XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2443
and perchloric acid (at 150°C), followed by hydrochlo-
ric acid (at 200°C). The digested solutions were filtered,
diluted, and analyzed for Cu by atomic absorption spec-
trophotometry in the SpectrAA 55 spectrometer (Varian
Inc., USA). Data reconciliation was performed based on
the measured Cu grades and the incremental mass recov-
eries, following the procedure described by Vinnett et al.
(2016). Consistent time-recovery data were thus obtained,
with kinetic responses calculated from the average of three
replicates.
MODEL FITTING AND SENSITIVITY
ANALYSIS
Non-linear regression was implemented to obtain the
model parameters. These regressions were performed in
Matlab, using the Statistics and Machine Learning Toolbox
version 12.3 (The MathWorks, USA). Once the model
parameters (e.g., R∞, k, aG, kG) were obtained for the entire
time-recovery dataset (baseline, with n datapoints), these
parameters were used as initial values to perform a sensitiv-
ity analysis under sub-sampled time-recovery datasets. This
analysis was carried out by omitting one data point at a
time, re-estimating the model parameters (Ni et al., 2018)
as illustrated in Figure 1. In this schematic, a cross indicates
that that specific datapoint was skipped from the dataset in
the new parameter estimation.
RESULTS AND DISCUSSION
The model structures of Equations (1), (2) and (4) were
fitted to the entire time-recovery dataset for the estimation
of R∞ and f(k) in the baseline. These R∞-f(k) pairs were used
as comparison references when subsampling the kinetic
responses. Figure 2 shows the R(t) modelling assuming the
Single Flotation Rate, Rectangular and Gamma models. A
poor characterization was obtained by assuming the deter-
ministic flotation rate of Equation (1).
Table 1 shows the model parameters along with the
mean squared errors (MSE) and the adjusted R2 (R2
Adj )
values. The Single Flotation Rate model proved to have
an erratic performance, with the Gamma model present-
ing the goodness-of-fit in terms of the MSE and R2
Adj .
This result is attributed to the additional parameter of the
Gamma model, which provides more flexibility, controlling
the overfitting. This flexibility allows the Gamma model to
represent high and low flotation rates in the same flotation
feed, with a variety of proportions.
Figure 1. Schematic of the procedure conducted in the sensitivity analysis
Table 1. Parameters of the Single Flotation Rate, Rectangular, and Gamma models
Single Flotation Rate Rectangular Gamma
R
∞ ,%76.98 R
∞ ,%79.81 R
∞ ,%80.17
k, 1/min 1.12 kmax, 1/min 2.70 kG, min 1.38
aG 1.19
MSE 12.0 MSE 0.940 MSE 0.327
R2
Adj 0.951 R2
Adj 0.996 R2
Adj 0.999
and perchloric acid (at 150°C), followed by hydrochlo-
ric acid (at 200°C). The digested solutions were filtered,
diluted, and analyzed for Cu by atomic absorption spec-
trophotometry in the SpectrAA 55 spectrometer (Varian
Inc., USA). Data reconciliation was performed based on
the measured Cu grades and the incremental mass recov-
eries, following the procedure described by Vinnett et al.
(2016). Consistent time-recovery data were thus obtained,
with kinetic responses calculated from the average of three
replicates.
MODEL FITTING AND SENSITIVITY
ANALYSIS
Non-linear regression was implemented to obtain the
model parameters. These regressions were performed in
Matlab, using the Statistics and Machine Learning Toolbox
version 12.3 (The MathWorks, USA). Once the model
parameters (e.g., R∞, k, aG, kG) were obtained for the entire
time-recovery dataset (baseline, with n datapoints), these
parameters were used as initial values to perform a sensitiv-
ity analysis under sub-sampled time-recovery datasets. This
analysis was carried out by omitting one data point at a
time, re-estimating the model parameters (Ni et al., 2018)
as illustrated in Figure 1. In this schematic, a cross indicates
that that specific datapoint was skipped from the dataset in
the new parameter estimation.
RESULTS AND DISCUSSION
The model structures of Equations (1), (2) and (4) were
fitted to the entire time-recovery dataset for the estimation
of R∞ and f(k) in the baseline. These R∞-f(k) pairs were used
as comparison references when subsampling the kinetic
responses. Figure 2 shows the R(t) modelling assuming the
Single Flotation Rate, Rectangular and Gamma models. A
poor characterization was obtained by assuming the deter-
ministic flotation rate of Equation (1).
Table 1 shows the model parameters along with the
mean squared errors (MSE) and the adjusted R2 (R2
Adj )
values. The Single Flotation Rate model proved to have
an erratic performance, with the Gamma model present-
ing the goodness-of-fit in terms of the MSE and R2
Adj .
This result is attributed to the additional parameter of the
Gamma model, which provides more flexibility, controlling
the overfitting. This flexibility allows the Gamma model to
represent high and low flotation rates in the same flotation
feed, with a variety of proportions.
Figure 1. Schematic of the procedure conducted in the sensitivity analysis
Table 1. Parameters of the Single Flotation Rate, Rectangular, and Gamma models
Single Flotation Rate Rectangular Gamma
R
∞ ,%76.98 R
∞ ,%79.81 R
∞ ,%80.17
k, 1/min 1.12 kmax, 1/min 2.70 kG, min 1.38
aG 1.19
MSE 12.0 MSE 0.940 MSE 0.327
R2
Adj 0.951 R2
Adj 0.996 R2
Adj 0.999