XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3719
with a side length of 9 mm, as depicted in Figure 2c. The
rotational speed was controlled using a variable-frequency
drive (VFD). It’s noteworthy that to mitigate the wall effect
on voidage measurements, the mill diameter should be at
least ten times larger than the diameter of the largest balls,
which was 11.5 mm in this study (Benyahia and O’Neill,
2005). Additionally, the Design-Expert software was uti-
lized for experiment design, model development, and graph
analysis. Moreover, Minitab software was employed to con-
struct a multiple regression model.
Experimental Procedure
Initially, the required weight of each ball size fraction was
determined based on the ball size distribution and frac-
tional mill filling, followed by charging them into the ball
mill. Subsequently, the mill’s rotational speed was adjusted
using the VFD. Once the load’s shape reached a steady
state, the water pipeline valve was opened, and water was
added until a measurable circular segment appeared on the
glass-fronted surface of the mill (Figure 3). Then, the height
(sagitta) of the circular segment (h) was measured using an
adjustable T-Square ruler (Figure 3). Finally, the water was
discharged and weighed. To prevent water from being car-
ried by the lifters, 26 holes with a diameter of 2 mm were
drilled in each lifter’s body, ensuring it was smaller than the
smallest ball’s diameter (refer to Figure 2c).
The volume of the cylindrical segment within the ball
mill, containing both water and balls, was determined for
each experiment using the following formulas (Harris and
Stocker, 1998):
V AL =(5)
A 2
R sinih
2 i =-^(6)
2arccosb R
R h i =-l (7)
where V is the volume of the cylindrical segment (m3), A
is the area of the circular segment (m2), L is the mill length
(m), R is the mill inside radius (m), and θ the central angle
(radians).
Ultimately, the dynamic voidage (Ø) of the grinding
media was obtained by calculating the cylindrical segment
volume (V) occupied by water and balls, along with the
water volume (VW), which corresponds to its weight, utiliz-
ing the following equation:
V
V
W Q =(8)
To further enhance the accuracy of the analysis regarding
the impact of grinding media size distribution on voidage
value, we also conducted measurements of the static void-
age for all BSDs listed in Table 1.
Design of Experiments
In this study, the surface response methodology (RSM),
which combines statistical and mathematical techniques,
was utilized to model and analyze processes where a
Figure 2. Equipment Utilized. (a) Balls used to prepare Bond’s suggested ball size distributions, (b) Schematic representation
of all equipment and instruments utilized. (c) Lifters’ general arrangement
with a side length of 9 mm, as depicted in Figure 2c. The
rotational speed was controlled using a variable-frequency
drive (VFD). It’s noteworthy that to mitigate the wall effect
on voidage measurements, the mill diameter should be at
least ten times larger than the diameter of the largest balls,
which was 11.5 mm in this study (Benyahia and O’Neill,
2005). Additionally, the Design-Expert software was uti-
lized for experiment design, model development, and graph
analysis. Moreover, Minitab software was employed to con-
struct a multiple regression model.
Experimental Procedure
Initially, the required weight of each ball size fraction was
determined based on the ball size distribution and frac-
tional mill filling, followed by charging them into the ball
mill. Subsequently, the mill’s rotational speed was adjusted
using the VFD. Once the load’s shape reached a steady
state, the water pipeline valve was opened, and water was
added until a measurable circular segment appeared on the
glass-fronted surface of the mill (Figure 3). Then, the height
(sagitta) of the circular segment (h) was measured using an
adjustable T-Square ruler (Figure 3). Finally, the water was
discharged and weighed. To prevent water from being car-
ried by the lifters, 26 holes with a diameter of 2 mm were
drilled in each lifter’s body, ensuring it was smaller than the
smallest ball’s diameter (refer to Figure 2c).
The volume of the cylindrical segment within the ball
mill, containing both water and balls, was determined for
each experiment using the following formulas (Harris and
Stocker, 1998):
V AL =(5)
A 2
R sinih
2 i =-^(6)
2arccosb R
R h i =-l (7)
where V is the volume of the cylindrical segment (m3), A
is the area of the circular segment (m2), L is the mill length
(m), R is the mill inside radius (m), and θ the central angle
(radians).
Ultimately, the dynamic voidage (Ø) of the grinding
media was obtained by calculating the cylindrical segment
volume (V) occupied by water and balls, along with the
water volume (VW), which corresponds to its weight, utiliz-
ing the following equation:
V
V
W Q =(8)
To further enhance the accuracy of the analysis regarding
the impact of grinding media size distribution on voidage
value, we also conducted measurements of the static void-
age for all BSDs listed in Table 1.
Design of Experiments
In this study, the surface response methodology (RSM),
which combines statistical and mathematical techniques,
was utilized to model and analyze processes where a
Figure 2. Equipment Utilized. (a) Balls used to prepare Bond’s suggested ball size distributions, (b) Schematic representation
of all equipment and instruments utilized. (c) Lifters’ general arrangement