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1
Development of an analytical model for copper heap leaching from
secondary sulfides in chloride media in an industrial environment
Manuel Saldaña1,2, Eleazar Salinas-Rodríguez3, Jonathan Castillo4, Felipe Peña-Graf5and Francisca Roldán6
1Faculty of Engineering and Architecture, Universidad Arturo Prat, Iquique 1110939, Chile
2Departamento de Ingeniería Química y Procesos de Minerales, Facultad de Ingeniería, Universidad de Antofagasta, Antofagasta
1240000, Chile
3Área Académica de Ciencias de la Tierra y Materiales, Universidad Autónoma del Estado de Hidalgo, Carretera Pachuca—
Tulancingo km. 4.5, C.P. 42184, Mineral de la Reforma, Hidalgo C.P. 42184, Mexico
4Departamento de Ingeniería en Metalurgia, Universidad de Atacama, Copiapó 1531772, Chile
5Escuela de Ingeniería, Universidad Católica del Norte, Coquimbo 1531772, Chile
6Centro de Investigación para la Gestión Integrada del Riesgo de Desastres (CIGIDEN), Departamento de Ciencias Geológicas.
Universidad Católica del Norte (UCN), Antofagasta, Chile
Abstract
In multivariate analysis, a predictive model is a mathematical/statistical model that relates a
set of independent variables to dependent or response variable(s). This work presents a
descriptive model that explains copper recovery from secondary sulfide minerals (chalcocite)
taking into account the effects of time, heap height, superficial velocity of leaching flow,
chloride concentration, particle size, porosity, and effective diffusivity of the solute within
particle pores. Copper recovery is then modelled by a system of first-order differential
equations. The results indicated that the heap height and superficial velocity of leaching flow
are the most critical independent variables while the others are less influential under
operational conditions applied. In the present study representative adjustment parameters are
obtained, so that the model could be used to explore copper recovery in chloride media as a
part of the extended value chain of the copper sulfides processing.
Keywords: copper extraction, phenomenological modeling, chloride leaching, modelling,
hydrometallurgy.
TECHNICAL PAPER
UDC: 669.3.053: 549.331.21:303.094
Hem. Ind. 00(0) 000-000 (2022)
Available on-line at the Journal web address: http://www.ache.org.rs/HI/
1. INTRODUCTION
Copper mining is an industry in constant growth [1]. Currently, 19.7 million tons of copper are produced world-
wide [2], 75 % of which are processed by pyrometallurgical processes, while the rest is processed by hydrometallurgical
routes [3–5]. Pyrometallurgical processes generate large environmental liabilities, such as tailings dams [6–8] produced
by flotation processes, which can affect acid rains and increase local pollution [9,10]. Hydrometallurgical processes,
together with copper bioleaching processes [11–13], have proven to be more environmentally friendly.
Leaching processes in recent years have been used to treat oxidized copper ores, being a useful technological
alternative to treat low-medium grade ores [14–17]. However, oxides available for treatment are becoming scarce
mainly due to overexploitation [18]. In Chile, for example, copper oxides that are processed by this route currently
represent 30.8 % of the country's production and are projected to decline to 12 % of the production by 2027 [2]. Despite
the problem, this option is currently being used not only for oxides, but also for secondary sulfides, especially low-grade
minerals [19,20], or copper sulfide minerals [21–26], like chalcocite or covellite, ores that are processed in acidic
environments with the addition of chlorides [27], found naturally in seawater. This resource has begun to be exploited
in recent decades in Chilean mining [28], mainly due to the situation of water scarcity in the country. It is worth
highlighting the rise in copper leaching in chloride media, finding applications, as stated in the literature, from processing
of secondary copper sulfide minerals [22,23,29,30], to copper smelting slag leaching [31,32]. On the other hand, though
Corresponding authors: Manuel Saldaña, Faculty of Engineering and Architecture, Universidad Arturo Prat, Iquique 1110939, Chile; Tel. +56 9 5383 4174
E-mail: masaldana@unap.cl
Paper received: 14 February 2022; Paper accepted: 16 August; Paper published: 2 September 2022.
https://doi.org/10.2298/HEMIND220214015S
Hem. Ind. 00(0) 000-000 (2022) M. SALDAÑA et al.: COPPER HEAP LEACHING FROM SECONDARY SULFIDES
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leaching of primary copper sulfides, such as chalcopyrite [33,34], is also possible, it is economically infeasible at
industrial scale.
In line with the above, considerable pressure is currently exerted on water resources required for population, urban
and economic growth [35]. Due to water scarcity and the lack of surface and subsurface water recharge, it has become
a priority focus for decision makers in the world [36]. Chile, for its part, has 1,251 rivers, which are in 101 main basins
of the country whose recharge comes mainly from rains. However, it is a country highly vulnerable to climate
change [37]. Reports indicate that there is an increase in temperature [38–40], increase in wind intensity [38,41],
appearance of flow-type landslides [42] and a decrease in rainfall [38,39,41]. Considering that a large part of the Chilean
economy is based on the economic contribution of the mining sector [43] and flotation metallurgical processes being
the ones that consume the most water in the country, promoting hydrometallurgy can help in this regard [44]. In turn,
the mining nucleus of Chile is located in the Atacama Desert, characterized as the driest desert in the world. This extreme
aridity is due to the fact that in the western regions there is a combination of the barrier effect of the high Andes
Mountain Range, permanence of the Southeast Pacific anticyclone and the existence of the Humboldt Current-coastal
upwelling system that prevents this region from receiving the moist Atlantic air masses. Its scant rainfall is due to climatic
anomalies, the main one being the ocean-climatic anomaly ENSO (El Niño-Southem Oscillation) [45,46]. On the other
hand, in the eastern highland regions, recharge is due to summer rainfall, fed by moisture of Amazonian origin, which
became, in certain years, very abundant [45]. The groundwater recharge comes from precipitation, melting ice or
lacustrine origin from the Andes Mountains and surrounding areas [47].
Mean annual rainfall in the Atacama Desert is less than 20 mm and has been a desert for 12000 ± 1000 years shifting
between arid and hyper-arid periods [48]. Accordingly, fluvial run-off is minimal and erosion rates are extremely low
ranging from 0.2 to 0.4 m Ma-1 [49]. Therefore, more environmentally friendly forms of copper extraction should be
sought in Chilean mining, providing water reuse, such as hydrometallurgical processes used in the country today.
Leaching processes include several stages. First, the ore (chalcocite) is crushed until it reaches a size under 1 in [14].
Then, the crushed ore is transported to the pile, having a height between 4 to 10 m [50]. Finally, the ore is irrigated by
a leaching solution distributed by sprinklers or drip emitters, flowing down through the heap by gravity [14]. At the end
of the process, cupric ions are obtained together with other ions dissolved in the PLS (Pregnant Liquid Solution), which
is deposited in leaching pools, and subsequently advanced to the solvent extraction stage [51].
In this work, an analytical model for estimation of copper recovery from chalcocite in the heap leaching process is
derived. Creation of analytical models (theoretical and/or empirical) representing dynamics of mining processes such as
heap leaching are of vital importance for study of the process performance at the operational level [52], since such
models can support development, verification, testing and application of new specialized technologies related to
process innovation, different modes of operation, or operational efficiency. This work presents the theoretical
framework supporting the analytical model for copper recovery estimation, fundamentals of the uncertainty analysis
and the model optimization process followed by application of the optimized analytical model, the uncertainty analysis,
and discussion of the model results.
2. MATERIALS AND METHODS
2. 1. Chalcocite Heap Leaching
Heap leaching data at the operational level were recovered from a mine with the copper ore grade of 0.6 %
approximately, located in the Antofagasta region, Chile. Samples of copper recovery were recorded for a period of two
years approximately, in processes with different levels of the factors: heap height (300 - 840 cm), particle size
(15 - 33 mm), porosity (1 -5.5), effective diffusivity of the solute within the particle pores (0.06 - 0.108 cm2 day-1),
superficial velocity of leaching flow (9 - 54 cm day-1) and chloride concentrations (20 - 50 g dm-3).
M. SALDAÑA et al.: COPPER HEAP LEACHING FROM SECONDARY SULFIDES Hem. Ind. 00(0) 000-000 (2022)
3
2. 2. Analytical models for heap leaching
There are several copper recovery models in literature [53,54], while the analytical model used in this work is given
by Eq. (1) [55–58], which is based on the hypothesis that the leaching process could be modelled by using a system of
first-order differential equation (1).
=−
n
yke
(1)
where y is a dynamic quantity, such as concentration or recovery R
, k
is the kinetic constant and n
is the order of the
reaction and
represent a time scale that depends on the phenomenon to model. To solve Eq. (1), an initial condition
is required, introducing a delay parameter
. Then, the general solution for this problem is known, as n
= 1, and the
solution is given by the Eq. (2).
R
= R
(1 - e-k
(
-) (2)
R
is the maximum expected recovery depending on the experimental conditions, and
is a factor of reaction delay
(associated with the activation time; generally, this period is minimal or is considered as 0). Dixon and Hendrix [59–61]
considered that the leaching process occurs at different scales of size and time with participation of different
phenomena [58]. It is possible to represent these phenomena by using Eq. (2) in conjunction with expressions associated
with the particle properties and the heap height in the leaching process. At the particle level
(see Eq. 3), the authors
considered that the process was dominated by the porosity
0 of the feeding material, the effective diffusivity of solute
within the particle pores DAe, particle size (radius, r) and t is time.
=Ae
2
0
Dt
r
(3)
At the bulk level
, the authors considered that the heap is porous, formed by particles through which the leaching
solution flows at a constant rate. Recovery can be defined based on the heap height as is presented in Eq. (4).
=s
b
t
Z
(4)
Where
s is the superficial velocity of the leaching flow,
b is the volumetric fraction of the leaching solution in the bed
and Z is the pile height. Eq. (2) could be rewritten to include both dependences of the reaction.
Then, considering the evident proportional relation between the copper recovery from secondary sulfides and
chloride concentration [20,26,27,62,63], the term
is incorporated into the analytical model as factors of both scales,
as shown in Equations (5) and (6). The term
i (chloride concentration, cCl) is defined as the potency of the fraction of
the sampled chloride concentration (xi) over the average concentration (x) raised to a mathematical adjustment
constant
i, where i are the particle levels and heap height.
−−
=−
Ae
12
0
ˆ
1
D
kt
r
R R e
(5)
−−
=−
s
2
b
1kt
Z
R R e
(6)
Including both scales in an aggregate model, Mellado et al. [58] assumed that the total recovery is the sum of both
recoveries (, R = R
+ R
, see Eq. 7), which shows asymptotic behaviour over time.
( )
−−
−−
= − − −
Ae
s1
22
b0
ˆ
11
D
kt
kt
Zr
R R e e
(7)
where
ˆ
represents the delay on the scale of
, and its relation to
is given by Eq. (8) [58], and
represents the kinetic
weight factor. The analytical model for R used by Mellado et al. [58] is presented by Eq. (9).
Hem. Ind. 00(0) 000-000 (2022) M. SALDAÑA et al.: COPPER HEAP LEACHING FROM SECONDARY SULFIDES
4
=Ae b
2s
0
ˆDZ
r
(8)
=+
RZ
(9)
Where
,
and
are mathematical adjustment coefficients, and Z is the heap height. Finally, the model for copper
heap leaching for copper sulfide minerals is given by Eq. (10).
( )
−−
−−
= − − −
+
Ae b
sb 1
2τ
θ2s
bs 0
11
DZ
Zkt
kt
Zr
R e e
Z
(10)
2. 3. Uncertainty analysis
The uncertainly analysis (UA) determines the uncertainty of output variables due to the uncertainty of input variables
[64]. It is generally performed by using the probability theory [65], where uncertainty is represented by probability
distribution functions (PDF) and can be realized in four steps. First, the PDF type and the uncertainty magnitude for each
input variable are determined, i.e. the input uncertainty is characterized. Second, for each input variable, a sample of
the PDF is generated. Third, output variable values are determined for each element sampled. Finally, the results are
analysed by using graphs, descriptive statistics, and statistical tests to characterize the behaviour of the output variables.
When the input variables have epistemic uncertainties, the uncertainty can be represented by a uniform distribution.
Design and operation variables present this type of uncertainty. When the input variables have stochastic uncertainties,
normal distribution is generally used to represent this type of uncertainty [66].
2. 4. Optimization of the experimental model
A multivariate function of seven independent variables and one dependent variable is defined from an inverse
exponential model, a statistical analysis, and the multiple regression adjustment is presented in Eq. (11).
R(X) = f(X) (11)
Formalizing the optimization model, the objective is to maximize copper recovery considering the range of sampled
values as domain of the input variables [67] expressed in Eq. (12).
Max {R(X)} (12)
The operational constraints for the
n
independent variables are shown in Eq. 13:
ximin ≤ xi ≤ ximax i X X = {x1, x2…, xn} (13)
Considering the nature of the analytical model developed, it is proposed to optimize it using the Lagrange multipliers
technique, which provides determination of the optimal values of a multivariate function when there are one or more
restrictions on the input parameters.
3. RESULTS AND DISCUSSIONS
3. 1. Analysis of the effects of main independent variables
Analysis of the data generated by the factorial model, also used for the fit of the analytical model presented by
Eq. (7), indicated only four factors that have main effects on the response variable: the heap height, superficial velocity
of leaching flow and chloride concentration. It is corroborated that the superficial velocity of leaching flow in the bed is
the variable with the greatest impact (of the analyzed variables) on the copper recovery as shown in Figure 1.
Figure 2 shows that copper recovery increases as superficial velocity of lixiviant flow and chloride concentration are
increased, while decreases at higher heap heights.
M. SALDAÑA et al.: COPPER HEAP LEACHING FROM SECONDARY SULFIDES Hem. Ind. 00(0) 000-000 (2022)
5
Figure 1. Main effects of independent variables heap height Z (a), superficial velocity of lixiviant flow
s (b), effective diffusivity DAe (c),
porosity e
0 (d), particle ratio r (e), leaching time t (f) and Cl concentration cCI (g), in copper extraction R from sulfide minerals
a
b
c
Figure 2. Contour plot of copper recovery versus the heap height and superficial velocity (a); heap height and chloride concentra-
tion (b); and, chloride concentration and superficial velocity (c).
3. 2. Uncertainty analysis
Eq. (10) included the copper recovery dependence on the following operational variables: leaching time, particle
size, heap height, particle porosity, superficial velocity of the leaching flow and effective diffusivity of the solute through
particle pores. A proper distribution function to represent epistemic uncertainties is the uniform PDF, taking the values
of x1, x2…, xn, as the central tendency values of the parameters p1, p2…, pn, respectively. The sensitivity analysis was
performed 3 times for leaching times of 30, 60 and 90 days (see Fig. 3).
Hem. Ind. 00(0) 000-000 (2022) M. SALDAÑA et al.: COPPER HEAP LEACHING FROM SECONDARY SULFIDES
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Figure 3. Probability density (a) and normal Q-Q plot (b) of UA at 30 days of leaching
Figure 4. Probability density (a) and normal Q-Q plot (b) of UA at 60 days of leaching
Figure 5. Probability density (a) and normal Q-Q plot (b) of UA at 90 days of leaching
UA shows that leaching time affects the kinetic uncertainty of the heap leaching phase. Histograms show that
uncertainty in the input variables produces lower uncertainty in recovery as the leaching time is increased from 30 to
90 days. The normal distribution plots (Q-Q plot) indicate that copper recovery normally behaves for all investigated
leaching times. The recovery presents a normal PDF with averages of 54.89 ± 1.76, 67.53 ± 1.03, and 71.29 ± 0.78 %. On
the other hand, the normal probability plot indicates that for the leaching time of 90 days the recovery has a tail that
deviates from the normal behaviour. The change in the recovery behaviour as a function of the time factor is due to the
tendency to asymptotic behaviour.
3. 3. Analytical model adjustment
The fit of the analytical model developed by Mellado et al. [57] by optimization algorithms [67] indicates that the
copper recovery from sulfide minerals can be explained by Eq. (14), where the volumetric fraction of the solution in the
bed is assumed as
b = 0.015, while the delay of the reaction is assumed as
= 1.
M. SALDAÑA et al.: COPPER HEAP LEACHING FROM SECONDARY SULFIDES Hem. Ind. 00(0) 000-000 (2022)
7
−−
−−
= − −
+
2.081 Ae
1.81 s
2s
s0
0.015
0.015 2.3
0.027 0.015
0.042
138.83 1 0.705 0.295
0.038
DZ
Zt
t
Zr
R e e
Z
(14)
The analytical model presented by this equation is validated by the goodness-of-fit statistics Mean Absolute Error
(MAD), Mean Squared Error (MSE) and Mean Absolute Percent Error (MAPE), with values of 1.147×10-2, 2.475×10-4 and
0.793 %, respectively. The error statistics indicates that the fitted model explains the system under the set of sampled
values. The Sensitivity analysis of the dependent variables that model the copper recovery for low, medium, and high
levels operational conditions, check again that the variables that influence the copper recovery are the heap height (see
Fig. 6a), chloride concentration (see Fig. 6b), and superficial velocity of lixiviant flow (see Fig. 6c). The impact of the
other variables at different levels is considered are negligible (see Figures 6d, 6e, and 6f).
Figure 6. Sensitivity analysis of copper recovery for low, medium, and high levels of the factors: heap height (a), chloride concen-
tration (b), superficial velocity of lixiviant flow (c), effective diffusivity within particle pores (d), porosity (e), and particle size (f)
Finally, comparison of the theoretical results in the literature and industrial heap processes, shows that despite the fact
that the heap height is inversely proportional to the recovery, the heights of industrial heaps are at high levels. The increase
in the height of commercial heaps is due to economic efficiency, which is the function of the available surface area [14]. The
factors that influence the process kinetics the most are the percolation rate and chloride concentration as the leaching agent,
of which, the percolation rate is directly related with the heap height and its permeability [51]. Also, the time factor has a
strong impact on the copper recovery, being initially rapid recovery and showing asymptotic behaviour as time increases.
The current low ore grades reduce profitability of other recovery technologies due to requirements of greater
comminution, temperature [68], or operating times. Those variables may change depending on the working material
and the recovery standards of a mining company [66] severely impacting the projected profits. Therefore, mathematical
models that provide optimization of the process would be an added value to the study and improvement of ore recovery
in industrial contexts.
3. 4. Optimization of the analytical model
Optimization of the analytical model adjusted in the previous section for the set of sampled values was performed
by using the Python library “SciPy Optimize” (Version 3.7.0). This library provides several methods to minimize/maximize
objective functions subject to constraints.
Hem. Ind. 00(0) 000-000 (2022) M. SALDAÑA et al.: COPPER HEAP LEACHING FROM SECONDARY SULFIDES
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Table 1 presents the operational restrictions associated with the independent variables in the form of lower and
upper limit values. Optimal values for each variable are also presented.
The graphical analysis of the copper recovery response surface (see Fig. 7), model the recovery over time in the form
of an inverse exponential function tending to become asymptotic between 80 and 100 days of leaching.
a
b
c
d
e
f
Figure 7. Response surface of copper recovery from sulfide minerals versus: time and the heap height (a); time and the superficial
velocity of the leaching flow (b); time and the chloride concentration (c); heap height and the superficial velocity (d); heap height
and the chloride concentration (e); superficial velocity and the chloride concentration (f)
M. SALDAÑA et al.: COPPER HEAP LEACHING FROM SECONDARY SULFIDES Hem. Ind. 00(0) 000-000 (2022)
9
Table 1 Limits and optimal values of the independent variables
Variable
Lower limit
Upper limit
Optimal value
Z / cm
300
900
900
s/ cm3 cm-2 d-1
9
54
54
DAe/ cm3 cm-1 d-1
0.06
0.108
0.108
0 / %
1
5.5
1
r / mm
15
33
15
t / day
0
90
90
cCl/ g dm-3
20
50
50
R / %
-
-
72.12
This value for leaching time is common in hydrometallurgical processes in the Chilean mining industry. On the other
hand, at the lower heap height, a greater recovery is observed, which is due to the improvement in the efficiency of the
percolation of the leaching flow through the heap. And finally, additional surface plots in Figure 7 model the dynamics
of response variable as functions of significant independent variables.
Additionally, future works could incorporate machine learning techniques for process modelling, simulation and
optimization [69]. Also, integration of the model presented here along with discrete events simulation models, (e.g.
[63,70]), aimed to optimize the mineral recovery, while incorporating the feeding mineralogical variation, different
process operation modes by variation of reagents or the levels of the variables and/or other operational parameters,
has the potential to add value to the study of leaching dynamics of sulfide copper minerals.
4. CONCLUSIONS
The present investigation presents the results of an analytical model of copper extraction from a sulfide mineral
(chalcocite) as a system of first-order differential equations through an inverse exponential function. The main findings
in this study are summarized below.
The behavior of heap leaching process of copper sulfide minerals (secondary sulfides), like chalcocite, can be
modelled as a system of first-order differential equations, which is validated by the goodness-of-fit statistics.
Particle size, porosity, and effective diffusivity of the solute within the particle pores are not as significant in the
copper extraction, as is leaching time, heap height, superficial velocity of the leaching flow and chloride concentration
in the leaching solution.
The best result predicted by optimizing the generated analytical model would be obtained by working at high
chloride concentration and a high leaching flow rate.
On the other hand, the fit of this type of analytical models can be extended to other variable levels or factors, such
as inclusion of different leaching agents. The model would be modified then according to the kinetics that describe or
dominate the operation.
Generation of analytical models to represent complex processes, such as mineral leaching, has the potential to be
used for analysis, generalization, and optimization tasks, since these models capture the essence of the modelled
process and could be used to predict the response under operational conditions, identifying the conditions that
maximize the aggregate process productivity.
Acknowledgements: Manuel Saldaña acknowledges the infrastructure and support of Doctorado en Ingeniería de
Procesos de Minerales of the Universidad de Antofagasta. Francisca Roldán acknowledges the support of
ANID/FONDAP/15110017 project.
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Razvoj analitičkog modela za izluživanje bakra iz sekundarnih sulfida u
hloridnim medijima u industrijskom okruženju
Manuel Saldaña1,2, Eleazar Salinas-Rodríguez 3, Jonathan Castillo 4, Felipe Peña-Graf 5 i Francisca Roldán6
1Faculty of Engineering and Architecture, Universidad Arturo Prat, Iquique 1110939, Chile
2Departamento de Ingeniería Química y Procesos de Minerales, Facultad de Ingeniería, Universidad de Antofagasta, Antofagasta
1240000, Chile
3Área Académica de Ciencias de la Tierra y Materiales, Universidad Autónoma del Estado de Hidalgo, Carretera Pachuca—
Tulancingo km. 4.5, C.P. 42184, Mineral de la Reforma, Hidalgo C.P. 42184, Mexico
4Departamento de Ingeniería en Metalurgia, Universidad de Atacama, Copiapó 1531772, Chile
5Escuela de Ingeniería, Universidad Católica del Norte, Coquimbo 1531772, Chile
6Centro de Investigación para la Gestión Integrada del Riesgo de Desastres (CIGIDEN), Departamento de Ciencias Geológicas.
Universidad Católica del Norte (UCN), Antofagasta, Chile
(Naučni rad)
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