3716 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
models like the Bond model and Morrell E-model are
widely used, while the Morrell C-model and Hogg and
Fuerstenau’s model are recognized as accurate fundamen-
tal and phenomenological models, respectively (Morrell,
1993 Rajamani et al., 2019). In these models, the mill
charge’s bulk density emerges as a significant influential
variable, estimated based on factors such as grinding media
density, ore density, and grinding media voidage (Morrell,
2016 Hilden et al., 2021).
Despite its critical role, determining grinding media
voidage in operational conditions remains challenging.
Presumed to be 40% across all models, this assumption lacks
direct empirical validation. While some studies on particle
bed voidage suggest a 40% value for man-sized spherical
particles under vibration (Yang, 2003), industrial mills
use a mixture of multi-sized particles, complicating void-
age estimation. Moreover, existing models assume constant
voidage during operation, overlooking dynamic changes
influenced by factors like fractional mill filling, rotational
speed, and grinding media size distribution. Recognizing
these limitations, Latchireddi (2002) attempted to predict
dynamic voidage, but mono-sized balls failed to reflect
real-world conditions. Hence, developing an equation to
estimate voidage under diverse operating conditions, con-
sidering realistic ball size distributions (BSD), is imperative.
The initial selection of BSD for the first-filling of
ball mills is typically based on Table 1 proposed by Bond
(1958). However, it’s important to note that the shape and
size distribution of balls can change over time due to vari-
ous factors like impact, erosion, corrosion, and crushing.
Therefore, during the commissioning of mineral processing
plants, the voidage associated with each Bond’s proposed
BSD must be taken into account when formulating equa-
tions for determining the grinding media voidage.
Design of Experiments (DOE) is a valuable method
that enables researchers to plan experiments and analyze the
cause-and-effect relationships between operating param-
eters in a process (Montgomery, 2005). Among the vari-
ous experimental designs, Response Surface Methodology
(RSM) stands out as an efficient tool of DOE. RSM is
extensively applied to establish mathematical relationships
between a dependent parameter and independent variables
(Soleymani Yazdi and Khorram, 2010 Golpayegani and
Abdollahzadeh, 2017).
Utilizing Design of Experiments (DOE) and Response
Surface Methodology (RSM), we aimed to enhance the
accuracy of the Morrel C-model and Hogg and Fuerstenau’s
model by investigating the static and dynamic voidage of
grinding media within ball mills. We employed a three-
level factorial design to model dynamic voidage, focusing
on fractional mill filling and mill rotating speed as input
parameters. Subsequently, a separate model was developed
for each BSD proposed by Bond. These models, along with
a general equation predicting dynamic voidage, were evalu-
ated using an industrial database of ball mills, assessing the
mean absolute percentage error (MAPE) of the models’
prediction results.
DESCRIPTION OF HOGG AND
FUERSTENAU’S MODEL AND MORELL
C-MODEL
To formulate a fundamental power equation for tumbling
mills, it’s essential to understand the motion of the charge
within the mill. Davis (1919) was the pioneer in establish-
ing a relationship between a ball mill’s charge motion and
its power draw, employing the energy balance approach.
Davis’ depiction closely resembles the earliest descriptions
provided by White (1905). According to this description,
particles within the mill move in a constrained manner
along a circular trajectory until reaching a point where
gravitational and centrifugal forces achieve equilibrium.
At this juncture, particles undergo a parabolic free fall
Table 1. The grinding media size distributions recommended by Bond for the initial filling of ball mills during the
commissioning stage (Bond, 1958)
Ball size
(mm)
Percentage Weight
BSD1 BSD2 BSD3 BSD4 BSD5 BSD6 BSD7
115 23.0 0 0 0 0 0 0
100 31.0 23 0 0 0 0 0
90 18.0 34 24 0 0 0 0
75 15.0 21 38 31 0 0 0
65 7.0 12 20.5 39 34 0 0
50 3.8 6.5 11.5 19.0 43.0 40.0 0.0
40 1.7 2.5 4.5 8.0 17.0 45.0 51.0
25 0.5 1.0 1.5 3.0 6.0 15.0 49.0
Sum 100 100 100 100 100 100 100
models like the Bond model and Morrell E-model are
widely used, while the Morrell C-model and Hogg and
Fuerstenau’s model are recognized as accurate fundamen-
tal and phenomenological models, respectively (Morrell,
1993 Rajamani et al., 2019). In these models, the mill
charge’s bulk density emerges as a significant influential
variable, estimated based on factors such as grinding media
density, ore density, and grinding media voidage (Morrell,
2016 Hilden et al., 2021).
Despite its critical role, determining grinding media
voidage in operational conditions remains challenging.
Presumed to be 40% across all models, this assumption lacks
direct empirical validation. While some studies on particle
bed voidage suggest a 40% value for man-sized spherical
particles under vibration (Yang, 2003), industrial mills
use a mixture of multi-sized particles, complicating void-
age estimation. Moreover, existing models assume constant
voidage during operation, overlooking dynamic changes
influenced by factors like fractional mill filling, rotational
speed, and grinding media size distribution. Recognizing
these limitations, Latchireddi (2002) attempted to predict
dynamic voidage, but mono-sized balls failed to reflect
real-world conditions. Hence, developing an equation to
estimate voidage under diverse operating conditions, con-
sidering realistic ball size distributions (BSD), is imperative.
The initial selection of BSD for the first-filling of
ball mills is typically based on Table 1 proposed by Bond
(1958). However, it’s important to note that the shape and
size distribution of balls can change over time due to vari-
ous factors like impact, erosion, corrosion, and crushing.
Therefore, during the commissioning of mineral processing
plants, the voidage associated with each Bond’s proposed
BSD must be taken into account when formulating equa-
tions for determining the grinding media voidage.
Design of Experiments (DOE) is a valuable method
that enables researchers to plan experiments and analyze the
cause-and-effect relationships between operating param-
eters in a process (Montgomery, 2005). Among the vari-
ous experimental designs, Response Surface Methodology
(RSM) stands out as an efficient tool of DOE. RSM is
extensively applied to establish mathematical relationships
between a dependent parameter and independent variables
(Soleymani Yazdi and Khorram, 2010 Golpayegani and
Abdollahzadeh, 2017).
Utilizing Design of Experiments (DOE) and Response
Surface Methodology (RSM), we aimed to enhance the
accuracy of the Morrel C-model and Hogg and Fuerstenau’s
model by investigating the static and dynamic voidage of
grinding media within ball mills. We employed a three-
level factorial design to model dynamic voidage, focusing
on fractional mill filling and mill rotating speed as input
parameters. Subsequently, a separate model was developed
for each BSD proposed by Bond. These models, along with
a general equation predicting dynamic voidage, were evalu-
ated using an industrial database of ball mills, assessing the
mean absolute percentage error (MAPE) of the models’
prediction results.
DESCRIPTION OF HOGG AND
FUERSTENAU’S MODEL AND MORELL
C-MODEL
To formulate a fundamental power equation for tumbling
mills, it’s essential to understand the motion of the charge
within the mill. Davis (1919) was the pioneer in establish-
ing a relationship between a ball mill’s charge motion and
its power draw, employing the energy balance approach.
Davis’ depiction closely resembles the earliest descriptions
provided by White (1905). According to this description,
particles within the mill move in a constrained manner
along a circular trajectory until reaching a point where
gravitational and centrifugal forces achieve equilibrium.
At this juncture, particles undergo a parabolic free fall
Table 1. The grinding media size distributions recommended by Bond for the initial filling of ball mills during the
commissioning stage (Bond, 1958)
Ball size
(mm)
Percentage Weight
BSD1 BSD2 BSD3 BSD4 BSD5 BSD6 BSD7
115 23.0 0 0 0 0 0 0
100 31.0 23 0 0 0 0 0
90 18.0 34 24 0 0 0 0
75 15.0 21 38 31 0 0 0
65 7.0 12 20.5 39 34 0 0
50 3.8 6.5 11.5 19.0 43.0 40.0 0.0
40 1.7 2.5 4.5 8.0 17.0 45.0 51.0
25 0.5 1.0 1.5 3.0 6.0 15.0 49.0
Sum 100 100 100 100 100 100 100