3718 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
fractional mill filling by steel balls, ρp indicates the slurry
density (tonnes/m3), J represents the total fractional filling,
and Jp signifies the interstitial slurry filling. It’s important
to note that Eq. (1) provides the power of the mill pinion,
thus to estimate the gross power draw, the energy losses
of the motor and gearbox must be accounted for. These
losses are typically reported at 4% and 3% for the motor
and gearbox, respectively (Morrell, 1993).
Literature suggests that Hogg and Fuerstenau’s model
can reasonably predict the power draw of mills (Rajamani
et al., 2019 Doll, 2016). Consequently, the Moly-Cop
tools, designed to evaluate grinding media performance at
an industrial scale, utilize Hogg and Fuerstenau’s model to
estimate the net power draw of industrial mills (Molycop,
2021)
Morrell (1993), through an extensive examination
of earlier models, asserted that the preceding approaches
failed to accurately predict the power draw across various
operational scenarios due to their unrealistic depiction
of the charge shape. To address this limitation, Morrell
employed a photographic technique alongside a glass-
fronted laboratory mill to introduce a novel charge descrip-
tion. In Morrell’s framework, the power draw of tumbling
mills is primarily influenced by a segment of the charge
known as the active charge, which imparts force onto the
mill shell. Notably, Morrell disregarded the contribution
of the mill charge in the free-flight region to the power
draw. Consequently, Morrell depicted crescent-like shapes,
as illustrated in Figures 1c and 1d representing the active
charge within grate discharge and overflow discharge mills,
while excluding the mill charge in the free-flight region.
Accordingly, Morrell employed three distinct approaches
involving torque, energy balance, and friction force to char-
acterize the net power draw, ultimately deriving the follow-
ing equation to calculate the gross power of tumbling mills
(Morrell, 2016):
P 11 LC
3 Zr
πgLN rm
3Zr ri r 2h-
(sini sin )(sin sini ),
3 Zr
N r π
Zrih4 ri4 1h4-
Gross m
2.5
s
m i
m
m
3
m
2
i
3
c S T P T TO
c
m i
m m
3
m
#
#"t
##^r
i t i
=
+--+-
-+-
+Lt -
---
^r
_r
^r
^Z
^3Z
h
i0.861
h (
#2r
2
(3)
In the provided equation, PGross
represents the gross
power (kW), rm is the radius of the mill inside liners (m),
L denotes the mill effective length (m), Cs stands for the
fractional speed, ρc signifies the density of the total charge
(tonnes/m3), ρP represents the slurry density (tonnes/m3),
Nm indicates the mill rotational rate (revs/sec), ri denotes
the radial distance of the charge’s inner surface from the
rotating axis (m), θT represents the toe angle (radians), θS
denotes the shoulder angle (radians), Z is a function of frac-
tional mill filling, and θTO represents the slurry toe angle
(refer to Figure 1b). For grate discharge mills, θTO equals
θT. Morrell specified that the value of 3.395 can be used for
θTO (Morrell, 1993).
The density of the pulp (ρP) is presumed to be identi-
cal to the discharge density of the slurry. The density of the
rock/ball charge (ρc) is approximated using the subsequent
equation (Morrell, 2016):
J
J US J
J U Sh
t
t t B b O
t t t
t Q Q
Q Q
tc
t
=
-++-
-+-^1
^1
^1
^t
h
h h )3
(4)
where Jt represents the total fractional mill filling by balls
and rocks (including voids), Jb denotes the fractional mill
filling by balls (including voids), ρo is the ore density
(tonnes/m3), ρb represents the material density of steel balls
(7.75 tonnes/m3), Øt indicates the fractional voidage of the
mill charge (0.4), U signifies the fraction of grinding media
voidage occupied by slurry, and S denotes the fractional sol-
ids content of discharge slurry (by volume). For ball mills,
Morrell considered the values of 1 and 0.5 for U and S,
respectively.
Based on Equations 2 and 4, the grinding media void-
age plays a crucial role in estimating the charge bulk den-
sity. It’s worth noting that all power equations assume the
voidage of the active charge during operation to mirror
the static voidage, conventionally set at a constant value
of 40%. However, the mill charge exhibits a dynamic state
during milling operation, leading to fluctuations in void-
age due to changing operating conditions such as rotating
speed, fractional mill filling, and variations in ball size dis-
tribution. Therefore, this study aims to propose an empiri-
cal equation to determine the grinding media’s dynamic
voidage under various operating conditions, taking into
account a more realistic ball size distribution.
MATERIALS AND METHODS
Materials
Different ball size distributions proposed by Bond for the
initial filling of ball mills during the commissioning stage,
scaled down to one-tenth of the sizes presented in Table 1,
were utilized as grinding media (see Figure 2a). A labora-
tory-scale glass-fronted ball mill, measuring 14 cm in length
and diameter, was employed (refer to Figure 2b). The mill
shell was equipped with 4 square cross-section lifters, each
fractional mill filling by steel balls, ρp indicates the slurry
density (tonnes/m3), J represents the total fractional filling,
and Jp signifies the interstitial slurry filling. It’s important
to note that Eq. (1) provides the power of the mill pinion,
thus to estimate the gross power draw, the energy losses
of the motor and gearbox must be accounted for. These
losses are typically reported at 4% and 3% for the motor
and gearbox, respectively (Morrell, 1993).
Literature suggests that Hogg and Fuerstenau’s model
can reasonably predict the power draw of mills (Rajamani
et al., 2019 Doll, 2016). Consequently, the Moly-Cop
tools, designed to evaluate grinding media performance at
an industrial scale, utilize Hogg and Fuerstenau’s model to
estimate the net power draw of industrial mills (Molycop,
2021)
Morrell (1993), through an extensive examination
of earlier models, asserted that the preceding approaches
failed to accurately predict the power draw across various
operational scenarios due to their unrealistic depiction
of the charge shape. To address this limitation, Morrell
employed a photographic technique alongside a glass-
fronted laboratory mill to introduce a novel charge descrip-
tion. In Morrell’s framework, the power draw of tumbling
mills is primarily influenced by a segment of the charge
known as the active charge, which imparts force onto the
mill shell. Notably, Morrell disregarded the contribution
of the mill charge in the free-flight region to the power
draw. Consequently, Morrell depicted crescent-like shapes,
as illustrated in Figures 1c and 1d representing the active
charge within grate discharge and overflow discharge mills,
while excluding the mill charge in the free-flight region.
Accordingly, Morrell employed three distinct approaches
involving torque, energy balance, and friction force to char-
acterize the net power draw, ultimately deriving the follow-
ing equation to calculate the gross power of tumbling mills
(Morrell, 2016):
P 11 LC
3 Zr
πgLN rm
3Zr ri r 2h-
(sini sin )(sin sini ),
3 Zr
N r π
Zrih4 ri4 1h4-
Gross m
2.5
s
m i
m
m
3
m
2
i
3
c S T P T TO
c
m i
m m
3
m
#
#"t
##^r
i t i
=
+--+-
-+-
+Lt -
---
^r
_r
^r
^Z
^3Z
h
i0.861
h (
#2r
2
(3)
In the provided equation, PGross
represents the gross
power (kW), rm is the radius of the mill inside liners (m),
L denotes the mill effective length (m), Cs stands for the
fractional speed, ρc signifies the density of the total charge
(tonnes/m3), ρP represents the slurry density (tonnes/m3),
Nm indicates the mill rotational rate (revs/sec), ri denotes
the radial distance of the charge’s inner surface from the
rotating axis (m), θT represents the toe angle (radians), θS
denotes the shoulder angle (radians), Z is a function of frac-
tional mill filling, and θTO represents the slurry toe angle
(refer to Figure 1b). For grate discharge mills, θTO equals
θT. Morrell specified that the value of 3.395 can be used for
θTO (Morrell, 1993).
The density of the pulp (ρP) is presumed to be identi-
cal to the discharge density of the slurry. The density of the
rock/ball charge (ρc) is approximated using the subsequent
equation (Morrell, 2016):
J
J US J
J U Sh
t
t t B b O
t t t
t Q Q
Q Q
tc
t
=
-++-
-+-^1
^1
^1
^t
h
h h )3
(4)
where Jt represents the total fractional mill filling by balls
and rocks (including voids), Jb denotes the fractional mill
filling by balls (including voids), ρo is the ore density
(tonnes/m3), ρb represents the material density of steel balls
(7.75 tonnes/m3), Øt indicates the fractional voidage of the
mill charge (0.4), U signifies the fraction of grinding media
voidage occupied by slurry, and S denotes the fractional sol-
ids content of discharge slurry (by volume). For ball mills,
Morrell considered the values of 1 and 0.5 for U and S,
respectively.
Based on Equations 2 and 4, the grinding media void-
age plays a crucial role in estimating the charge bulk den-
sity. It’s worth noting that all power equations assume the
voidage of the active charge during operation to mirror
the static voidage, conventionally set at a constant value
of 40%. However, the mill charge exhibits a dynamic state
during milling operation, leading to fluctuations in void-
age due to changing operating conditions such as rotating
speed, fractional mill filling, and variations in ball size dis-
tribution. Therefore, this study aims to propose an empiri-
cal equation to determine the grinding media’s dynamic
voidage under various operating conditions, taking into
account a more realistic ball size distribution.
MATERIALS AND METHODS
Materials
Different ball size distributions proposed by Bond for the
initial filling of ball mills during the commissioning stage,
scaled down to one-tenth of the sizes presented in Table 1,
were utilized as grinding media (see Figure 2a). A labora-
tory-scale glass-fronted ball mill, measuring 14 cm in length
and diameter, was employed (refer to Figure 2b). The mill
shell was equipped with 4 square cross-section lifters, each