874 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
process variables for fitting the empirical model are consid-
ered based on the experimental data and are given below:
0.25 ≤ J ≤ 0.4 0.415 ≤ c ≤ 0.79 and 0.2 ≤ Q' ≤ 17.8
The dynamic holdup of the ball mill is modelled as a func-
tion of the independent variables such as slurry flow rate,
ball filling and slurry concentration. The following model
is obtained to predict the holdup (H) of the ball mill as a
fraction of mill volume:
..7
'
H
J c Q Ql
J c Q'
0 2 1
8
.59
..5
1
0
2
1
0 59 0
1 1
1
a a
a a2Q
=
+
+
l
)(19)
where, J, c, and Q' are ball filling, solids concentration in
the slurry and modified slurry flow rate (slurry flow rate
scaled by mill volume), respectively. α1 and α2 are tunable
parameters that vary from system to system.
Figure 2 shows the comparison of experimental data
with the model for various cases. In the regime defined
by 0.2Q'1.7, the model is linear in terms of the scaled
flow rate Q' whereas in the regime defined by Q'8, the
volumetric holdup is proportional to Q' 0.5. This is consis-
tent with the observations in the literature (Songfack and
Rajamani 1999 Kinneberg and Herbst 1984 Austin et al.,
1989 Klimpel et al., 1989). It is observed that a regime
shift happens somewhere in 1.7Q'8 as the dependency
of volumetric holdup on scaled flow rate changes from
linear to Q' 0.5. The tuned model parameters are provided
in Table 1.
Grinding Circuit Optimization
Grinding is usually carried out in a closed or open circuit.
A typical closed circuit grinding operation (Figure 3) is
employed to demonstrate utility of the grinding model
presented above. The circuit consists of a ball mill, a sump,
a pump, and a system of cyclone separators (cyclone). In
this case, the slurry from the sump is subjected to classifi-
cation at the cyclone and the underflow (oversize) is fed to
the ball mill. After grinding, the slurry from the ball mill
is sent back to the sump where it mixes with fresh feed,
FF and sump water, W. The sump is treated as a perfect
mixer for the fresh feed and the ball mill output streams.
A centrifugal pump is used to transport the slurry from
the sump to the cyclone and provide the necessary pres-
sure head required for the classification. The pump power
is assumed to be proportional to volumetric flow rate and
the proportionality constant is calculated based on pipe
friction for a NPS 3/4 steel pipe of length 5 m. Cyclone
Figure 2. Comparison of predicted holdup with experimental data (Kinneberg et al., 1984 Songfack and Rajamani 1999 Shi
2016). A single model is fitted for Shi 2016, due to lack of data points for each steady state. For these systems, J=0.25 and 0.42
c 0.79
Table 1. Numerical values of parameters in the Holdup model
Dataset α1 α2
Songfack and Rajamani 1999 0.7 0.06
Kinneberg et al.1984 0.15 0.24
Shi 2016 0.17 0.47
process variables for fitting the empirical model are consid-
ered based on the experimental data and are given below:
0.25 ≤ J ≤ 0.4 0.415 ≤ c ≤ 0.79 and 0.2 ≤ Q' ≤ 17.8
The dynamic holdup of the ball mill is modelled as a func-
tion of the independent variables such as slurry flow rate,
ball filling and slurry concentration. The following model
is obtained to predict the holdup (H) of the ball mill as a
fraction of mill volume:
..7
'
H
J c Q Ql
J c Q'
0 2 1
8
.59
..5
1
0
2
1
0 59 0
1 1
1
a a
a a2Q
=
+
+
l
)(19)
where, J, c, and Q' are ball filling, solids concentration in
the slurry and modified slurry flow rate (slurry flow rate
scaled by mill volume), respectively. α1 and α2 are tunable
parameters that vary from system to system.
Figure 2 shows the comparison of experimental data
with the model for various cases. In the regime defined
by 0.2Q'1.7, the model is linear in terms of the scaled
flow rate Q' whereas in the regime defined by Q'8, the
volumetric holdup is proportional to Q' 0.5. This is consis-
tent with the observations in the literature (Songfack and
Rajamani 1999 Kinneberg and Herbst 1984 Austin et al.,
1989 Klimpel et al., 1989). It is observed that a regime
shift happens somewhere in 1.7Q'8 as the dependency
of volumetric holdup on scaled flow rate changes from
linear to Q' 0.5. The tuned model parameters are provided
in Table 1.
Grinding Circuit Optimization
Grinding is usually carried out in a closed or open circuit.
A typical closed circuit grinding operation (Figure 3) is
employed to demonstrate utility of the grinding model
presented above. The circuit consists of a ball mill, a sump,
a pump, and a system of cyclone separators (cyclone). In
this case, the slurry from the sump is subjected to classifi-
cation at the cyclone and the underflow (oversize) is fed to
the ball mill. After grinding, the slurry from the ball mill
is sent back to the sump where it mixes with fresh feed,
FF and sump water, W. The sump is treated as a perfect
mixer for the fresh feed and the ball mill output streams.
A centrifugal pump is used to transport the slurry from
the sump to the cyclone and provide the necessary pres-
sure head required for the classification. The pump power
is assumed to be proportional to volumetric flow rate and
the proportionality constant is calculated based on pipe
friction for a NPS 3/4 steel pipe of length 5 m. Cyclone
Figure 2. Comparison of predicted holdup with experimental data (Kinneberg et al., 1984 Songfack and Rajamani 1999 Shi
2016). A single model is fitted for Shi 2016, due to lack of data points for each steady state. For these systems, J=0.25 and 0.42
c 0.79
Table 1. Numerical values of parameters in the Holdup model
Dataset α1 α2
Songfack and Rajamani 1999 0.7 0.06
Kinneberg et al.1984 0.15 0.24
Shi 2016 0.17 0.47