1510 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
The test product was dry screened for 20 min in a
Tyler RoTap using the same closing screen aperture as
the BBMWI test. The torque data obtained during the
test was used to calculate the specific energy according to
Equation 2.
SE m 3600
2 rNT =(2)
Where SE is the specific energy (kWh/t), N is the total mill
revolutions, T is the average net torque (Nm), and m is the
ore mass tested (kg)
The size-specific energy was then calculated using
Equation 3 (Ballantyne et al., 2015).
New size SSE generated
SE
size =-(3)
where SSEsize represents the size-specific energy (kWh/t)
for the chosen Work Index closing screen aperture (mm)
for each set of samples, SE represents the specific energy
(kWh/t), and the new size generated is the product passing
through the selected screen size (%).
In this study, we implemented a multi-regression
model with quadratic terms to examine the complex rela-
tionships between our dependent variable and a set of inde-
pendent variables. We utilized Python’s sci-kit-learn library,
renowned for its robust and efficient machine-learning
tools. Our approach involved transforming the original
features into their polynomial counterparts, including qua-
dratic terms, using the `PolynomialFeatures` module. This
transformation allowed us to explore not only the linear
but also the non-linear associations within our dataset. To
address the potential issue of multicollinearity arising from
the inclusion of these polynomial terms and to enhance the
model’s generalizability, we applied Lasso (Least Absolute
Shrinkage and Selection Operator) regression with cross-
validation using the ‘LassoCV’ class. LassoCV combines
Lasso regression with efficient hyperparameter tuning,
automatically determining the optimal regularization
strength through cross-validation. This regularization tech-
nique effectively shrinks less significant feature coefficients
to zero, thus performing feature selection and yielding a
more interpretable model. The resulting regression model,
therefore, encapsulates both the linear and quadratic rela-
tionships in the data while maintaining parsimony and
robustness against overfitting.
RESULTS AND DISCUSSION
Correlation Between BBMWI and SSE in the
Hardgrove Test
The results of the BBMWI and Hardgrove tests, consider-
ing only the SSE, are illustrated in Figure 3. The BBMWI
ranged from 3.3 to 31.7, thus covering a wide range of
values. The database organized by Doll, 2023 includes
2384 results of BBMWI tests, 99.9% of which are within
these ranges. The SE parameter ranged from 1.2 kWh/t to
3.0 kWh/t, thus confirming the variation in this parameter
as reported by Austin et al., 1981, Mucsi, 2008, Mucsi et
al., 2019, Shi and Zuo, 2014 and Palaniandy, 2017.
Figures 1a and 1b illustrate the linear and quadratic
correlations obtained with their 95% confidence interval
for the regression as a blue band. The correlation equation
is shown in each plot, along with their respective coefficient
of correlation (R2) and the MAPE (mean absolute predic-
tion error). The correlation is used to predict the BBMWI
as illustrated in Figures 1c and 1d, showing the 5% and
10% error bands in green and red around the 1:1 line.
The predicted BBMWI based solely on the SSE had a
12.2% error on average for the linear model and a 9.9%
error on average for the quadratic model.
Correlation Between BBMWI and the Hardgrove Test
with the Multi-Regression Model
The correlation from the Hardgrove with BBMWI was
then calculated based on SSE and other parameters from
the test’s feed and product. Figures 2a and 2b illustrate the
correlation obtained, showing the 5% and 10% error bands
in green and red around the 1:1 line.
Equations 4 and 5 describe the correlation using the
multi-regression model. The correlations are for the SSE
interval from 2.2 kWh/t to 24.2 kWh/t.
Multi-linear model:
*
*
.*
WI A F
SSE 0 486
75 µm Passing
80 =+-B
+-C
+
(4)
Quadratic model:
**
***
**
**
**sin
WI A B C SSE
D F802 E F80
F F
G
H Pas g SSE
75
5 7
Passing Feed
Passing Feed
Passing
Passing Feed Passing
80
=+-+
+-+
+-
+-
+-
(5)
The average error for BBMWI prediction with the linear
model is 6.7%. In the case of the quadratic model, the aver-
age absolute error is 6.1%. This result contrasts the 12.2
and 9.9% average error obtained in the Hardgrove test con-
sidering only the SSE. The prediction errors are similar to
those reported in the literature using the Hardgrove mill
The test product was dry screened for 20 min in a
Tyler RoTap using the same closing screen aperture as
the BBMWI test. The torque data obtained during the
test was used to calculate the specific energy according to
Equation 2.
SE m 3600
2 rNT =(2)
Where SE is the specific energy (kWh/t), N is the total mill
revolutions, T is the average net torque (Nm), and m is the
ore mass tested (kg)
The size-specific energy was then calculated using
Equation 3 (Ballantyne et al., 2015).
New size SSE generated
SE
size =-(3)
where SSEsize represents the size-specific energy (kWh/t)
for the chosen Work Index closing screen aperture (mm)
for each set of samples, SE represents the specific energy
(kWh/t), and the new size generated is the product passing
through the selected screen size (%).
In this study, we implemented a multi-regression
model with quadratic terms to examine the complex rela-
tionships between our dependent variable and a set of inde-
pendent variables. We utilized Python’s sci-kit-learn library,
renowned for its robust and efficient machine-learning
tools. Our approach involved transforming the original
features into their polynomial counterparts, including qua-
dratic terms, using the `PolynomialFeatures` module. This
transformation allowed us to explore not only the linear
but also the non-linear associations within our dataset. To
address the potential issue of multicollinearity arising from
the inclusion of these polynomial terms and to enhance the
model’s generalizability, we applied Lasso (Least Absolute
Shrinkage and Selection Operator) regression with cross-
validation using the ‘LassoCV’ class. LassoCV combines
Lasso regression with efficient hyperparameter tuning,
automatically determining the optimal regularization
strength through cross-validation. This regularization tech-
nique effectively shrinks less significant feature coefficients
to zero, thus performing feature selection and yielding a
more interpretable model. The resulting regression model,
therefore, encapsulates both the linear and quadratic rela-
tionships in the data while maintaining parsimony and
robustness against overfitting.
RESULTS AND DISCUSSION
Correlation Between BBMWI and SSE in the
Hardgrove Test
The results of the BBMWI and Hardgrove tests, consider-
ing only the SSE, are illustrated in Figure 3. The BBMWI
ranged from 3.3 to 31.7, thus covering a wide range of
values. The database organized by Doll, 2023 includes
2384 results of BBMWI tests, 99.9% of which are within
these ranges. The SE parameter ranged from 1.2 kWh/t to
3.0 kWh/t, thus confirming the variation in this parameter
as reported by Austin et al., 1981, Mucsi, 2008, Mucsi et
al., 2019, Shi and Zuo, 2014 and Palaniandy, 2017.
Figures 1a and 1b illustrate the linear and quadratic
correlations obtained with their 95% confidence interval
for the regression as a blue band. The correlation equation
is shown in each plot, along with their respective coefficient
of correlation (R2) and the MAPE (mean absolute predic-
tion error). The correlation is used to predict the BBMWI
as illustrated in Figures 1c and 1d, showing the 5% and
10% error bands in green and red around the 1:1 line.
The predicted BBMWI based solely on the SSE had a
12.2% error on average for the linear model and a 9.9%
error on average for the quadratic model.
Correlation Between BBMWI and the Hardgrove Test
with the Multi-Regression Model
The correlation from the Hardgrove with BBMWI was
then calculated based on SSE and other parameters from
the test’s feed and product. Figures 2a and 2b illustrate the
correlation obtained, showing the 5% and 10% error bands
in green and red around the 1:1 line.
Equations 4 and 5 describe the correlation using the
multi-regression model. The correlations are for the SSE
interval from 2.2 kWh/t to 24.2 kWh/t.
Multi-linear model:
*
*
.*
WI A F
SSE 0 486
75 µm Passing
80 =+-B
+-C
+
(4)
Quadratic model:
**
***
**
**
**sin
WI A B C SSE
D F802 E F80
F F
G
H Pas g SSE
75
5 7
Passing Feed
Passing Feed
Passing
Passing Feed Passing
80
=+-+
+-+
+-
+-
+-
(5)
The average error for BBMWI prediction with the linear
model is 6.7%. In the case of the quadratic model, the aver-
age absolute error is 6.1%. This result contrasts the 12.2
and 9.9% average error obtained in the Hardgrove test con-
sidering only the SSE. The prediction errors are similar to
those reported in the literature using the Hardgrove mill