18th European Symposium on Computer Aided Process Engineering – ESCAPE 18
Bertrand Braunschweig and Xavier Joulia (Editors)
© 2008 Elsevier B.V./Ltd. All rights reserved.
An analytical-numerical method for solving a heap
leaching problem of one or more solid reactants
from porous pellets
Mario E. Mellado, Luis A. Cisternas
Depto. Ing. Química, Universidad de Antofagasta, Antofagasta, Chile and Centro de
Investigación Científico Tecnológico para la Minería (CICITEM), Antofagasta, Chile
Abstract
In this paper we present an alternative method based upon analytical and numerical
solutions for the ordinary differential and partial differential equations which describe
the heap leaching problem of one or more solid reactants from porous pellets. We
propose to use analytical solutions for the differential equations which describe rate
dissolutions along the pores and the surface of the particles under suitable regularity
conditions. Moreover, we propose the use of continuous-discontinuous solutions for the
continuity partial differential equations describing balances. All of this, allows to obtain
a numerical scheme which is fast and accurate for simulating the heap leaching
problem. The model includes the effects of particle scales kinetic factors, heap scales
and several operation variables. Finally, numerical experiments are presented.
Keywords: Heap leaching, Differential equations, Analytical and numerical methods.
1. Introduction
Heap leaching is a hydrometallurgical process which has been used since a long time
which was originally designed for oxides ores but today have several applications
including sulphide ores and caliche minerals. Heap leaching has been a matter of wide
research and several mathematical models which allows to simulate, design, optimize
and analyze the phenomena have been developed. Some contributions we consider
relevant to this work are Andrade, 2004, Bouffard and Dixon, 2001, Bouffard and
Dixon, 2006, Cross et. al., 2006, Dixon and Hendrix, 1993ab, Lizama et. al., 2005,
Mousavi et. al., 2006, Sheikhzadeh et. al., 2005, Sidborn et. al., 2003, Wu et. al. The
main goal of this work is to develop an efficient analytical-numerical scheme for
solving the differential equations that describe the heap leaching phenomena.
2. Mathematical framework
Following Dixon and Hendrix, 1993ab, we present a system of Partial Differential
Equations (PDE) and Ordinary Differential Equations (ODE) which describes the heap
leaching process. The ODE to compute the dimensionless grade of solid reactant within
the particle  p is given by,
d p
dt

kp
(1   )
p
 p ,
 p ( x, 0)  1,
(1)
where t is the time, x is the space coordinate, k p is the Damkohler II number for solid
reactant within the particle, dimensionless,  is a dimensionless stoichiometric ratio,
 is the fraction of solid reactant residing on the particle surfaces initially,  is the
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M. Mellado & L. Cisternas
dimensionless reagent concentration and  p denotes the order of reaction. The ODE to
compute the dimensionless grade of solid reactant on the particle surface  s is given by
d s
k
  s  ss  b ,
 s (0)  1,
(2)
dt

where k s is the Damkohler II number for solid reactant on the particle surface,
dimensionless b stands for the dimensionless concentration of the reagent in the bulk
solution external to the particle. The continuity equation for  is
  2 2 
 2 
 k p p ,
(3)
t
x x
x
together with,

 ( x, 0)  0,
 (1, t )   b ,
(0, t )  0,
x
Moreover, it follows that,

0 (1   0 )k p C pp R 2
3k C s R
 0bi C A0
ks  s s , k p 
, 
,
bi DAe
bi DAe
0 (1   0 )CE 0
where 0 is the ore density [g/cm3],  0 is the ore porosity, C p is the grade of solid
reactant within a particle [mol/g], bi is the stoichiometric number [mol/mol], DAe is the
effective pore diffusivity of reagent A [cm3/cm s], C A0 is the reference reagent
concentration, CE 0 is the initial extractable grade of solid reactant [mol/g],  b is the
bulk solution volume fraction,  h is the heap void fraction and u s is the superficial
bulk flow velocity [cm3/cm2 s]. We now replace the dimensionless diffusion time t in all
the particular model equations by the dimensionless bulk flow time  . The PDE for the
dimensionless concentration of dissolved species  i , is given by
3  i  i  2  i 2  i



 k p p ,
 
t
x x
x 2
(4)
together with i ( x,0)  0, i (1, t )  b , i x (0, t )  0 . Here, we have the design
parameters,
b
3(1   h ) DAe Z
ut

, 
,  s ,
2
 0 (1   h )
b Z
us R
where R is the particle radius [cm] and Z is the heap depth [cm]. The PDE for the
dimensionless bulk concentration of dissolved species  ib is given by,
 ib

  
  ib    i 
,  ib ( z, 0)  0,  ib (0,  )  0

z
 x  ( x 1)
The balance equation in the bulk lixiviant is given by,
 b

  
  b   
,  b ( z, 0)  0,  b (0,  )  1

z
 x  ( x 1)
(5)
(6)
The differential equations (1)-(6) complete the ODE-PDE model for the heap leaching
process. We use the notion of extraction of reactant from the heap given by

E ( )     ib (1, t )dt (7)
0
An analytical-numerical method for solving a heap leaching problem…
3
3. Analytical-numerical scheme for the model problem
Now, we discuss in detail the analyticalnumerical aspects for solving the system (1)-(6).
We first concentrate our attention in (1)-(2). Certainly, at a first glance, these equations
can not be solved analytically. As we will solve the system of differential equations in
an iterative way, we can assume that the functions  and b have enough regularity. If
we assume the latter, the equations (1)-(2) can be solved analytically. Certainly, and
considering reaction order for both equations as equal to the unity, we find that,
k 
 p ( x, )  exp( p
),
(8)
3(1   )
and
 k 
 s ( x,  )  exp( s
),
(9)
3
It is important to recall that if one need to consider reactions orders  p and  s different
to the unity, the differential equations can be solved analytically by using the Bernoulli
1
transformation    p p and    s1s , respectively. Now we describe our approach
for the partial differential equations (3)-(4). We solve these equations with a CrankNicholson fully implicit finite difference scheme (Hackbush, 1992). It is very important
to note that the matrices generated are sparse, non symmetric and stable. The systems of
equations are solved with a preconditioned LU method.
Now we present our approach for the partial differential equations (5)-(6). We solve
these equations in an analytical manner. We recall that in the bulk lixiviant of the heap,
the mass balance for dissolved species is given by (5). To obtain analytical solutions,
we consider the transformation,   z   and   z   . We find that,
ib ( z, )   ,
  z,
(10)
and
ib ( z, )   z,
  z,
(11)
  i 
where     . By using the same arguments, we find the following analytical
2  x  ( x1)
solution for equation (6),
(12)
b ( z, )   ,
  z,
and
(13)
 b ( z,  )  1   z,
  z,
where       ,. We note that (10)-(11) is continuous in R2 while (12)-(13) is
2  x  ( x1)
discontinuous when z   . This discontinuity reveals the important fact that, for b , the
PDE must be modified in order to have a consistent model. This can be achieved, for
instance, by introducing a second order term which allows to have a continuous
solution. The discontinuity is just when z   which suggests the use of this
discontinuous solution because the regularizing effect of the algorithm sequence of
integrations.
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M. Mellado & L. Cisternas
4. Heap leaching with distribution function for different particles sizes
To deal with heap leaching with a distribution function for different particles sizes, we
define a dimensionless particle radius as   R R , where R is a reference radius, and
making the scaling k p  2 k p ,   2 ,   2 and using the Gates-GaudinSchuhmann distribution function of parameter m, the algorithm is as follows: for a
 Z space step, solve:
 Given  p and b , obtain  by solving
  2 2 
 2 
 k p p ,
t
x x
x

k p 
3(1   )
),
 s ( x, )  exp(

(0, t )  0,
x
 k s 
),
3
Obtain  i by solving
3  i  i  2  i 2  i



 k p p ,
 
t
x x
x 2

 (1, t )   b ,
With  obtain  p and  s through
 p ( x, )  exp(

 ( x, 0)  0,
i ( x,0)  0, i (1, t )  b , i x (0, t )  0 .
Compute  ib and b through
 ib ( z , )  

2
  i 
m 3
 x  m d   

 ( x 1)
  z,
  i 
m 3
 x  m d   z

 ( x 1)
  z,
  
m 3
 x  m d   
  ( x 1)
  z,
1
0
and
 ib ( z,  )  

2
1
0
also,
 b ( z, )  

2
1
0
and
 b ( z, )  1 


2
1
0
  
m 3
 x  m d   z
  ( x 1)
  z,

Compute and store E ( )     ib (1, t )dt
0
Do the same up to Z=1. Then, increase  t . Once the desired simulation time is
reached, stop the iterations.
We use Gauss quadrature for computing E. We note that, even for dense simulations,
the main time cost is due to the numerical solution of (3)-(4).

5. Numerical experiments
First, we need to point out that our approach is able to reproduce the results shown in
Dixon and Hendrix, 1993ab, by considering the operation conditions indicated there. A
complete agreement with that simulations was used to test our computational code. We
An analytical-numerical method for solving a heap leaching problem…
now present an industry simulation by considering a standard heap leaching process in
the north of Chile. We present numerical results for the following situation: 0 :=1.3,
 0 :=0.03, k p :=0.000157, C p :=0.000157,r:=2.5, bi :=1.0, DAe :=10^{-6}, C A0 :=0.00333,
CE 0 :=0.000157,  b :=0.03,  h :=0.03, u s :=0.000333, z:=1000.0.
Figure 1.  i against days.
Figure 2. E against days.
The above pictures show an agreement with the industry standard heap leaching process
which validates our approach and allows to conclude this simple modification of the
5
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M. Mellado & L. Cisternas
numerical algorithm for solving (1)-(6) can produce good results at a low computational
effort.
6. 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). Also, this
methodology can be applied to obtain totally analytical models by considering the
presented analytical solutions together with analytical solutions for the equations (3)(4), for instance, by some reasonable simplifications of the differential equations. A
complete agreement with industry processes validates our method from the engineering
point of view.
Acklowledgments
LC wishes to thank CONICYT for financial support, through Fondecyt Project
1060342.
References
de Andrade Lima, L. R. P., (2004). Brazilian Journal of Chemical Engineering 21, 435-447.
S. Bouffard. & Dixon, D. G., (2001). Metallurgical and Material Transactions B 32B, 763-776.
S. Bouffard, & Dixon, D. G., (2006). Hydrometallurgy 86, 63-71.
M. Cross, Bennett, C. R., Croft, T. N., McBride, D. & Gebhardt, J. E., (2006). Minerals
Engineering 19, 1098-1108.
D. Dixon, & Hendrix, J. L., (1993). Metallurgical Transactions 24B, 1087-1102.
D. Dixon, & Hendrix, J. L., (1993). Metallurgical Transactions 24B, 157-168.
W. Hackbusch, (1992). Elliptic Differential Equations. Springer-Verlag.
H. Lizama, Harlamovs, J. R., McKay, D. J. & Dai, Z., (2005). Minerals Engineering 18 , 623630.
S. Mousavi, Jafari, A., Yaghmaei, S., Vossoughi, M. & Sarkomaa, P., (2006). Minerals
Engineering 19, 1077-1083.
G. Sheikhzadeh, Mehrabian, M. A., Mansouri, S. H. & Sarrafi A., (2005). International Journal of
Heat and Mass Transfer 48, 279-292.
M. Sidborn, Casas, J., Martinez, J. & Moreno, L., (2003) Hydrometallurgy 71, 67-74.
Wu Ai-xiang, Liu Jin-zhi & Tang Ling-yan, (2007). Applied Mathematics and Mechanics 28,
327-335.