XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3639
to the Ga circuit, which includes three cleaner stages and
one scavenger. Finally, the tail of the Pb circuit is fed to the
third circuit to recover Sp.
Design Based on Optimization
The procedure used by Cisternas et al. (2015) and Botero et
al. (2024) was used to design the circuits. Basically, a super-
structure representing a set of possible circuits is defined
using an origin-destination matrix of the concentrate and
tail streams. The superstructure is represented mathemati-
cally by mass balances of all components, including binary
variables (0 or 1) that represent the destination selection of
the concentrate and tailings streams. The objective func-
tion used is the profits from the sale of concentrate, which
include refining and treatment costs of the concentrates.
An important aspect is the assumption that the recoveries
at each stage are known, which allows obtaining the global
optimum. In this case study, the recoveries are indicated
in Table 1. This assumption is justified by the studies of
Botero et al. (2024) and Cisternas et al. (2015), which
show that few circuits are optimal despite the uncertainty
in the recovery of each stage. Then, it is possible to identify
a set of optimal circuits (for example, the first four) with a
high probability of selecting the best ones for the analyzed
problem. Only the global optimal circuit is discussed here
for simplicity.
The optimization problem is solved with three different
strategies. First, each circuit is designed at a time that is,
the Cp circuit is designed first, and the tail obtained is used
in the design of the Ga circuit, and then, the Ga circuit tail
in the design of the Sp circuit. We call it a “One-at-time
design” for this strategy. The second strategy is to solve the
three circuits at the same time, optimizing the total rev-
enues. However, the connection between the circuits is the
tail streams: the Cp circuit tail feeds the Ga circuit, and
the Ga circuit tail feeds the Sp circuit. We call this strat-
egy “Simultaneous design” to this second strategy. Finally,
a simultaneous design is applied to a superstructure where
several stream connections are possible, including stream
recycling from the downstream plant to the upstream plant.
This strategy is called an “integrated design.”
Bottleneck Identification Based on Uncertainty and
Sensitivity Analyses
Global sensitivity analysis (GSA) is used to identify bottle-
necks. The GSA consists of identifying which input vari-
ables (recoveries in the flotation stages) are those that most
affect the recovery and grade of the valuable species (output
variables). To do this, uncertainty is assigned to the recov-
eries of each stage, represented by uniform distribution
functions, U(recovery lower bound, recovery upper
bound). Here a variation of ± 5% of the sulfur species and
±3.5% of the non-sulfur gangue in the stage recovery val-
ues indicated in Table 1 was used to determine those upper
and lower bounds. These ranges in recovery values repre-
sent the possible values to be obtained depending on the
adjustment of the operational conditions (pH, residence
time, airflow, etc.) and/or design (type and number of cells,
type of reagents). The procedure presented by Sepúlveda
et al. (2014) is used with the Sobol-Jensen method. In the
Sobol-Jensen method, the Sobol indices are identified for
each input variable, the interpretation of which is simple:
the higher its value, the greater the effect of the uncertainty
of that variable on the output variable. Thus, the variables
with the highest Sobol index values are the critical variables
or bottlenecks to improve the circuit. Here, the procedure
is applied to the circuit designed under the “integrated
design” strategy.
RESULTS
Optimal Design
The optimization results are provided in Table 2. The one-
at-time design and simultaneous design delivered the same
results. This was not expected, given that the one-at-time
design is a local optimum of the simultaneous design prob-
lem. The fact that the copper concentrate had a better value
and that the recoveries of lead and zinc were low in the cop-
per circuit helped achieve this result. The circuit obtained is
similar to the circuit in Figure 1 without optimization, the
copper circuit being the same. For the Ga and Sp circuits,
small differences were obtained in the direction of the tail
streams of the second cleaning stages, which were sent to
the rougher stage instead of the first cleaning stage. Also,
the concentrate from the scavenger stage of the Ga circuit
was sent to the second cleaning stage instead of the first
stage.
The integrated design presents revenues of 131.6 MM
USD/year, which represents an increase of over 9% com-
pared to the one-at-time and the simultaneous designs.
This increase in revenues is due to an improvement in the
recoveries of the polymetallic circuit. This is explained, as
shown in Figure 2, by the recirculation of the streams of the
non-float products of the first cleaner stage from the lead
and zinc circuits to the copper circuit. This type of recycling
is not common in practice due to the possible problems
caused by the chemistry of these systems. Moving forward,
part of the future challenge is to understand how to manage
those complexities. For example, when comparing with the
circuit in Figure 1, it is possible to consider regrinding of
to the Ga circuit, which includes three cleaner stages and
one scavenger. Finally, the tail of the Pb circuit is fed to the
third circuit to recover Sp.
Design Based on Optimization
The procedure used by Cisternas et al. (2015) and Botero et
al. (2024) was used to design the circuits. Basically, a super-
structure representing a set of possible circuits is defined
using an origin-destination matrix of the concentrate and
tail streams. The superstructure is represented mathemati-
cally by mass balances of all components, including binary
variables (0 or 1) that represent the destination selection of
the concentrate and tailings streams. The objective func-
tion used is the profits from the sale of concentrate, which
include refining and treatment costs of the concentrates.
An important aspect is the assumption that the recoveries
at each stage are known, which allows obtaining the global
optimum. In this case study, the recoveries are indicated
in Table 1. This assumption is justified by the studies of
Botero et al. (2024) and Cisternas et al. (2015), which
show that few circuits are optimal despite the uncertainty
in the recovery of each stage. Then, it is possible to identify
a set of optimal circuits (for example, the first four) with a
high probability of selecting the best ones for the analyzed
problem. Only the global optimal circuit is discussed here
for simplicity.
The optimization problem is solved with three different
strategies. First, each circuit is designed at a time that is,
the Cp circuit is designed first, and the tail obtained is used
in the design of the Ga circuit, and then, the Ga circuit tail
in the design of the Sp circuit. We call it a “One-at-time
design” for this strategy. The second strategy is to solve the
three circuits at the same time, optimizing the total rev-
enues. However, the connection between the circuits is the
tail streams: the Cp circuit tail feeds the Ga circuit, and
the Ga circuit tail feeds the Sp circuit. We call this strat-
egy “Simultaneous design” to this second strategy. Finally,
a simultaneous design is applied to a superstructure where
several stream connections are possible, including stream
recycling from the downstream plant to the upstream plant.
This strategy is called an “integrated design.”
Bottleneck Identification Based on Uncertainty and
Sensitivity Analyses
Global sensitivity analysis (GSA) is used to identify bottle-
necks. The GSA consists of identifying which input vari-
ables (recoveries in the flotation stages) are those that most
affect the recovery and grade of the valuable species (output
variables). To do this, uncertainty is assigned to the recov-
eries of each stage, represented by uniform distribution
functions, U(recovery lower bound, recovery upper
bound). Here a variation of ± 5% of the sulfur species and
±3.5% of the non-sulfur gangue in the stage recovery val-
ues indicated in Table 1 was used to determine those upper
and lower bounds. These ranges in recovery values repre-
sent the possible values to be obtained depending on the
adjustment of the operational conditions (pH, residence
time, airflow, etc.) and/or design (type and number of cells,
type of reagents). The procedure presented by Sepúlveda
et al. (2014) is used with the Sobol-Jensen method. In the
Sobol-Jensen method, the Sobol indices are identified for
each input variable, the interpretation of which is simple:
the higher its value, the greater the effect of the uncertainty
of that variable on the output variable. Thus, the variables
with the highest Sobol index values are the critical variables
or bottlenecks to improve the circuit. Here, the procedure
is applied to the circuit designed under the “integrated
design” strategy.
RESULTS
Optimal Design
The optimization results are provided in Table 2. The one-
at-time design and simultaneous design delivered the same
results. This was not expected, given that the one-at-time
design is a local optimum of the simultaneous design prob-
lem. The fact that the copper concentrate had a better value
and that the recoveries of lead and zinc were low in the cop-
per circuit helped achieve this result. The circuit obtained is
similar to the circuit in Figure 1 without optimization, the
copper circuit being the same. For the Ga and Sp circuits,
small differences were obtained in the direction of the tail
streams of the second cleaning stages, which were sent to
the rougher stage instead of the first cleaning stage. Also,
the concentrate from the scavenger stage of the Ga circuit
was sent to the second cleaning stage instead of the first
stage.
The integrated design presents revenues of 131.6 MM
USD/year, which represents an increase of over 9% com-
pared to the one-at-time and the simultaneous designs.
This increase in revenues is due to an improvement in the
recoveries of the polymetallic circuit. This is explained, as
shown in Figure 2, by the recirculation of the streams of the
non-float products of the first cleaner stage from the lead
and zinc circuits to the copper circuit. This type of recycling
is not common in practice due to the possible problems
caused by the chemistry of these systems. Moving forward,
part of the future challenge is to understand how to manage
those complexities. For example, when comparing with the
circuit in Figure 1, it is possible to consider regrinding of