O

Ma

b

c

d

a

ARR1AA

IM

KHADS

1

ubcNctlmHidaccUcc

me

0d

Computers and Chemical Engineering 35 (2011) 220–225

Contents lists available at ScienceDirect

Computers and Chemical Engineering

journa l homepage: www.e lsev ier .com/ locate /compchemeng

n scalable analytical models for heap leaching

ario E. Melladoa,∗, María P. Casanovab, Luis A. Cisternasa,c, Edelmira D. Gálveza,d

Centro de Investigación Científico Tecnológico para la Minería, (CICITEM), ChileUniversidad de Concepción, ChileUniversidad de Antofagasta, ChileUniversidad Católica del Norte, Chile

r t i c l e i n f o

rticle history:eceived 23 June 2009eceived in revised form0 September 2010ccepted 19 September 2010vailable online 29 September 2010

a b s t r a c t

In this paper we present analytical models suitable for scaling up the heap leaching process of solidreactants from porous pellets. The models are based on first order ordinary differential equations togetherwith some constitutive relations derived from models based on ordinary and partial differential equationsand other relations based on insight. The models are suitable for applications in which the scale-up isneccesary. This approach allows to obtain accurate solutions for actual industry heap leaching operations.Novelty of this approach is the simple form of the models and its accuracy as compared with more complex

n loving memory of our friend Dr. David A.éndez 1979–2008

eywords:eap leachingnalytical models

models. Due to the models simplicity, they can be used for analysis, design, control and optimization ofheap leaching processes without mathematical complexities. The models include the effect of heap height,particle sizes, flow rates, and several operation-design variables. Finally, some numerical experimentswhich confirm our theory are presented.

© 2010 Elsevier Ltd. All rights reserved.

ifferential equations-based modelscale-up

. Introduction

Heap leaching is a hydrometallurgical process which has beensed since a long time which was originally designed for oxides oresut today have several applications including sulphide ores andaliche minerals (see Valencia, Méndez, Cueto, & Cisternas, 2007).owadays, the main interest to use hydrometallurgical processesames from avoiding ambient contamination caused by conven-ional pyro-metallurgical processes and economical advantages forow grade minerals. The heap leaching process is an operation of

ass solid–liquid transfer than can occur at ambient conditions.eap leaching has been a matter of wide research on develop-

ng models to describe accurately the phenomena. We now brieflyescribe the three main modelling strategies used in heap leachingnd for sure in other fields. The first approach, and the more simpler,onsists in the empirical models. They are based just in to adjust a

urve to experimental data to have an idea how the system behaves.sually is used where no accurate and complex predictions andomputations are needed because this approach does not allow tohange the input variables, do not give an idea what is happening

∗ Corresponding author. Tel.: +56 55 637313; fax: +56 55 240152.E-mail addresses: mario.mellado@cicitem.cl (M.E. Mellado),

acasanova@udec.cl (M.P. Casanova), lcisternas@uantof.cl (L.A. Cisternas),galvez@ucn.cl (E.D. Gálvez).

098-1354/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2010.09.009

inside the system and of course, its description of the phenomenais rather poor (see for instance Roman, Benner, & Becker, 1974).The second modelling approach is the phenomenological one andit is based in the physics equations which describe the system. Usu-ally one encounter systems of differential equations which describeconsistently and accurately the system under consideration. How-ever, they are more difficult to deal with in industry applicationsdue to sometimes, one needs to deal with complex mathematicsto use them (see for instance Dixon & Hendrix, 1993a,b; Bouffard& Dixon, 2009; Mellado & Cisternas, 2008; Reverberi, Esposito,& Vegliò, 2002). The phenomenological models often use a largenumber of parameters which need to be obtained by means ofexpensive experiments. Also, the phenomenological models usu-ally, suppose an ideal behaviour because complex conditions suchas canalizations and particle imperfections are difficult to include inthe model. A third modelling approach is also possible. It consists inthe combination of the two latter approaches. For instance, with theknowledge that some system have a particular behaviour (say, forinstance, exponential) the physical equations can be used to iden-tify time constants, asymptotic behaviour, delay, etc. This approachthen leads to rather simple models but enough accurate for some

applications (see, for instance, Mellado, Cisternas, & Gálvez, 2009).We now give a brief idea of what applications the third approachcan be more suitable as compared with the other two. First, it is thestochastic analysis of heap leaching. As starting point, is somehowobvious that Montecarlo simulations with these kind of models can

Chemical Engineering 35 (2011) 220–225 221

bldmelwbtnppaetppt

wiomsoac(aarDlt(ttsiSsAoiacDidbttttfaLphalsac

i

R

R

2nR

q0

qn−1

M.E. Mellado et al. / Computers and

e done almost straightforward. Another application can be heapeaching planning operations. To have twenty or more heaps ofifferent ages, height, ore characteristics, flow rates, under an opti-ization framework can be a heavy task which can be done in an

asy way with this kind of modelling. Finally, we mention the heapeaching operation optimization itself (flow rates, heights, etc). Also

e can mention that the topic of recuperation at infinity has note gained much attention in the literature. Many models assume aotal recuperation of the valuable species. This is not true and also, isot just simple as scale by a constant the curves obtained even withhenomenological models. For instance, the dissolution equationsresented in Dixon and Hendrix (1993a,b), and which are widelyccepted, establish that at a enough large time, the valuable min-ral is totally recovered. In this paper we present an extension ofhe hybrid models already presented in Mellado et al. (2009), androve again its accuracy and scalability. Novelty here is that weresent two new models and a function which allows to scale-uphe recovery at infinity.

A complete literature review is outside the objective of thisork, but a brief revision is given hereafter. In Andrade (2004),

s presented a model developed to simulate the transient evolutionf the dissolved chemical species in the heap and column isother-al leaching processes. In Bouffard and Dixon (2001), is presented a

tudy about the heap leaching planning from an optimization pointf view. In Bouffard and Dixon (2006), the rates of pore diffusionnd cyanide gold dissolution in coarse, porous gold oxide ore parti-les are compared. In Cross, Bennett, Croft, McBride, and Gebhardt2006), a computational modelling framework is described for thenalysis of multi-phase flows in reactive porous media targetedt the metals recovery through stockpile leaching and in envi-onmental recovery processes. In Lizama, Harlamovs, McKay, andai (2005), it is shown that for sphalerite and pyrite heap bio-

eaching kinetics were proportional to the irrigation rate divided byhe height. In Mousavi, Jafari, Yaghmaei, Vossoughi, and Sarkomaa2006), is provided a detailed information of momentum and massransfer phenomena in a granular bed. The results obtained suggesthat the liquid phase distribution in the bed is mainly controlled byurface tension and particle induced turbulence appears to havensignificant effects. In Sheikhzadeh, Mehrabian, Mansouri, andarrafi (2005), unsaturated flow of liquid in a bed of uniform andpherical ore particles is studied numerically and experimentally.n unsteady and two-dimensional model is developed on the basisf the mass conservation equations of liquid phase in the bed andn the particles. In Sidborn, Casas, Martinez, and Moreno (2003),

two-dimensional dynamic model for bio-leaching of secondaryopper minerals from a pile has been developed. In Petersen andixon (2007), a comprehensive modelling study of the HydroZ-

nc heap bioleach process, using the HeapSim modelling tool, isescribed. The model was calibrated on the basis of a small num-er of column leach experiments and compared against pilot heapest results. The model calibration thus confirmed, a detailed sensi-ivity study was conducted in order to establish the key parametershat determine the overall rate of Zn extraction. In the present casehese were found to be oxygen gas-liquid mass transfer, variousactors affecting the delivery of acid into the heap, and factorsffecting the temperature distribution within the heap. In Wu,iu, and Tang (2007), the governing equations for a fully cou-led flowing-reaction-deformation behavior with mass transfer ineap-leaching are developed. In Mellado and Cisternas (2008), annalytical–numerical scheme for solving the heap leaching prob-em was developed. In Mellado et al. (2009) was presented a hybrid

cheme which combines phenomenology with insight to gener-te analytical models with good results as compared with moreomplex models.

The main goal of this work is to develop totally scalable leach-ng analytical models. Moreover, some parts of our approach can

Fig. 1. Heap section of height Z.

be used to scale up other models already presented in the litera-ture. Also, our approach is simple but enough accurate to analyze,design, control and optimize the heap leaching process. An outlineof the paper is as follows. In Section 2, we present our mathemat-ical modelling. In Section 3, we present a model which scale-upone kinetic and the recovery when time tends to infinity. In Sec-tion 4, we present a model based on two kinetics and the scale-upof the final recovery. In Section 5, we present, a generalization ofthe model already presented in Section 4. In Section 6, we presentsome numerical experiments and, finally, in Section 7, we end withsome concluding remarks.

2. Mathematical framework

In what follows, we present a mathematical framework to com-pute the kinetics, delays and the recovery when the time tends toinfinity for the heap leaching process. First, we deal with the recov-ery when the time tends to infinity, namely, E∞. In fact, we considern ore particles, of radius R, numbered from 0 to n − 1, which formsa heap section of height Z (see Fig. 1).

We denote by qi the quantity of valuable specie which can berecovered from the particle i when time tends to infinity, 0 ≤ ˛i ≤ 1,the fraction recovered from qi, i = 0, n − 1. We consider that totalrecovery of each particle it is not possible and does not happen

in practice. The reason for that is that in the heap the things doesnot behaves ideally as phenomenological models assume. In fact, inthe heap there exists canalizations, clusters, and particles can notbehave ideally because they are difficult to reach for diffusions.

2 Chemi

E

wˇ

f

a

˛

T

E

If

E

T

E

wtnvb

t

E

Itttes(viS

E

To7q

aethmw

at

22 M.E. Mellado et al. / Computers and

Therefore,

∞ =n−1∑i=0

˛i qi = q̄

n−1∑i=0

˛i,

here q̄ is the expected average value for the set of particles. Let ˛,, � ∈ R. We define a real valued function f : R4 → R by

(˛, ˇ, �, Z):=Z� + ˇ

˛, (1)

nd define,

i = [1 − f ||˛, ˇ, �, Z||]i, i = 0, n − 1.

he above choice will be clear in what follows. Then,

∞ = q̄

n−1∑i=0

[1 − f (˛, ˇ, �, Z)]i = q̄

n−1∑i=0

[1 − Z� + ˇ

˛

]i

.

t is straightforward to show, by using the basic finite summationormula, that,

∞(˛, ˇ, �, Z) = q̄

[1 − (1 − (Z� + ˇ/˛))Z/2R

(Z� + ˇ/˛)

].

he above equation can be normalized to obtain,

∞(˛, ˇ, �, Z) = ˛

Z� + ˇ

[1 −

(1 − Z� + ˇ

˛

)Z/2R]

,

hich is our proposed constitutive equation for the recovery whenime tends to infinity, E∞. This equation includes five parameters,amely, three numbers ˛, ˇ, � , the heap height Z and the averagealue of the particle radius R. In practice, ˛, ˇ, � can be computedy optimization techniques to fit and scale the recovery.

Also, we mention that, in actual operations, Z � R which allowso obtain a simpler expression for E∞, this is,

∞(˛, ˇ, �, Z) = ˛

Z� + ˇ. (2)

t can be seen the above expression is a simple formula that allowso scale-up a model in what concerning the recovery when the timeends to infinity. We consider this formula as a generalization forhe adjustment which is done in phenomenological models. It isnough to consider that � = 0 and ˛ = ˇ + 1. In fact, by using leastquares in the Non-linear regression module of Infostat softwaresee footnote 1), we adjusted three data sets of experimental obser-ations which involves recovery E(t) and time t, the recovery atnfinity E∞ and the height Z. Description of this data is presented inection 6. The obtained model is

(t) = E∞(Z) (1 − q e−k (t−t0)).

he estimates for q and k coincides (q = 1, k = 0.14), but E∞ dependsn Z. This is, for Z = 3, Z = 6 and Z = 9, the values are 78.77, 73.57 and0.08, respectively. The effect on the recovery E(t) of the parameteris not significant.

The choice of the function f is not so arbitrary. For particles athigher height, 1 − f is less than at lower level. This offers sev-

ral choices for f. On the other hand, the experimental data shows

hat the relation between the recovery at infinity E∞ and the heapeight Z, is decreasing and convex. With the objective to deter-ine which function represents in an adequate way this relation,e used the non-linear regression module of the Infostat software1,

1 Infostat software in this module utilizes the Downhill Simplex method, Neldernd Mead (1965), to then use the Levenberg–Marquardt method, 1986, starting fromhe previous solution.

cal Engineering 35 (2011) 220–225

and proved several functions. The best adjustment in terms of sta-tistical significance was the function (2) above. The function f leadsto that expression and corresponds to (1). The estimates for theparameters are ˛ = 90, � = 0.1 and ˇ = 0.03. So, the experimentaladjustments corroborate theoretical results, which are presentedbelow.

Now, we comment on two kinetics and the corresponding delaytime of the whole leaching process. First of all, we note that thetime scale for all the differential equations, after the dimension-less procedure, for the heap leaching model in Dixon and Hendrix(1993a,b), are given in terms of the dimensionless bulk time �,which is given by,

� = ust

�bZ,

and consistently, we consider here that scaling. Here, us is thesuperficial bulk flow velocity [cm3/cm2/s], Z is the heap depth [cm],�b is the bulk solution volume fraction and t is the time [s]. This sim-ple observation allows to obtain one kinetic constant for our modelin terms of us, Z and �b.

Also, from Dixon and Hendrix (1993a,b), the dimensionless dif-fusion time is given by

� = DAet

�oR2,

where DAe is the effective pore diffusivity of the reagent [cm3/cm/s],�0 is the ore porosity and R is the particle radius [cm]. This factallows to obtain another kinetic constant in terms of DAe, �0 and R.

Moreover, in what concerning the delay time for the dominantkinetic �, we refer to the appendix in Mellado et al. (2009), whereall the computations are presented. Indeed, it is proved that,

tw = c�bZ

us

Up to now we have the final recovery E∞, two kinetics for the leach-ing process, i.e., the particle kinetics and the heap kinetics, andthe delay time w in the dominant kinetic �. In the following, weshow how to use these results to obtain analytical models for heapleaching.

3. The first analytical model suitable for scale-up

In what follows, we use a Bernoulli-type model, because it is thecorresponding behaviour of the heap leaching process we are inter-ested in, together with the constitutive relations deduced in the lastSection to obtain our analytical models. Now, we consider just thedominant kinetic � with a reaction order equal to the unity, n = 1.Let D the derivative operator and state the differential equation,

D(D + k�w�)E = 0,

where E is the recovery at time t and k� is a constant related withw� = (us/�bZ). The initial and radiation condition for this equationare,

E(w) = 0

and

E = E∞(˛, ˇ, �, Z) when t → ∞.

It is easy to obtain that,

E(t) = E∞(˛, ˇ, �, Z) (1 − e−k((us/�bZ)t−k̄)), (3)

where k, k̄ and ˛, ˇ, � , are constants to be computed. When the reac-tion order is different to the unity, we need to state the Bernoulli

Chemi

e

wticioa

t

t

E

T

E

Aoaft

E

wbac

4

e

D

wwr

E

a

E

W

E

w

btrt

M.E. Mellado et al. / Computers and

quation,

dy

d�= −k� yn� , (4)

here y is a dynamic variable and n� stands for a generalized reac-ion order with respect to �. It is important to remark that, fornstance, in Petersen and Dixon (2007), the reaction order of chemi-al reactions has been established trough a chemical analysis of thenner reactions. Those formulae can be replaced here also. More-ver we point out that also that model needs to be calibrated, but were interested in the global behaviour and overall in the scaling-up.

Without loss of generality, we define y := E∞(˛, ˇ, � , Z) − E. Then,he differential equation for modelling heap leaching is given by

dd�

(E∞(˛, ˇ, �, Z) − E) = −k� (E∞(˛, ˇ, �, Z) − E)n� . (5)

ogether with

(w) = 0.

he above equation have an analytical solution given by

(�) = E∞(˛, ˇ, �, Z)

−[k� (n� − 1)(� − w) + E∞(˛, ˇ, �, Z)1−n�

]1/(1−n� ). (6)

n important issue from the mathematical and engineering pointf view is the fact that, if we take limit when n� tends to one in thebove equation, one recovers exactly the model for n� = 1. There-ore, we have a family of models which is continuous with respecto n� . Therefore, in actual variables, we have,

(t) = E∞(˛, ˇ, �, Z)

−[

k (n� − 1)(

us

�bZt − k̄

)+ E∞(˛, ˇ, �, Z)1−n�

]1/(1−n� ), (7)

here k, k̄, n� and ˛, ˇ, � , are constants to be computed. As cane seen, the above model can scale-up the height of the heap, thecid flux and the final recovery. The unknown parameters can beomputed by using optimization techniques.

. The second analytical model suitable for scale-up

Now, we consider both � and � kinetics with a reaction orderqual to the unity, n� = n� = 1. We state the differential equation

(D + k�w�)(D + k�w�)E = 0,

here E is the recovery at time t, k� is a constant related with

� = (us/�bZ) and k� is related to w� = (DAe/�oR2). The initial andadiation condition are,

(w) = 0

nd

= E∞(˛, ˇ, �, Z) when t → ∞.

e obtain that,

(t) = E∞(˛, ˇ, �, Z)[1 − � e−k�

(us

�bZ t−w)

− (1 − �) (1 − e−k�

DAeR2�0

(t− �bZus

w))

], (8)

here k� , k� , ˛, ˇ, � and � are constants to be computed.

We now present the case when the reaction orders n� and n� can

e different to the unity. We include the order of reaction relatedo �, n� and the order of reaction n� as the induced by �. We have aecuperation E� , with asymptotic behavior E�,∞, which is due to thehe kinetic induced by �. Also, a recuperation E� , with asymptotic

cal Engineering 35 (2011) 220–225 223

behavior E�,∞, which is induced by �. Moreover, we consider thetotal recovery E given by

E = E� + E�

and, consequently,

E∞ = E�,∞ + E�,∞.

Here, E� and E� are the solutions of the ordinary differential equa-tions,

dd�

(E�,∞ − E�) = −k� (E�,∞ − E�)n� , E�(w) = 0

and

dd�

(E�,∞ − E�) = −k� (E�,∞ − E�)n� , E�

(DAe

R2�0

�bZ

usw

)= 0.

The solutions of these differential equations are given by,

E� = E�,∞ −[

k� (n� − 1)(

us

�bZt − w

)+ (E�,∞)1−n�

]1/(1−n� )(9)

and

E� = E�,∞ −[

k� (n� − 1)DAe

R2�0(t − �bZ

usw) + (E�,∞)1−n�

]1/(1−n� ). (10)

Therefore, by adding the equations we find a heap leaching modelwhere we have assumed that the generalized reaction orders areunknown. This is as follows,

E(t) = E∞(˛, ˇ, �, Z)

−[

k� (n� − 1)(

us

�bZt − w

)+ (� E∞(˛, ˇ, �, Z))1−n�

]1/(1−n� )

−[

k� (n�−1)DAe

R2�0

(t−�bZ

usw

)+((1−�) E∞(˛, ˇ, �, Z))1−n�

]1/(1−n� ). (11)

Finally, we observe that with a sample of observations, nonlinearoptimization methods can be used to adjust this scale-up analyticalmodel.

5. The third analytical model suitable for scale-up

As in the previous section, we consider both � and � kineticswith a reaction order equal to the unity, n� = n� = 1. We can statethe differential equation,

D(D + k�w�)(D + k�w�)E = 0,

where E, k� , w� = (us/�bZ), k� , w� = (DAe/�oR2) have the samemeaning as above. Now, we just only consider the radiation condi-tion to obtain a degree of freedom in the model, i.e.,

E = E∞(˛, ˇ, �, Z) when t → ∞.

It is easy to obtain that,

E(t) = E∞(˛, ˇ, �, Z) [1 + � e−k�((us/�bZ)t−w)

+� e−k� (DAe/R2�0)(t−(�bZ/us)w)]

, (12)

where ˛, ˇ, � , k� , k� , �, � are constants to be computed.We now present the case when the reaction orders n� and n� can

be different to the unity. We include the order of reaction relatedto �, n� and the order of reaction n� as the induced by �. We havea recuperation E� , with asymptotic behavior E�,∞, which is due to

the the kinetic induced by �. Also, a recuperation E� , with asymp-totic behavior E�,∞, which is caused by the kinetic induced by �.Moreover, we consider the total ore recovery E given by

E = E� + E�

2 Chemical Engineering 35 (2011) 220–225

a

E

Ht

a

T

E

a

E

Twu

E

Fo

ttcemeoa

6

mpfptd

E

Ticppts

e

Fig. 2. Simple model prediction.

presents SSE = 94.8, � = 1.540 and R̃2 = 99.75%, i.e., the model (3)explains the 99.75% of the variability produced in the process.

Fig. 3 shows the results for the three heights where, ˛ = 90.00,

24 M.E. Mellado et al. / Computers and

nd, consequently,

∞ = E�,∞ + E�,∞.

ere, E� and E� are the solutions of the ordinary differential equa-ions

dd�

(E�,∞ − E�) = −k� (E�,∞ − E�)n� ,

nd

dd�

(E�,∞ − E�) = −k� (E�,∞ − E�)n� .

he solutions of these differential equations are given by,

� = E�,∞ −[

k� (n� − 1)(

us

�bZt)

+ (E�,∞)1−n�

]1/(1−n� )(13)

nd

�=E�,∞−[

k� (n�−1)DAe

R2�0

(t − �bZ

usw

)+ (E�,∞)1−n�

]1/(1−n� ). (14)

herefore, by adding the equations we find a heap leaching modelhere we have assumed that the generalized reaction orders arenknown. This is as follows,

(t) = E∞(˛, ˇ, �, Z) −[

k� (n� − 1)(

us

�bZt − w

)+(� E∞(˛, ˇ, �, Z))1−n�

]1/(1−n� )

−[

k� (n� − 1)DAe

R2�0

(t − �bZ

usw

)+(� E∞(˛, ˇ, �, Z))1−n�

]1/(1−n� ). (15)

inally, we observe that with a sample of observations, nonlinearptimization methods can be used to adjust this third model.

Before the applications, we summarize that we have proposedhree models. Model 1 described by Eqs. (3) and (7) considers justhe heap kinetic, model 2 presented by Eqs. (8) and (11) whichonsiders heap and particle kinetics and finally model 3, as a gen-ralization of model 2, by means of Eqs. (12) and (15). For eachodel we have stated the equations by considering reactions orders

qual to the unity (exponential behaviour) and also, with reactionrders as different to the unity. In the next Section, we present somepplication for each model.

. Application examples

In this section, present simulations by using our three analyticalodels under similar conditions to standard copper heap leaching

rocess in the north of Chile. We present numerical results for theollowing situation: ore porosity �0 := 0.03, r := 2.5 [cm], effectiveore diffusivity of reagent DAe := 10−6, bulk solution volume frac-ion �b := 0.03, superficial bulk flow velocity us := 0.000333, heapepth Z := 3, 6, 9[m]. For all numerical experiments, we use

∞(˛, ˇ, �, Z) = ˛

Z� + ˇ.

he sample of observations corresponds to a standard heap leach-ng process under operation in the northern part of Chile. Weonsider for the experiments, three heaps of 3, 6 and 9 [m]. Let= 40 and Rp, p = 1, n be the sample of observations for the recu-eration of each heap. Let R̄p, p = 1, n, the values obtained throughhe model. The function to be minimized, which is the sum of the

quared discrete errors, is given by

:=p∑

i=1

(Ri − R̄i)2.

Fig. 3. Second model prediction.

As a performance measure we consider the deviation function,

d:=√

e

p.

The Fig. 2 shows the least squares adjustment by using Excel Solvertool2. Some guess values of the parameters k� , w, ˛, � and ˇ, withZ = 6, were used as initial values in the iterative process for the esti-mation of the parameters in the first model (3), which requires tofind 5 independent parameters (see Fig. 2). The estimated valuesfor that model were used to adjust the second model that requiresto find 7 independent parameters (see Fig. 3). Finally, these lastestimations were used to find the 8 independent parameters forthe third model (see Fig. 4). Due to the scalability of the models,in each case the parameter estimations were done just with thedata corresponding with Z = 6 and the interpolations and extrapola-tions were made by preserving the same set of parameters. Finally,for each model we report the sum of the squared errors SSE, thestandard deviation � = (SSE/n) and the determination coefficient

R̃2 = [(R(t), ˆR(t))]2. The Fig. 2 shows the scalability of the model

(3). We strongly recall the behaviour at infinity is no 100% as thepure phenomenological models predicts.

Fig. 2 shows the results for the three heights where ˛ = 91.76,ˇ = 0.03, � = 0.101, k = 1999.47 and k̄ = 0.000462. This adjustment

ˇ = 0.03, � = 0.091, k� = 1976.63, w = 0.000462, k� = 0.043486 and

2 Microsoft Excel uses the non-linear optimization code GRG2, Leon Lasdon, Uni-versity of Texas, Austin and Alan Waren, Cleveland State University.

M.E. Mellado et al. / Computers and Chemi

�s

�9ivt

mwige

ditoeiocosdpcc

7

btsaitte

model of a copper sulphide ore bed. Hydrometallurgy, 71, 67–74.

Fig. 4. Third model prediction.

= 0.99999. With respect to the model (3), it has been reduced theum of squared errors to SSE = 90.7 and the standard deviation to

= 1.506. The determination coefficient is almost not affected R̃2 =9.74%. This shows that the model explain 99.74% of the variability

n the recovery. It is important to mention that the reduction in theariability implies more reliable estimates where, the inclusion ofhe second exponential is the key point.

Finally, Fig. 4 shows the results for three heights, where the esti-ations corresponds to ˛ = 90.00, ˇ = 0.03, � = 0.091, k� = 1976.63,= 0.000461, k� = 0.043471, � = 0.99999 and � = 0.00861. It is

mportant to remark that, although the model (8) works in a veryood manner, the model (12) reduces even more the sum of squaredrrors SSE = 86.9 and the standard deviation � = 1.474 (although

oes not affect the determination coefficient R̃2 = 99.74%, explain-ng the 99.74% of the variability in the recovery). In this case,he variability reduction implies reliable estimates than the onebtained with the other two previous models. This is almostntirely explained as far one note that there is not included thenitial condition for the delay time giving to the model one degreef freedom which benefits the optimization procedure. Finally, weonclude remarking that all three models behaves very good, andne can think that is enough with the first model. The reason to con-ider the three models here is that our industrial data is extremelyominant in the heap and not at a particle level. The results are com-letely different, for instance, ROM heap leaching operations. Wean cite Mellado et al. (2009), for other proofs in different operationonditions for instance, the radius of the particles.

. Concluding remarks

A scale-up methodology for the heap leaching modelling haseen proposed. The mathematical theory is consistent and showhat one can scale-up by parts several kinds of models. The pre-ented three models were developed by using the combination of

n underlying Bernoulli-type model together with phenomenolog-cal relations in order to obtain analytical models able to scale-uphe heap leaching process. The results in predicting and scalinghe recovery under rather big changes in the height are goodnough to consider these models for the analysis, design, control

cal Engineering 35 (2011) 220–225 225

and optimization of the heap leaching process. Usually, in industryapplications, one have to deal with a number of heaps. The heapleaching planning, to reduce costs and increase utilities, must bedone with optimization techniques. In this sense, the analyticalmodels can be used because they are simple but enough accurate.Moreover, depending on the choice of the optimization algorithm,the model structure is a key point to be considered because a notsuitable choice of the model structure can lead to non global opti-mal solutions.

Acknowledgments

Mario E. Mellado, Luis A. Cisternas and Edelmira D. Gálvezwish to thank CONICYT for financial support, through FondecytProject 1090406. María P. Casanova wishes to thank Universi-dad de Concepción for financial support, through DIUC Project208.014.016-1.0.

References

de Andrade Lima, L. R. P. (2004). A mathematical model for isothermal heap andcolumn leaching. Brazilian Journal of Chemical Engineering, 21, 435–447.

Bouffard, S. C., & Dixon, D. G. (2001). Investigative study into the hydrodynamics ofheap leaching processes. Metallurgical and Material Transactions B, 32B, 763–776.

Bouffard, S. C., & Dixon, D. G. (2006). Evaluation of kinetic and diffusion phenomenain cyanide leaching of crushed and run-of-mine gold ores. Hydrometallurgy, 86,63–71.

Bouffard, S. C., & Dixon, D. G. (2009). Modeling pyrite bioleaching in isothermal testcolumns with the HeapSim model. Hydrometallurgy, 95(3–4), 215–226.

Cross, M., Bennett, C. R., Croft, T. N., McBride, D., & Gebhardt, J. E. (2006). Computa-tional modeling of reactive multi-phase flows in porous media: Applications tometals extraction and environmental recovery processes. Minerals Engineering,19, 1098–1108.

Dixon, D. G., & Hendrix, J. L. (1993a). A mathematical model for heap leaching of oneor more solid reactants from porous ore pellets. Metallurgical Transactions, 24B,1087–1102.

Dixon, D. G., & Hendrix, J. L. (1993b). General model for leaching of one or more solidreactants from porous ore pellets. Metallurgical Transactions, 24B, 157–168.

Lizama, H. M., Harlamovs, J. R., McKay, D. J., & Dai, Z. (2005). Heap leaching kineticsare proportional to the irrigation rate divided by heap height. Minerals Engineer-ing, 18, 623–630.

Mellado, M. E., & Cisternas, L. A. (2008). An analytical–numerical method for solvingheap leaching of one or more solid reactants from porous pellets. Computers andChemical Engineering, 32(10), 2395–2402.

Mellado, M. E., Cisternas, L. A., & Gálvez, E. D. (2009). An analytical approach to heapleaching. Hydrometallurgy, 95(1–2), 33–38.

Mousavi, S. M., Jafari, A., Yaghmaei, S., Vossoughi, M., & Sarkomaa, P. (2006). Com-puter simulation of fluid motion in a porous bed using a volume of fluid method:Application in heap leaching. Minerals Engineering, 19, 1077–1083.

Petersen, J., & Dixon, D. G. (2007). Modelling zinc heap bioleaching. Hydrometallurgy,85, 127–143.

Reverberi, A. P., Esposito, A., & Vegliò, F. (2002). Solid dissolution in a batch processwith intraparticle diffusion and lumped kinetics. Chemical Engineering Science,57(17), 3733–3744.

Roman, R. J., Benner, B. R., & Becker, G. W. (1974). Diffusion model for heap leachingand its application to scale-up. AIME Transactions, 256, 247–252.

Sheikhzadeh, G. A., Mehrabian, M. A., Mansouri, S. H., & Sarrafi, A. (2005). Compu-tational modelling of unsaturated flow of liquid in heap leaching – using theresults of column tests to calibrate the model. International Journal of Heat andMass Transfer, 48, 279–292.

Sidborn, M., Casas, J., Martinez, J., & Moreno, L. (2003). Two-dimensional dynamic

Valencia, J. A., Méndez, D. A., Cueto, J. Y., & Cisternas, L. A. (2007). Saltpeter extractionand modelling of caliche mineral heap leaching. Hydrometallurgy, 90, 103–114.

Wu, Ai-xiang., Liu, Jin-zhi., & Tang, Ling-yan. (2007). Simulation of coupled flowing-reaction-deformation with mass transfer in heap leaching processes. AppliedMathematics and Mechanics, 28, 327–335.