XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 917
into the plant model, enabling to simulate the impact of
the short-loop recycling on plant performance, as well as
to determine the type and sizing of water treatment units
that should be considered to maintain or even optimize
this performance. Finally, the results of the simulations are
analyzed and discussed, leading to the identification of the
limits of the modeling approach and possible improvement
perspectives.
MODELLING APPROACH
Dissolution
The experimental results obtained by Le, Schreitofer and
Dahl (2020) are fitted by simulating a series of 8 batch reac-
tors. To simulate dissolution, the first order kinetic model
of Noyes and Whitney is used. It has two parameters: the
kinetic constant k and the maximum concentration Cmax
(Dokoumetzidis and Macheras 2006). The rate of dissolu-
tion is proportional to the difference between the instan-
taneous concentration, C at time t, and the saturation
solubility, Cmax.
C C C C kx
kx
1 max output input input =+-+^h (1)
Thus, integration provides the equation for a dissolution
test loop, considering a batch reactor:
R R e =-
3
-kt ^th ^1 h (2)
where τ is the total residence time. For this experiment,
27 min are spent for grinding stage, and 50 min for dis-
solution stage simulating the residence time in the flotation
circuit. In this paper, focus is made on sulfates, magnesium,
and calcium.
The dissolution model calibration back-calculates the
kinetic constant and the maximum concentration fitting
as well as possible the experimental observations. But, in
the current case, there is a large range of possible values
giving suitable fit. Table 1 gives some pair of such opti-
mal parameters (the errors correspond to the mean square
deviation) and Figure 3 displays the corresponding fits for
sulfates. Compared to Figure 1, the fitting curves are only
linking the concentration values at the end of each cycle,
for a better comparison.
Parameters corresponding to the different fits and
related error are summed up on Table 1.
This parameter determination difficulty can be math-
ematically explained from the model equation. Indeed,
as the observed concentrations remain significantly lower
than the maximum, and as long as kτ 1, the equation
for batch reactor is simplified to:
C C k
max output input xC =+(3)
This linear form explains coupling between k and Cmax.
Same products of these two values provide similar fitting
but different impact when applied to the plant simulation.
One important highlight of the project was the difficulty
to reliably assess process water quality (Le et al., 2020).
Water quality analyses are thus associated to uncertainty,
and so are the calibrated parameters of the model. It seems
consequently worth keeping all these pairs of parameters
Figure 2. Sulfate concentration measured during dissolution test (dots) and
calculated sulfate concentration using dissolution model (dot-line). Test conditions:
20°C—Distilled Water (DW)
into the plant model, enabling to simulate the impact of
the short-loop recycling on plant performance, as well as
to determine the type and sizing of water treatment units
that should be considered to maintain or even optimize
this performance. Finally, the results of the simulations are
analyzed and discussed, leading to the identification of the
limits of the modeling approach and possible improvement
perspectives.
MODELLING APPROACH
Dissolution
The experimental results obtained by Le, Schreitofer and
Dahl (2020) are fitted by simulating a series of 8 batch reac-
tors. To simulate dissolution, the first order kinetic model
of Noyes and Whitney is used. It has two parameters: the
kinetic constant k and the maximum concentration Cmax
(Dokoumetzidis and Macheras 2006). The rate of dissolu-
tion is proportional to the difference between the instan-
taneous concentration, C at time t, and the saturation
solubility, Cmax.
C C C C kx
kx
1 max output input input =+-+^h (1)
Thus, integration provides the equation for a dissolution
test loop, considering a batch reactor:
R R e =-
3
-kt ^th ^1 h (2)
where τ is the total residence time. For this experiment,
27 min are spent for grinding stage, and 50 min for dis-
solution stage simulating the residence time in the flotation
circuit. In this paper, focus is made on sulfates, magnesium,
and calcium.
The dissolution model calibration back-calculates the
kinetic constant and the maximum concentration fitting
as well as possible the experimental observations. But, in
the current case, there is a large range of possible values
giving suitable fit. Table 1 gives some pair of such opti-
mal parameters (the errors correspond to the mean square
deviation) and Figure 3 displays the corresponding fits for
sulfates. Compared to Figure 1, the fitting curves are only
linking the concentration values at the end of each cycle,
for a better comparison.
Parameters corresponding to the different fits and
related error are summed up on Table 1.
This parameter determination difficulty can be math-
ematically explained from the model equation. Indeed,
as the observed concentrations remain significantly lower
than the maximum, and as long as kτ 1, the equation
for batch reactor is simplified to:
C C k
max output input xC =+(3)
This linear form explains coupling between k and Cmax.
Same products of these two values provide similar fitting
but different impact when applied to the plant simulation.
One important highlight of the project was the difficulty
to reliably assess process water quality (Le et al., 2020).
Water quality analyses are thus associated to uncertainty,
and so are the calibrated parameters of the model. It seems
consequently worth keeping all these pairs of parameters
Figure 2. Sulfate concentration measured during dissolution test (dots) and
calculated sulfate concentration using dissolution model (dot-line). Test conditions:
20°C—Distilled Water (DW)