optimal design and planning of heap leaching process. application to copper oxide leaching

7/27/2019 Optimal Design and Planning of Heap Leaching Process. Application to Copper Oxide Leaching

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Title: Optimal design and planning of heap leaching process.application to copper oxide leaching

Author: Jorcy Y. Trujillo Luis A. Cisternas Edelmira D.

Galvez Mario E. Mellado

PII: S0263-8762(13)00323-7

DOI: http://dx.doi.org/doi:10.1016/j.cherd.2013.07.027

Reference: CHERD 1328

To appear in:

Received date: 18-7-2012

Revised date: 9-6-2013

Accepted date: 25-7-2013

Please cite this article as: Trujillo, J.Y., Cisternas, L.A., Galvez, E.D., Mellado,

M.E., OPTIMAL DESIGN AND PLANNING OF HEAP LEACHING PROCESS.

APPLICATION TO COPPER OXIDE LEACHING, Chemical Engineering Research

and Design (2013), http://dx.doi.org/10.1016/j.cherd.2013.07.027

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OPTIMAL DESIGN AND PLANNING OF HEAP

LEACHING PROCESS. APPLICATION TO COPPER

OXIDE LEACHING

Jorcy Y. Trujillo3, Luis A. Cisternas2,31, Edelmira D. Glvez1,2, Mario E.

Mellado2

1Department of Metallurgical Engineering, Universidad Catlica del Norte, Antofagasta,

Chile2Centro de Investigacin Cientfico Tecnolgico para la Minera (CICITEM), Antofagasta,

Chile3Department of Chemical Engineering, Universidad de Antofagasta, Antofagasta, Chile

Submitted to: Chemical Engineering Research and Design

Date: July 12, 2012

ABSTRACT

Although the process of heap leaching is an established technology for treating minerals,

such as copper, gold, silver, uranium and saltpeter, as well as remediating soil, no studies to

date have investigated process optimization. This work presents a methodology for the

design and planning of heap leaching systems to optimize the process. This methodology

consists of the creation of a superstructure that represents a set of alternatives to search for

the optimal solution; from this superstructure, a mixed integer nonlinear programming

(MINLP) model was generated, and a BARON-GAMS solver was used to find the optimal

solution. This method was applied to the extraction of copper from systems with one, two

and three heaps, and the effects of copper price, ore grade and other variables were

analyzed for each system. From the results, it can be concluded that this methodology can

be used to optimize heap leaching processes, including planning and design issues.

1Corresponding author: L.A.Cisternas, e-mail: [email protected]


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Keywords: Heap leaching, process design, process optimization

NOMENCLATURE

Sets

process unit}.

, heap leaching unit}.

, unit of solvent extraction}.

, species to be extracted}

Variables and Parameters

Area of heap. [m2]

Recovery constants for disjunction model.

Acid consumption constants for disjunction model.

Costs. [MUS$]

Cost of building a heap . [MUS$/cycle]

Total fixed cost of heapj. [MUS$/cycle]

Acid consumption at infinite time. [kg/ton]

Acid consumption of heapj. [kg/ton]

Initial acid consumption. [kg/ton]

Acid consumption of heapj. [kg/cycle]

Total acid consumption. [kg/cycle]

Variable cost of heapj. [MUS$/ton]

Operational cost before leaching of heapj. [MUS$/ton]

Operational cost after leaching. [MUS$/ton]

Linear availability of species in heap . [ton/m]

Planning time horizon. [days]


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Income. [MUS$]

Mass flow from process unit to of species . [ton/cycle]

Input mass flow to process unit of species . [ton/cycle]

Output mass flow from process unit of species . [ton/cycle]

Molecular weight of acid. [ton/ton-mol]

Molecular weight of species k. [ton/ton-mol]

Mass of mineral loaded on heap of species . [ton]

Constant in the big M method.

Number of cycles.

Production of species . [ton/cycle]

Price of species . [MUS$/ton]

Price of acid. [MUS$/kg]

Volumetric flow from process unit to . [m3/cycle]

Output volumetric flow from the SX unit. [m3/cycle]

Recovery from heap of species . [%]

Recovery of species in disjunctive model. [%]

Recovery at infinite time. [%]

Cycle time of the heap system. [days]

End time of leaching of heap . [days]

Profits. [MUS$]

Weight factor for incomeI.

Weight factor for cost C.

Concentration of stream from process unit to of species . [ton/m3]

Concentration of output stream from the SX unit of species . [ton/m3]

Binary variable of the cycle times.

Disjunctive binary variable.

Height of heap. [m]


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Greek letters

Recovery constants for Mellado et al. (2011) model.

Acid consumption constant for the analytical model.

Ore density. [ton/m3]

Grade of species k in the ore. [%]

Superscript

Lower bound .

Upper bound.

Subscript

Disjunctive.


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1. INTRODUCTION

Heap leaching is a hydrometallurgical process that is widely used in the mining industry, in

which low-grade metals, such as copper, gold, silver and uranium, are extracted; the

process also has new applications in the treatment of non-metallic minerals, such as

saltpeter (Valencia et al., 2008), and soil remediation (Carlsson et al., 2005). Several

studies have been performed with the goal of improving the leaching process; these works

include searching for the ideal operational conditions to achieve better extraction of metals.

However, most modeling-based research has focused on the search for models based on

first principles (Dixon and Hendrix, 1993a; Dixon and Hendrix, 1993b) and semi-empirical

models (Mellado et al., 2009; Mellado et al., 2011). An analysis of the relationship between

the operational time and costs and the identification of an optimal operation time and heap

size have been performed (Padilla et al., 2008); however, despite the importance of heap

operations, no previous studies have analyzed the optimization of the entire system.

Heap leaching is a mineral processing technology in which piles of ore (crushed or run-of-

mine rock) are leached with various chemical solutions to extract valuable minerals. Large

tonnages are involved, and metal is recovered over a long period. As shown in Figure 1, a

barren solution is pumped to the pile surface and sprayed, and as it percolates downward,the extraction of metal begins. Usually an aqueous cyanide solution is used for precious

metals, and an aqueous sulfuric acid solution is used for copper ores. The metal is

recovered from the pregnant solution by a recovery process that depends on the metal to be

extracted. For example, solvent extraction (SX) followed by electrowinning is used for

copper, and activated carbon or precipitation with zinc are used for gold (Gupta and

Mukherjee, 1990). In this work, an SX process was considered as the recovery process. The

barren solution from the recovery process is recirculated to the heap system. A heap system

can include several heaps, and several irrigation networks can be used.

Figure 1: Hydrometallurgical process based on heap leaching

The objective of this research was to develop a methodology that allows design and

planning to simultaneously find the optimal operational conditions, such as recovery and

leach time, and the optimal design parameters, such as heap height. The methodology is


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based on a superstructure that represents the flowsheet alternatives and a mathematical

model that is solved through a mixed integer nonlinear programming (MINLP) approach.

The development of the model with its superstructure and corresponding equations is

provided in section 2; the application of the model to systems with one, two and three heaps

is provided in section 3, with the corresponding characteristics and sensitivity analysis.

Finally, the last section includes the conclusions.

2. MATHEMATICAL MODEL

The design and planning of heap leaching systems is a coupled problem. Padilla et al.

(2008) studied the economic optimization of this process by analyzing variables such as the

leaching time and heap height in a copper mineral treatment plant and found that the design

(heap height) and operational planning (leaching time) are coupled problems from an

economic standpoint because these variables affect both the recovery and operational

capacity of the leaching process.

The purpose of this work is to develop a methodology to design and plan heap leach

systems. To accomplish this goal, a superstructure representing a set of flowsheet

alternatives to look for the optimum solution was constructed, based on which amathematical model was developed using mass balances and their corresponding

operational conditions to obtain a model that can determine the values of variables and

parameters affecting the process. The resulting model corresponds to an MINLP.

The superstructure was built based on heap leaching and solvent extraction units. The

schematic in Figure 2 represents these units, where represents a process unit. In this

superstructure, a mixer is used at the input of the unit (square) and a divider is used at the

output of the unit (triangle) to represent the set of connection possibilities between the

different units. The rectangle in the middle of the figure represents the unit itself, either a

heap leaching or SX unit. This type of representation has been used in other problems, e.g.,

water networks (Castro and Teles, 2013) and reactor networks (Silva and Salcedo, 2011).

Figure 2: Process units used in the modeling


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To develop the mathematical model, the following sets are defined. First, the set of units is

defined as is a process unit}, following which the subsetsJ , heap

leaching unit} and , solvent extraction unit} are defined. The set of

species to be extracted is represented as .

To perform the corresponding balances, the following assumptions were considered. a) The

evaporative water losses are compensated for with fresh water so that the water losses and

fresh water flow rates are null. The water balance and the cost of water losses were not

considered because these balances and costs depend on weather conditions, heap irrigation

technology, and evaporation losses/evaporation mitigation technologies in solution pools.

b) The heaps are considered to be in a series, where heap has been operated longer than

heap . c) During solvent extraction, the output concentration of the valuable species is

assumed to be known and constant, . This concentration is assumed constant and known

because it depends on downstream operations, in this case from the electrowinning stage.

Additionally, it is assumed that this stage is ideal and does not represent entrainment of

solution or loss of valuable species. d) A constant density is assumed in the dissolutions.

2.1 Mass Balances

Mass balance in j heaps

Equation 1 shows the mass balance of species in heapj in the liquid phase as a function

of the input and output mass flow rates and the recovery in the heap:

(1)

where is the output mass flow of species in heap , [ton/cycle], and represents

the recovery of the valuable species in the leaching heap process, which depends on the

leaching time and the height of the heap , along with other variables. The term

, [%/cycle] is included in equation 1 because the heaps operate in series,

and the recovery of the heap j, , begins where the recovery of the heapj-1, , ends.


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can be calculated using an adequate model, for example, the model proposed by

Lizama et al. (2005) for bioleaching, Mellado et al. (2011) for low grade ores, or Glvez et

al. (2012 ) for Caliche ores. Note that ; therefore, the superscript represents the

input solution of heap , the superscript o represents the output solution of heap and

represents the mass of the valuable species in heap , given by the following:

(2)

In equation 2, is the linear availability of species , [ton/m] which corresponds to the

area of the heap multiplied by the density of the ore with a corresponding ore

grade .

The balance of acid consumption is given by equation (3), where the first term represents

the consumption of acid in heap j, and the second term is the credit for the recovery of acid

in the electrowinning stage.

(3)

where is the acid consumption of heapj by cycle [kg/ton of ore cycle] andMWk is the

molecular weight of species k.

The total acid consumption is the sum of the acid consumption for each heap, that is,

(4)

Mass balance in the mixers and dividers in unit

For the mixers and dividers, the mass balance per component was performed based on the

input and output mass flows rates of the different process units. There is a mixer and a

divider on each process unit.


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(5)

(6)

Mass balance in SX

The mass balance for species in the solvent extraction process unit was performed based

on the production of the valuable species [ton/cycle], and the input and output mass

flows to the process unit, as shown in the following equation:

(7)

2.2 Planning and operation constraints

To design and plan the heap leaching process, certain restrictions must be considered. First,

the number of cycles throughout the planned time horizon must be obtained as follows:

(8)

where is the planned time horizon [days], is the number of cycles and represents the

cycle time. A cycle consists of all operation phases of a heap, e.g., if a plant uses two heaps,

the cycle consists of the operation of both heaps. The time horizon is the length of time

used to evaluate the alternatives over the same period of time. The cycle time is the period

required to complete one cycle of heap operation. In this work it is assumed that

overlapping operation is used, this is, simultaneous heaps can be operated. Then, the cycle

time is the maximal operation time between all heap units, calculated as follows:


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(9)

Note that if a non-overlapping operation is used, then the cycle time is the sum of the

operation times of all heap units.

Heap starts and finishes the operation at times and , respectively. Equation (9) can

be written as a disjunction expression:

(10)

In this work, equation (10), the disjunctive expression for the cycle time, is represented by

the big M method (Biegler et al. 1997) as follows:

(11)

(12)

(13)

(14)

where , is a binary variable that represents the selection of the heap with the maximal

operation time. Here, the constraint in equation (12) only allows one choice of . M is a

large parameter introduced in the right side of equations (11), (13) and (14), which renders

the inequalities redundant if and enforced if .


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The following operational bounds must be considered to achieve satisfactory results. These

restrictions include lower and upper bounds in the mass flows, the height of the heap, the

recovery, the number of cycles and the cycle time as follows:

(15)

(16)

(17)

(18)

(19)

(20)

(21)

where . Then, based on equation (17), a good value of M in equations (11), (13) and

(14) is tUP.

2.3 Specifications


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In the following discussion, the expressions to calculate some bounds and parameters are

given. First, the model uses the mass flow rates of species k; however, in industry,

concentration and volumetric flow rates are used. The relationship between these variables

is as follows:

(22)

where represents the concentration of species in mass per volume [ton/m3] in the

stream moving from process unit to unit , and represents the volumetric flow rate

from process unit to [m3/cycle]. Because is the maximal flow of species , this

parameter can be calculated as the sum of the output flows of all of the SX units as follows:

(23)

In addition, from equation (22), the balance for the solvent extraction unit as a function of

the mass flows is given by

(24)

where corresponds to the output volumetric flow rate from the SX unit and ,

corresponds to the concentration of species at the output of the SX unit.

2.4 Objective Function

The expression to be maximized is an economic optimization that must be simultaneously

maximized in terms of income and minimized in terms of costs. The income is represented

as the total production multiplied by the price, and the cost is the number of cycles

produced multiplied by the cost of each heap over the time horizon.


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A possible expression is represented in equation (25), where is the income; represents

the costs; and are weight factors; and represents the profits when .

Income, cost and profits can give different optimal solutions (Novak Pintari and Kravanja,

2006), and therefore different weight factor values can be analyzed.

Maximize (25)

The income, [MUS$], can be represented in terms of production, the number of cycles,

the price of the species to be extracted and the post-leaching cost

[MUS$/ton of metal], which includes the cost of the purification process (e.g., solvent

extraction or ion exchange) and the product recovery process (e.g., electrowinning or

crystallization), and represents the cost after the leaching is performed. The income is given

by

(26)

In addition, the cost, [MUS$], is given by the number of cycles, the cost of building and

operating heap , and acid consumption, as shown in the following equation:

(27)

where is the acid price in [MUS$/kg].

Then, the cost of building and operating heap [MUS$/cycle], is defined as a function

of the different costs of the heap leaching system according to the following expression:

(28)


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where represents the total fixed cost, including the cost per square meter and per cycle

in the process; is the variable cost per ton of ore and per cycle [MUS$/ton of ore cycle]

in the process that includes the power, water consumption, land preparation, maintenance

and labor; and is the cost before leaching per ton of ore and per cycle in the

process [MUS$/ton of ore cycle], including costs such as mine operation, crushing and

agglomeration.

2.5 Recovery Expressions

In this work, two methods are used to express the recovery of copper, but different models

can be used depending on the leaching technology and the leached ore. First, a disjunctive

expression is used to approximate the recovery using straight lines, and an analytical model

developed by Mellado et al. (2011) is then used. The disjunctive expression is written in the

following manner:

(29)

The recovery is expressed by straight lines for different time ranges and heap heights.

The disjunctive expression of recovery in equation (29) is expressed using the Convex Hull

method (Biegler et al. 1997):

(30)

(31)


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(32)

(33)

(34)

(35)

where are constants used to approximate the recovery from the straight line

equation, is the binary variable, and and indicate the lower and upper bounds of

the operation time for each disjunction, respectively.

Mellado et al. (2011) developed an empirical knowledge-based model, which considers two

time and size kinetics scales and includes variables such as particle radius, heap height,

irrigation rate and porosities. Here, only the heap height and leaching time are considered

as variables, keeping other variables constant. Then, the analytical expression proposed by

Mellado et al. (2011) can be written as follows:

(36)

Where represents the recovery at t= infinity, which is a function of the height of the

heap as follows:

(37)

The parameters of equations (29) and (37) are shown in Table 1.


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Table 1: Parameters considered in the analytical model by Mellado et al. (2011) for copper

recovery and the analytical model (eq. 39) for acid consumption.

Figure 3 shows the graphic representation of equations (27) and (34) for the disjunctive

model and the model by Mellado et al. (2011).

Similarly as for copper recovery, two ways of expressing the acid consumption were used.

First, a disjunction expression based on straight lines for different time ranges and heap

heights was used, and then an analytical model was used. The disjunctive expression is

written as follows:

(38)

The disjunctive expression for acid consumption in equation (38) is expressed using the

Convex Hull method (Biegler et al. 1997):

(39)

(40)

where are constants.

The analytical model is

(41)

The parameters of equations 39 and 41 are given in Table 1.

The disjunctive model of acid consumption (Eqs. 39 and 40) is used together with the

disjunctive model of copper recovery (Eqs. 30 to 35). In the same way, the analytical model


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of acid consumption, Eq. 41, is used together with the analytical model of Mellado et al.

(2011), Eq. 36.

3. CASE STUDIES

This section illustrates the application of the model for the copper heap leaching process in

systems of one, two and three heaps to analyze the effects of variables such as copper price,

heap size, ore grade, acid price and variable cost on the process. The case studies consider a

heap of 200,000 m2 with an ore grade of 0.9% copper. The copper price is assumed to be

7,700 US$/ton. A time horizon of 360 days was used to analyze, design, and plan the

leaching system. Other ore characteristics and heap costs are indicated in Table 2, and they

are shared by all examples. The costs used in this work are those normally observed in the

context of copper production in Chile and were updated from the work of Schmidt (2001).

Table 2: Parameters considered in the case studies.

We must consider that the variable cost, the pre-leaching cost and the total cost in systems

with two and three heaps apply only to the first heap. In other words, the costs of heaps two

and three are zero because the heaps operate in series and because their costs have been

considered in the first heap; furthermore, it is assumed that the operational cost of heaps

two and three can be neglected.

In the following sections, the three studied cases are shown, which were implemented in

GAMS and solved by BARON using an AMD Athlon II Dual-Core M300 2.00 GHz

processor. In all studied cases only one parameter was changed, while all other parameters

were kept constant and equal to the values in table 2.

Figure 3: Recovery as a function of time with heap heights of 6 and 9 m.

3.1 One-heap leaching system


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Table 4: Cycle time fluctuations based on the ore grade and different leaching systems

using the disjunctive model (DM) and Mellado et al. (2011) model.

Table 5: Recovery fluctuations as a function of the ore grade for the different leaching

systems using the disjunctive model (DM) and Mellado et al. (2011) model (M). NC=Does

not converge.

The effect of the variable cost (assuming constant expenses for acid consumption),

including expenses for power, water consumption, ore transportation, maintenance and

labor, is significant. If the variable cost is increased, the cycle time and recovery increase

proportionally, and profits and income decrease. Figure 5 shows the effect of the variable

cost on the cycle time (Figure 5a) and recovery (Figure 5b), calculated for both the

disjunctive model and the Mellado et al. (2011) model. Additionally, the second and third

columns in Table 6 show the profits for both models.

Figure 5 Fluctuations in cycle times (a) and recovery (b) at 0.005, 0.01 and 0.02

kUS$/ton of ore.

Table 6: Profit fluctuations based on the variable cost and different leaching systems using

the disjunctive model (DM) and Mellado et al. (2011) model.

The effect of the acid price was studied separately from other variable costs. When the

effect of the acid price on the system is analyzed, an impact on the cost and profit is

observed: cost increases and profit decreases as the acid price increases. Nevertheless, no

change in the height of the heap is observed (9 m for the disjunctive model and 9.5 m for

the Mellado et al. model, for all values of acid price). For cycle time and recoveries, nochanges are observed for disjunctive model. On the other hand, cycle time and recoveries

increase as acid price increases for the Mellado et al. model. More details are given in table

7.

Table 7: Effect of acid price for the Disjunction and Mellado et al. models.


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3.2 Two- and three-heap leaching systems

The models are applied to a system formed by two- and three-heap leaching units to

compare its behavior to the single heap system and to observe the effect of parameters such

as copper price, heap area, ore grade, acid price and variable cost. The results obtained in

these systems show a trend similar to that obtained for the single heap leaching system. For

example, when the copper price increases, the profits increase in systems with one, two and

three heaps. Table 3 shows some of the results obtained for the three systems based on the

effect of the copper price, using the disjunctive model and the model by Mellado et al.

(2011).

In addition to the copper price, the variation of profits with other parameters is important to

characterize. Table 6 shows some of the results obtained for the three systems based on the

effect of the variable cost for the disjunctive and Mellado et al. (2011) models; the results

show that in a system with more than one heap, the profits are higher with each additional

heap; this is true for most cases, except for the Mellado model with three heaps.

As in the single heap system, if the ore grade is increased the cycle time decreases, and,

therefore, the number of cycles increases; the only difference in these cases is the shorter

leaching time. With more heap leaching units in a system, the leaching time in each heap is

shorter. Table 4 shows the results obtained for the three systems based on the cycle times

using both the disjunctive model and the Mellado et al. (2011) model. As shown in Tables

3, 4 and 5, there are convergence problems in both models.

Table 4 shows that for a two-heap system and an ore grade of 0.5%, the cycle time for each

heap is 40 and 49 days, and the entire two-heap system operates for approximately 80 and

98 days, for the disjunctive model and the Mellado et al. model, respectively. These values

are similar to the values for a one-heap system.

In addition, the recovery decreases as the ore grade increases in the different systems. Table

5 shows the results obtained for the three systems based on the ore grade for the disjunctive

model and the model by Mellado et al. (2011).

As shown in Table 5, in all systems, the recovery is low. For example, the recovery in heap

two for the two-heap system with an ore grade of 0.9% is 44 and 42% for the disjunctive


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model and the Mellado et al. model, respectively. To analyze this situation, a minimal

recovery value of 70% to 90% was assigned to the model. The results of this test indicate

that the model obtains high recoveries; however, there are variations in the cycle time and

profits. The cycle times increase significantly, and the profits decrease compared to the

results shown in Table 5. Therefore, in some cases the maximal recovery is not necessarily

the best measure of operational efficiency, based on economic considerations.

As in the single heap system, if the acid price is increased the cost increases, and therefore

the profit decreases. Table 8 shows the results obtained for the three systems using both the

disjunctive model and the Mellado et al. (2011) model. It can be observed that the number

of heaps has a greater effect than the acid price on the total leaching time and recoveries. Inboth models, leaching times and recoveries are approximately constant with the acid price.

Table 8. Effect of acid price for the disjunction model and Mellado et al. (2011) model.

NC=Does not converge.

In all of the systems, the heap height calculated using the disjunctive model was 9 m, the

maximal height allowed. With the Mellado et al. (2011) model, the maximal height allowed

in the model was obtained, which was not affected by parameters such as the copper price

and ore grade. However, a test showed that if the upper bound of the height is increased in

the Mellado et al. (2011) model, the optimal height obtained is below the upper bound

because if the heap height is increased, the recovery decreases.

Finally, Figure 6 shows the effect of the weight factors on the income and costs calculated

with the Mellado et al. (2011) model. The values in the graph indicate the weight factors

. The point indicates the minimal cost, and as a result, the

minimal income. Additionally, indicates the maximal income, and as a

result, the maximal cost. The income increases significantly as the income weight factor

increases until . For an income weight between 0.5 and 0.6, the cost increases but

the income is almost constant. Then, for between 0.6 and 0.7, the income increases as

the income weight factor increases; however, the cost also increases.


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As shown in these studies, parameters such as copper price, ore grade and variable cost

have an important effect on the profit as well as on some process variables. In addition, the

results of the disjunctive and Mellado et al. (2011) models are generally similar. The results

of the Mellado et al. (2011) model are slightly better than those of the disjunctive model

because the disjunctive model is an approximation to the Mellado model, and the

disjunctive model underestimates the recovery. However, the Mellado et al. (2011) model

has more convergence problems than does the disjunctive model.

Convergence problems and local optima can be attributed to the fact that the model is a

mixed integer nonlinear programming problem; it is nonconvex due to exponential terms

(recovery in the Mellado et al., 2011 model) and bilinear terms in the mass balance (e.g.,equations 1 and 2). To solve these problems, linearization (Quesada and Grossmann, 1995)

and discretization (Pham et. al, 2009) techniques can be used. This may explain why the

disjunctive model presents fewer convergence problems compared to the Mellado model.

The disjunctive model is a linearization of the latter. For bilinear expressions, the

discretization approach can be used.

Figure 6 Income and cost fluctuations based on the weight functions and .

4. CONCLUSIONS

In this work, a method has been developed to plan and design a heap leaching system. Two

models were used to calculate the recovery, one based on disjunction and the other based

on the model by Mellado et al. (2011). The developed mathematical model is a MINLP.

As determined in the copper leaching cases that were studied, one of the primary variablesaffecting the profit of the process is the copper price, as expected. Additionally, certain

other variables are shown to be important, such as the ore grade, acid price and the variable

cost, as they significantly affect the operation planning and the profit. The cycle time and

recovery decrease, but the profit increases with increasing ore grade. In addition, the cycle

time and recovery increase, but the profit decreases with increases in variable costs. The

cycle time decreases in systems with more than one heap. Moreover, the use of more than


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one heap allows better control of the irrigation rate in each heap. These results show that as

the number of heaps increases, there is more flexibility in planning and a greater production

capacity.

This study has shown that the design (heap height) and planning of a heap leaching

operation are coupled problems from an economic perspective, and therefore, these

variables must be included in the model. This coupling occurs because these are interactive

factors, and these variables affect both the recovery and the capacity of the operation in the

heap leaching process.

In addition, the results of the analytical model by Mellado et al. (2011) show higher

precision and convergence than those obtained using the disjunctive model, despite the fact

that the trend in the results is similar. Additionally, the CPU time required to obtain results

is less for the Mellado et al. (2011) model than for the disjunctive model. Because the

system converges to local optima, there is room for improvement in the computing

algorithm.

Finally, this study should be considered as a new step in the development of methodologiesfor the design and planning of heap leaching systems, and future studies must include the

effect of other variables, such as the particle size, irrigation rate and acid concentration, to

predict the optimal operational conditions.

Acknowledgments

The authors wish to thank CONICYT for its support through the Fondecyt Project 1090406

and 1090592.

REFERENCES

Biegler, L. T., Grossmann, I. E., & Westerberg, A. W., (1997). Systematic methods of

chemical process design. NJ: Prentice Hall.


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Carlsson, E., Buchel, G., (2005). Screening of residual contamination at a former uranium

heap leaching site. Thuringia.,Germany Chemie der Erde, 65S1, 7595.

Castro, P.M., Teles, J.P. (2013). Comparison of global optimization algorithms for the

design of water-using networks. Computers & Chemical Engineering 52, 249 261

Dixon, D.G. and Hendrix, J.L., (1993a). A Mathematical model for heap leaching of one or

more solid reactants from porous ore pellets. Metallurgical Transactions, 24B, 157-168.

Dixon, D.G. and Hendrix, J.L., (1993b). General model for leaching of one or more solid

reactants from porous ore pellets. Metallurgical Transactions, 24B, 157-168.

Glvez, E.D., Moreno, L., Mellado, M.E., Ordez, J.I., Cisternas, L.A. (2012). Heap

leaching of caliche minerals: Phenomenological and analytical models Some

comparisons, Minerals Engineering, 33, 46-53

Gupta C.K., Mukherjee T.K. (1990). Hydrometallurgy in Extraction Processes, Vol I, CRC

Press.

Lizama, H.M., Harlamovs, J.R., McKay, D.J., Dai, Z. (2005). Heap leaching kinetics are

proportional to the irrigation rate divided by heap height. Minerals Engineering, 18, 623-

630.

Mellado, M.E., Casanova, M.P., Cisternas, L.A., Glvez E.D. (2011). On scalable

analytical models for heap leaching. Computers and Chemical Engineering, 35, 220-225.

Mellado, M.E., Cisternas, L.A., Glvez E.D. (2009). An analytical model approach to heap

leaching. Hydrometallurgy. 95, 33-38.


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Table 1: Parameters considered in the analytical model of Mellado et al. (2011) for copper

recovery and analytical model (eq. 39) for acid consumption.

Parameters

Copper recovery model (Eq. 36)Recovery Constant

Recovery Constant

Infinite Recovery Constant

Infinite Recovery Constant 1

Infinite Recovery Constant 0.03

Acid consumption model (Eq. 39)

Acid consumption constant 0.0125Acid consumption constant 1

Infinite acid consumption 70.96 kg/ton ore

Initial acid consumption 25 kg/ton ore

Table 2: Parameters considered in the case studies.

Parameters ValuesApparent ore density

Ore grade

Time horizon

Price of copper

Area of heap

Price of acid 0.162 [kUS$/ton of acid]

Post leaching cost, 0.16675 [MUS$/ton of copper]

Pre leaching cost, 0.006074 [MUS$/ton of ore]

Variable cost, 0.0028554[MUS$/ton of ore]

Total fixed cost, 0.00095 [MUS$]

Copper mass flow rate at the output of SX 3,500


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Table 3: Variation of profits based on the copper price and the different leaching systems

using the disjunctive model (DM) and Mellado et al. (2011) model. NC=Does not

converge.

Profits[MUS$]

One Heap Two Heaps Three HeapsCopper

Price[MUS$/ton]DM Mellado DM Mellado DM Mellado

0.0055 174.8 181.6 603.0 663.7 NC NC0.0077 344.0 369.2 907.6 1,001.9 1,032.0 NC0.0099 513.2 566.5 1,212.3 1,340.6 1,539.5 NC

Table 4: Variation of the cycle times based on the ore grade and different leaching systems

using the disjunctive model (DM) and Mellado et al. (2011) model. NC=Does not

converge.

Cycle time[Days]

One Heap Two Heaps Three HeapsOre grade[%]

DM Mellado DM Mellado DM Mellado

0.5 80 97 40 49 NC 420.9 80 65 40 31 27 221.3 40 55 31 28 NC NC

Table 5: Variation of recovery as a function of the ore grade for the different leaching

systems using the disjunctive model (DM) and Mellado et al. (2011) model (M). NC=Does

not converge.

Recovery %

One Heap Two Heaps Three Heaps

Heap 1 Heap 1 Heap 2 Heap 1 Heap 2 Heap 3

Ore

grade[%]

DM M DM M DM M DM M DM M DM M

0.5 62 68 20 22 44 45 22 NC 46 NC 62 NC

0.9 62 55 20 19 44 42 22 NC 46 NC 62 NC

1.3 38 49 20 19 44 42 13 NC 38 NC 50 NC


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Table 6: Variation of profits based on the variable cost and different leaching systems using

the disjunctive model (DM) and Mellado et al. (2011) model.

Profits[MUS$]One Heap Two Heaps Three Heaps

Variable Cost

[kUS$/ton]DM Mellado DM Mellado DM Mellado

0.005 400.1 446.2 799.3 892.4 1,198.3 1,198.30.01 330.2 351.5 479.4 703.0 990.7 1.054.50.02 192.5 203.3 385.1 406.5 462.7 303.2

Table 7: Effect of acid price for Disjunction and Mellado et al. models.

Acid Price [kUS$/ton of acid]

0.11 0.162 0.21 0.11 0.162 0.21

Disjunction Model Mellado et al. Model

Heap height 9.00 9.00 9.00 9.50 9.50 9.50

Leaching cycles 4.50 4.50 4.50 5.81 5.53 5.31

Profits [MUS $] 379,458 344,004 311,278 411,925 369,118 330,635

Recovery [%] 62 62 62 53 55 56

Cycle time [days] 80.00 80.00 80.00 61.99 65.09 67.81

Income [MUS$] 579,241 579,241 579,241 673,520 662,199 652,159

Table 8. Effect of acid price for the disjunction model and Mellado et al. (2011) model.

NC=Does not converge.

DM Mellado et al.

Acid Price One Heap Two Heaps Three Heaps One Heap Two Heaps Three Heaps

[kUS$/ton] Profits[MUS$]

0.11 379.5 926.0 1,138.4 411.9 1,022.8 NC

0.162 344.0 907.6 1,032.0 369.1 1,002.0 NC

0.210 311.3 890.7 933.8 330.6 982.8 NC

Total Leaching Time [Days]

0.11 80.0 50.8 80.0 62.0 45.5 NC

0.162 80.0 50.8 80 65.1 45.5 NC

0.210 80.0 50.8 80.0 67.8 45.5 NC

Total Recovery [%]

0.11 62 44 62 53 42 NC

0.162 62 44 62 55 42 NC

0.210 62 44 62 56 42 NC


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Figure 1: Hydrometallurgy process based on heap leaching

Mine Ore

Recovery Process

Pregnantsolution

Heaps

Spent

Ore

Barrensolution

Metal


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Figure 2: Process units used in the modeling


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Figure 3: Recovery as a function of time with heap heights of 6 and 9 m.


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Figure 4: Variation of profits for ore grades of 0.5, 0.9 and 1.3%.


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a)

b)

Figure 5: Variation of cycle times (a) and recovery (b) at 0.005, 0.01 and 0.02 kUS$/ton of

ore.


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Figure 6: Variation of income and costs based on the weight functions and .


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This work presents a methodology for the design and planning of heap leaching systems.

The method used a superstructure that represents a set of alternatives to search for the

solution.

A MINLP model was generated and a BARON-GAMS solver was used to find the optimal

solution

It can be concluded that this method can be used to optimize heap leaching planning and

design.

optimal design and planning of heap leaching process. application to copper oxide leaching

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