56 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
obtaining a value of ~325 kgm–3 for suspensions at low
viscosity.
Laminar Flow Model
The gravity separation achieved by the Reflux Classifier
improved significantly with the establishment of the lam-
inar-shear separation mechanism (Galvin and Liu, 2011).
In low density coal applications, the inclined channel was
eventually set at 6 mm, but more recent work involving
the separation of dense minerals has involved a spacing of
3 mm and more recently 1.8 mm. Figure 2 provides a sche-
matic representation of the hydrodynamics. Laminar flow
leads to a parabolic velocity field across the inclined chan-
nel of perpendicular distance z. Thus, for a superficial fluid
velocity, U', the local fluid velocity, u, at a distance, x, from
the wall is given by,
u z z
x 6Ulx 1 =-`j (4)
The laminar flow field carries several important features.
Firstly, particles effectively escape the flow by progressively
settling towards the upward facing surfaces where the no-
slip boundary condition applies. This feature of the Boycott
Effect delivers a significant capacity advantage over con-
ventional fluidized bed systems given the particles become
subject to relatively low fluid velocities. Conversely, those
particles that remain suspended are transported rapidly in
the upwards tangent direction of the inclined plates. The
high shear rate at the wall in turn leads to shear induced
inertial lift which, if sufficient, leads to physical lift, mean-
ing the lower density particles migrate from the upward
facing surface of the inclined plates to join the faster flow-
ing fluid.
The critical condition defining whether a particle,
residing at the wall, migrates upwards or slides downwards,
can be established by equating the local fluid velocity, u,
acting on the particle (at x =d/2) with the particle termi-
nal velocity in the tangent direction, uTt. It follows from
Equation 4 that the ratio of the superficial fluid velocity
within the channel to the terminal velocity in the tangent
direction is approximately (Galvin et al., 2011),
u
Ul z
3d
Tt
=(5)
Application of Equation 5 to particles of the same
diameter, with different density, gives identical capacity
advantages, U'/uTt, as the right-hand side of the Equation
is constant. In practice, however, the system operates with
a fixed superficial fluid velocity. Conventional dilute elu-
triators applied to ultrafine particles that settle in accor-
dance with the well-known Stokes law, give m=2. Thus,
assuming Stokes law applies to the particle terminal veloc-
ity, Equation 5 predicts that the buoyant density will scale
inversely with the particle diameter. That is,
d
p w +t t t =--1 ^h (6)
It is evident the laminar flow model under dilute con-
ditions gives m=1. Thus, the velocity gradient within the
inclined channels reduces the value of m=2 to a value of
m=1, greatly improving the separation efficiency based on
particle density. This dependence however falls well short
of what is observed in practice, where m~0.26 to 0.28.
Therefore, we need to consider other factors that contribute
to the separation performance.
We now introduce a useful constitutive model to
describe the hindered slip velocity, us, of a particle in a
suspension containing particles of different density (Asif,
1997 Galvin et al., 1999). That is,
u utf ut
s
p w
p susp
n- 1
t t
t t
==-
-
d n (7)
where ut is the terminal settling velocity of the particle
in water, and f a hindered settling factor, dependent on
p t ,the density of the particle, tsusp ,the density of the
local suspension, and tw ,the density of water, and n, the
Richardson and Zaki exponent. Note for very low particle
Figure 2. Schematic representation of the laminar flow
velocity field and critical condition for particle transport up
through an inclined channel
obtaining a value of ~325 kgm–3 for suspensions at low
viscosity.
Laminar Flow Model
The gravity separation achieved by the Reflux Classifier
improved significantly with the establishment of the lam-
inar-shear separation mechanism (Galvin and Liu, 2011).
In low density coal applications, the inclined channel was
eventually set at 6 mm, but more recent work involving
the separation of dense minerals has involved a spacing of
3 mm and more recently 1.8 mm. Figure 2 provides a sche-
matic representation of the hydrodynamics. Laminar flow
leads to a parabolic velocity field across the inclined chan-
nel of perpendicular distance z. Thus, for a superficial fluid
velocity, U', the local fluid velocity, u, at a distance, x, from
the wall is given by,
u z z
x 6Ulx 1 =-`j (4)
The laminar flow field carries several important features.
Firstly, particles effectively escape the flow by progressively
settling towards the upward facing surfaces where the no-
slip boundary condition applies. This feature of the Boycott
Effect delivers a significant capacity advantage over con-
ventional fluidized bed systems given the particles become
subject to relatively low fluid velocities. Conversely, those
particles that remain suspended are transported rapidly in
the upwards tangent direction of the inclined plates. The
high shear rate at the wall in turn leads to shear induced
inertial lift which, if sufficient, leads to physical lift, mean-
ing the lower density particles migrate from the upward
facing surface of the inclined plates to join the faster flow-
ing fluid.
The critical condition defining whether a particle,
residing at the wall, migrates upwards or slides downwards,
can be established by equating the local fluid velocity, u,
acting on the particle (at x =d/2) with the particle termi-
nal velocity in the tangent direction, uTt. It follows from
Equation 4 that the ratio of the superficial fluid velocity
within the channel to the terminal velocity in the tangent
direction is approximately (Galvin et al., 2011),
u
Ul z
3d
Tt
=(5)
Application of Equation 5 to particles of the same
diameter, with different density, gives identical capacity
advantages, U'/uTt, as the right-hand side of the Equation
is constant. In practice, however, the system operates with
a fixed superficial fluid velocity. Conventional dilute elu-
triators applied to ultrafine particles that settle in accor-
dance with the well-known Stokes law, give m=2. Thus,
assuming Stokes law applies to the particle terminal veloc-
ity, Equation 5 predicts that the buoyant density will scale
inversely with the particle diameter. That is,
d
p w +t t t =--1 ^h (6)
It is evident the laminar flow model under dilute con-
ditions gives m=1. Thus, the velocity gradient within the
inclined channels reduces the value of m=2 to a value of
m=1, greatly improving the separation efficiency based on
particle density. This dependence however falls well short
of what is observed in practice, where m~0.26 to 0.28.
Therefore, we need to consider other factors that contribute
to the separation performance.
We now introduce a useful constitutive model to
describe the hindered slip velocity, us, of a particle in a
suspension containing particles of different density (Asif,
1997 Galvin et al., 1999). That is,
u utf ut
s
p w
p susp
n- 1
t t
t t
==-
-
d n (7)
where ut is the terminal settling velocity of the particle
in water, and f a hindered settling factor, dependent on
p t ,the density of the particle, tsusp ,the density of the
local suspension, and tw ,the density of water, and n, the
Richardson and Zaki exponent. Note for very low particle
Figure 2. Schematic representation of the laminar flow
velocity field and critical condition for particle transport up
through an inclined channel