XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2741
the viscous terms. Since computational effort limits the
size of the simulation domain, a representative region of a
flotation cell was simulated that exhibits statistically con-
stant behaviour over the entire domain. Background tur-
bulence, such as that caused by a mechanical stirrer, was
generated using a forced turbulence generator for homoge-
neous, isotropic turbulence (Chouippe &Uhlmann, 2015)
(Chouippe &Uhlmann, 2015).
With the present approach, the bubbles are modelled
as fully resolved, rigid spheres with a no-slip boundary con-
dition. Coupling to the fluid is achieved by a direct forc-
ing Immersed-Boundary Method (Tschisgale, et al., 2018)
(Tschisgale, et al., 2018). The motion of the bubbles is
governed by the conservation of translational and angular
momentum, and collisions are modelled using a soft sphere
collision model (Heitkam, et al., 2013)(Heitkam, et al.,
2013).
Mineral particles are modelled as two-way coupled
point particles. Their motion is described by the Maxey-
Riley equations (Maxey &Riley, 1983)(Maxey &Riley,
1983). The forces acting on the particles are calculated by
linearly interpolating fluid quantities to the particle centre.
They are also inserted in the fluid equations to realize two-
way coupling.
Collisions between particles and bubbles are treated as
ghost collisions. A collision is counted if the surfaces of the
collision partners touch, meaning that distance between
their centres, r =|xb – xp|, is less than the collision radius
rc. The attachment efficiency is assumed to be Ea =100%.
A particle is removed from the simulation domain after a
collision. To maintain constant statistics over time, a new
particle is randomly seeded into the fluid domain Ωf at a
random position with up =uf. The mass of the bubbles
remains unchanged.
Simulation Setup
To determine the impact of turbulence on particle-bubble
collisions, two three-phase DNS were conducted on a rep-
resentative region of the flotation cell. The region of the
first simulation is located in the mixing and aeration zone
of a mechanical flotation cell, utilizing externally forced
turbulence. The region of the second simulation case is
located in the separation and concentration zone, relying
solely on gravity as the driving force for particle and bubble
motion. Otherwise, both simulation cases use identical
physical parameters. In the following, the case with forced
turbulence will be referred to as TREF, and the latter case
with gravity as the only driving force will be referred to as
GREF. The details and results of the GREF case have been
presented by Tiedemann and Fröhlich (2023).
The simulations involve water as the fluid, monodisperse
air bubbles, and monodisperse fine mineral particles. The
representative control volume used in the simulations has
the dimensions of ..5hd L L L 5 5 11 5
x y z b
3 ####=^.
Both cases were simulated accounting for the influence of
gravity, with tri-periodic boundary conditions employed to
replicate the presence of a large swarm with locally homog-
enous conditions. Figure 1a displays a snapshot of the sim-
ulation domain and its dimensions.
The physical parameters chosen for this study are based
on representative flotation conditions (Norori-McCormac,
et al., 2017 Mesa, et al., 2020 Hadler &Cilliers, 2019
Ostadrahimi, et al., 2020 Ran, et al., 2019). They are sum-
marised in Table 1 together with some nondimensional
numbers characterising the simulations. For case TREF,
the turbulence generator was tuned to obtain a Taylor
Reynolds number for the single-phase turbulence of Reλ =
53. The GREF simulation case does not use any externally
forced turbulence.
Following the bubble shape regime outlined by Clift
(1978), the assumption of spherical bubbles is considered
valid for both simulated cases.
RESULTS
Bubble, Particle, and fluid regime
Collisions between particles and bubbles are influenced by
several factors depending on the regime of fluid motion,
bubble and particle motion. The situation here is character-
ized by the nondimensional number reported in Table 2.
These include the particle Stokes number, Stp, as defined
in the previous section, and the bubble Reynolds number
,Reb
U db
b o =(10)
where Ub is the bubble velocity realtive to the fluid.
Furthermore, the Froude number, as defined in (6),
describes the ratio of fluid interaction to gravitational
effects (Chan, et al., 2023)(Chan, et al., 2023).
Due to the forced background turbulence, case TREF
exhibits a significantly higher turbulence level, compared
to the simulation GREF driven only by gravity. Case TREF
represents a moderate level of turbulence when compared
to turbulence levels commonly found in mechanical flota-
tion cells CITATION Ngo18 \l 1031 (Ngo-Cong, et al.,
2018)(Ngo-Cong, et al., 2018). The turbulence generator
for the single-phase system was tuned to a Taylor Reynolds
number of Reλ =53. However, only a Taylor Reynolds
number of Reλ =48 was obtained for the three-phase case.
Under the present conditions, the presence of particles and
the viscous terms. Since computational effort limits the
size of the simulation domain, a representative region of a
flotation cell was simulated that exhibits statistically con-
stant behaviour over the entire domain. Background tur-
bulence, such as that caused by a mechanical stirrer, was
generated using a forced turbulence generator for homoge-
neous, isotropic turbulence (Chouippe &Uhlmann, 2015)
(Chouippe &Uhlmann, 2015).
With the present approach, the bubbles are modelled
as fully resolved, rigid spheres with a no-slip boundary con-
dition. Coupling to the fluid is achieved by a direct forc-
ing Immersed-Boundary Method (Tschisgale, et al., 2018)
(Tschisgale, et al., 2018). The motion of the bubbles is
governed by the conservation of translational and angular
momentum, and collisions are modelled using a soft sphere
collision model (Heitkam, et al., 2013)(Heitkam, et al.,
2013).
Mineral particles are modelled as two-way coupled
point particles. Their motion is described by the Maxey-
Riley equations (Maxey &Riley, 1983)(Maxey &Riley,
1983). The forces acting on the particles are calculated by
linearly interpolating fluid quantities to the particle centre.
They are also inserted in the fluid equations to realize two-
way coupling.
Collisions between particles and bubbles are treated as
ghost collisions. A collision is counted if the surfaces of the
collision partners touch, meaning that distance between
their centres, r =|xb – xp|, is less than the collision radius
rc. The attachment efficiency is assumed to be Ea =100%.
A particle is removed from the simulation domain after a
collision. To maintain constant statistics over time, a new
particle is randomly seeded into the fluid domain Ωf at a
random position with up =uf. The mass of the bubbles
remains unchanged.
Simulation Setup
To determine the impact of turbulence on particle-bubble
collisions, two three-phase DNS were conducted on a rep-
resentative region of the flotation cell. The region of the
first simulation is located in the mixing and aeration zone
of a mechanical flotation cell, utilizing externally forced
turbulence. The region of the second simulation case is
located in the separation and concentration zone, relying
solely on gravity as the driving force for particle and bubble
motion. Otherwise, both simulation cases use identical
physical parameters. In the following, the case with forced
turbulence will be referred to as TREF, and the latter case
with gravity as the only driving force will be referred to as
GREF. The details and results of the GREF case have been
presented by Tiedemann and Fröhlich (2023).
The simulations involve water as the fluid, monodisperse
air bubbles, and monodisperse fine mineral particles. The
representative control volume used in the simulations has
the dimensions of ..5hd L L L 5 5 11 5
x y z b
3 ####=^.
Both cases were simulated accounting for the influence of
gravity, with tri-periodic boundary conditions employed to
replicate the presence of a large swarm with locally homog-
enous conditions. Figure 1a displays a snapshot of the sim-
ulation domain and its dimensions.
The physical parameters chosen for this study are based
on representative flotation conditions (Norori-McCormac,
et al., 2017 Mesa, et al., 2020 Hadler &Cilliers, 2019
Ostadrahimi, et al., 2020 Ran, et al., 2019). They are sum-
marised in Table 1 together with some nondimensional
numbers characterising the simulations. For case TREF,
the turbulence generator was tuned to obtain a Taylor
Reynolds number for the single-phase turbulence of Reλ =
53. The GREF simulation case does not use any externally
forced turbulence.
Following the bubble shape regime outlined by Clift
(1978), the assumption of spherical bubbles is considered
valid for both simulated cases.
RESULTS
Bubble, Particle, and fluid regime
Collisions between particles and bubbles are influenced by
several factors depending on the regime of fluid motion,
bubble and particle motion. The situation here is character-
ized by the nondimensional number reported in Table 2.
These include the particle Stokes number, Stp, as defined
in the previous section, and the bubble Reynolds number
,Reb
U db
b o =(10)
where Ub is the bubble velocity realtive to the fluid.
Furthermore, the Froude number, as defined in (6),
describes the ratio of fluid interaction to gravitational
effects (Chan, et al., 2023)(Chan, et al., 2023).
Due to the forced background turbulence, case TREF
exhibits a significantly higher turbulence level, compared
to the simulation GREF driven only by gravity. Case TREF
represents a moderate level of turbulence when compared
to turbulence levels commonly found in mechanical flota-
tion cells CITATION Ngo18 \l 1031 (Ngo-Cong, et al.,
2018)(Ngo-Cong, et al., 2018). The turbulence generator
for the single-phase system was tuned to a Taylor Reynolds
number of Reλ =53. However, only a Taylor Reynolds
number of Reλ =48 was obtained for the three-phase case.
Under the present conditions, the presence of particles and