XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2743
GREF. The root mean square of the particle and bubble
velocity fluctuations, ub,rms and up,rms, respectively, also
highlight the effect. Case TREF exhibits significantly higher
fluctuations of the bubble and particle velocities compared
to case GREF. The added background turbulence is a sig-
nificant driver of particle and bubble motion. Additionally,
it causes an increase in particle Stokes number due the
decrease in the Kolmogorov time scale τη.
Collision Kernel &Collision Angle
To investigate the particle-bubble collision behaviour over
700,000 collision events were analysed in both simulations
and the particle-bubble collision kernel determined. The
results are provided in Table 3. They show that the addition
of turbulence in the present setup results in an increase of
the collision kernel by approximately 70%. This increase is
caused by several effects. Firstly, the presence of turbulence
leads to an increase in the radial relative velocity between
particles and bubbles. Forced background turbulence sig-
nificantly affects particle and bubble motion, resulting in
higher bubble and particle velocities and increased veloc-
ity fluctuations, as demonstrated in the previous section.
Secondly, in gravity driven flows particles and bubbles
exhibit a preferred direction of motion which is not the
case in homogeneous and isotropic turbulence. Although
the flow in TREF is not perfectly isotropic, the dominance
of the vertical fluctuations against the horizontal ones is
reduced to 9% so that the amount of isotropy is substan-
tially reduced. As a result, particles approach the bubbles
more or less from all sides, increasing the number of poten-
tial collision partners.
It is important to note that the normalised particle-
bubble collision kernel, Γpb τη/rc3, is decreased for case
TREF compared to case GREF. As case TREF uses forced
background turbulence, the overall turbulence level is
higher than in case GREF. Therefore, there is a significant
decrease in the Kolmogorov time scale, τη, between the two
cases from 0.584 s to 0.136 s. However, the collision ker-
nel, Γpb, does not increase in the same way, resulting in a
decrease of the normalised collision kernel.
One factor causing this increase in the particle-bubble
collision kernel is the distribution of the particle-bubble
collisions along the bubble surface. Furthermore, this
aspect plays a crucial role in modelling and predicting
collision events, with potential implications for attachment
probability (Hassanzadeh, et al., 2018 Dai, et al., 2000).
The position of the particle collision on the bubble surface
is characterized by the collision angle, denoted Θ. The colli-
sion angle is defined using the spherical coordinate θ, where
θ =0° represents the top of the bubble and θ =180° repre-
sents the bottom. It is important to note that the collision
angle is referenced against the vertical direction, opposing
gravity, regardless of the direction of motion of the bubbles.
Figure 2 depicts the PDF of the collision angle Θ, obtained
by averaging over rings around the θ =0° axis. For both
cases, the distribution of the collision angle starts off at low
values around Θ =0°. This is caused by the small bubble
surface area associated with small angles. Upon investigat-
ing the particle-bubble collision angle per area, the distri-
bution, indeed, peaks around Θ =0° (not shown here). The
PDFs shown in Figure 2 for both cases peak around Θ =
Table 2. Non-dimensional characteristic number for fluid, bubbles, and particles resulting from the simulation. All values are
spatial and temporal averages
k/(dbg) εdb/(dbg)3/2 Reλ λ/db η/dp Fr Reb ub,rms/(dbg)1/2 Rep Stp up,rms/(dbg)1/2
TREF 1.41 0.56 48.0 0.50 1.23 2.0 145 0.91 0.28 0.13 1.07
GREF 0.09 0.05 9.8 0.42 2.35 0.3 62,8 0.16 0.04 0.04 0.23
Figure 2. Comparison of probability density of particle-
bubble collision angle Θ. The upper and lower bubble halves
are marked by the vertical broken line
Table 3. Nondimensional particle-bubble collision kernels
and fraction of collisions on upper and lower bubble half
Γ
pb τ
η /r
c 3 Γ
pb
Collisions in
[0°, 90°]
Collisions in
[90°, 180°]
TREF 0.018 1.83 10 m s 9 3 #--64.5%1 35.5%
GREF 0.034 .m s 1 08 10 9 3 #--82.2%1 17.8%
GREF. The root mean square of the particle and bubble
velocity fluctuations, ub,rms and up,rms, respectively, also
highlight the effect. Case TREF exhibits significantly higher
fluctuations of the bubble and particle velocities compared
to case GREF. The added background turbulence is a sig-
nificant driver of particle and bubble motion. Additionally,
it causes an increase in particle Stokes number due the
decrease in the Kolmogorov time scale τη.
Collision Kernel &Collision Angle
To investigate the particle-bubble collision behaviour over
700,000 collision events were analysed in both simulations
and the particle-bubble collision kernel determined. The
results are provided in Table 3. They show that the addition
of turbulence in the present setup results in an increase of
the collision kernel by approximately 70%. This increase is
caused by several effects. Firstly, the presence of turbulence
leads to an increase in the radial relative velocity between
particles and bubbles. Forced background turbulence sig-
nificantly affects particle and bubble motion, resulting in
higher bubble and particle velocities and increased veloc-
ity fluctuations, as demonstrated in the previous section.
Secondly, in gravity driven flows particles and bubbles
exhibit a preferred direction of motion which is not the
case in homogeneous and isotropic turbulence. Although
the flow in TREF is not perfectly isotropic, the dominance
of the vertical fluctuations against the horizontal ones is
reduced to 9% so that the amount of isotropy is substan-
tially reduced. As a result, particles approach the bubbles
more or less from all sides, increasing the number of poten-
tial collision partners.
It is important to note that the normalised particle-
bubble collision kernel, Γpb τη/rc3, is decreased for case
TREF compared to case GREF. As case TREF uses forced
background turbulence, the overall turbulence level is
higher than in case GREF. Therefore, there is a significant
decrease in the Kolmogorov time scale, τη, between the two
cases from 0.584 s to 0.136 s. However, the collision ker-
nel, Γpb, does not increase in the same way, resulting in a
decrease of the normalised collision kernel.
One factor causing this increase in the particle-bubble
collision kernel is the distribution of the particle-bubble
collisions along the bubble surface. Furthermore, this
aspect plays a crucial role in modelling and predicting
collision events, with potential implications for attachment
probability (Hassanzadeh, et al., 2018 Dai, et al., 2000).
The position of the particle collision on the bubble surface
is characterized by the collision angle, denoted Θ. The colli-
sion angle is defined using the spherical coordinate θ, where
θ =0° represents the top of the bubble and θ =180° repre-
sents the bottom. It is important to note that the collision
angle is referenced against the vertical direction, opposing
gravity, regardless of the direction of motion of the bubbles.
Figure 2 depicts the PDF of the collision angle Θ, obtained
by averaging over rings around the θ =0° axis. For both
cases, the distribution of the collision angle starts off at low
values around Θ =0°. This is caused by the small bubble
surface area associated with small angles. Upon investigat-
ing the particle-bubble collision angle per area, the distri-
bution, indeed, peaks around Θ =0° (not shown here). The
PDFs shown in Figure 2 for both cases peak around Θ =
Table 2. Non-dimensional characteristic number for fluid, bubbles, and particles resulting from the simulation. All values are
spatial and temporal averages
k/(dbg) εdb/(dbg)3/2 Reλ λ/db η/dp Fr Reb ub,rms/(dbg)1/2 Rep Stp up,rms/(dbg)1/2
TREF 1.41 0.56 48.0 0.50 1.23 2.0 145 0.91 0.28 0.13 1.07
GREF 0.09 0.05 9.8 0.42 2.35 0.3 62,8 0.16 0.04 0.04 0.23
Figure 2. Comparison of probability density of particle-
bubble collision angle Θ. The upper and lower bubble halves
are marked by the vertical broken line
Table 3. Nondimensional particle-bubble collision kernels
and fraction of collisions on upper and lower bubble half
Γ
pb τ
η /r
c 3 Γ
pb
Collisions in
[0°, 90°]
Collisions in
[90°, 180°]
TREF 0.018 1.83 10 m s 9 3 #--64.5%1 35.5%
GREF 0.034 .m s 1 08 10 9 3 #--82.2%1 17.8%