XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2733
The viscosities used in this equation include both a pulp
viscosity based on empirical relationships as well as turbu-
lent eddy viscosity based on the Smagorinsky model. As
the simulation is dynamic, the above grid scale turbulent
dynamics will be directly resolved, with the simulations
thus taking the form of large eddy simulations (LES).
We can now introduce the velocities of each of the
phases relative to this refence frame:
*u vc u
c =-(8)
*u v u
,,d k d k =-(9)
With a bit of simplification and by assuming that the con-
tinuous phase and the individual particle phases are incom-
pressible (though the mixture will have a density that varies
in both space and time), the following equations can be
derived for the motion of the discrete phases relative to ref-
erence frame, as well as the evolution of their concentra-
tions within this reference frame:
g
F Fd
,,,,,
,
,,
,
,,
,
d k d k
dc,k lk
d k
d k d k d k
d k
d k d k
d k
d k d k
l
l k
2t
2u
du dP dP
dx
t a t
a t a t
=-u --
+++
-+
!/(10)
u
,,d k d k 2t
2ad,k
=-d^a h (11)
We do not explicitly need to solve for the motion of the
continuous phase, but can instead obtain its velocity and
the evolution of the continuous phase volume fraction by
difference:
u
u
1
,k
,,k ,k
c k
n
d
k
n
d k d d
c
1
1
t a
a t
=
-
-
=
=/
/
b l
(12)
1
,c
k
n
d k
1
a a =-
=
/(13)
These equations can be solved within an SPH framework,
where the integration points (SPH “particles”) move at the
reference frame velocity, u*, with additional fields for the
discrete phase relative velocities and volume fractions being
solved for each of these points.
Discrete-Continuous and Discrete-Discrete Coupling
In addition to the overall governing equations, relationships
are required for the forces that the discrete and continuous
phases exert on one another, as well as for the forces exerted
within and between the discrete phases. For the interactions
between the discrete and continuous phases we consider 3
forces, namely drag, lift and added mass.
The drag force arises due to the relative motion of the
fluid and particles. In order to account for a wide range of
solids contents and particle Reynolds numbers, the model
of (Gobin et al., 2003) is used. This model, in turn uses the
Ergun equation at high solids contents and the model of
(Wen &Yu, 1966) at lower solids contents.
Lift forces arise due to the rotation of the fluid and/or
the particles. In this work we follow the modelling approach
of Yeoh et al. (2013) in the modelling of lift.
The added mass force arises due to fact that fluid is dis-
placed and needs to be accelerated in the direction opposite
to the direction of acceleration of discrete phase. This force
is especially important in the behaviour of discrete phases
that are of much lower density than the continuous phase,
such as when predicting the motion of air bubbles within
a liquid flow.
In addition to the interactions between the continu-
ous and discrete phases, there is also interaction within the
discrete phases. There are two main types of interactions,
namely granular shear stresses and granular pressures. These
can be modelled by means of a granular temperature as they
both arise due to random motion of the individual par-
ticles relative to the average motion of the discrete phases.
The granular pressure and shear stress are most important
at high solids contents, with the granular pressure, in par-
ticular, ensuring that the solids content does not exceed the
maximum packing fraction. The models used in this simu-
lator are based on the work of (Syamlal, 1987).
Flotation Interactions
In flotation, in addition to the conventional interactions
between the discrete phases described above, the discrete
particle phases also have the propensity to attach to the
discrete bubble phases. In this modelling we therefore
introduce an additional set of continuity and momentum
equations for the bubble-particle aggregates. Each of these
aggregate types is characterised by means of a single velocity
field and associated momentum equation as the attached
bubbles and particles cannot move relative to one another,
but there are multiple concentration fields and associated
continuity equations as the composition of these aggregates
in terms of relative number and types and particles and
bubbles involved will vary both spatially and temporally.
The key feature of these flotation interactions is the
rate at which the particles attach and detach to and from
the bubbles. The attachment model is based on a model
for the rate at which bubbles and particles will interact
due to both their both average relative motions and their
The viscosities used in this equation include both a pulp
viscosity based on empirical relationships as well as turbu-
lent eddy viscosity based on the Smagorinsky model. As
the simulation is dynamic, the above grid scale turbulent
dynamics will be directly resolved, with the simulations
thus taking the form of large eddy simulations (LES).
We can now introduce the velocities of each of the
phases relative to this refence frame:
*u vc u
c =-(8)
*u v u
,,d k d k =-(9)
With a bit of simplification and by assuming that the con-
tinuous phase and the individual particle phases are incom-
pressible (though the mixture will have a density that varies
in both space and time), the following equations can be
derived for the motion of the discrete phases relative to ref-
erence frame, as well as the evolution of their concentra-
tions within this reference frame:
g
F Fd
,,,,,
,
,,
,
,,
,
d k d k
dc,k lk
d k
d k d k d k
d k
d k d k
d k
d k d k
l
l k
2t
2u
du dP dP
dx
t a t
a t a t
=-u --
+++
-+
!/(10)
u
,,d k d k 2t
2ad,k
=-d^a h (11)
We do not explicitly need to solve for the motion of the
continuous phase, but can instead obtain its velocity and
the evolution of the continuous phase volume fraction by
difference:
u
u
1
,k
,,k ,k
c k
n
d
k
n
d k d d
c
1
1
t a
a t
=
-
-
=
=/
/
b l
(12)
1
,c
k
n
d k
1
a a =-
=
/(13)
These equations can be solved within an SPH framework,
where the integration points (SPH “particles”) move at the
reference frame velocity, u*, with additional fields for the
discrete phase relative velocities and volume fractions being
solved for each of these points.
Discrete-Continuous and Discrete-Discrete Coupling
In addition to the overall governing equations, relationships
are required for the forces that the discrete and continuous
phases exert on one another, as well as for the forces exerted
within and between the discrete phases. For the interactions
between the discrete and continuous phases we consider 3
forces, namely drag, lift and added mass.
The drag force arises due to the relative motion of the
fluid and particles. In order to account for a wide range of
solids contents and particle Reynolds numbers, the model
of (Gobin et al., 2003) is used. This model, in turn uses the
Ergun equation at high solids contents and the model of
(Wen &Yu, 1966) at lower solids contents.
Lift forces arise due to the rotation of the fluid and/or
the particles. In this work we follow the modelling approach
of Yeoh et al. (2013) in the modelling of lift.
The added mass force arises due to fact that fluid is dis-
placed and needs to be accelerated in the direction opposite
to the direction of acceleration of discrete phase. This force
is especially important in the behaviour of discrete phases
that are of much lower density than the continuous phase,
such as when predicting the motion of air bubbles within
a liquid flow.
In addition to the interactions between the continu-
ous and discrete phases, there is also interaction within the
discrete phases. There are two main types of interactions,
namely granular shear stresses and granular pressures. These
can be modelled by means of a granular temperature as they
both arise due to random motion of the individual par-
ticles relative to the average motion of the discrete phases.
The granular pressure and shear stress are most important
at high solids contents, with the granular pressure, in par-
ticular, ensuring that the solids content does not exceed the
maximum packing fraction. The models used in this simu-
lator are based on the work of (Syamlal, 1987).
Flotation Interactions
In flotation, in addition to the conventional interactions
between the discrete phases described above, the discrete
particle phases also have the propensity to attach to the
discrete bubble phases. In this modelling we therefore
introduce an additional set of continuity and momentum
equations for the bubble-particle aggregates. Each of these
aggregate types is characterised by means of a single velocity
field and associated momentum equation as the attached
bubbles and particles cannot move relative to one another,
but there are multiple concentration fields and associated
continuity equations as the composition of these aggregates
in terms of relative number and types and particles and
bubbles involved will vary both spatially and temporally.
The key feature of these flotation interactions is the
rate at which the particles attach and detach to and from
the bubbles. The attachment model is based on a model
for the rate at which bubbles and particles will interact
due to both their both average relative motions and their