2110 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
/tan
K
D
D
D
L
Dc
D
R g
V
i
f
f
2h
1
180 10
.2
.27
.82
.72
max
ci ag
c
u
f
si f
m
c
c
o t
v
v
0.567 0.908
0.127 0.182 0
0 2 0.016
0.868
1
0
#b
#`cos`
a
t
t
nw
n
=
-
-
c
c
e
d
^t
c ^
e
^1
c
m
m
l
jj
n
h o
m
h2
o
m
(10)
The multi-component mathematical model developed
in this study evolved from the concept of the single aver-
age density-based non-dimensional approach models. The
interactions of the fully liberated particles in the feed mix-
ture are considered through macroscopic hydrodynamic
equations. To satisfy the data requirement for the mathe-
matical model development, data from historical databases
on natural ores where analyses was conducted to obtained
deportment of various components in the feed to the prod-
uct streams [1,34] and from studies with artificial mixtures
was used. As the literature does not have much data repre-
senting the multi-component particle separations, labora-
tory experiments were performed using 2, 3, and 4-inch
hydrocyclone at various operating conditions and compu-
tational simulations were carried out for selected conditions
as presented in Figure 1 [4,5]. Both the laboratory experi-
ments and computational simulations indicated that there
was interaction between the multi-density particles in the
hydrocyclone during separation.
The interactions observed from the experimental and
computational simulations results provided motivation to
include slurry macroscopic behaviour, which shows the
potential to improve the prediction of the multicomponent
performance indices in the model.
Considering all the practical hydrocyclone models
developed by other researchers, together with the current
state-of-art models and latest test results on bi-component
&multi-components, the new set of the improved multi-
component model is fitted. The relationships were inves-
tigated using the EXCEL Solver using the multiple linear
regression approach. This model is generally used when
the mathematical equations have more than one predictor.
The least-square regression approach estimates the model
coefficients by minimizing the sum of square error between
the experimental and predicted values. The model fit statis-
tics such as R2 and relative error are utilized to understand
how close the predicted value is to the given data set. R2
is a measure of the variability in the response explained by
the model. A second important measure of model fit is the
standard error, which is the standard deviation of its sam-
pling distribution. To account for the input variable holistic
effect of the material and machine size-dependent constants
‘Ki’ is introduced. To minimize the variability, a few empiri-
cal constants for geometrical parameters are directly takewn
from the (Narasimha et al., 2012) equation [29]. The fol-
lowing mathematical models are obtained representing the
multi-component classification:
The cut-size equation presented by Narasimha et
al., 2014 [2] was modified by including the parameters
Figure 6.(a) Comparison of multi-component model predictions with experimental data, (b) Validation of multi-component
cut-size model predictions with experimental data
/tan
K
D
D
D
L
Dc
D
R g
V
i
f
f
2h
1
180 10
.2
.27
.82
.72
max
ci ag
c
u
f
si f
m
c
c
o t
v
v
0.567 0.908
0.127 0.182 0
0 2 0.016
0.868
1
0
#b
#`cos`
a
t
t
nw
n
=
-
-
c
c
e
d
^t
c ^
e
^1
c
m
m
l
jj
n
h o
m
h2
o
m
(10)
The multi-component mathematical model developed
in this study evolved from the concept of the single aver-
age density-based non-dimensional approach models. The
interactions of the fully liberated particles in the feed mix-
ture are considered through macroscopic hydrodynamic
equations. To satisfy the data requirement for the mathe-
matical model development, data from historical databases
on natural ores where analyses was conducted to obtained
deportment of various components in the feed to the prod-
uct streams [1,34] and from studies with artificial mixtures
was used. As the literature does not have much data repre-
senting the multi-component particle separations, labora-
tory experiments were performed using 2, 3, and 4-inch
hydrocyclone at various operating conditions and compu-
tational simulations were carried out for selected conditions
as presented in Figure 1 [4,5]. Both the laboratory experi-
ments and computational simulations indicated that there
was interaction between the multi-density particles in the
hydrocyclone during separation.
The interactions observed from the experimental and
computational simulations results provided motivation to
include slurry macroscopic behaviour, which shows the
potential to improve the prediction of the multicomponent
performance indices in the model.
Considering all the practical hydrocyclone models
developed by other researchers, together with the current
state-of-art models and latest test results on bi-component
&multi-components, the new set of the improved multi-
component model is fitted. The relationships were inves-
tigated using the EXCEL Solver using the multiple linear
regression approach. This model is generally used when
the mathematical equations have more than one predictor.
The least-square regression approach estimates the model
coefficients by minimizing the sum of square error between
the experimental and predicted values. The model fit statis-
tics such as R2 and relative error are utilized to understand
how close the predicted value is to the given data set. R2
is a measure of the variability in the response explained by
the model. A second important measure of model fit is the
standard error, which is the standard deviation of its sam-
pling distribution. To account for the input variable holistic
effect of the material and machine size-dependent constants
‘Ki’ is introduced. To minimize the variability, a few empiri-
cal constants for geometrical parameters are directly takewn
from the (Narasimha et al., 2012) equation [29]. The fol-
lowing mathematical models are obtained representing the
multi-component classification:
The cut-size equation presented by Narasimha et
al., 2014 [2] was modified by including the parameters
Figure 6.(a) Comparison of multi-component model predictions with experimental data, (b) Validation of multi-component
cut-size model predictions with experimental data