XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3717
trajectory until impacting the bed, after which they resume
their circular path. The locus of these free-fall points forms
a circle with a radius of g/2ω2, termed the “Davis Circle”
(refer to Figure 1a).
Hogg and Fuerstenau (1972) identified two primary
shortcomings in Davis’ description: firstly, the omission
of the quantity of material (ball filling), and secondly, the
unrealistic assumption that the circular paths depicted in
Figure 1a do not intersect at any point. In response, they
introduced a novel description wherein the mill charge
comprises two distinct components: a ‘static’ portion that
moves with the mill shell, and a ‘shear’ region where par-
ticles flow along the charge’s surface (refer to Figure 1b).
According to this description, the mill motor must provide
energy to lift the balls in each circular path through the
static portion from the toe to the shoulder. As the milling
system operates in a dynamic equilibrium state, particles
at the same location within the mill should possess iden-
tical energy levels. Consequently, the potential energy of
particles (mgλt) in the static portion of the bed is entirely
dissipated within the shear zone. Utilizing these assump-
tions, they derived the following simple power equation:
P 3
2 g LN 2
D sin sina
bulk
2
3
c
2.5 3
0 t i =`j (1)
where P represents the power draw (kW), D denotes the
mtill effective diameter (ft.), L signifies the mill effective
length (ft.), ρbulk represents the mill charge’s bulk density
(tonnes/m3), α denotes the charge’s angle of repose (typi-
cally falling within the range of 30° to 35°), Nc indicates the
fractional speed, and θ0 represents the filling angle associ-
ated with the fractional mill filling (measured in radians).
Additionally, they provided the following equation to cal-
culate the mill charge’s bulk density (ρap):
ρap =[(1–φ) ρb Jb +ρp Jp φ Jb +ρp (J–Jb)]/J (2)
where φ represents the grinding media voidage, ρb stands
for the steel balls density (tonnes/m3), Jb denotes the
Figure 1. Schematics depicting the descriptions of charge motion. (a) Davis’ description (Davis 1919), (b)
Hogg and Fuerstenau’s description (Hogg and Fuerstenau 1972)
trajectory until impacting the bed, after which they resume
their circular path. The locus of these free-fall points forms
a circle with a radius of g/2ω2, termed the “Davis Circle”
(refer to Figure 1a).
Hogg and Fuerstenau (1972) identified two primary
shortcomings in Davis’ description: firstly, the omission
of the quantity of material (ball filling), and secondly, the
unrealistic assumption that the circular paths depicted in
Figure 1a do not intersect at any point. In response, they
introduced a novel description wherein the mill charge
comprises two distinct components: a ‘static’ portion that
moves with the mill shell, and a ‘shear’ region where par-
ticles flow along the charge’s surface (refer to Figure 1b).
According to this description, the mill motor must provide
energy to lift the balls in each circular path through the
static portion from the toe to the shoulder. As the milling
system operates in a dynamic equilibrium state, particles
at the same location within the mill should possess iden-
tical energy levels. Consequently, the potential energy of
particles (mgλt) in the static portion of the bed is entirely
dissipated within the shear zone. Utilizing these assump-
tions, they derived the following simple power equation:
P 3
2 g LN 2
D sin sina
bulk
2
3
c
2.5 3
0 t i =`j (1)
where P represents the power draw (kW), D denotes the
mtill effective diameter (ft.), L signifies the mill effective
length (ft.), ρbulk represents the mill charge’s bulk density
(tonnes/m3), α denotes the charge’s angle of repose (typi-
cally falling within the range of 30° to 35°), Nc indicates the
fractional speed, and θ0 represents the filling angle associ-
ated with the fractional mill filling (measured in radians).
Additionally, they provided the following equation to cal-
culate the mill charge’s bulk density (ρap):
ρap =[(1–φ) ρb Jb +ρp Jp φ Jb +ρp (J–Jb)]/J (2)
where φ represents the grinding media voidage, ρb stands
for the steel balls density (tonnes/m3), Jb denotes the
Figure 1. Schematics depicting the descriptions of charge motion. (a) Davis’ description (Davis 1919), (b)
Hogg and Fuerstenau’s description (Hogg and Fuerstenau 1972)