspensions. In
Hyperbolic Conservation Laws and Related Analysis
with Applications: Edinburgh, September 2011
(pp. 23–68). Springer Berlin Heidelberg 2014.
[7] Betancourt, F. Bürger, R. Diehl, S. Farås, S.
Modelling and controlling clarifier-thickeners fed by
suspensions with time-dependent properties. Miner.
Eng. 2014, 62, 91–101.
close to, but above, the feed level zF, i.e., almost the entire
zone 3, is filled with froth. The theoretical model does not
fully capture this behavior and predicts that, for points (qU,
qF) below the white region, the entire zone 3 is filled with
foam, and there are possibly bubbles dragged down to the
underflow. This discrepancy between the model and experi-
ments near the location of the pulp–froth interface when
wash water is applied should be further investigated.
In the model development in Bürger et al 2022, sev-
eral reasonable assumptions (partially verified by reported
experiments) for the drainage in the froth were assumed
to hold for volume fractions close to but above the criti-
cal concentration ϕC in order to obtain a unified model. It
appears that further modelling is needed for the behavior
near the pulp–froth interface. That said, we suggest that
that the foam model and the description of that interface
by a critical concentration is consistent with the approach
by Neethling and Cilliers in 2002 and 2003 (which is fur-
ther elaborated, e.g., in Neethling et al. 2018).
CONCLUSIONS
From the results showed in previous sections it is possible
to say that the model presented in section 2, plus consti-
tutive equations and specifications of control functions, is
based on several existing theories (the drift–flux and sol-
ids–flux models as well as the model of foam drainage, see
for instance Stevenson et al. 2008 and Vandenberghe et al.
2005) and sets the ground for a complete simulator of a
flotation column in one space dimension without the need
to impose boundary conditions or track, for instance, the
pulp–froth interface. We once again refer to Bürger et al.
2022 for an exposition of all technical details
The comparison with experimental results conducted
in this work are the first results that indicate that the model
is consistent with experimental observations. That said,
further experiments and comparisons with simulations
should be conducted with a focus on transient behavior and
involving solids.
With respect to the potential use of the model in real
flotation practice (for instance, to optimize the flotation
performance) we mention that the model presented is an
advancement of the phenomenological models currently
reported in relation to the description of the foam level
within the column as well as the gas hold-up. Although
it does not yet consider the attachment and detachment
mechanism of particles to bubbles, the model is reasonably
accurate in determining the stable operating zones, which
would allow its use as a complement to current control sys-
tems [Tian et al. 2018, Azhin et al. 2021, Quintanilla et al.
2021a]. Examples of control systems based on the involved
phenomenological models (coupled PDEs) have been
reported and used in other unit operations [Diehl 2001,
Diehl 2008, Diehl &Farås 2013, Betancourt et al., 2013,
2014, Torfs et al. 2015].
The present approach captures the multiphase hydro-
dynamics of aggregates (bubbles) and gangue particles in
the column but does not model the aggregation process
itself (that is, the attachment of hydrophobic (valuable)
particles to gas bubbles). That process usually takes place
in the collection zone (zone 2 in Figure 1). To add realism
and to explore the interdependence of velocities and reac-
tion kinetics, the flotation model (3) should be extended
to include the process of attachment of hydrophobic (valu-
able) particles. One option consists of considering the
valuable and gangue particles as two independent disperse
solid phases (while the present approach only includes the
gangue) and adding another field variable that describes the
local state of aggregation. This procedure leads to two addi-
tional PDEs for the two new variables and likely involves
spatial variants of known kinetic models for the adhesion
of particles (as reviewed, for instance, in Wang et al. 2018).
REFERENCES
[1] Azhin, M. Popli, K. Prasad, V. Modelling and
boundary optimal control design of hybrid column
flotation. Can. J. Chem. Eng. 2021, 99 (Suppl. 1),
S369–S388.
[2] Bascur, O.A. A unified solid/liquid separation frame-
work. Fluid/Part. Sep. J. 1991, 4, 117–122.
[3] Bergh, L.G. Yianatos, J.B. Experimental studies on
flotation column dynamics. Miner. Eng. 1994, 7,
345–355
[4] Bergh, L.G. Yianatos, J.B. Flotation column auto-
mation: State of the art. Control Eng. Pract. 2003,
11, 67–72.
[5] Bergh, L.G. Yianatos, J.B. The long way to multivari-
ate predictive control of flotation processes. J. Process
Control 2022, 21, 226–234.
[6] Betancourt, F., Bürger, R., Ruiz-Baier, R., Torres,
H., &Vega, C. A. On numerical methods for hyper-
bolic conservation laws and related equations mod-
elling sedimentation of solid-liquid suspensions. In
Hyperbolic Conservation Laws and Related Analysis
with Applications: Edinburgh, September 2011
(pp. 23–68). Springer Berlin Heidelberg 2014.
[7] Betancourt, F. Bürger, R. Diehl, S. Farås, S.
Modelling and controlling clarifier-thickeners fed by
suspensions with time-dependent properties. Miner.
Eng. 2014, 62, 91–101.