3884 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
The Functional Performance equation, calculated at
75 microns as the target production size for the circuit in
Appendix A is calculated as follows (Clark et al., 2023):
–75 µm production rate =Circuit tonnage
× ((%–75 µm in circuit product
– (%– 75 µm in circuit feed))
=719.6 t/h × (82.6% – 42.8%) =286.4 t/h
The ball mill power (measured at the pinion) is 9,315 kW
As above, averaging the %+75 µm in the mill feed and
discharge, CSE =76.1%
Using Equation 2, above:
286.4 t/h =9,315 kW × 76.1%
× Mill grinding rate of +75 µm
Mill grinding rate of +75 µm =0.404 t/kWh
The Bond test using a closing screen of 75 microns pro-
vided a grindability of the ore of 1.29 g/rev.
Therefore, the ball mill grinding efficiency is:
0.404 t/kWh /1.29 g/rev =0.313 (t/kWh)/(g/rev)
The Functional Performance equation for this survey is:
286.4 t/h new –75 µm =9,315 kW × 76.1%
× 1.29 g/v × 0.313 (t/kWh)/(g/rev)
APPARENT CUMULATIVE GRINDING
RATES (ACGR) AND RELATIONSHIP TO
FUNCTIONAL PERFORMANCE MILL
GRINDING RATE
When survey data is collected to measure Functional
Performance Equation parameters, the ball mill power
and its feed and product size distributions can also be
used to calculate ball mill Apparent Cumulative Grinding
Rates (ACGR) at each sieve size using the first order rate
Equation 5.
Quantity of plus size
in mill discharge
quantity of plus size
in mill feed e kE #=-(5)
E is the energy input to all the material during each pass
through the mill in kWh/t.
E =Mill kW /t/h of solids through the mill (6)
Values of ‘k’ are calculated in this manner. These provide a
complete ball mill size specific grinding rates model, rep-
resented by a single column matrix of cumulative grinding
rates at each sieve size.
To give an example of the ACGR calculation and plot,
we refer the reader to the paper by Clark, et al. (2023)
for complete ball mill circuit survey data conducted at
the Fekola Mine. In the 2018 baseline survey, the solids
tonnage flow through the ball mill was 3,184 t/h. With
mill power equal to 9,315 kW, E =9,315 /3,184 =2.926
kWh/t. The grinding rates, k, at each mesh size, given the
mill feed (cyclone underflow) size distribution and the mill
discharge (including scats) size distribution, are calculated
using Equation 5 and are shown in Table 1 and Figure 3.
These are correctly designated as “apparent” cumulative
grinding rates as the underlying assumption is plug flow of
all particle sizes and their equal exposure to energy input
as they pass through the mill. However, plant testing has
recently shown that coarser particles are retained in a ball
mill longer than finer ones (McIvor et al., 2021). In addi-
tion to being able to reliably mathematically model the ball
mill with ACGRs, the shape of the ACGR curve (Figure 3)
can provide other mill performance insights such as the
appropriateness of the media size in use.
This method of modelling ball mill grinding was first
proposed by Roberts (1950) and verified by Bowdish
(1960) using time rather than energy-based grinding rates.
It has been and now continues to be used by a growing
number of investigators (Finch &Ramirez-Castro, 1981
Laplante et al., 1987), including its extension into stirred
Table 1. Apparent cumulative grinding rate (by size class) for
Fekola ball mill, 2018 survey
Microns
Ball Mill Feed
Cum. Retained.
(t/h)
Ball Mill Disch.
Cum. Retained.
(t/h)
ACGR (‘k’)
(t/kWh)
9500 8.58 0.64 0.8852
6300 38.71 2.49 0.9373
4750 71.72 19.67 0.4422
3350 104.77 32.44 0.4007
2360 141.74 49.82 0.3573
1700 170.82 65.02 0.3301
1180 220.43 92.08 0.2983
850 281.04 131.32 0.2600
600 390.95 213.90 0.2061
425 579.57 369.14 0.1542
300 862.97 612.73 0.1170
212 1239.87 954.12 0.0895
150 1761.82 1449.45 0.0667
106 2234.33 1918.20 0.0521
75 2569.42 2284.41 0.0402
53 2735.98 2484.70 0.0329
38 2836.59 2614.88 0.0278
pan 3183.62 3183.62 0.0000
The Functional Performance equation, calculated at
75 microns as the target production size for the circuit in
Appendix A is calculated as follows (Clark et al., 2023):
–75 µm production rate =Circuit tonnage
× ((%–75 µm in circuit product
– (%– 75 µm in circuit feed))
=719.6 t/h × (82.6% – 42.8%) =286.4 t/h
The ball mill power (measured at the pinion) is 9,315 kW
As above, averaging the %+75 µm in the mill feed and
discharge, CSE =76.1%
Using Equation 2, above:
286.4 t/h =9,315 kW × 76.1%
× Mill grinding rate of +75 µm
Mill grinding rate of +75 µm =0.404 t/kWh
The Bond test using a closing screen of 75 microns pro-
vided a grindability of the ore of 1.29 g/rev.
Therefore, the ball mill grinding efficiency is:
0.404 t/kWh /1.29 g/rev =0.313 (t/kWh)/(g/rev)
The Functional Performance equation for this survey is:
286.4 t/h new –75 µm =9,315 kW × 76.1%
× 1.29 g/v × 0.313 (t/kWh)/(g/rev)
APPARENT CUMULATIVE GRINDING
RATES (ACGR) AND RELATIONSHIP TO
FUNCTIONAL PERFORMANCE MILL
GRINDING RATE
When survey data is collected to measure Functional
Performance Equation parameters, the ball mill power
and its feed and product size distributions can also be
used to calculate ball mill Apparent Cumulative Grinding
Rates (ACGR) at each sieve size using the first order rate
Equation 5.
Quantity of plus size
in mill discharge
quantity of plus size
in mill feed e kE #=-(5)
E is the energy input to all the material during each pass
through the mill in kWh/t.
E =Mill kW /t/h of solids through the mill (6)
Values of ‘k’ are calculated in this manner. These provide a
complete ball mill size specific grinding rates model, rep-
resented by a single column matrix of cumulative grinding
rates at each sieve size.
To give an example of the ACGR calculation and plot,
we refer the reader to the paper by Clark, et al. (2023)
for complete ball mill circuit survey data conducted at
the Fekola Mine. In the 2018 baseline survey, the solids
tonnage flow through the ball mill was 3,184 t/h. With
mill power equal to 9,315 kW, E =9,315 /3,184 =2.926
kWh/t. The grinding rates, k, at each mesh size, given the
mill feed (cyclone underflow) size distribution and the mill
discharge (including scats) size distribution, are calculated
using Equation 5 and are shown in Table 1 and Figure 3.
These are correctly designated as “apparent” cumulative
grinding rates as the underlying assumption is plug flow of
all particle sizes and their equal exposure to energy input
as they pass through the mill. However, plant testing has
recently shown that coarser particles are retained in a ball
mill longer than finer ones (McIvor et al., 2021). In addi-
tion to being able to reliably mathematically model the ball
mill with ACGRs, the shape of the ACGR curve (Figure 3)
can provide other mill performance insights such as the
appropriateness of the media size in use.
This method of modelling ball mill grinding was first
proposed by Roberts (1950) and verified by Bowdish
(1960) using time rather than energy-based grinding rates.
It has been and now continues to be used by a growing
number of investigators (Finch &Ramirez-Castro, 1981
Laplante et al., 1987), including its extension into stirred
Table 1. Apparent cumulative grinding rate (by size class) for
Fekola ball mill, 2018 survey
Microns
Ball Mill Feed
Cum. Retained.
(t/h)
Ball Mill Disch.
Cum. Retained.
(t/h)
ACGR (‘k’)
(t/kWh)
9500 8.58 0.64 0.8852
6300 38.71 2.49 0.9373
4750 71.72 19.67 0.4422
3350 104.77 32.44 0.4007
2360 141.74 49.82 0.3573
1700 170.82 65.02 0.3301
1180 220.43 92.08 0.2983
850 281.04 131.32 0.2600
600 390.95 213.90 0.2061
425 579.57 369.14 0.1542
300 862.97 612.73 0.1170
212 1239.87 954.12 0.0895
150 1761.82 1449.45 0.0667
106 2234.33 1918.20 0.0521
75 2569.42 2284.41 0.0402
53 2735.98 2484.70 0.0329
38 2836.59 2614.88 0.0278
pan 3183.62 3183.62 0.0000