An Analytical-Numerical Method for Solving a Heap

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Please cite this article in press as: Mellado, M. E., Cisternas, L. A., An analyticalnumerical method for solving a heap leaching problem of

one or more solid reactants from porous pellets, Comput Chem Eng (2008), doi:10.1016/j.compchemeng.2008.01.001

ARTICLE IN PRESS+Model

CACE 3589 18

Available online at www.sciencedirect.com

Computers and Chemical Engineering xxx (2008) xxxxxx

An analyticalnumerical method for solving a heap leaching problemof one or more solid reactants from porous pellets2

Mario E. Mellado a,, Luis A. Cisternas b3a Centro de Investigacion Cientfico Tecnologico para la Minera (CICITEM), Mar del Plata 02485, Antofagasta, Chile4

bDepartamento de Ingeniera Qumica, Universidad de Antofagasta, Centro de Investigacion Cientfico Tecnologico

para la Minera (CICITEM), Chile

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Received 4 September 2007; received in revised form 26 December 2007; accepted 1 January 2008

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Abstract8

In this paper we present an alternative method based on analytical and numerical solutions for solving the differential equations which describe

heap leaching of one or more solid reactants from porous pellets. We propose to useanalytical solutions forthe differential equationswhich describe

rate dissolutions along the pores and the surface of the particles under suitable regularity conditions. Moreover, we propose to use continuous and

discontinuous solutions for the continuity partial differential equations describing balances. We comment on the efficient numerical solving of the

remaining partial differential equations within the proposed numerical scheme. All of this, allows to obtain a numerical algorithm which is fast and

accurate for the heap leaching problem. Also, we include particle size distributions on the proposed numerical methodology. This method applies

to the case where the rate-controlling reagent is a component of the lixiviation solution only and not of the gas phase. The model includes the

effects of particle scales, kinetic factors, heap scales and several operation variables. Finally, numerical experiments are presented.

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2007 Published by Elsevier Ltd.7

Keywords: Heap leaching; Differential equations; Discontinuous solutions; Analytical solutions; Numerical methods8

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1. Introduction0

Heapleachingis a hydrometallurgicalprocess whichhas been

used since a long time which was originally designed for oxides2

ores but today have several applications including sulphide ores3

and caliche minerals. Heap leaching has been a matter of wide4

research and several mathematical models which allows to sim-5

ulate, analyze, design and optimize these operations. Indeed,6

the extractive metallurgy by chemical dissolution is one of the7

metal recuperation techniques most used in modern industry.8

Nowadays, the maininterest to use hydrometallurgicalprocesses9

arises from avoiding ambient contamination produced by con-0

ventional pyro-metallurgical processes and the treatment of low

grade minerals. Also, heap leaching allows low operational cost2

from the energetic and production point of view. Theheap leach-3

ing can use a pre-treatment of the ore. This operation generally4

is based upon the trituration of the ore up to a finer level and5

Corresponding author. Tel.: +56 55 637313; fax: +56 55 240152.Q1E-mail addresses: [email protected] (M.E. Mellado),

[email protected] (L.A. Cisternas).

its granulation by means of water and concentrated sulphuric

acid. Throughout this process, it is possible to increase the heap

material resistance while keeping good enough permeability.

All of this with the objective of constructing heaps with a suit-

able height, improve the recuperation and decrease the leaching

time. The solution obtained in heap leaching is sent to recupera-

tion operations, usually solvent extraction. A complete literature

review is outside the objective of this work, but a brief revision

is given hereafter.

In de Andrade Lima (2004), is presented a model to simulate

the transient evolution of the dissolved chemical species in the

heap andcolumn isothermal leaching processes. In Bouffard and

Dixon (2001), a study about the heap leaching planning from a

optimization point of view is presented. In Bouffard and Dixon

(2006), the rates of pore diffusion and cyanide gold dissolution

in coarse, porous gold oxide ore particles are compared. Respect

to thenumerical solution of a twodimensionalpartial differential

equations model for a two-phase flow in a porous media repre-

senting the leaching process we refer to Cariaga, Concha, and

Sepulveda (2005). With respect a two-dimensional partial dif-

ferential equations model in bioleaching we can cite the work of

Casas, Martnez, Moreno, and Vargas (1998). In Cross, Bennett,

0098-1354/$ see front matter 2007 Published by Elsevier Ltd.

doi:10.1016/j.compchemeng.2008.01.001
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2 M.E. Mellado, L.A. Cisternas / Computers and Chemical Engineering xxx (2008) xxxxxx

Nomenclature

bi stoichiometric number (mol/mol)

CA0 reference reagent concentration

CE0 initial extractable grade of solid reactant (mol/g)

Cp is the grade of solid reactant within particle

(mol/g)Cs is the grade of solid reactant on particle surface

(mol/g)

DAe effective pore diffusivity of reagent (cm3/(cm s))

E extraction

kp is therate constant of solid reactant withinparticle

(mol/(g s))

ks is the rate constant of solid reactant on particle

surface (mol/(cm2 s))

R particle radius (cm)

t time (s)

us superficial bulk flow velocity (cm3/(cm2 s))

x space coordinate (cm)

z space coordinate (cm)

Z heap depth (cm)

Greek letters

dimensionless concentration of reagent

b dimensionless concentration of the reagent exter-

nal to the particle

reagent strength parameter relative to solid reac-

tant

b bulk solution volume fraction

h heap void fraction

0 ore porosity

dimensionless flow timep Damkohler II number for solid reactant within the

particle

s Damkohler II number for solid reactant on the

particle surface

surface fraction of solid reactant

0 ore density (g/cm3)

p dimensionless grade of solid reactant within the

particle

s dimensionless grade of solid reactant on the par-

ticle surface

p reaction order for solid reactant within particle

s

reaction order for solid reactant on the particle

surface

i dimensionless concentration of dissolved species

ib dimensionless bulk concentration of dissolved

species

Croft, McBride, and Gebhardt (2006), a framework was devel-

oped for multi-phase flows, gasliquidsolid, in reactive porous

media, capturing several key chemico-physicalbacterial phe-

nomena and their interactions. In Lizama, Harlamovs, McKay,

and Dai (2005), it is shown that for sphalerite and pyrite heap

bio-leaching kinetics were proportional to the irrigation rate

divided by the height. In Mousavi, Jafari, Yaghmaei, Vossoughi,

and Sarkomaa (2006), is provided a detailed information of

momentum and mass transfer phenomena in a granular bed. In

Sheikhzadeh, Mehrabian, Mansouri, and Sarrafi (2005), unsatu-

rated flow of liquidin a bedof uniform andspherical oreparticles

is studied numerically and experimentally. An unsteady and

two-dimensional model is developed on the basis of the mass

conservation equations of liquid phase in the bed and in the par-

ticles. The model equationsare solved using a fully implicit finite

difference method giving the distribution of the degree of satu-

ration in the particles and in the bed and the vertical velocity of

flow in the bed, as well as the effect of periodic infiltration on the

above distributions. In Sidborn, Casas, Martinez, and Moreno

(2003), a two-dimensional dynamic model for bio-leaching of

secondary copper minerals from a pile has been developed. The

set of partial differential equations is solved using commercial

software. In Wu, Liu, and Tang (2007), the governing equations

for a fully coupled flowing-reaction-deformation behaviour with

mass transfer in heap-leaching are developed. The model equa-

tions are solved using an explicit finite difference methodunder the conditions of invariable application rate and constant

hydraulic head.

In the workDixon and Hendrix (1993a, 1993b), is presented

a mathematical model, in dimensionless form, for heap leaching

of one or more solid reactants from nonreactive, porous, spher-

ical ore particles. These contributions are significant to model

the complete heap leaching process (Cross et al., 2006). How-

ever, the works of Dixon and collaborators require of expensive

numeric procedures to solve their equations, situation that lim-

its their applicability, for example, in optimization. The main

goal of this work is to develop an efficient analyticalnumerical

scheme for solving differential equations that describe heapleaching. An outline of the paper is as follows. In Section 2 we

present a mathematical framework already presented in the lit-

erature. In Section 3 we show our analyticalnumerical scheme

for the model problem. Some numerical essays are shown in

Section 4. Finally, we conclude with some remarks in Section 5. 1

2. Mathematical framework 1

Following Dixon and Hendrix (1993a, 1993b), we present 1

a system of partial and ordinary differential equations which 1

describes heap leaching. For further details we refer to Dixon 1

and Hendrix (1993a, 1993b). First, we define the following 1

parameters: 1

s =3ksC

ss R

biDAe, p =

0(1 0)kpCpp R

2

biDAe, 1

=0biCA0

0(1 0)CE0, =

b

0(1 h)and 1

=3(1 h)DAeZ

usR2, 1

where ks is the rate constant of solid reactant on the particle 1

surface (mol/(cm2 s)), Cs the grade of solid reactant on par- 1

ticle surface (mol/g), s denotes the reaction order for solid1
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M.E. Mellado, L.A. Cisternas / Computers and Chemical Engineering xxx (2008) xxxxxx 3

reactant on particle surface, R the particle radius (cm), bi the3

stoichiometric number (mol/mol), DAe the effective pore dif-4

fusivity of reagent A (cm3/(cm s)), 0 the ore density (g/cm3),5

0 the ore porosity, kp the rate constant of solid reactant within6

particle (mol/(g s)), Cp the grade of solid reactant within particle7

(mol/g), p stands for the reaction order for solid reactant within8

particle, CA0 the reference reagent concentration, CE0 the initial9

extractable grade of solid reactant (mol/g), b the bulk solution0

volume fraction, h the heap void fraction, us the superficial bulk

flow velocity (cm3/(cm2 s)) and Zdenotes the heap depth (cm).2

With this, the ordinary differential equation to compute p,3

the dimensionless grade of solid reactant within the particle, is4

given as follows:5

dp

dt=

p

(1 )pp (1)6

together with the initial condition:7

p(x, 0) = 1,8

where tis the time, xthe space coordinate, p the Damkohler II9

number for solid reactant within the particle, a dimensionless0

stoichiometric ratio which indicates reagent strength relative to

the grade of solid reactant, the fraction of solid reactant resid-2

ing on the particle surfaces initially and is the dimensionless3

reagent concentration.4

The ordinary differential equation to compute s, the dimen-5

sionless grade of solid reactant on the particle surface, is given6

by7

ds

dt=

s

ss b (2)8

together with the initial condition:

9

s(0) = 1,0

where s is the ratio of the reaction rate of solid reactant on the

particle surface and b stands for the dimensionless concentra-2

tion of the reagent in the bulk solution external to the particle.3

The differential Eqs. (1) and (2) are complemented with four4

partial differential equations to have a model for heap leach-5

ing. In fact, the continuity equation for , the dimensionless6

concentration of reagent A, is given by7

t=

2

x2+

2

x

x pp (3)8

together with the initial condition:9

(x, 0) = 00

and the boundary conditions:

(1, t) = b,

x(0, t) = 0.2

It is important to remark that the differential Eqs. (1)(3)3

describe the leaching of one particle. It will be clear in the fol-4

lowing that ouranalysisis also applicable to this situation(Dixon5

& Hendrix, 1993a).6

There exist two time scales involved in this heap leaching7

model. One time scale describes the chemical reactions within a8

particle andthe other onedescribes theinteraction of all particles

through the heap. Of course, the particle time scale is faster than

the time scale which involves all particles. Therefore, the time

t in all the particular model equations must be replaced with a

dimensionless bulk flow time to solve the equations with a

common time.

Now, we present the partial differential equation to compute

the dimensionless concentration of dissolved species i, this is

3

i

=

i

t=

2i

x2+

2

x

i

x+ pp (4)


i(x, 0) = 0

and the boundary conditions:

i(1, t) = b,i

x(0, t) = 0.

We note that

=ust

bZ.

In the bulk lixiviant of the heap, the partial differential

equation to compute the dimensionless bulk concentration of

dissolved species ib is given by

ib

=

ib

z

i

x

(x=1)

(5)


ib(z, 0) = 0

and the boundary condition:

ib(0, ) = 0.

The balance equation for a strictly aqueous phase reagent in

the bulk lixiviant is expressed by means of

b

=

b

z

x

(x=1)

(6)


b(z, 0) = 0

and the boundary condition:

b(0, ) = 1.

The differential Eqs. (1)(6) complete a partialordinary dif-

ferential equation model for heap leaching. We use the notion of

extraction of reactant from the heap, which represents the total

fraction of extractable reactant which hasissued from the heap at

any given time. Except for very short leaching times or in heaps

with an unusually high holdup, extraction takes approximately

the same value as conversion. Therefore, as a heap performance

index we use the extraction notion, which can be written as

E() =

0

ib(1, t) dt. (7)


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Unlike conversion, extraction may be evaluated directly from

chemical analysis of theheap effluent and therefore, is far a more

convenient measure of the heap performance.

3. The analyticalnumerical scheme for the model

Now, we discuss in detail the analyticalnumerical aspects for

solving the system (1)(6). First, it is clear that there exist cou-

plings between the differential equations through the unknown

variables and the initial and boundary conditions. Moreover, the

model consists in two ordinary differential Eqs. (1) and (2), two

partial differential equations of first order (5) and (6) and two

equations of second order (3) and (4).

We first concentrate our attention in (1) and (2). Certainly,

at a first glance, these equations cannot be solved analytically

because the term and b. Moreover, we remark that the terms

and b in the right-hand side of both equationsare discrete andin

certain sense, the only way to solve these equations is via numer-

ical methods, such as Euler, RungeKutta, among others (Stoer

& Bulirsh, 1996). As we will solve the system of differentialequations in an iterative way, we can assume that the functions

and b are constants through a small enough time step, i.e.,

these functions are considered to have enough regularity. If we

assume the latter, Eqs. (1) and (2) have the structure of a general

first order ordinary differential equation. All of this, have the

obvious consequence of improve the speed and accuracy of the

numerical code.

Certainly, by using ordinary differential equations theory

(Boyce&DiPrima,1993), we findthat, andconsideringreaction

order for both equations as equal to the unity:

p(x, )=

expp

3(1 )

(8)

and

s(x, ) = exp

s

3

. (9)

It is important to recall that if it is necessary to consider

reactions orders p and s different to the unity, the differen-

tial equations can be solved analytically by using the Bernoulli

transformation (Boyce & Di Prima, 1993):

v = 1pp and v = 1ss ,

respectively.

Now we describe our approach for the partial differential Eqs.(3) and (4). These equations are actually the radial part of the

Helmholtz equation written in spherical coordinates. Although

its is possible to solve (3) and (4), under several assumptions, in

an analytical manner, we note that this is quite expensivefrom the

theoretical and numerical point of view. The analytical solution

of this equation needs to consider a finite number of terms in an

infinite sum of terms and for each term, it is necessary to solve

a nonlinear equation. This procedure is, at least, so expensive

from the computational point of view than using, for instance,

finite differences. Moreover, it is easier to deal with a numerical

method than sum truncation in what concerns error studies of

the proposed methodology.

There exista numberof ways to solvenumericallytheseequa- 2

tions. We note that the number 2 in Eqs. (3) and (4) corresponds 2

to spherical diffusion and corresponds to the radial part of the 2

Helmholtz equation in spherical coordinates (see Bouffard & 2

Dixon, 2001, Eq. (5)). 2

We can mention the finite difference method (Hackbusch, 2

1992), thefiniteelementmethod(Brenner & Scott, 1994;Ciarlet, 2

1978; Hackbusch, 1992; Zienkiewicz & Taylor, 2000), finite 2

volume methods, among others. We recall that we use uniform 2

discretizations for solving Eqs. (3) and (4). In this sense, and 2

by considering standard versions, the use of the finite element 2

method or finite differences is equivalent. Also, as the finite 2

differences are easier to handle as compared with finite ele- 2

ments, we solve these equations with a CrankNicholson fully 2

implicit finite difference scheme (Hackbusch, 1992). Both equa- 2

tions, after the discretization procedure, require the solving of 2

a linear tridiagonal system of equations whose entries depends 2

entirely of the discretization in time, space and coordinates. It 2

is very important to note that the generated matrices are sparse, 2

non symmetric and stable. Therefore, we have to deal with a 2linear system of equations that must be solved in each step of 2

the iterations either in space or time. The linear systems can be 2

solved with a number of methods (Stoer & Bulirsh, 1996). We 2

use here a preconditioned LU solver and we recall other solvers 2

can be used, specially taking into account the sparse structure of 2

the matrices. For programming purposes, we note that it is only 2

necessary to assemble the matrices just once, but update their 2

respective right-hand sides each iteration. Therefore, there is no 2

computational effort in assembling the stiffness matrices. 2

Now we present our approach for the partial differential Eqs. 2

(5) and (6). First, we note that both equations have exactly the 2

same structure and they differs just on the initial and bound-2

ary conditions. Again, we need to mention that these equations 2

can be solved numerically by using, for example, the tracking of 2

fluid elements, finite volume methods or other similar numerical 2

methods. In despite of that, we solve these equations in an ana- 2

lytical manner. We recall that in the bulk lixiviant of the heap, the 2

mass balance for dissolved species is given by Eq. (5). To obtain 2

analytical solutions, we consider the transformation (Kesavan, 2

1989): 2

= z and = z + . 2

With this, and after some computations, we find that 2

ib

=

2

ix

(x=1)

. 2

Now, by using the initial and boundary value conditions, we 2

have 2

ib(z, ) = , < z (10) 2

and 2

ib(z, ) = z, > z, (11) 2

where2

=

2

i

x

(x=

1)

. 2


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By using the same arguments, we find the analytical solution8

for (6), this is9

b(z, ) = , < z (12)0

and

b

(z, ) = 1 z, > z, (13)2

where3

=

2

x

(x=1)

.4

We note that (10) and (11) is continuous in R2 while (12) and5

(13) is discontinuous when z = . This discontinuity reveals6

the important fact that, for b, the partial differential equation7

could be modified in order to have a consistent model. This can8

be achieved, for instance, by introducing a second order term9

which allows to have a continuous solution.0

On the other hand, the discontinuity is just when = z which

isa zero measure set in R2 (Kufner, John, & Fucik, 1977), which2suggests the use of this discontinuous solution because the regu-3

larizing effect of the sequence of integrations. This fact is proved4

via numerical experiments which shows not only good results5

in what concerning the agreement with reality but also giving6

suitable continuous global solutions.7

Now, in what concerning the use of size distribution of par-8

ticles functions, as the GatesGaudinSchumann distribution,9

our analyticalnumerical approach applies without any modifi-0

cation which allows to deal with distribution of particles just by

taking under consideration the corresponding scaling for p, 2

and by means of defining a reference particle size (Dixon &3

Hendrix, 1993a, 1993b).4

Now, we discuss on the algorithmic aspects itself. In fact,5

once we know how to solve each differential equation there6

exists mainly two approaches to compute the solution of the sys-7

tem, namely, uncoupled and totally coupled. The totally coupled8

approach consists on solving a non-linear system of algebraic9

equations per iteration which means to deal with nonlinear0

solvers, uniqueness of solution studies, non feasible solutions,

the conditions to achieve correct solutions and take care about2

the cost of the computational implementation. But, if we follow3

this approach correctly, we would have a very accurate solution.4

It depends on the application if such accurate solution is needed5

or not.6

On the other hand, the uncoupled algorithm means that the7

solution of the previous equationsare introduced as source terms8

to the other equations producing a sequence of equations solv-9

ing. Of course, this method introduces numerical errors if one no0

take care about discretizations in time and space. But the imple-

mentation cost is lower, the errors can be controlled and with our2

approach, one can infer that the errors are not high because we3

have analytical solutions. Also, it is much easier to understand4

from the engineering and numerical point of view.5

Therefore, we solve the model with the uncoupled scheme6

and the order of solving Eqs. (1)(6) is within two sweeps of7

iterations. The inner loop is in space and the outer loop is in8

time.9

3.1. Heap leaching with homogeneous particles sizes

Now, we present the algorithm by considering heap leach-

ing with homogeneous particle sizes. The algorithm reads as

follows, for a Z space step, solve:

Given p and b, obtain by solving

t=

2

x2+

2

x

x pp, (x, 0) = 0,

(1, t) = b,

x(0, t) = 0.

With obtain p through

p(x, ) = exp

p

3(1 )

.

With obtain s through

s(x, ) = exp

s

3

.

Obtain i by solving

3

i

=

2i

x2+

2

x

i

x+ kpp, i(x, 0) = 0,

i(1, t) = b,i

x(0, t) = 0.

Compute the corresponding fluxes for and i.

Compute ib and b through

ib(z, ) =

2

i

x

(x=1)

, < z

and

ib(x, t) =

2

i

x

(x=1)

z, > z.

Also

b(z, ) =

2

x

(x=1)

, < z

and

b(x, t) = 1

2

x

(x=1)

z, > z.

Compute and store

E() =

0

ib(1, t) dt.

Do the same up to Z = 1. Then, increase t. Once the desired

simulation time is reached, stop the iterations.


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3.2. Heap leaching with distribution function for different

particles sizes

To deal with heap leaching with a distribution function fordif-

ferent particles sizes, we define a dimensionless particle radius

as

= RR

,

where R is a reference radius, and making the scaling

p = 2p , =

2, w = 2w

and using the GatesGaudinSchuhmann distribution function

of parameter m, the algorithm is as follows: for a Z space step,

solve:

Given p and b, obtain by solving

t =

2

x2 +

2

x

x pp, (x, 0) = 0,

(1, t) = b,

x(0, t) = 0.

With obtain p through

p(x, ) = exp

p

3(1 )

.

With obtain s through

s(x, ) = exps

3

.

Obtain i by solving

3

i

=

2i

x2+

2

x

i

x+ pp,

i(x, 0) = 0, i(1, t) = b,i

x(0, t) = 0.

Compute the corresponding fluxes for and i.

Compute ib and b through

ib(z, ) = 2

10

ix

(x=1)

m m3 d , < z

and

ib(z, ) =

2

10

i

x

(x=1)

m m3 d z, > z.

Also

b(z, ) =

2

10

x

(x=

1)

m m3 d , < z

and 4

b(z, ) = 1

2

10

x

(x=1)

m m3 d z, > z. 4

Compute and store4

E() =

0

ib(1, t) dt. 4

Do the same up to Z = 1. Then, increase t. Once the desired 4

simulation time is reached, stop the iterations. 4

We use Gauss quadrature for computing E and recall other 4

methods can be used (Stoer & Bulirsh, 1996). We finally note 4

that, even for dense simulations, the only time cost is due to the 4

numerical solution of the two differential Eqs. (3) and (4). Also, 4

every time the inner loop is finished, the extraction (7) must be 4

stored for further study. 4

4. Numerical experiments 4

Before to start with a numerical experimentation of our 4

approach we need to point out that our approach is able to repro- 4

duce the results shown in Dixon and Hendrix (1993a, 1993b), 4

by considering the operation conditions indicated there for the 4

cases of oneparticle, leaching of oneparticleand leaching of par- 4

ticle distributions. A complete agreement with that simulations 4

was used to test our computational code. Again, we remark that 4

our approach generates a simple computational code with time 4

efficiency because analytical solutions are available and allows4

to have an good control of the numerical error. No additional 4

iterations are needed to solve differential Eqs. (3)(6) which, in 4

fact, need to be within two sweeping loops which is suitable for 4

computational purposes. We now present, in Figs. 1 and 2, and 4

just for the sake of completeness, an abstract result presented 4

in Dixon and Hendrix (1993a, 1993b), Figs. 3 and 4, with the Q2 4

design parameters w = 1, p = 1, = 1, = 1. 4

Fig. 1. Behaviour ofi against days. Q3


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CACE 3589 18


Fig. 2. Behaviour of Eagainst days.

Fig. 3. Behaviour ofi against days.

We now present a simulation, using our numerical scheme,3

under similar conditions to standard copper heap leaching pro-4

cess in the north of Chile. We present numerical results for5

the following situation: ore density 0 := 1.3, ore porosity6

Fig. 4. Behaviour of Eagainst days.

0 := 0.03, p := 0.000157, grade of solid reactant within par-

ticle Cp := 0.000157, particle radius r := 2.5, stoichiometric

number bi := 1.0, effective pore diffusivity of reagent DAe :=

106, reference reagent concentration CA0 := 0.00333, initial

extractable grade of solid reactant CE0 := 0.000157, bulk solu-

tion volume fraction b := 0.03, heap void fraction h := 0.03,

superficial bulk flow velocity us := 0.000333, heap depth Z :=

1000.0.

Figs. 3 and 4show a qualitative agreement with the standard Q4

copper heap leaching process which validates, in some way, our

approach and allows to conclude this simple modification of

the numerical algorithm for solving (1)(6) can produce good

results at a low computational effort.

5. Conclusions

An analyticalnumerical approach method for solving the

heap leaching problem has been developed. The mathematical

theory is consistent and shows the possibility of improvement

the model by including, for instance, a second order term in (5)in order to get a continuous solution. Also, this methodology

can be applied to obtain totally analytical models by consider-

ing the analytical solutions presented together with analytical

solutions for Eqs. (3) and (4), for instance, by some reasonable

simplifications of the differential equations. A qualitative agree-

ment with industry processes validates our method from the

engineering point of view. Further research may be developed

in what concerning the mathematical aspects of the proposed

method, for instance, convergence of the algorithm, order of

convergence, finite volume methods for the differential equa-

tions, modification of (5). Also, this numerical approach can

be used in others models. Our modification of the numericalsolving for this model allows to obtain accurate solutions in

an efficient manner from a computational point of view. More-

over, the fact to use analytical solutions reveals several facts in

what leaching process concerns. In fact, there are two kinetics

involved in the process which can be used to develop totally

analytical models. It is possible to use discontinuous solutions

with continuous and good results. The fact that there is an equa-

tion which is discontinuous reveals the fact that, in particular,

this model could be corrected. The obtention of analytical solu-

tions would allow to obtain hybrid models suitable for heap

leaching planning and design using techniques as mathematical

programming.

Acknowledgment

Luis A. Cisternas wishes to thank CONICYT for financial

support, through Fondecyt Project 1060342.

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An Analytical-Numerical Method for Solving a Heap

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