XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2105
to define the relative viscosity model effect as a function of
solids concentration as noted by Mangesana et al., (2008)
[16] and Narasimha et al., (2005) [17].
The dimensionless slurry viscosity for the mixture
accounting multi-components developed byPadhi et
al.,2019 [19] is used as shown in equation 1.
**kl and
f L
f
f H- lw
f h@ h z
h h
h z
h h
=
-
+
-
=6 6h@1.5 (1)
Further, it is expanded to describe the viscosities of the
lighter and heavier components in a lighter slurry compo-
nent as shown in Equation 2 and 3, respectively.
*.62 k1
L wh
0
w h
hL atL
=-
+d1
^n (2)
*.62 k
L wh H
0
w
tL H
2 h
h a
=-
++-1.55 d1 ^n (3)
where, ηrmix, ηlw or ηHr and ηH–lw represents the relative
viscosity for mixture, lighter component slurry viscosity,
and heavier component viscosity in a lighter component
slurry, respectively. The interaction parameter is defined by
the total intrinsic viscosity, [η] by the lighter and heavier
particle slurry which represents the light and heavy particle
solute concentration effect on the mixture slurry viscosity
in total, given in Equation 4.
**kl and
f L
f
f H- lw
f h@ h z
h h
h z
h h
=
-
+
-
=6 6h@1.5 (4)
k', a function of intrinsic viscosity is incorporated in k2 =k'
*k'2, and the power ‘1.5’ is fitted by regression analysis, as
an exponential fitting constant. k1, represents the interac-
tion parameter between lighter and heavier particle as the
total intrinsic viscosity the bi-component particles’ solute
concentration effect on the overall mixture slurry viscosity.
Similarly, k', and k'2 are constants representing the fraction
F .39
38
0 (fines below 38 microns).
Relative Setting Velocities
The suspension hindered settling is influenced by the sol-
ids concentrations in mono and poly-size systems for sin-
gle-density [18–21]. However, as discussed earlier, in the
bi-density particle system, the density of particles have a
significant influence on the settling phenomenon. For a
suspension of particles having mixed sizes, the total con-
centration of the particles controls the sedimentation in the
case of a single density system [22]. Richardson and Meikle
1961 [23] studied the sedimentation of suspension consist-
ing of a two-component mixture using glass ballotine and
polystyrene particles in a 22% by mass ethanol-water mix-
ture. The settling of two-component can arise due to the
buoyancy force and the flow pattern in the proximity of the
particle. The effect of flow pattern is considered through
the solid concentration however, the buoyant force may
be altered for the independent component in the mixture.
In a review by Zeidan et al. 2003 [24], the Richardson &
Zaki [25] coupled with Masliyah’s [26] interaction model
corrections is proven as the best fit model of bimodal sys-
tems. This equation is further evaluated using new data
from experiments performed for this study. The final form
of the mixture – bimodal –polydisperse hindered settling
equation is given in Equation 5.
ga
*v
d
d 18n
.7
i
f
f i i
j j j susp
2 2
2
a tsusph
a t
zhn =
--
--
-
^1
^
^1
h^t
h^ti
h
H (5)
The volume fraction, density, and diameter of each com-
ponent account for the interaction between the particles
at different proportions. Whereas the Richardson &Zaki
[25], formulation accounts for the solids concentration
effect in total based on the ‘n’ values. Richardson &Zaki
(1954) [25] proposes the terminal velocity as a function of
the variables given in Equation 6.
f ,Re, v
vs
D
d
o
s z= b l (6)
where, Re is the Reynolds number and ds/D is the ratio of
particles diameter to vessel and at high solids, the concen-
tration (ϕ) is another key factor that account for the viscous
regimes where resistance due to friction is more significant.
Hence, these factors are defined in terms of n for different
Reynolds number ranges. To represent the effect of high
solids the ϕ factor is considered, and the settling velocity
was given as shown in Equations 7 and 8.
V
V
1
o
s zhn =-^(7)
Re, n f D
d
s =b l (8)
RESULTS AND DISCUSSIONS
Multicomponent Hydrocyclone Experiments
The initial hydrocyclone classification experiments were
Performed for pure components to analyse the mono-den-
sity separation behaviour as presented in most of the litera-
ture reviewed. The two-component density systems’ actual
grade efficiencies obtained in this work exhibited the stan-
dard S-shaped sigmoidal curve. Figure 3(a) represents the
mixture classification test for the 80:20 (silica: magnetite)
in a 3-inch hydrocyclone.
The efficiency curve for the low-density component (sil-
ica) was at the coarse size end while the heavier component
to define the relative viscosity model effect as a function of
solids concentration as noted by Mangesana et al., (2008)
[16] and Narasimha et al., (2005) [17].
The dimensionless slurry viscosity for the mixture
accounting multi-components developed byPadhi et
al.,2019 [19] is used as shown in equation 1.
**kl and
f L
f
f H- lw
f h@ h z
h h
h z
h h
=
-
+
-
=6 6h@1.5 (1)
Further, it is expanded to describe the viscosities of the
lighter and heavier components in a lighter slurry compo-
nent as shown in Equation 2 and 3, respectively.
*.62 k1
L wh
0
w h
hL atL
=-
+d1
^n (2)
*.62 k
L wh H
0
w
tL H
2 h
h a
=-
++-1.55 d1 ^n (3)
where, ηrmix, ηlw or ηHr and ηH–lw represents the relative
viscosity for mixture, lighter component slurry viscosity,
and heavier component viscosity in a lighter component
slurry, respectively. The interaction parameter is defined by
the total intrinsic viscosity, [η] by the lighter and heavier
particle slurry which represents the light and heavy particle
solute concentration effect on the mixture slurry viscosity
in total, given in Equation 4.
**kl and
f L
f
f H- lw
f h@ h z
h h
h z
h h
=
-
+
-
=6 6h@1.5 (4)
k', a function of intrinsic viscosity is incorporated in k2 =k'
*k'2, and the power ‘1.5’ is fitted by regression analysis, as
an exponential fitting constant. k1, represents the interac-
tion parameter between lighter and heavier particle as the
total intrinsic viscosity the bi-component particles’ solute
concentration effect on the overall mixture slurry viscosity.
Similarly, k', and k'2 are constants representing the fraction
F .39
38
0 (fines below 38 microns).
Relative Setting Velocities
The suspension hindered settling is influenced by the sol-
ids concentrations in mono and poly-size systems for sin-
gle-density [18–21]. However, as discussed earlier, in the
bi-density particle system, the density of particles have a
significant influence on the settling phenomenon. For a
suspension of particles having mixed sizes, the total con-
centration of the particles controls the sedimentation in the
case of a single density system [22]. Richardson and Meikle
1961 [23] studied the sedimentation of suspension consist-
ing of a two-component mixture using glass ballotine and
polystyrene particles in a 22% by mass ethanol-water mix-
ture. The settling of two-component can arise due to the
buoyancy force and the flow pattern in the proximity of the
particle. The effect of flow pattern is considered through
the solid concentration however, the buoyant force may
be altered for the independent component in the mixture.
In a review by Zeidan et al. 2003 [24], the Richardson &
Zaki [25] coupled with Masliyah’s [26] interaction model
corrections is proven as the best fit model of bimodal sys-
tems. This equation is further evaluated using new data
from experiments performed for this study. The final form
of the mixture – bimodal –polydisperse hindered settling
equation is given in Equation 5.
ga
*v
d
d 18n
.7
i
f
f i i
j j j susp
2 2
2
a tsusph
a t
zhn =
--
--
-
^1
^
^1
h^t
h^ti
h
H (5)
The volume fraction, density, and diameter of each com-
ponent account for the interaction between the particles
at different proportions. Whereas the Richardson &Zaki
[25], formulation accounts for the solids concentration
effect in total based on the ‘n’ values. Richardson &Zaki
(1954) [25] proposes the terminal velocity as a function of
the variables given in Equation 6.
f ,Re, v
vs
D
d
o
s z= b l (6)
where, Re is the Reynolds number and ds/D is the ratio of
particles diameter to vessel and at high solids, the concen-
tration (ϕ) is another key factor that account for the viscous
regimes where resistance due to friction is more significant.
Hence, these factors are defined in terms of n for different
Reynolds number ranges. To represent the effect of high
solids the ϕ factor is considered, and the settling velocity
was given as shown in Equations 7 and 8.
V
V
1
o
s zhn =-^(7)
Re, n f D
d
s =b l (8)
RESULTS AND DISCUSSIONS
Multicomponent Hydrocyclone Experiments
The initial hydrocyclone classification experiments were
Performed for pure components to analyse the mono-den-
sity separation behaviour as presented in most of the litera-
ture reviewed. The two-component density systems’ actual
grade efficiencies obtained in this work exhibited the stan-
dard S-shaped sigmoidal curve. Figure 3(a) represents the
mixture classification test for the 80:20 (silica: magnetite)
in a 3-inch hydrocyclone.
The efficiency curve for the low-density component (sil-
ica) was at the coarse size end while the heavier component