XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3721
effects of variables on the response. In this procedure, the
significance of a model or a parameter effect hinges on the
F and P values, with a lower p-value (P 0.05) and a higher
F-value indicating significance at a 95% confidence level
(Montgomery, D.C., 2005). The F and P values for the
models and significant parameters are presented in Table 4.
Analysis of the table suggests that the fractional mill filling
(Jt), rotating speed (Cs), their interaction (Jt*Cs), and the
quadratic term of fractional mill filling (Jt2) are significant
factors in all models. Consequently, Eqs. 10 to 16 were
derived to estimate the grinding media voidage of BSD1 to
BSD7, respectively.
0.462313J 0.038590C
0.000804J C 0.006424J
1 t s
t s t
2
Q =+43.12158 -+
-+
(10)
0.414306J 0.060327Cs
0.001087J C 0.005812J
t
t s t
2
Q2 =+42.28959 -+
-+(11)
0.417540J 0.060085Cs
0.001070J C 0.005776J
t
t s t
2
Q3 =+43.51516 -+
-+(12)
0.455356J 0.074383C
0.001087J C 0.006311J
4 t s
t s t
2
Q =+44.61249 -+
-+(13)
0.441269J 0.094564C
0.001171J C 0.006204J
5 t s
t s t
2
Q =+44.34998 -+
-+(14)
0.409801J 0.085930C
0.001161J C 0.005708J
6 t s
t s t
2
Q =+44.88569 -+
-+
(15)
0.413476J 0.098013C
0.001340J C 0.005793J
7 t s
t s t
2
Q =+46.32825 -+
-+
(16)
These equations utilize actual values for all variables. By
utilizing these equations, one can predict the grinding
media voidage and subsequently the bulk density of the
ball mills’ charge during the ball mills’ commissioning
stage. Moreover, considering the alterations in the grinding
media’s shape and size due to various factors such as impact,
attrition, corrosion, and chipping, it can be assumed that
the actual ball size distribution inside ball mills closely
resembles BSD1. Therefore, by substituting the response of
Eq. 10 into Eq. 2 and Eq. 7 as the value of grinding media
voidage (Øt), the mill charge’s bulk density can be deter-
mined for a given operational condition of ball mills.
Ensuring the accuracy of model outputs is crucial for
reliable data analysis, as an inadequate model could result
in inaccurate results. Therefore, validating a model based on
the precision of its outputs is essential. The adequate preci-
sion ratio serves as a critical indicator of a model’s accuracy,
with a value exceeding 4 indicating validity. Analysis of the
data reported in Table 4 reveals that all models possess an
adequate precision ratio greater than 4, signifying the high
precision of the provided models.
Table 3. Designed experiments by three-level factorial design and obtained results
Run
No.
J
(%)
Q
(%)
Grinding Media Voidage (%)
BSD1 BSD2 BSD3 BSD4 BSD5 BSD6 BSD7
1 30 65 36.43 36.43 37.43 37.44 38.73 38.62 39.61 39.54 40.30 40.34 41.26 41.42 43.23 43.22
2 45 40 35.61 35.60 36.14 36.08 37.21 37.13 38.08 38.00 38.72 38.65 39.6 39.49 41.22 41.08
3 30 90 37.24 37.21 38.62 38.63 39.97 39.88 40.86 40.77 41.70 41.62 43.19 43.04 45.08 44.96
4 30 65 36.40 36.43 37.46 37.44 38.69 38.62 39.44 39.54 40.39 40.34 41.48 41.42 43.35 43.22
5 45 65 35.98 35.94 36.64 36.7 37.80 37.81 38.69 38.76 39.53 39.56 40.54 40.56 42.14 42.23
6 30 65 36.40 36.43 37.54 37.44 38.55 38.62 39.59 39.54 40.30 40.34 41.5 41.42 43.22 43.22
7 30 40 35.68 35.79 36.26 36.41 37.33 37.54 38.27 38.38 38.85 38.99 39.68 39.91 41.34 41.57
8 45 90 36.36 36.42 37.48 37.48 38.60 38.67 39.56 39.57 40.37 40.40 41.66 41.75 43.42 43.46
9 30 65 36.50 36.43 37.50 37.44 38.61 38.62 39.60 39.54 40.40 40.34 41.47 41.42 43.14 43.22
10 15 65 39.69 39.81 40.69 40.79 41.92 42.03 43.22 43.17 43.89 43.91 44.79 44.85 46.79 46.82
11 30 65 36.51 36.43 37.40 37.44 38.64 38.62 39.50 39.54 40.36 40.34 41.48 41.42 43.26 43.22
12 15 40 38.96 38.87 39.43 39.35 40.68 40.54 41.63 41.60 42.19 42.12 43.03 42.9 44.77 44.67
13 15 90 40.92 40.89 42.40 42.38 43.67 43.69 44.73 44.81 45.59 45.64 46.84 46.9 48.98 49.06
Experimental Predicted Experimental Predicted Experimental Predicted Experimental Predicted Experimental Predicted Experimental Predicted Experimental Predicted
effects of variables on the response. In this procedure, the
significance of a model or a parameter effect hinges on the
F and P values, with a lower p-value (P 0.05) and a higher
F-value indicating significance at a 95% confidence level
(Montgomery, D.C., 2005). The F and P values for the
models and significant parameters are presented in Table 4.
Analysis of the table suggests that the fractional mill filling
(Jt), rotating speed (Cs), their interaction (Jt*Cs), and the
quadratic term of fractional mill filling (Jt2) are significant
factors in all models. Consequently, Eqs. 10 to 16 were
derived to estimate the grinding media voidage of BSD1 to
BSD7, respectively.
0.462313J 0.038590C
0.000804J C 0.006424J
1 t s
t s t
2
Q =+43.12158 -+
-+
(10)
0.414306J 0.060327Cs
0.001087J C 0.005812J
t
t s t
2
Q2 =+42.28959 -+
-+(11)
0.417540J 0.060085Cs
0.001070J C 0.005776J
t
t s t
2
Q3 =+43.51516 -+
-+(12)
0.455356J 0.074383C
0.001087J C 0.006311J
4 t s
t s t
2
Q =+44.61249 -+
-+(13)
0.441269J 0.094564C
0.001171J C 0.006204J
5 t s
t s t
2
Q =+44.34998 -+
-+(14)
0.409801J 0.085930C
0.001161J C 0.005708J
6 t s
t s t
2
Q =+44.88569 -+
-+
(15)
0.413476J 0.098013C
0.001340J C 0.005793J
7 t s
t s t
2
Q =+46.32825 -+
-+
(16)
These equations utilize actual values for all variables. By
utilizing these equations, one can predict the grinding
media voidage and subsequently the bulk density of the
ball mills’ charge during the ball mills’ commissioning
stage. Moreover, considering the alterations in the grinding
media’s shape and size due to various factors such as impact,
attrition, corrosion, and chipping, it can be assumed that
the actual ball size distribution inside ball mills closely
resembles BSD1. Therefore, by substituting the response of
Eq. 10 into Eq. 2 and Eq. 7 as the value of grinding media
voidage (Øt), the mill charge’s bulk density can be deter-
mined for a given operational condition of ball mills.
Ensuring the accuracy of model outputs is crucial for
reliable data analysis, as an inadequate model could result
in inaccurate results. Therefore, validating a model based on
the precision of its outputs is essential. The adequate preci-
sion ratio serves as a critical indicator of a model’s accuracy,
with a value exceeding 4 indicating validity. Analysis of the
data reported in Table 4 reveals that all models possess an
adequate precision ratio greater than 4, signifying the high
precision of the provided models.
Table 3. Designed experiments by three-level factorial design and obtained results
Run
No.
J
(%)
Q
(%)
Grinding Media Voidage (%)
BSD1 BSD2 BSD3 BSD4 BSD5 BSD6 BSD7
1 30 65 36.43 36.43 37.43 37.44 38.73 38.62 39.61 39.54 40.30 40.34 41.26 41.42 43.23 43.22
2 45 40 35.61 35.60 36.14 36.08 37.21 37.13 38.08 38.00 38.72 38.65 39.6 39.49 41.22 41.08
3 30 90 37.24 37.21 38.62 38.63 39.97 39.88 40.86 40.77 41.70 41.62 43.19 43.04 45.08 44.96
4 30 65 36.40 36.43 37.46 37.44 38.69 38.62 39.44 39.54 40.39 40.34 41.48 41.42 43.35 43.22
5 45 65 35.98 35.94 36.64 36.7 37.80 37.81 38.69 38.76 39.53 39.56 40.54 40.56 42.14 42.23
6 30 65 36.40 36.43 37.54 37.44 38.55 38.62 39.59 39.54 40.30 40.34 41.5 41.42 43.22 43.22
7 30 40 35.68 35.79 36.26 36.41 37.33 37.54 38.27 38.38 38.85 38.99 39.68 39.91 41.34 41.57
8 45 90 36.36 36.42 37.48 37.48 38.60 38.67 39.56 39.57 40.37 40.40 41.66 41.75 43.42 43.46
9 30 65 36.50 36.43 37.50 37.44 38.61 38.62 39.60 39.54 40.40 40.34 41.47 41.42 43.14 43.22
10 15 65 39.69 39.81 40.69 40.79 41.92 42.03 43.22 43.17 43.89 43.91 44.79 44.85 46.79 46.82
11 30 65 36.51 36.43 37.40 37.44 38.64 38.62 39.50 39.54 40.36 40.34 41.48 41.42 43.26 43.22
12 15 40 38.96 38.87 39.43 39.35 40.68 40.54 41.63 41.60 42.19 42.12 43.03 42.9 44.77 44.67
13 15 90 40.92 40.89 42.40 42.38 43.67 43.69 44.73 44.81 45.59 45.64 46.84 46.9 48.98 49.06
Experimental Predicted Experimental Predicted Experimental Predicted Experimental Predicted Experimental Predicted Experimental Predicted Experimental Predicted