2732 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
The simulations include full coupling between the
phases, integrating drag, lift and added mass forces with
continuum and discrete phase behaviour. Additionally, a
granular temperature model accounts for particle-particle
and bubble-particle interactions, ensuring accurate simula-
tion even at very high solids contents. Moreover, the frame-
work incorporates detailed flotation models involving the
interactions between particles and bubbles, encompassing
collision, attachment and detachment events. The bubbles
can have any number of different size classes, whereas the
particles can be divided into an arbitrary number of classes
based on a combination of size, density, and hydrophobicity.
Furthermore, the simulator can make use of compo-
nent geometries sourced from CAD files, enabling the
specification of motion for these components. Notably, as
this is a meshless method – other than for the component
surfaces – there are no issues with maintaining a computa-
tional mesh in the presence of rapidly moving and poten-
tially interlocking components.
THE MATHEMATICAL MODELLING
A Semi-Lagrangian Approach
The standard Eulerian-Eulerian approach to modelling
flows containing both a continuous (subscript c) and a
number of discrete phases (subscript d, with different
phases indicated by different values of k) is described by
means of mass continuity equations (equations 1 and 3)
and momentum equations (equations 2 and 4).
0
c c 2t
2 d^ac a t tcvch +=^h (1)
v v v
Fdc,k
c c c c c c c c c
c
k
n
1
2t
2 d dP dx a t t a
a tcg
+=-+
++
=
/
^^a h h
(2)
where α is the volume fraction of the phase, ρ is the density
of the phase, vc is the velocity of the continuous phase in
an Eulerian reference frame, P is the pressure, g is the vec-
tor for the acceleration due to body forces, τc is the shear
stress tensor for the continuous phase and Fdc,k is the force
exerted by the kth discrete phase on the continuous phase.
For the discrete phases the following continuity and
momentum balance equations hold:
vd 0
,,,,,d k k d k k k 2t
2 d^a a td td +=^h h (3)
vd,kh
F
g
,
,,
,,k ,k
,k ,lk
d k
dc
d k
d k
d k d d
l
l
d
2t
2
d dPd,k
dx
,vd,kh
td,k
td,kv ad,kdP
a t
+=--
++
-+/kF!
^a
^a
(4)
where Pd,k is the granular pressure of phase k and τd,k is the
granular shear stress for phase k. Both of these will involve
the sum of the interactions with all the other phases. When
more than one discrete phase is being modelled Fd,lk is the
force that phase l exerts on phase k.
These sets of equations are commonly used to simu-
late multiphase flows, with finite volume or finite element
approaches being the most commonly used. They can,
though, suffer from excessive numerical diffusion under
conditions typical of flotation cells. This is due to the large
velocity magnitudes for both the continuous and discrete
phases, but comparatively small relative velocities between
the phases. This numerical dispersion can mask details in the
flow and concentration behaviour that can have important
implications for our understanding of flotation behaviour.
In this work we eliminate most of the numerical dif-
fusion by using a moving reference with both the continu-
ous and discrete phases moving relative to this reference
frames, but with relative velocities that are typically much
smaller than the velocity of the reference frame itself. In
order to eliminate the advection term in the equation for
the motion of the reference frame, the momentum aver-
aged velocity is used for the motion of the reference frame:
*=u
v v
,,k
,,k ,k
c c k
n
d k d
c c c k
n
d k d d
1
1
a t a t
a t a t
+
+
=
=/
/
(5)
It is also useful to define an average density for the mixture
of the discrete and continuous phases:
*=
,,k c c
k
n
d k d
1
t a t a t +
=
/(6)
The evolution of the velocity of this reference can be
obtained by summing the momentum equations for each
of the phases and rewriting in the new reference frame:
*
**
*
*
*
g
u
1
,
,
,
,
k
n
d k
d k
c k
n
d k
d k
1
1
2t
2u
2t
2t
dP
dxc dx
t
t
a
t a a
=-
-
+
+++
=
=
/
/
ddP
d
n
n
(7)
The simulations include full coupling between the
phases, integrating drag, lift and added mass forces with
continuum and discrete phase behaviour. Additionally, a
granular temperature model accounts for particle-particle
and bubble-particle interactions, ensuring accurate simula-
tion even at very high solids contents. Moreover, the frame-
work incorporates detailed flotation models involving the
interactions between particles and bubbles, encompassing
collision, attachment and detachment events. The bubbles
can have any number of different size classes, whereas the
particles can be divided into an arbitrary number of classes
based on a combination of size, density, and hydrophobicity.
Furthermore, the simulator can make use of compo-
nent geometries sourced from CAD files, enabling the
specification of motion for these components. Notably, as
this is a meshless method – other than for the component
surfaces – there are no issues with maintaining a computa-
tional mesh in the presence of rapidly moving and poten-
tially interlocking components.
THE MATHEMATICAL MODELLING
A Semi-Lagrangian Approach
The standard Eulerian-Eulerian approach to modelling
flows containing both a continuous (subscript c) and a
number of discrete phases (subscript d, with different
phases indicated by different values of k) is described by
means of mass continuity equations (equations 1 and 3)
and momentum equations (equations 2 and 4).
0
c c 2t
2 d^ac a t tcvch +=^h (1)
v v v
Fdc,k
c c c c c c c c c
c
k
n
1
2t
2 d dP dx a t t a
a tcg
+=-+
++
=
/
^^a h h
(2)
where α is the volume fraction of the phase, ρ is the density
of the phase, vc is the velocity of the continuous phase in
an Eulerian reference frame, P is the pressure, g is the vec-
tor for the acceleration due to body forces, τc is the shear
stress tensor for the continuous phase and Fdc,k is the force
exerted by the kth discrete phase on the continuous phase.
For the discrete phases the following continuity and
momentum balance equations hold:
vd 0
,,,,,d k k d k k k 2t
2 d^a a td td +=^h h (3)
vd,kh
F
g
,
,,
,,k ,k
,k ,lk
d k
dc
d k
d k
d k d d
l
l
d
2t
2
d dPd,k
dx
,vd,kh
td,k
td,kv ad,kdP
a t
+=--
++
-+/kF!
^a
^a
(4)
where Pd,k is the granular pressure of phase k and τd,k is the
granular shear stress for phase k. Both of these will involve
the sum of the interactions with all the other phases. When
more than one discrete phase is being modelled Fd,lk is the
force that phase l exerts on phase k.
These sets of equations are commonly used to simu-
late multiphase flows, with finite volume or finite element
approaches being the most commonly used. They can,
though, suffer from excessive numerical diffusion under
conditions typical of flotation cells. This is due to the large
velocity magnitudes for both the continuous and discrete
phases, but comparatively small relative velocities between
the phases. This numerical dispersion can mask details in the
flow and concentration behaviour that can have important
implications for our understanding of flotation behaviour.
In this work we eliminate most of the numerical dif-
fusion by using a moving reference with both the continu-
ous and discrete phases moving relative to this reference
frames, but with relative velocities that are typically much
smaller than the velocity of the reference frame itself. In
order to eliminate the advection term in the equation for
the motion of the reference frame, the momentum aver-
aged velocity is used for the motion of the reference frame:
*=u
v v
,,k
,,k ,k
c c k
n
d k d
c c c k
n
d k d d
1
1
a t a t
a t a t
+
+
=
=/
/
(5)
It is also useful to define an average density for the mixture
of the discrete and continuous phases:
*=
,,k c c
k
n
d k d
1
t a t a t +
=
/(6)
The evolution of the velocity of this reference can be
obtained by summing the momentum equations for each
of the phases and rewriting in the new reference frame:
*
**
*
*
*
g
u
1
,
,
,
,
k
n
d k
d k
c k
n
d k
d k
1
1
2t
2u
2t
2t
dP
dxc dx
t
t
a
t a a
=-
-
+
+++
=
=
/
/
ddP
d
n
n
(7)