XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 873
parameters A and µ depend on the scale of mill while the
other parameters are specific to the material. Equations (1
to 8) can be applied for mills of different sizes by tuning
these two parameters.
Power Draw Model
Many factors influence power consumption in the ball mill,
for example, media charge level (ball filling, J ),slurry filling,
slurry concentration, and mill speed (Morrell 1996a). We
have used the following semi-empirical equation for esti-
mating the power drawn by the ball mill (Morrell 1996b):
Lt
D Lzh0.804i Gross power .5
.5
c
2
2 cdi
=
+_KD
_2.62 ^
(9)
J J
t t
2 c b
b
=
-^h (10)
..49267h 2 2.9863 2 2129 0 2 b z z =--^(11)
*.42_z
max
19 *
max d z`1 z =-_1 ---zi ie j (12)
..135J 0 954 0 *
max t z =-(13)
.
.
J
J
J
U
0.4 1h@
0 2Uh
0 2UJ
1
t
t o
b b o t
2
tc
t
t
=+-
+
+-+
^0.6
^t
^U
h0.6 =
61
G
(14)
.2Uh
.
Jt
J
J
U
0
0 2UJt
1
c
t o
b b
1
t
t
toh0.6
=
+
+-+^t
^0.6
=G
(15)
.66,
.K
7
8 81, Calibration constant
Overflow discharge
Grate discharge =((16)
where, δ and ϒ are nonlinear functions of mill speed and
filling, respectively D and L are mill diameter and length
(effective grinding length), respectively Jt is fraction of
the mill volume occupied by the entire charge (includ-
ing voids), JB is fraction of the mill volume occupied by
balls (including voids), ϕ and ϕmax are critical speed frac-
tion and critical speed fraction at max power, respectively
U and ρo are slurry filling fraction, specific gravity of ore,
respectively ρb is specific gravity of ball material (e.g., 7.8
for steel, assumed for calculation), and ρc is charge density
(ton/m3). In the calculation of ρc void fraction of 0.4 and
the fractional solid content in discharge slurry of 0.5 are
assumed (Morrell 1996a). Also, it is assumed that Jt =JB for
estimating power drawn by the ball mill.
Holdup Model
During grinding, a variety of operating regimes can be
encountered under different operating conditions for vari-
ous mill designs. Having a heuristics-based tool or soft sen-
sor for estimating the holdup is important for controlling
the targeted throughput and to avoid abnormal filling states
of the mill. On the contrary, overfilling of the ball mill
results in reduced power draw. This is because the slurry
mass on the opposite side of the rising ball media counters
the torque required to rotate the mill (Shi 2016). The major
consequence of overfilling is backflow of slurry from the
inlet resulting in choking of the ball mill. However, even
minor instances of sub-optimal holdup levels can lead to a
loss of productivity as grinding efficiency and throughput
are adversely affected, which can go unnoticed until a feed-
back response of product particle size is received. Several
attempts have been made to model holdup and empirical
and phenomenological expressions were proposed in the
literature (Rogers and Austin 1984 Moys 1986 Klimpel
et al., 1989 Austin and Klimpel 1989 Schneider 1995
Shi 2016). According to Rogers and Austin (1984), the
holdup in an overflow ball mill varies as square root of flow
rate. Similar dependency on square root of slurry flow rate
(F) has been proposed in a few other studies (Weller 1980
Marchand et al., 1980 Gardner et al., 1982). However, a
linear dependence of holdup on slurry flow rate has been
reported in other works (Kinneberg et al., 1984 Songfack
and Rajamani 1999). For grate discharge ball mill opera-
tion, holdup has been found proportional to F 0.6 (Moys
1986). The disagreement in models may be attributed to
different regimes of operation in each model.
In the present work, an empirical model for holdup
is fitted for steady state overflow operational data of ball
mills published in the literature (Kinneberg et al., 1984 Shi
2016 Songfack and Rajamani 1999). The process variables
considered for developing an empirical model for hold–up
are flow rate (Q), ball filling (J) and solids concentration in
the slurry (c). To enable broader applicability of our model,
the variables are used in a non–dimensional or scaled form.
The ball mill holdup is expressed as a volume fraction of
the mill. The ratio of volume of the ball bed, including the
interstitial volume to the total volume of the ball mill is
known as ball filling, J, while slurry concentration, c, is
mass fraction of solids in the slurry. Flow rate may be scaled
with the volume of slurry holdup in the mill to obtain mean
residence time but since the holdup is not known before-
hand, we scale the flowrate with the ball mill volume itself
to obtain the modified flow rate, Q' =Q/V with the units of
min–1. Only steady state data points of the overflow opera-
tion of ball mill are considered for modelling. The ranges of
parameters A and µ depend on the scale of mill while the
other parameters are specific to the material. Equations (1
to 8) can be applied for mills of different sizes by tuning
these two parameters.
Power Draw Model
Many factors influence power consumption in the ball mill,
for example, media charge level (ball filling, J ),slurry filling,
slurry concentration, and mill speed (Morrell 1996a). We
have used the following semi-empirical equation for esti-
mating the power drawn by the ball mill (Morrell 1996b):
Lt
D Lzh0.804i Gross power .5
.5
c
2
2 cdi
=
+_KD
_2.62 ^
(9)
J J
t t
2 c b
b
=
-^h (10)
..49267h 2 2.9863 2 2129 0 2 b z z =--^(11)
*.42_z
max
19 *
max d z`1 z =-_1 ---zi ie j (12)
..135J 0 954 0 *
max t z =-(13)
.
.
J
J
J
U
0.4 1h@
0 2Uh
0 2UJ
1
t
t o
b b o t
2
tc
t
t
=+-
+
+-+
^0.6
^t
^U
h0.6 =
61
G
(14)
.2Uh
.
Jt
J
J
U
0
0 2UJt
1
c
t o
b b
1
t
t
toh0.6
=
+
+-+^t
^0.6
=G
(15)
.66,
.K
7
8 81, Calibration constant
Overflow discharge
Grate discharge =((16)
where, δ and ϒ are nonlinear functions of mill speed and
filling, respectively D and L are mill diameter and length
(effective grinding length), respectively Jt is fraction of
the mill volume occupied by the entire charge (includ-
ing voids), JB is fraction of the mill volume occupied by
balls (including voids), ϕ and ϕmax are critical speed frac-
tion and critical speed fraction at max power, respectively
U and ρo are slurry filling fraction, specific gravity of ore,
respectively ρb is specific gravity of ball material (e.g., 7.8
for steel, assumed for calculation), and ρc is charge density
(ton/m3). In the calculation of ρc void fraction of 0.4 and
the fractional solid content in discharge slurry of 0.5 are
assumed (Morrell 1996a). Also, it is assumed that Jt =JB for
estimating power drawn by the ball mill.
Holdup Model
During grinding, a variety of operating regimes can be
encountered under different operating conditions for vari-
ous mill designs. Having a heuristics-based tool or soft sen-
sor for estimating the holdup is important for controlling
the targeted throughput and to avoid abnormal filling states
of the mill. On the contrary, overfilling of the ball mill
results in reduced power draw. This is because the slurry
mass on the opposite side of the rising ball media counters
the torque required to rotate the mill (Shi 2016). The major
consequence of overfilling is backflow of slurry from the
inlet resulting in choking of the ball mill. However, even
minor instances of sub-optimal holdup levels can lead to a
loss of productivity as grinding efficiency and throughput
are adversely affected, which can go unnoticed until a feed-
back response of product particle size is received. Several
attempts have been made to model holdup and empirical
and phenomenological expressions were proposed in the
literature (Rogers and Austin 1984 Moys 1986 Klimpel
et al., 1989 Austin and Klimpel 1989 Schneider 1995
Shi 2016). According to Rogers and Austin (1984), the
holdup in an overflow ball mill varies as square root of flow
rate. Similar dependency on square root of slurry flow rate
(F) has been proposed in a few other studies (Weller 1980
Marchand et al., 1980 Gardner et al., 1982). However, a
linear dependence of holdup on slurry flow rate has been
reported in other works (Kinneberg et al., 1984 Songfack
and Rajamani 1999). For grate discharge ball mill opera-
tion, holdup has been found proportional to F 0.6 (Moys
1986). The disagreement in models may be attributed to
different regimes of operation in each model.
In the present work, an empirical model for holdup
is fitted for steady state overflow operational data of ball
mills published in the literature (Kinneberg et al., 1984 Shi
2016 Songfack and Rajamani 1999). The process variables
considered for developing an empirical model for hold–up
are flow rate (Q), ball filling (J) and solids concentration in
the slurry (c). To enable broader applicability of our model,
the variables are used in a non–dimensional or scaled form.
The ball mill holdup is expressed as a volume fraction of
the mill. The ratio of volume of the ball bed, including the
interstitial volume to the total volume of the ball mill is
known as ball filling, J, while slurry concentration, c, is
mass fraction of solids in the slurry. Flow rate may be scaled
with the volume of slurry holdup in the mill to obtain mean
residence time but since the holdup is not known before-
hand, we scale the flowrate with the ball mill volume itself
to obtain the modified flow rate, Q' =Q/V with the units of
min–1. Only steady state data points of the overflow opera-
tion of ball mill are considered for modelling. The ranges of