XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2683
[Smith &Warren 1989, Savassi et al. 1998, Gutiérrez et
al. 2020].
Modelling flotation and developing strategies to con-
trol this process are research areas that have generated many
contributions [Maldonado et al. 2009, Quintanilla et al.
2021a, 2021b, Bürger et al. 2020, Dickinson &Galvin
2014, Galvin &Dickinson 2014, Galvin et al. 2014, Bergh
&Yianatos 1994, 2003, 2022]. The development of con-
trol strategies requires dynamic models along with a classifi-
cation of steady-state (stationary) solutions of such models.
These models should focus on the separation process aligned
with gravity and are, therefore, spatially onedimensional. In
fact, we wish to avoid the additional computational effort
associated with spatially two- or threedimensional models,
mostly based on computational fluid dynamics (CFD) that
also involve the solution of additional equations for the
motion of the mixture we refer to [Wang et al. 2018] for a
review on CFD-based models of flotation. Instead of CFD
models, we consider in this contribution a phenomeno-
logical model for three-phase systems with bubbles (aggre-
gates) rising and particles settling (hydrophilic) in a liquid
which are derived from the physical laws of conservation
of mass and momentum. By making certain simplifying
assumptions on the stress tensor and partial pressure of the
bubbles/solids, one can obtain first- or second-order partial
differential equations (PDEs) involving one or two consti-
tutive functions that are specific to the material (Bürger et
al. 2022). The main objective of this model is to determine
the volume fractions of gas (bubbles), liquid, and poten-
tially solid particles as functions of both time and spatial
position. This information is obtained from the numerical
resolution of the systems of governing equations that are
partial differential equations.
The rest of the paper is organized as follow: Section 2
describes the model equations and constitutive assumptions.
Section 3 report the material, methods and experimental
conditions. The numerical results and the experimental
validation are reported in section 4. A Conclusions section
summaries the main goals of the article as well as future
work on the topic.
MODEL EQUATIONS
The model is formulated as a three-phase model formed of
the gas bubbles (hydrophobic aggregates) and (hydrophilic
gangue) solid particles as primary and secondary disperse
phases that move in the fluid that forms the continuous
phase. The three phases and their (dimensionless) volume
fractions are the fluid ϕF, solid ϕS and bubbles ϕB, where
ϕF +ϕS +ϕB =1 .A mixture of fluid and solid particles
is addressed as a suspension. The volume fraction of sol-
ids within the suspension that fills the interstices between
bubbles is given by:
1
F S
S
B
S {:=z z
z
z
z
+=-(0.1)
which will be used as an auxiliary variable. The system of
governing equations can be written as a system of 2 PDE
for the solids and the bubbles fraction as follows: (for details
we refer Bürger et al. 2022):
z
z
,
,,z,th
/^1
A
J z,th
D^z
Q z
1
t
B
S
B
S B
z
S B
B
i
i G,F,W
B
i
S
i i
2
2
z
z z
z z
z
z
+-Fu^z
=--
+-
!
^zh2
dA^zhc^z
d
^the
dA^zhd
^th
^th
^z
n
hd
od^z
h
n2
h
nn
hn
",
/
(0.2)
Here A(z) is the sectional cross-area as a function of the
height, J and F are convective flux function depending dis-
continuously on the location of the gas inlet zG, pulp feed
inlet zF, wash water zW, underflow outlet zU and overflow
outlet zE (see Figure 1).
Regarding to the convective fluxes, we consider:
in the underflow zone.
,
,
,
,
,
J
j th:= q
j th:= q j
j th:= q j
j th:= q j
j th:= q
in zone 3,
in zone 2,
in zone 1,
E B E B
B B b B
B B b B
B B b B
U B B
3 3
2 2
1 1
1
=
+
+
+
in the effluent zone,
^zB,z,th
^z
^z
^z
^z
^z
^thz
^thz
^thz
^thz
^thz
^z
^z
^z
h
h
h
Z
[
\
]
]
]
]
]]
]
with
,th
:
:
/A :=-Q
qE:=
Q Q
Q q^
QG Q QWh/A
q q
in the effluent zone,
in zone 3,
in zone 2,
in zone 1 in the underflow and
U G F
U G
U
U F
U
3
2
1
=
+++
=++
=+
=
^-Q
^-Q
^-Q
h/A
h/A
Z
[
\
]]q
]q
]]
(0.3)
[Smith &Warren 1989, Savassi et al. 1998, Gutiérrez et
al. 2020].
Modelling flotation and developing strategies to con-
trol this process are research areas that have generated many
contributions [Maldonado et al. 2009, Quintanilla et al.
2021a, 2021b, Bürger et al. 2020, Dickinson &Galvin
2014, Galvin &Dickinson 2014, Galvin et al. 2014, Bergh
&Yianatos 1994, 2003, 2022]. The development of con-
trol strategies requires dynamic models along with a classifi-
cation of steady-state (stationary) solutions of such models.
These models should focus on the separation process aligned
with gravity and are, therefore, spatially onedimensional. In
fact, we wish to avoid the additional computational effort
associated with spatially two- or threedimensional models,
mostly based on computational fluid dynamics (CFD) that
also involve the solution of additional equations for the
motion of the mixture we refer to [Wang et al. 2018] for a
review on CFD-based models of flotation. Instead of CFD
models, we consider in this contribution a phenomeno-
logical model for three-phase systems with bubbles (aggre-
gates) rising and particles settling (hydrophilic) in a liquid
which are derived from the physical laws of conservation
of mass and momentum. By making certain simplifying
assumptions on the stress tensor and partial pressure of the
bubbles/solids, one can obtain first- or second-order partial
differential equations (PDEs) involving one or two consti-
tutive functions that are specific to the material (Bürger et
al. 2022). The main objective of this model is to determine
the volume fractions of gas (bubbles), liquid, and poten-
tially solid particles as functions of both time and spatial
position. This information is obtained from the numerical
resolution of the systems of governing equations that are
partial differential equations.
The rest of the paper is organized as follow: Section 2
describes the model equations and constitutive assumptions.
Section 3 report the material, methods and experimental
conditions. The numerical results and the experimental
validation are reported in section 4. A Conclusions section
summaries the main goals of the article as well as future
work on the topic.
MODEL EQUATIONS
The model is formulated as a three-phase model formed of
the gas bubbles (hydrophobic aggregates) and (hydrophilic
gangue) solid particles as primary and secondary disperse
phases that move in the fluid that forms the continuous
phase. The three phases and their (dimensionless) volume
fractions are the fluid ϕF, solid ϕS and bubbles ϕB, where
ϕF +ϕS +ϕB =1 .A mixture of fluid and solid particles
is addressed as a suspension. The volume fraction of sol-
ids within the suspension that fills the interstices between
bubbles is given by:
1
F S
S
B
S {:=z z
z
z
z
+=-(0.1)
which will be used as an auxiliary variable. The system of
governing equations can be written as a system of 2 PDE
for the solids and the bubbles fraction as follows: (for details
we refer Bürger et al. 2022):
z
z
,
,,z,th
/^1
A
J z,th
D^z
Q z
1
t
B
S
B
S B
z
S B
B
i
i G,F,W
B
i
S
i i
2
2
z
z z
z z
z
z
+-Fu^z
=--
+-
!
^zh2
dA^zhc^z
d
^the
dA^zhd
^th
^th
^z
n
hd
od^z
h
n2
h
nn
hn
",
/
(0.2)
Here A(z) is the sectional cross-area as a function of the
height, J and F are convective flux function depending dis-
continuously on the location of the gas inlet zG, pulp feed
inlet zF, wash water zW, underflow outlet zU and overflow
outlet zE (see Figure 1).
Regarding to the convective fluxes, we consider:
in the underflow zone.
,
,
,
,
,
J
j th:= q
j th:= q j
j th:= q j
j th:= q j
j th:= q
in zone 3,
in zone 2,
in zone 1,
E B E B
B B b B
B B b B
B B b B
U B B
3 3
2 2
1 1
1
=
+
+
+
in the effluent zone,
^zB,z,th
^z
^z
^z
^z
^z
^thz
^thz
^thz
^thz
^thz
^z
^z
^z
h
h
h
Z
[
\
]
]
]
]
]]
]
with
,th
:
:
/A :=-Q
qE:=
Q Q
Q q^
QG Q QWh/A
q q
in the effluent zone,
in zone 3,
in zone 2,
in zone 1 in the underflow and
U G F
U G
U
U F
U
3
2
1
=
+++
=++
=+
=
^-Q
^-Q
^-Q
h/A
h/A
Z
[
\
]]q
]q
]]
(0.3)