XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2745
Radial Relative Velocity
The radial relative velocity between particles and bubbles
wr constitutes a major impact on the collision kernel.
According to the analytical derivation by Saffman and
Turner (1956) for homogeneous and isotropic turbulence
the collision kernel can be modelled by
.w 2rr
pb c r
2 C =(12)
Hence, in a first place the factors affecting the collision
kernel reduce to a geometric property, the collision radius
rc, and the radial component of the relative particle-bubble
velocity defined as
,w r
w r
r
$=(13)
where w =ub – up is the relative velocity between a bub-
ble and a particle and r =xp – xb is the connecting vector
between their centres and averaging is performed over dVΩ.
The radial velocity, in turn, depends on various proper-
ties. For the present simulations the mean particle-bubble
radial relative velocity was analysed as a function of the
polar coordinates r and θ around a bubble. Figure 4 illus-
trates the normalised magnitude of the radial relative veloc-
ity between the particles and the bubbles, wr, with respect
to the fluid velocity fluctuations, urms, for both cases.
The comparison of these distributions demonstrates
the impact of turbulence and gravity on particle and bub-
ble motion and their collision behaviour. In case GREF the
highest particle-bubble radial relative velocities occur around
θ =0°…45° at a far distance from the bubble. Gravity acts
in the vertical direction, causing a motion of particles and
bubbles in that direction. Furthermore, in these areas, the
direction of gravity aligns to a large extent with the radial
component of the relative velocity. On the other hand, for
θ =135°…180° the same effect causes particles and bub-
bles to move away from each other. Therefore, particles are
not approaching the bubble from this side greatly reduc-
ing the number of potential collision partners and avail-
able collision angles. For θ =45°…135°, there is almost no
radial relative motion, as the motion of the bubbles caused
by gravity is tangential to the radial direction between the
particles at this location. An area of lower absolute radial
relative velocity exists near the bubble for r =(0.5…0.8)
db. This area has an increased radial extent for polar angles
of θ =45°…135°, coinciding with the bubble wake.
Particles in this area are influenced by the fluid motion of
the bubble wake transporting them upwards counteracting
the downwards motion of gravity. Furthermore, in close
vicinity to the bubble surface the magnitude of the relative
radial velocity is also low. Near the bubble surface the fluid
streamlines are bending around the bubble. The particles,
at least partially, follow these streamlines. Hence their rela-
tive velocity is mostly tangential to the bubble surface.
While qualitatively similar, case TREF, employing
forced background turbulence, shows a reduced magnitude
(a) (b)
Figure 3. Particle-bubble 2D pair correlation function g
pb (r,θ). a) TREF, b) GREF
Radial Relative Velocity
The radial relative velocity between particles and bubbles
wr constitutes a major impact on the collision kernel.
According to the analytical derivation by Saffman and
Turner (1956) for homogeneous and isotropic turbulence
the collision kernel can be modelled by
.w 2rr
pb c r
2 C =(12)
Hence, in a first place the factors affecting the collision
kernel reduce to a geometric property, the collision radius
rc, and the radial component of the relative particle-bubble
velocity defined as
,w r
w r
r
$=(13)
where w =ub – up is the relative velocity between a bub-
ble and a particle and r =xp – xb is the connecting vector
between their centres and averaging is performed over dVΩ.
The radial velocity, in turn, depends on various proper-
ties. For the present simulations the mean particle-bubble
radial relative velocity was analysed as a function of the
polar coordinates r and θ around a bubble. Figure 4 illus-
trates the normalised magnitude of the radial relative veloc-
ity between the particles and the bubbles, wr, with respect
to the fluid velocity fluctuations, urms, for both cases.
The comparison of these distributions demonstrates
the impact of turbulence and gravity on particle and bub-
ble motion and their collision behaviour. In case GREF the
highest particle-bubble radial relative velocities occur around
θ =0°…45° at a far distance from the bubble. Gravity acts
in the vertical direction, causing a motion of particles and
bubbles in that direction. Furthermore, in these areas, the
direction of gravity aligns to a large extent with the radial
component of the relative velocity. On the other hand, for
θ =135°…180° the same effect causes particles and bub-
bles to move away from each other. Therefore, particles are
not approaching the bubble from this side greatly reduc-
ing the number of potential collision partners and avail-
able collision angles. For θ =45°…135°, there is almost no
radial relative motion, as the motion of the bubbles caused
by gravity is tangential to the radial direction between the
particles at this location. An area of lower absolute radial
relative velocity exists near the bubble for r =(0.5…0.8)
db. This area has an increased radial extent for polar angles
of θ =45°…135°, coinciding with the bubble wake.
Particles in this area are influenced by the fluid motion of
the bubble wake transporting them upwards counteracting
the downwards motion of gravity. Furthermore, in close
vicinity to the bubble surface the magnitude of the relative
radial velocity is also low. Near the bubble surface the fluid
streamlines are bending around the bubble. The particles,
at least partially, follow these streamlines. Hence their rela-
tive velocity is mostly tangential to the bubble surface.
While qualitatively similar, case TREF, employing
forced background turbulence, shows a reduced magnitude
(a) (b)
Figure 3. Particle-bubble 2D pair correlation function g
pb (r,θ). a) TREF, b) GREF