XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3727
To assess the efficacy of Eqs. 10 and 18 in enhancing the
accuracy of the Hogg and Fuerstenau model and Morrell
C-model, we utilized the ball mills’ dataset from Morrell’s
(1993) studies (see Table 7). Notably, a mean absolute
deviation (MAD) value of 24.62 for BSD1 was employed
to calculate the dynamic voidage using Eq. 18. Figure 10
illustrates the comparison between predicted and actual
power draw of the mills for three scenarios: the original
Morrell C-model and Hogg and Fuerstenau model (assum-
ing a voidage of 40%), and the modified models using Eqs.
10 and 18 to estimate the voidage of the balls (Modified
versions of Morrell C-model and Hogg and Fuerstenau
model). Additionally, detailed actual power draw and pre-
dicted values by both the original and modified versions of
models presented in Table 7. The results demonstrate that
employing Eqs. 10 and 18 for voidage estimation leads to a
closer alignment between predicted and actual power draw
values. Furthermore, the Mean Absolute Percentage Error
(MAPE), a reliable measure for evaluating prediction accu-
racy in statistical models, was utilized to assess the impact
of Eqs. 10 and 18 on the precision of Morrell C-model and
Hogg and Fuerstenau’s model. A lower MAPE indicates
higher predictive accuracy, with a MAPE below 10 indicat-
ing high accuracy. The MAPE is defined as follows (Kim
and Kim, 2016):
MAPE 100 N
1
A
A P
t 1
n
t
t t #=
-
=
/(19)
where N represents the number of data points, At denotes
the actual value, and Pt stands for the predicted value.
Table 7 presents the MAPE values, showing that utiliz-
ing Equations 10 and 18 for grinding media voidage cal-
culation leads to a reduction in the MAPE of Hogg and
Fuerstenau’s model by approximately 3.9% and 3.2%,
respectively. These findings signify a notable enhancement
in the prediction accuracy of Hogg and Fuerstenau’s model.
Furthermore, it is evident that the MAPE of the Morrell
C-model decreases by about 2.61% and 2.29% when
employing Equations 10 and 18 for grinding media void-
age calculation, respectively, resulting in improved model
prediction accuracy.
CONCLUSION
This study aimed to enhance the predictive accuracy of the
Morrel C-model and the Hogg and Fuerstenau model by
investigating both static and dynamic voidage of grind-
ing media within ball mills. Through a three-level facto-
rial design, we analyzed the effects of fractional mill filling
and mill rotating speed on grinding media voidage across
various size distributions. ANOVA analysis identified frac-
tional mill filling (Jt), rotating speed (Cs), their interaction
(Jt*Cs), and the quadratic term of fractional mill filling
(Jt2) as significant factors influencing dynamic voidage
across all tested ball size distributions recommended by
Bond. Dynamic voidage generally increased with decreas-
ing fractional mill filling and increasing mill rotating speed.
Additionally, both static and dynamic voidage decreased
Figure 9. Comparison of actual voidage with predicted values from developed models. (a) Prediction for BSD1 using Eq. 10.
(b) General multiple regression model prediction for all BSDs using Eq. 18
To assess the efficacy of Eqs. 10 and 18 in enhancing the
accuracy of the Hogg and Fuerstenau model and Morrell
C-model, we utilized the ball mills’ dataset from Morrell’s
(1993) studies (see Table 7). Notably, a mean absolute
deviation (MAD) value of 24.62 for BSD1 was employed
to calculate the dynamic voidage using Eq. 18. Figure 10
illustrates the comparison between predicted and actual
power draw of the mills for three scenarios: the original
Morrell C-model and Hogg and Fuerstenau model (assum-
ing a voidage of 40%), and the modified models using Eqs.
10 and 18 to estimate the voidage of the balls (Modified
versions of Morrell C-model and Hogg and Fuerstenau
model). Additionally, detailed actual power draw and pre-
dicted values by both the original and modified versions of
models presented in Table 7. The results demonstrate that
employing Eqs. 10 and 18 for voidage estimation leads to a
closer alignment between predicted and actual power draw
values. Furthermore, the Mean Absolute Percentage Error
(MAPE), a reliable measure for evaluating prediction accu-
racy in statistical models, was utilized to assess the impact
of Eqs. 10 and 18 on the precision of Morrell C-model and
Hogg and Fuerstenau’s model. A lower MAPE indicates
higher predictive accuracy, with a MAPE below 10 indicat-
ing high accuracy. The MAPE is defined as follows (Kim
and Kim, 2016):
MAPE 100 N
1
A
A P
t 1
n
t
t t #=
-
=
/(19)
where N represents the number of data points, At denotes
the actual value, and Pt stands for the predicted value.
Table 7 presents the MAPE values, showing that utiliz-
ing Equations 10 and 18 for grinding media voidage cal-
culation leads to a reduction in the MAPE of Hogg and
Fuerstenau’s model by approximately 3.9% and 3.2%,
respectively. These findings signify a notable enhancement
in the prediction accuracy of Hogg and Fuerstenau’s model.
Furthermore, it is evident that the MAPE of the Morrell
C-model decreases by about 2.61% and 2.29% when
employing Equations 10 and 18 for grinding media void-
age calculation, respectively, resulting in improved model
prediction accuracy.
CONCLUSION
This study aimed to enhance the predictive accuracy of the
Morrel C-model and the Hogg and Fuerstenau model by
investigating both static and dynamic voidage of grind-
ing media within ball mills. Through a three-level facto-
rial design, we analyzed the effects of fractional mill filling
and mill rotating speed on grinding media voidage across
various size distributions. ANOVA analysis identified frac-
tional mill filling (Jt), rotating speed (Cs), their interaction
(Jt*Cs), and the quadratic term of fractional mill filling
(Jt2) as significant factors influencing dynamic voidage
across all tested ball size distributions recommended by
Bond. Dynamic voidage generally increased with decreas-
ing fractional mill filling and increasing mill rotating speed.
Additionally, both static and dynamic voidage decreased
Figure 9. Comparison of actual voidage with predicted values from developed models. (a) Prediction for BSD1 using Eq. 10.
(b) General multiple regression model prediction for all BSDs using Eq. 18