XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3685
The data sets include from small mills to the biggest
mills: nowadays drawing up to 22,800 kW for a SAG
mill of 40×25 feet and up to 21,500 kW for a Ball Mill of
28×44 feet both in Peru.
The main factors influencing the power draw, such as
ball load and critical speed percent, also vary considerably
from 0 to 48%, and from 48 to 89%, respectively.
Therefore, the following discussion focuses on some of
the more well-known power draw models.
F. Bond (1961) proposed a model to determine power
draw for ball mills which is expressed in Equation 1:
..937 .1 m AJzc Jhc1 7 33 0
2
0 .3
p 9 10
2
c
tbLD =--^1 m (1)
where:
mp =Power draw at pinion, kW
A =Constant, 1.00 for overflow discharge and 1.16
for grate discharge
J =Ball load, fraction
c z =Critical speed, fraction
b t =Ball density, t/m3
L =Internal mill length, m
D =Internal mill diameter, m
If it is applied the Bond equation (1) for the 67 data sets
from Table 1, it is possible to see from Figure 1 that the
model under predicts the actual power draw.
The Bond model does not apply to SAG and AG mills,
so it is limited to ball mills, however it has a systematic error
as the mill gets larger.
L. Austin (1990) proposed a model to determine power
draw for SAG mills expressed in Equation 2:
.03
.6Jb .1
m D L Jh^1 Wc
W
10.6 1
0
2
0
.5
p B
s
c
s
2
9 10 c
f
t
t
=--
+--ctb
^1
c1 mzc
hc
m
mJ
(2)
where:
mp =Power draw at pinion, kW
D =Internal mill diameter, m
L =Internal mill length, m
J =Total load, fraction
B f =Porosity of the charge, fraction (taken as 0.3)
Wc =Solids, fraction (taken as 0.8)
Jb =Ball load, fraction
s t =Ore density, t/m3
b t =Ball density, t/m3
c z =Critical speed, fraction
Note that the Austin model, in equation 2, does not con-
sider power for conical ends however according to Alex
Doll (Procemin 2013), Austin proposed a geometric factor
for use in cone-ended mills. Alex Doll’s experience from fit-
ting this model to published surveys suggests that Austin’s
geometric factor adds too much power and proposes that
5% extra power should be added for better accuracy.
Figure 2 shows the parity graph of 108 industrial data sets
from Table 1, versus the Austin model by equation 2, plus
5%.
Furthermore, the Austin model does not apply for ball
mills (mostly overflow mills), in which the over prediction
is much more pronounced.
Note that the author’s observation, that Bond model
under predicts for ball mills, and Austin model over pre-
dicts for SAG mills, was also pointed out by other authors,
as demonstrated in Figure 3.
METHODOLOGY
To overcome the bias of previous equations the author pro-
poses an enhanced equation, that applies for grate and over-
flow discharge mills (SAG, AG and Ball Mills), as shown in
the equation (3):
Table 1. Range of industrial data sets for tumbling mills
Measurement
SAG (108 data sets) AG (16 data sets) Ball (67 data sets)
Min Max Min Max Min Max
Diameter inside liners (m) 1.70 12.05 1.75 12.02 0.85 8.33
Belly length inside liners (m) 0.45 9.76 0.45 6.10 1.52 13.11
Ball load (%)3 40 0 0 20 48
Total load (%)7.0 40.7 10.0 42.5 20 48
Speed (rpm) 7.6 26.8 9.6 24.9 10.1 32.6
Critical speed (%)48.0 89.0 72.0 79.0 60.5 86.5
Ore density (t/m3) 2.52 4.10 2.60 4.60 2.57 4.50
Aspect ratio (D/L) 0.5 3.9 1.0 3.9 0.6 1.2
Gross power (kW) 10 22,800 9 11,189 6 21,500
The data sets include from small mills to the biggest
mills: nowadays drawing up to 22,800 kW for a SAG
mill of 40×25 feet and up to 21,500 kW for a Ball Mill of
28×44 feet both in Peru.
The main factors influencing the power draw, such as
ball load and critical speed percent, also vary considerably
from 0 to 48%, and from 48 to 89%, respectively.
Therefore, the following discussion focuses on some of
the more well-known power draw models.
F. Bond (1961) proposed a model to determine power
draw for ball mills which is expressed in Equation 1:
..937 .1 m AJzc Jhc1 7 33 0
2
0 .3
p 9 10
2
c
tbLD =--^1 m (1)
where:
mp =Power draw at pinion, kW
A =Constant, 1.00 for overflow discharge and 1.16
for grate discharge
J =Ball load, fraction
c z =Critical speed, fraction
b t =Ball density, t/m3
L =Internal mill length, m
D =Internal mill diameter, m
If it is applied the Bond equation (1) for the 67 data sets
from Table 1, it is possible to see from Figure 1 that the
model under predicts the actual power draw.
The Bond model does not apply to SAG and AG mills,
so it is limited to ball mills, however it has a systematic error
as the mill gets larger.
L. Austin (1990) proposed a model to determine power
draw for SAG mills expressed in Equation 2:
.03
.6Jb .1
m D L Jh^1 Wc
W
10.6 1
0
2
0
.5
p B
s
c
s
2
9 10 c
f
t
t
=--
+--ctb
^1
c1 mzc
hc
m
mJ
(2)
where:
mp =Power draw at pinion, kW
D =Internal mill diameter, m
L =Internal mill length, m
J =Total load, fraction
B f =Porosity of the charge, fraction (taken as 0.3)
Wc =Solids, fraction (taken as 0.8)
Jb =Ball load, fraction
s t =Ore density, t/m3
b t =Ball density, t/m3
c z =Critical speed, fraction
Note that the Austin model, in equation 2, does not con-
sider power for conical ends however according to Alex
Doll (Procemin 2013), Austin proposed a geometric factor
for use in cone-ended mills. Alex Doll’s experience from fit-
ting this model to published surveys suggests that Austin’s
geometric factor adds too much power and proposes that
5% extra power should be added for better accuracy.
Figure 2 shows the parity graph of 108 industrial data sets
from Table 1, versus the Austin model by equation 2, plus
5%.
Furthermore, the Austin model does not apply for ball
mills (mostly overflow mills), in which the over prediction
is much more pronounced.
Note that the author’s observation, that Bond model
under predicts for ball mills, and Austin model over pre-
dicts for SAG mills, was also pointed out by other authors,
as demonstrated in Figure 3.
METHODOLOGY
To overcome the bias of previous equations the author pro-
poses an enhanced equation, that applies for grate and over-
flow discharge mills (SAG, AG and Ball Mills), as shown in
the equation (3):
Table 1. Range of industrial data sets for tumbling mills
Measurement
SAG (108 data sets) AG (16 data sets) Ball (67 data sets)
Min Max Min Max Min Max
Diameter inside liners (m) 1.70 12.05 1.75 12.02 0.85 8.33
Belly length inside liners (m) 0.45 9.76 0.45 6.10 1.52 13.11
Ball load (%)3 40 0 0 20 48
Total load (%)7.0 40.7 10.0 42.5 20 48
Speed (rpm) 7.6 26.8 9.6 24.9 10.1 32.6
Critical speed (%)48.0 89.0 72.0 79.0 60.5 86.5
Ore density (t/m3) 2.52 4.10 2.60 4.60 2.57 4.50
Aspect ratio (D/L) 0.5 3.9 1.0 3.9 0.6 1.2
Gross power (kW) 10 22,800 9 11,189 6 21,500