2740 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
between a particle and a bubble occurs due to a sufficiently
close encounter between the surfaces of these two collision
partners. The number of attached particles, which directly
translates into flotation performance, is strongly influenced
by the number of particle-bubble collisions. An important
measure to quantify the effect is the collision frequency per
unit volume, Zpb (Abrahamson, 1975 Schubert, 1999),
which is given by
,Z N N
pb pb p b C= (1)
Np and Nb are the number of particles and bubbles, respec-
tively, in a given domain Ω with volume VΩ and Γpb the
so-called collision kernel. It is defined as(Rosa, et al., 2013
Wang, et al., 2005).
,N N
N V
pb
p b
c C =X
o
(2)
where N
c
o is the number of collisions per time in Ω.
Research on the collision kernel, and ultimately its
prediction, has been a major focus of research for several
decades. Several models for this collision kernel have been
proposed so far (Abrahamson, 1975 Yuu, 1984 Kostoglou,
et al., 2020 Dai, et al., 2000). One of the earliest models
developed for monodisperse point particles in the limit of
zero Stokes number was proposed by Saffman and Turner
(1956). Their collision kernel for two particles of classes i
and j reads
,r r
15o
8rf
15
8 //2
ij i j
c 1 2 1 3 r
x C =+=
h
^r a a ^STh h3 k k (3)
where rc =rp +rb is the so-called collision radius and τ𝜂
=(ν/ε)1/2 is the Kolmogorov time scale, with ε the mean
turbulent dissipation rate. The associated length scale is
given by 𝜂 =(ν3/ε)1/4 and ν the kinematic viscosity. This
collision kernel is mainly determined by the radii of the
collision partners and the small scales of turbulence. The
prefactor is approximately 1.29, which is close to 1. As the
collision kernel is an inherently dimensional quantity, this
expression without a prefactor is widely used to normalise
collision kernel.
However, real particles and bubbles in flotation do not
in general obey the zero Stokes number limit. Therefore,
a realistic collision kernel is influenced by a wider range
of factors, including parameters describing the dispersed
particles and bubbles, as well as fluid properties such as
turbulence. With the normalisation shown in (3) Chan et
al. (2023) and Hassanzadeh et al. (2018) list the following
general dependencies
/x ,,,,,,,,Re r f St
d d d Fr
c
pb
p
f
b
f
p p b b
3 t
t
t
t
h h m
C
=
h
m
e o (4)
where db=2rb and dp=2rp are the bubble and particle diam-
eters, respectively. The Stokes number of the particles Stp is
defined as the ratio of the particle time scale τp to the fluid
time scale τf, thus
/t
St
d
36ox
2 1i
p
f
p p p f x
x t
==
+
h
_
(5)
For isotropic turbulence (Chan, et al., 2023)(Chan, et al.,
2023).The densities of the particles, bubbles, and fluid are
given by ρp, ρb, and ρf, respectively. The Froude number Fr
describes the ratio of flow inertia to gravity and is defined,
here, as
/x
.Fr g
2 h
=h (6)
Turbulence has a major influence on the frequency of colli-
sions and micro-processes in flotation, such as the collision
process, are mainly controlled by small-scale turbulence.
Turbulence strongly influences this relative motion of par-
ticles and bubbles, leading to an increase in the collision
kernel (Kostoglou, et al., 2020 Meyer &Deglon, 2011). A
measure to describe the intensity of the fluid turbulence is
the Taylor Reynolds number
.Re 15^o/ 2
rms
fhu =
m (7)
where /3 u k 2
rms =and k the turbulent kinetic energy.
METHODOLOGY
Numerical Method
The three-phase Direct Numerical Simulations (DNS)
reported here were performed with the in-house multi-
phase flow solver PRIME (Tschisgale, et al., 2018 Kempe
&Fröhlich, 2012 Uhlmann, 2005). The fluid is described
by the unsteady Navier-Stokes equations with constant
density
u uh u f f f 1
f IBM p T
2 $2t
2u d dp t od +=-++++^(8)
u 0, $d =(9)
where u is the fluid velocity vector, p the pressure, fIBM a
volume force vector used to couple fluid and bubbles, fp the
force resulting from the two-way coupling of particles, and
fT a volume force generating sustained turbulence.
The spatial discretisation of these equations is accom-
plished by a second-order finite-volume method on a stag-
gered, equidistant Cartesian grid. The time discretisation is
based on a three-step second-order Runge-Kutta method,
with a pressure correction term solved for each substep. A
Crank-Nicholson scheme is used in each substep to solve
between a particle and a bubble occurs due to a sufficiently
close encounter between the surfaces of these two collision
partners. The number of attached particles, which directly
translates into flotation performance, is strongly influenced
by the number of particle-bubble collisions. An important
measure to quantify the effect is the collision frequency per
unit volume, Zpb (Abrahamson, 1975 Schubert, 1999),
which is given by
,Z N N
pb pb p b C= (1)
Np and Nb are the number of particles and bubbles, respec-
tively, in a given domain Ω with volume VΩ and Γpb the
so-called collision kernel. It is defined as(Rosa, et al., 2013
Wang, et al., 2005).
,N N
N V
pb
p b
c C =X
o
(2)
where N
c
o is the number of collisions per time in Ω.
Research on the collision kernel, and ultimately its
prediction, has been a major focus of research for several
decades. Several models for this collision kernel have been
proposed so far (Abrahamson, 1975 Yuu, 1984 Kostoglou,
et al., 2020 Dai, et al., 2000). One of the earliest models
developed for monodisperse point particles in the limit of
zero Stokes number was proposed by Saffman and Turner
(1956). Their collision kernel for two particles of classes i
and j reads
,r r
15o
8rf
15
8 //2
ij i j
c 1 2 1 3 r
x C =+=
h
^r a a ^STh h3 k k (3)
where rc =rp +rb is the so-called collision radius and τ𝜂
=(ν/ε)1/2 is the Kolmogorov time scale, with ε the mean
turbulent dissipation rate. The associated length scale is
given by 𝜂 =(ν3/ε)1/4 and ν the kinematic viscosity. This
collision kernel is mainly determined by the radii of the
collision partners and the small scales of turbulence. The
prefactor is approximately 1.29, which is close to 1. As the
collision kernel is an inherently dimensional quantity, this
expression without a prefactor is widely used to normalise
collision kernel.
However, real particles and bubbles in flotation do not
in general obey the zero Stokes number limit. Therefore,
a realistic collision kernel is influenced by a wider range
of factors, including parameters describing the dispersed
particles and bubbles, as well as fluid properties such as
turbulence. With the normalisation shown in (3) Chan et
al. (2023) and Hassanzadeh et al. (2018) list the following
general dependencies
/x ,,,,,,,,Re r f St
d d d Fr
c
pb
p
f
b
f
p p b b
3 t
t
t
t
h h m
C
=
h
m
e o (4)
where db=2rb and dp=2rp are the bubble and particle diam-
eters, respectively. The Stokes number of the particles Stp is
defined as the ratio of the particle time scale τp to the fluid
time scale τf, thus
/t
St
d
36ox
2 1i
p
f
p p p f x
x t
==
+
h
_
(5)
For isotropic turbulence (Chan, et al., 2023)(Chan, et al.,
2023).The densities of the particles, bubbles, and fluid are
given by ρp, ρb, and ρf, respectively. The Froude number Fr
describes the ratio of flow inertia to gravity and is defined,
here, as
/x
.Fr g
2 h
=h (6)
Turbulence has a major influence on the frequency of colli-
sions and micro-processes in flotation, such as the collision
process, are mainly controlled by small-scale turbulence.
Turbulence strongly influences this relative motion of par-
ticles and bubbles, leading to an increase in the collision
kernel (Kostoglou, et al., 2020 Meyer &Deglon, 2011). A
measure to describe the intensity of the fluid turbulence is
the Taylor Reynolds number
.Re 15^o/ 2
rms
fhu =
m (7)
where /3 u k 2
rms =and k the turbulent kinetic energy.
METHODOLOGY
Numerical Method
The three-phase Direct Numerical Simulations (DNS)
reported here were performed with the in-house multi-
phase flow solver PRIME (Tschisgale, et al., 2018 Kempe
&Fröhlich, 2012 Uhlmann, 2005). The fluid is described
by the unsteady Navier-Stokes equations with constant
density
u uh u f f f 1
f IBM p T
2 $2t
2u d dp t od +=-++++^(8)
u 0, $d =(9)
where u is the fluid velocity vector, p the pressure, fIBM a
volume force vector used to couple fluid and bubbles, fp the
force resulting from the two-way coupling of particles, and
fT a volume force generating sustained turbulence.
The spatial discretisation of these equations is accom-
plished by a second-order finite-volume method on a stag-
gered, equidistant Cartesian grid. The time discretisation is
based on a three-step second-order Runge-Kutta method,
with a pressure correction term solved for each substep. A
Crank-Nicholson scheme is used in each substep to solve