XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3725
voidage (ranging from 35.61% to 40.92%), across all values
of Cs and Jt. Since the range of variation in the diameter of
spherical particles has a significant impact on the bed void-
age, it’s crucial to select a statistical index that represents
the dimensional dispersion of the balls to analyze these
results effectively. To this end, the mean absolute deviation
(MAD) of the balls’ diameter was chosen as a suitable mea-
sure of dispersion. MAD, a dimensionless statistical index,
quantifies how much the values in a dataset are likely to
deviate from their mean (Christine et al., 2007). Therefore,
the following equation, which calculates the MAD of the
balls’ diameter, was utilized:
MAD n
D D
i 0
n
i m =
-
=
/
(17)
where Di represents the diameter of each ball in a specific
BSD (in millimeters), Dm is the mean diameter of the balls
(in millimeters), and n denotes the total number of balls in
the distribution.
Table 5 presents the Mean Absolute Deviation (MAD)
of the balls’ diameter for Bond’s proposed BSDs. BSD1
exhibits the maximum MAD, corresponding to the mini-
mum voidage, while BSD7 shows the minimum MAD,
associated with the maximum voidage. These findings
highlight a notable correlation between the voidage of balls
and the dispersion of their diameter: higher MAD values
lead to a decrease in voidage and a denser load. This is
because smaller balls can fill the gaps between larger balls
more effectively in systems with greater dimensional disper-
sion. Similar observations have been reported previously,
indicating that the density of a multi-sized particle system
increases with a broader particle size distribution, attrib-
uted to increased dimensional dispersion of balls (Sohn and
Moreland, 1968 Bierwagen and Sanders, 1974 Desmond
and Weeks, 2014).
Figures 7a, 7b and 7c depict the rise in dynamic void-
age of the grinding media relative to its static voidage with
increasing mill rotation speed for fractional mill fillings of
15%, 30%, and 45%, respectively. BSD7 exhibits the high-
est percentage increase in voidage, while BSD1 shows the
lowest. This difference is attributed to small balls becoming
trapped between moving ball layers, reducing displacement
amplitude between them. Using the conducted experi-
ments (Table 3) and the calculated MAD for various BSDs
(Table 5), we aimed to create a comprehensive equation,
beyond Eqs. 10 to 16, to predict dynamic voidage for dif-
ferent BSDs based on ball diameter MAD. Thus, we devised
the following quadratic equation through multiple regres-
sion analysis in Minitab software:
Ø =48.309 – 0.4511 Jt +0.11010 Cs
– 0.3459 MAD +0.006095 Jt2
+0.00332 MAD2 – 0.001102 JtCs
(18)
+0.000982 JtMAD – 0.001763 CsMAD
where Ø represents the dynamic voidage (%),Jt stands for
the fractional mill filling (%),Cs denotes the fractional
speed (%),and MAD refers to the mean absolute deviation
of the balls’ diameter. In addition to the factors outlined
in Eqs. 10 to 16 (Jt, Cs, Jt2, and JtCs), the MAD of the
balls’ diameter, its quadratic term (MAD2), the interaction
of MAD and fractional mill filling (JtMAD), and the inter-
action of rotating speed and MAD (CsMAD) are deemed
significant for predicting dynamic voidage.
The statistical results from the Minitab software,
including the p-value and R-squared of the developed
regression model, are depicted in Figure 8. The reported
p-value, significantly below 0.05, confirms the significance
of the model. Moreover, the adjusted R-squared of 99.21%
suggests a strong relationship between the model and the
dependent variables, indicating a good agreement between
observed and predicted responses.
Validity and Performance of Models
To represent the diverse ball size distribution in industrial
ball mills, we selected Bond’s first proposed ball size dis-
tribution (BSD1) as it closely mirrors actual operational
conditions. This choice was validated against data from the
Sarcheshmeh copper complex processing plant, where the
static voidage of balls in a 5 m diameter ball mill was mea-
sured at 36%, aligning closely with BSD1s static voidage of
35.6% (Hesami, 2016 Table 6).
Figure 9a shows the predicted dynamic voidage of balls
using Eq. 10 for BSD1, indicating a strong agreement with
measured values. Additionally, Figure 9b confirms the effec-
tiveness of Eq. 18 in estimating the dynamic voidage across
all Bond’s proposed BSDs. These findings validate the accu-
racy of Eqs. 10 and 18 in predicting grinding media void-
age, enabling the calculation of dynamic charge density by
substituting their output into Eqs. 2 and 4.
Table 5. Mean Absolute Deviation (MAD) of Balls’ Diameter for Bond’s Proposed BSDs
Parameter BSD
1 BSD
2 BSD
3 BSD
4 BSD
5 BSD
6 BSD
7
MAD 24.621 22.071 18.460 15.622 11.773 8.972 5.660
voidage (ranging from 35.61% to 40.92%), across all values
of Cs and Jt. Since the range of variation in the diameter of
spherical particles has a significant impact on the bed void-
age, it’s crucial to select a statistical index that represents
the dimensional dispersion of the balls to analyze these
results effectively. To this end, the mean absolute deviation
(MAD) of the balls’ diameter was chosen as a suitable mea-
sure of dispersion. MAD, a dimensionless statistical index,
quantifies how much the values in a dataset are likely to
deviate from their mean (Christine et al., 2007). Therefore,
the following equation, which calculates the MAD of the
balls’ diameter, was utilized:
MAD n
D D
i 0
n
i m =
-
=
/
(17)
where Di represents the diameter of each ball in a specific
BSD (in millimeters), Dm is the mean diameter of the balls
(in millimeters), and n denotes the total number of balls in
the distribution.
Table 5 presents the Mean Absolute Deviation (MAD)
of the balls’ diameter for Bond’s proposed BSDs. BSD1
exhibits the maximum MAD, corresponding to the mini-
mum voidage, while BSD7 shows the minimum MAD,
associated with the maximum voidage. These findings
highlight a notable correlation between the voidage of balls
and the dispersion of their diameter: higher MAD values
lead to a decrease in voidage and a denser load. This is
because smaller balls can fill the gaps between larger balls
more effectively in systems with greater dimensional disper-
sion. Similar observations have been reported previously,
indicating that the density of a multi-sized particle system
increases with a broader particle size distribution, attrib-
uted to increased dimensional dispersion of balls (Sohn and
Moreland, 1968 Bierwagen and Sanders, 1974 Desmond
and Weeks, 2014).
Figures 7a, 7b and 7c depict the rise in dynamic void-
age of the grinding media relative to its static voidage with
increasing mill rotation speed for fractional mill fillings of
15%, 30%, and 45%, respectively. BSD7 exhibits the high-
est percentage increase in voidage, while BSD1 shows the
lowest. This difference is attributed to small balls becoming
trapped between moving ball layers, reducing displacement
amplitude between them. Using the conducted experi-
ments (Table 3) and the calculated MAD for various BSDs
(Table 5), we aimed to create a comprehensive equation,
beyond Eqs. 10 to 16, to predict dynamic voidage for dif-
ferent BSDs based on ball diameter MAD. Thus, we devised
the following quadratic equation through multiple regres-
sion analysis in Minitab software:
Ø =48.309 – 0.4511 Jt +0.11010 Cs
– 0.3459 MAD +0.006095 Jt2
+0.00332 MAD2 – 0.001102 JtCs
(18)
+0.000982 JtMAD – 0.001763 CsMAD
where Ø represents the dynamic voidage (%),Jt stands for
the fractional mill filling (%),Cs denotes the fractional
speed (%),and MAD refers to the mean absolute deviation
of the balls’ diameter. In addition to the factors outlined
in Eqs. 10 to 16 (Jt, Cs, Jt2, and JtCs), the MAD of the
balls’ diameter, its quadratic term (MAD2), the interaction
of MAD and fractional mill filling (JtMAD), and the inter-
action of rotating speed and MAD (CsMAD) are deemed
significant for predicting dynamic voidage.
The statistical results from the Minitab software,
including the p-value and R-squared of the developed
regression model, are depicted in Figure 8. The reported
p-value, significantly below 0.05, confirms the significance
of the model. Moreover, the adjusted R-squared of 99.21%
suggests a strong relationship between the model and the
dependent variables, indicating a good agreement between
observed and predicted responses.
Validity and Performance of Models
To represent the diverse ball size distribution in industrial
ball mills, we selected Bond’s first proposed ball size dis-
tribution (BSD1) as it closely mirrors actual operational
conditions. This choice was validated against data from the
Sarcheshmeh copper complex processing plant, where the
static voidage of balls in a 5 m diameter ball mill was mea-
sured at 36%, aligning closely with BSD1s static voidage of
35.6% (Hesami, 2016 Table 6).
Figure 9a shows the predicted dynamic voidage of balls
using Eq. 10 for BSD1, indicating a strong agreement with
measured values. Additionally, Figure 9b confirms the effec-
tiveness of Eq. 18 in estimating the dynamic voidage across
all Bond’s proposed BSDs. These findings validate the accu-
racy of Eqs. 10 and 18 in predicting grinding media void-
age, enabling the calculation of dynamic charge density by
substituting their output into Eqs. 2 and 4.
Table 5. Mean Absolute Deviation (MAD) of Balls’ Diameter for Bond’s Proposed BSDs
Parameter BSD
1 BSD
2 BSD
3 BSD
4 BSD
5 BSD
6 BSD
7
MAD 24.621 22.071 18.460 15.622 11.773 8.972 5.660