2742 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
bubbles in this case causes a damping of the turbulence.
This effect was observed in the literature (Fabre, et al.,
1987 Rashidi &Banerjee, 1990). As case GREF is gravity-
driven only, the turbulence values obtained are low. The
Froude number, for example, is small in case GREF, the
Froude number is above one in case TREF, indicating that
turbulence dominates over gravity.
This is further illustrated in Figures 1b,c which depict
a slice through the domain for both cases. For case TREF, a
higher fluid velocity magnitude is visible compared to case
GREF. Additionally, turbulent flow structures of large scale
dominate case TREF, while the effects of the bubble wakes
on the flow field are less apparent.
For both cases the Taylor micro-scale is of the same
order of magnitude as the bubble diameter, and the
Kolmogorov scale is slightly smaller than the particle diam-
eter, but still of a similar order of magnitude. It is impor-
tant to note that the resulting turbulence is anisotropic in
the y-direction due to the presence of gravity in both cases,
which causes particles and bubbles to have a preferred direc-
tion of motion. Their wakes generate anisotropy in the fluid
velocity. It can be observed qualitatively that these wakes
are overshadowed by the superimposed turbulence, so that
they are less visible with the level of turbulence motion
being increased.
The presence of turbulence in case TREF case leads to
higher particle and bubble velocities, thus increasing the
particle and bubble Reynolds numbers compared to case
Table 1. Physical parameters of the performed simulations
Fluid
Density ρf =998 kg/m3
Viscosity µf =1 mPas, νf =1 mm2/s
Gravitational acceleration g =9.81 m/s2
Bubbles
Number of bubbles Nb =56
Diameter db =1 mm
Density ρ
b =1.225 kg/m3
Surface Tension σ
b =0.073 kg/s2
Particles
Number of particles N
p =2.35 × 106
Diameter d
p =30 µm
Density ρ
p =3,000kg/m3
Nondimensional Numbers
Archimedes number /_ρ Ar gd 9753
b f f
3 2 ν Δρ ==i
Bond number /.134 Eo gd 0
b b
2 σ Δρ ==
Galilei number /Δρ Ga Ar 9765
f
ρ ==
Gas hold-up (bubble
volume fraction)
/.8% V 8
g b
/V ε ==
Ω
Particle volume fraction /VΩ 10
b p /V ε ==
(a) (b) (c)
Figure 1. Instantaneous snapshots of the simulations showing. a) Three-dimensional domain b) Slice through the simulation
domain for case TREF at z
s =const. where the colour represents the instantaneous vertical velocity. The local bubble extent in
the shown slice is depicted. Particles with their centre in the interval z
s − r
p z z
s +r
p are shown. c) Similar slice for the case
GREF. In all graphs air bubbles (light) and particles (dark) enlarged and their concentration reduced for better visibility
bubbles in this case causes a damping of the turbulence.
This effect was observed in the literature (Fabre, et al.,
1987 Rashidi &Banerjee, 1990). As case GREF is gravity-
driven only, the turbulence values obtained are low. The
Froude number, for example, is small in case GREF, the
Froude number is above one in case TREF, indicating that
turbulence dominates over gravity.
This is further illustrated in Figures 1b,c which depict
a slice through the domain for both cases. For case TREF, a
higher fluid velocity magnitude is visible compared to case
GREF. Additionally, turbulent flow structures of large scale
dominate case TREF, while the effects of the bubble wakes
on the flow field are less apparent.
For both cases the Taylor micro-scale is of the same
order of magnitude as the bubble diameter, and the
Kolmogorov scale is slightly smaller than the particle diam-
eter, but still of a similar order of magnitude. It is impor-
tant to note that the resulting turbulence is anisotropic in
the y-direction due to the presence of gravity in both cases,
which causes particles and bubbles to have a preferred direc-
tion of motion. Their wakes generate anisotropy in the fluid
velocity. It can be observed qualitatively that these wakes
are overshadowed by the superimposed turbulence, so that
they are less visible with the level of turbulence motion
being increased.
The presence of turbulence in case TREF case leads to
higher particle and bubble velocities, thus increasing the
particle and bubble Reynolds numbers compared to case
Table 1. Physical parameters of the performed simulations
Fluid
Density ρf =998 kg/m3
Viscosity µf =1 mPas, νf =1 mm2/s
Gravitational acceleration g =9.81 m/s2
Bubbles
Number of bubbles Nb =56
Diameter db =1 mm
Density ρ
b =1.225 kg/m3
Surface Tension σ
b =0.073 kg/s2
Particles
Number of particles N
p =2.35 × 106
Diameter d
p =30 µm
Density ρ
p =3,000kg/m3
Nondimensional Numbers
Archimedes number /_ρ Ar gd 9753
b f f
3 2 ν Δρ ==i
Bond number /.134 Eo gd 0
b b
2 σ Δρ ==
Galilei number /Δρ Ga Ar 9765
f
ρ ==
Gas hold-up (bubble
volume fraction)
/.8% V 8
g b
/V ε ==
Ω
Particle volume fraction /VΩ 10
b p /V ε ==
(a) (b) (c)
Figure 1. Instantaneous snapshots of the simulations showing. a) Three-dimensional domain b) Slice through the simulation
domain for case TREF at z
s =const. where the colour represents the instantaneous vertical velocity. The local bubble extent in
the shown slice is depicted. Particles with their centre in the interval z
s − r
p z z
s +r
p are shown. c) Similar slice for the case
GREF. In all graphs air bubbles (light) and particles (dark) enlarged and their concentration reduced for better visibility