2744 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
60°…80°. For GREF, gravity is the main driver of particle
and bubble motion, resulting in most collisions occurring
on the upper half of the bubble (82.2%). This is the natu-
ral location of collision as particles settle and bubbles rise.
Since most particles either collide with the bubble or devi-
ate around it, only a few particles get trapped in the par-
ticle wake and are transported upwards by the recirculating
flow to collide with the lower half of the bubble. This is
mostly taking place at collision angles of Θ =90°…120°.
At even higher collision angles collisions are rare. In con-
trast, the collision angle distribution for case TREF is a lot
more uniform. Ideal isotropic turbulence, as generated by
the turbulence generator used, would results in no prefer-
ential direction of particle and bubble motion, leading to a
uniform distribution of the particle-bubble collision angle
per unit area. However, case TREF is a superposition of
the effects of isotropic turbulence and gravity. Therefore,
the collision angle distribution is more uniform than the
one for GREF. Still 64.5% of the collisions take place on
the upper half of the bubble. A significant number of par-
ticles collide on the lower half of the bubble, with collisions
occurring at collision angles close to Θ =180°. Overall, tur-
bulence increases the surface area of the bubble exposed to
possible encounters, therefore increasing the overall likeli-
hood of particle collision reflected by an increase of around
80% in Γpb.
Particle Accumulation
Particle-bubble collisions can be influenced significantly by
variations of the particle concentration around the bubble.
An important aspect of turbulence in multiphase flow is the
phenomenon of preferential concentration, where particles
and bubbles tend to form clusters along certain features of
the fluid field. Specifically, particles tend to move towards
areas of low vorticity and high strain rates CITATION
Max87 \l 1031 (Maxey, 1987)(Maxey, 1987). This phenom-
enon is most pronounced when the particle Stokes number
is around one. Hence, particles and bubbles become sepa-
rated, reducing the likelihood of collisions. However, this
effect enhances collisions between pairs of particles (Fayed
&Ragab, 2013 Zaichik, et al., 2010 Zaichik, et al., 2006).
Additionally, the local modulation of the fluid field by the
bubbles can cause particle accumulation or depletion.
The average particle concentration with respect to the
bubble can be analysed using the 2D particle-bubble pair
correlation function gpb(r,θ). The volume surrounding the
bubble is divided into rings having a volume dVrθ, each
with its centre located at specific values of r and θ with
respect to the centre of the bubble and width Δr and Δθ
in r and θ, respectively. The number of particles with their
centre in such a ring, dNp,rθ, is counted and an average is
taken over all bubbles. With these quantities the pair cor-
relation function (PCF) is defined as
.g N
VΩ
dVri
dN
,ri
pb
p
p =^r,ih ^r,ih
^r,ih
(11)
For a homogenous distribution of particles in the
domain, the PCF takes a value of one. Figure 3 shows
gpb(r,θ) for both simulation cases. There are no particles
present within the collision radius r rc =rb +rp, as they
are removed from the simulation domain once colliding
with the bubble. In case TREF, an area of increased particle
concentration is visible for 0.5db r db. The presence of
a bubble disturbs the fluid field around it. Inertia-less par-
ticles would ideally follow the fluid streamlines as they are
bending around the bubble. However, as the particle Stokes
number is greater than zero, particles tend to move along
lines of lower curvature than the streamlines. Hence, above
the bubble they deviate inwards leading to an increase in
particle concentration near the bubble. As a result of the
collision process only very few particles are observed in very
close vicinity of the bubble surface.
On the other hand, case GREF illustrates the impact
of gravity on particle concentration. Due to the opposing
direction of motion of particles and bubbles, particles accu-
mulate on the upper bubble half for 0.5db r db. From
there particles are deflected around the bubble creating a
region of high g(r,θ) going almost straight in vertical direc-
tion for θ 90°. In the bubble wake for θ 90° and r db,
there is a region of low particle concentration due to the
bubble blocking particles from reaching this area. As case
TREF is a superposition of the effects of turbulence and
gravity, a small influence of the aforementioned effects of
gravity seen for case GREF is still visible. For example, for
angles less than 90°, the area of increased particle accumu-
lation has a higher magnitude compared to angles larger
than 90°. Additionally, a small area of low concentration is
visible along the lower boundary of the bubble.
For larger radii, both cases have, on average, an almost
homogeneous distribution of particles with gpb ≈ 1. For
case TREF the highest magnitude and lowest magnitudes
of g(r,θ) are closer to one, hence indicating a more homo-
geneous particle distribution around the bubbles on aver-
age. Therefore, particles are more capable of utilizing the
entire bubble surface for possible collisions, contributing
to the increase of the particle-bubble collision frequency as
seen above.
60°…80°. For GREF, gravity is the main driver of particle
and bubble motion, resulting in most collisions occurring
on the upper half of the bubble (82.2%). This is the natu-
ral location of collision as particles settle and bubbles rise.
Since most particles either collide with the bubble or devi-
ate around it, only a few particles get trapped in the par-
ticle wake and are transported upwards by the recirculating
flow to collide with the lower half of the bubble. This is
mostly taking place at collision angles of Θ =90°…120°.
At even higher collision angles collisions are rare. In con-
trast, the collision angle distribution for case TREF is a lot
more uniform. Ideal isotropic turbulence, as generated by
the turbulence generator used, would results in no prefer-
ential direction of particle and bubble motion, leading to a
uniform distribution of the particle-bubble collision angle
per unit area. However, case TREF is a superposition of
the effects of isotropic turbulence and gravity. Therefore,
the collision angle distribution is more uniform than the
one for GREF. Still 64.5% of the collisions take place on
the upper half of the bubble. A significant number of par-
ticles collide on the lower half of the bubble, with collisions
occurring at collision angles close to Θ =180°. Overall, tur-
bulence increases the surface area of the bubble exposed to
possible encounters, therefore increasing the overall likeli-
hood of particle collision reflected by an increase of around
80% in Γpb.
Particle Accumulation
Particle-bubble collisions can be influenced significantly by
variations of the particle concentration around the bubble.
An important aspect of turbulence in multiphase flow is the
phenomenon of preferential concentration, where particles
and bubbles tend to form clusters along certain features of
the fluid field. Specifically, particles tend to move towards
areas of low vorticity and high strain rates CITATION
Max87 \l 1031 (Maxey, 1987)(Maxey, 1987). This phenom-
enon is most pronounced when the particle Stokes number
is around one. Hence, particles and bubbles become sepa-
rated, reducing the likelihood of collisions. However, this
effect enhances collisions between pairs of particles (Fayed
&Ragab, 2013 Zaichik, et al., 2010 Zaichik, et al., 2006).
Additionally, the local modulation of the fluid field by the
bubbles can cause particle accumulation or depletion.
The average particle concentration with respect to the
bubble can be analysed using the 2D particle-bubble pair
correlation function gpb(r,θ). The volume surrounding the
bubble is divided into rings having a volume dVrθ, each
with its centre located at specific values of r and θ with
respect to the centre of the bubble and width Δr and Δθ
in r and θ, respectively. The number of particles with their
centre in such a ring, dNp,rθ, is counted and an average is
taken over all bubbles. With these quantities the pair cor-
relation function (PCF) is defined as
.g N
VΩ
dVri
dN
,ri
pb
p
p =^r,ih ^r,ih
^r,ih
(11)
For a homogenous distribution of particles in the
domain, the PCF takes a value of one. Figure 3 shows
gpb(r,θ) for both simulation cases. There are no particles
present within the collision radius r rc =rb +rp, as they
are removed from the simulation domain once colliding
with the bubble. In case TREF, an area of increased particle
concentration is visible for 0.5db r db. The presence of
a bubble disturbs the fluid field around it. Inertia-less par-
ticles would ideally follow the fluid streamlines as they are
bending around the bubble. However, as the particle Stokes
number is greater than zero, particles tend to move along
lines of lower curvature than the streamlines. Hence, above
the bubble they deviate inwards leading to an increase in
particle concentration near the bubble. As a result of the
collision process only very few particles are observed in very
close vicinity of the bubble surface.
On the other hand, case GREF illustrates the impact
of gravity on particle concentration. Due to the opposing
direction of motion of particles and bubbles, particles accu-
mulate on the upper bubble half for 0.5db r db. From
there particles are deflected around the bubble creating a
region of high g(r,θ) going almost straight in vertical direc-
tion for θ 90°. In the bubble wake for θ 90° and r db,
there is a region of low particle concentration due to the
bubble blocking particles from reaching this area. As case
TREF is a superposition of the effects of turbulence and
gravity, a small influence of the aforementioned effects of
gravity seen for case GREF is still visible. For example, for
angles less than 90°, the area of increased particle accumu-
lation has a higher magnitude compared to angles larger
than 90°. Additionally, a small area of low concentration is
visible along the lower boundary of the bubble.
For larger radii, both cases have, on average, an almost
homogeneous distribution of particles with gpb ≈ 1. For
case TREF the highest magnitude and lowest magnitudes
of g(r,θ) are closer to one, hence indicating a more homo-
geneous particle distribution around the bubbles on aver-
age. Therefore, particles are more capable of utilizing the
entire bubble surface for possible collisions, contributing
to the increase of the particle-bubble collision frequency as
seen above.