Macroscopic transport mechanisms as a rate-limiting factor in dump leaching of pyritic ores

Macroscopic transport mechanisms as a ratelimiting factor in dump leaching of pyritic ores G. Pantelis and A. I. M. Ritchie Environmental Organization,

Science Program, Australian Nuclear Menai, New South Wales, Australia

Science

and Technology

The oxidation of mining heaps containing pyrite is modelled as a coupling of macroscopic transport and the microscopic particle reaction kinetics. A numerical model is developed incorporating oxygen and heat transport based on the two major mechanisms of convection and diffusion. We examine separately the oxidation of single particles and compare these rates with that of an entire heap to establish the rate limitation imposed by macroscopic transport. Keywords:

pyrite oxidation,

bio-leaching,

mathematical

The leaching of metals such as copper, nickel, zinc, and uranium from some ores depends on the conversion (oxidation) of largely insoluble metal sulphides to the much more soluble sulphates. The process is usually carried out in large heaps of sulphidic material through which water is passed. The water dissolves the metal sulphates and passes out through the base of the heap, where it is processed to remove the metal of economic interest. The process is one of concentration and is generally applied to low-grade ores for which conventional hydrometallurgical and pyrometallurgical techniques are uneconomic. Crucial to the solubilization process is bacterially catalyzed oxidation of iron pyrite or other iron containing pyrite.‘,2 There have been a number of studies on oxidant transport and oxidation of pyritic material within particles3-6 and a number of models flowing from these studies. These models largely attempt to predict oxidation of entire heaps based on the reaction kinetics within individual particles with little or no consideration of the macroscopic transport mechanisms. Commonly used is the so-called “shrinking core model,“’ which indicates that the characteristic time scale for oxidizing all the pyrite in a particle is t, = la2p,,l(3@2esc0)r which is about 0.09 years for a 0.002m particle and typical values for the other parameters. This time scale is typical of leaching rates observed in many reactor and shake flask experiments. The characteristic time scale for convection of a gas through a heap with a competing chemical reaction is t,. = Address reprint requests to Dr. Pantelis at the Environmental ence Program, Australian Nuclear Science and Technology nization, Mail Bag I, Menai, NSW 2234, Australia.

136

I9 February

Appl.

which, for typical parameter values, is about 0.43 year, somewhat longer than t,, while td = E~,,L~I(c~D~), the time scale for diffusion through a heap with a competing chemical reaction, is much longer at about 83 years. This means that in many heaps, oxygen transport is likely to be rate limiting and that high overall leach rates are likely to occur in heaps in which convection is effective. It follows that heap performance may well reflect more the macroscopic physical properties than the microscopic kinetics of pyrite oxidation. In an industrial heap, which can measure tens to hundreds of meters in height, oxygen will initially enter the heap from the atmosphere/heap interface by the process of molecular diffusion. At this time the oxidation reaction is confined to the surface layers of the heap. The heat released from the pyritic reaction will cause small changes in the air density within the heap. This in turn will induce a convective air current within the heap and, in so doing, increase the rate of atmospheric oxygen transport into the heap and permit oxidation throughout the heap. Davis and Ritchie’-’ considered the case in which transport was a two-stage process: diffusion through the pore space of the dump followed by diffusion into reaction sites within particles comprising the dump. This model is applicable to columns and thin heaps in which convection is negligible. Cathles and Schlitt’” considered a similar case but assumed convection through the dump to be the only macroscopic transport process. We shall attempt here to combine convection and diffusion and to show that the initial period (or startup period) preceding the onset of convection is sufficiently long to warrant the inclusion of the two macroscopic transport mechanisms. Also of considerable interest is the extent to which oxidation rates in the dump are dominated by macroscopic transport processes and the impact that the E~.,Lpl,l(Kpf;gpToco),

Introduction

Received

model, convection-diffusion

Math.

1990; accepted

Modelling,

31 August

1991,

SciOrga-

1990

Vol.

15, March

0

1991 Butterworth-Heinemann

Rate-limiting

factor in dump leaching of pyritic ores: G. Pantelis and A. I. M. Ritchie

oxidation rate at the particle or microscopic level has on the global oxidation rate (GOR) of the dump. We shall model the heaps as a two-phase system consisting of a rigid, solid porous phase made up of spherical particles through which an air phase flows. We shall use a shrinking core model to describe oxidation at the microscopic level, since, as is shown below, it is then easy to change the oxidation rate at the microscopic level. We consider dumps composed of only one-sized particles at the moment, since extension to include the effect of a range of particle sizes is straightforward if computationally tedious.’ It is unlikely that the additional computational effort will change conclusions on the relative importance of diffusive compared to convective processes or macroscopic compared to the microscopic transport processes. Similarly, we shall omit the water phase. We note, however, that the macroscopic transport mechanisms can be significantly affected by the presence of the water phase, as is demOur model heaps will be onstrated elsewhere.” cylindrical, which is somewhat of an idealization but sufficiently representative of many real heaps. Mathematical

+

formulation

v . (pp”)

= 0

(x, t) E 0’

(1)

where the intrinsic density p” is related to pu by the relationship Pa = EPa

CY = a,s

-

PT)

(4)

v=pa = p;gpg

(x, t) E R’

with the associated

initial and boundary

P” = -P%z

n-v”=0

(5)

XELn,

conditions

t = 0;

(x,0=;;

p” = -pijgz

(6)

(x, t) E 6,

We consider only the oxygen species in the air phase and set c = pawU, where ma is the mass fraction of the oxygen in the air phase. Employing the approximation P‘I = p;, we have & •~~

+

GJV ".Vc

-

V.(E,DVC)

=

-vS(c,f,T)

(X,t)Ea'

(7)

where u = 3yD2e,la2. Here, C = pSws,where o.’ is the mass fraction of the reactant (sulphur) in the solid phase. We adopt here the shrinking core model so that

We consider a heap occupying a domain fl lying on a relatively impermeable and insulating ground surface r,, z = 0, and bounded above by a surface TZ, z = z(,. The sides of the heap will be denoted by r3. Let to > 0, 0 < t 5 to be some time over which the problem is considered, and let R’ = R x (0, to] and r! = Ti x (0, to] (i = 1, 2, 3). We consider two phases, (Y = (I, s (air and solid, respectively), where the solid phase is assumed to be rigid. The mathematical description that follows is based on the macroscopic equations that describe multicomponent flows through porous media, which are derived in Ref. 12. The mass balance of the air phase is given by 2

P” = Pa

By using equations (l)-(4) the mass balance equation for air under the Boussinesq approximation becomes

(2)

The air velocity v” is assumed to be governed by Darcy’s law through (3) The pressure is taken in relation to the atmospheric pressure at ground level (Z = 0). The temperature on the entire atmosphere/heap interface is assumed to be constant at the ambient atmospheric temperature. The air density within the dump will remain close to its ground-level atmospheric value but will deviate from it owing to heat released from reactions. We assume that the temperature T, which is taken relative to the ambient atmospheric temperature, is in local equilibrium with all phases, and we introduce the relationship

cp3/(p;,;3

S(c, f, T) =

_

c;l/3)

o

T<

To

T 2 T,,

(8)

where CltCo = p_. As was mentioned in the introduction, microorganisms play a crucial part in stepping up the oxidation rate of pyrite present in the particles. For simplicity we have included a temperature ceiling of To > 0 at which the microorganisms cease to be effective as catalysts for the oxidation of pyrite. We note that in reality the temperature dependence is more complicated and that there is a range of microorganisms that come into play at different temperatures.‘,= Detailed data on temperature dependence are not available, but it seems important to include the point that microbial activity will cease in heaps at temperatures approaching the boiling point of water. The initial and boundary conditions associated with (7) are XELn, t = 0; c = co n . (vc - DVc) = 0 (x,t)=;; c = co (x, t) E r;,,

(9)

The depletion of solid reactant is expressed dc’ -_= dt

- aS(c,

as

6, T)

where v = ~YD~cJ(cu~). To avoid the singularity at t = 0 in the computations, we set the somewhat artificial initial condition E = 0.99p,, x E R, t = 0. Since we are assuming that the temperature T is in local equilibrium with all phases, the heat equation for all phases combined becomes x paca $ + p,c#

. VT - V . (D,VT)

a = hS(c, 6, T)

Appl.

Math.

Modelling,

1991,

(x,t>Efl’

Vol. 15, March

(II)

137

Rate-limiting

factor in dump leaching of pyritic ores: G. Pantelis and A. 1. M. Ritchie

where A = 3&yD+J(&). For (11) we impose the initial and boundary conditions T=Q

x E R, t = 0;

(x, t) E r;;

T=O

Outline of numerical

scheme

n,VT=Q (x9 t) E G.3

(12)

The system of equations (5)-( 12) is highly coupled and of a mixed type. The air pressure equation (5) is of the elliptic type, whereas the oxygen transport equation (7) and the heat equation (11) are parabolic but take on hyperbolic properties as convection becomes dominant. Because of these points it may well be expected that solving the entire system simultaneously in one implicit scheme will result in excessive computations and possibly a failure to converge. Of the several options at our disposal we found that convergence failures could be avoided through a decoupling finite difference scheme by first solving explicitly, in each time step, the oxygen transport equation (7), the heat equation (1 l), and the solid reactant depletion equation (10). The updated temperatures were then used in the finite difference approximation of (5) to obtain the air pressures iteratively. For the sake of the outline presented here the numerical solution of (5)-(12) will be discussed in the context of a planar geometry x = (x, y) (y being the vertical coordinate) with a uniform grid system of intervals Ax and by. The modification to cylindrical coordinates should then be straightforward. The incorporation of the boundary conditions (6), (9), and (12) into the finite difference systems presented will not be explicitly described, since the details are cumbersome to write down but straightforward to implement. We depart for the moment from the nomenclature of the previous sections. The oxygen transport equation (7) and the heat equation (11) take the form of the convection-diffusion equation dll

;)t+v.Vu-DV2u=S

(13)

where u stands for c or T. The convection coefficients v and the diffusion coefficient D are different for the oxygen transport and heat equations, and we have linearized the diffusion term, although this is not an essential part of the scheme to be described. A simple difference scheme for (13) that involves a forward time central space scheme (FTCS scheme) can be obtained by using a forward difference in time and central finite differences for the spatial derivatives. Such a scheme is well known to require a necessary and sufficient stability condition in time” Atsmin

1

Ax2Ay2 20 2D(Ax* + Ay*)’ 11~1~

(14)

Along with this stability condition must be attached an accuracy limitation At << ~D/~zJ[~.While the stability and accuracy conditions are restrictive in the requirement of very small time steps, this explicit scheme was a practical consideration and performed well for

138

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Modelling,

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Vol. 15, March

most cases. However, in the case of very high permeabilities (K 2 10Py m2), spurious oscillations occurred when convection became very large. To overcome this problem, a procedure was adopted that has become popular in recent years for convection-dominant problems. I4 This scheme involves decoupling convection from diffusion by rewriting (13) in its Lagrangian form du - = DV*u + S dt

(15)

where the substantial derivative dldt indicates the time rate of change along the characteristic curves associated with pure convection. These convection flow paths are defined by dx -_=v dt

Let xcj denote the position of the space grid point at (xi, yj) on a prescribed fixed grid system of uniform intervals Ax and Ay. The origin of a fluid particles -nXU ’ = (X8- ‘, Y;- ‘) at the previous time step t”- ’ reachmg the grid point xti at time t” is obtained from a simple discretization of (16) through K;-’ = xij - v;jAt

(17)

where vij = v(xJ and we have assumed that the velocities are effectively constant in time within the relatively small time step At. It is noticed that the origins of fluid particles sZ;- ’ at time t”- ’ are found by tracking particles backward in time from each grid point x0 at time I”. An explicit finite difference scheme for (15) can now be written in the form

u; -

--n-l

ujj

= mj-’

+ gj-’

(18)

At where

u;

=

u(xti,

fl),

@F’

=

u(x$~‘,

f-l),

3j-l

=

S(sZ$-‘, t”- ‘), and [DV2u];P I denotes some spatial discretization of the diffusion term centered at the point Since the points X;-’ will generally sZ;-’ at time T’. lie off the prescribed fixed grid points, some interpolation scheme must be employed to obtain the values of Uz-’ and ~j-’ from the known grid point values of u and S, respectively, at the time t”- I. The simplest interpolation that can be taken involves the four surrounding grid point values through bilinear interpolation. Higher-order interpolation schemes could of course be used, but we found that bilinear interpolation was adequate. A discretization of the diffusion term [ DV ‘uj$- ’ may be achieved by using the standard central difference scheme of the spatial derivatives centered at the point X;-’ using the surrounding values u(X$~’ 2 Ax, y;-‘, t” - ‘) and u(??;- ’ , j$-’ k Ay, F’), which must also be obtained by interpolation. In such a scheme it can be shown that a necessary and sufficient condition for stability of (18) is given byI At+2D(&+&)]-’

(19)

Rate-limiting

factor in dump leaching of pyritic ores: G. Pantelis and A. I. M. Ritchie

It should be remembered that two such stability conditions of the form (19) exist, one for the oxygen transport equation and one for the heat equation. The selected time step size is the minimum of the two. Apart from the fact that (19) is less restrictive than (14), the scheme (17)-(18) was found to perform very well in all cases considered, and the spurious oscillations observed in using the FTCS scheme described above were avoided. The progressive stages of the computations can then be summarized as follows. Given the grid point values of c, c v”, and T at times t”-’ the discretized convection-diffusion equations for c and T given in the form (17)-(18) described above are used to obtain the grid point values of c and Tat the new time step t”. Using the grid point values of c, c, and T at time t”-’ and the grid point values of c and T at time t”, the grid point values of C:at the new time t” were computed by using a predictor-corrector difference scheme (improved Euler scheme). Within the same time step the new temperatures are then employed in the finite difference approximation of the air pressure equation (5) given by PY+l,j +

PK1.j

-

2P$

+

P!lj+l

+

P:lj-I

-

2P$

Ay?

AX’ T” =

pp

‘.J+ ’

-

2Ay

T!‘. r.,

I

(20)

where p; = p”(x,, t’l). The scheme (20) is implicit, and the new pressures are obtained after iteration on the corresponding system of difference equations. Having obtained the air pressures at the new time step t”, the new air velocities are obtained from a finite difference approximation of (3), and the entire procedure is repeated. Numerical

Figure 1. Fraction of reactant in a spherical particle as a function of time using the shrinking core model of diffusing oxygen for two particle sizes. The oxygen at the surface of the particle is maintained at a constant value

results and discussion

Our main goal here is a comparative study between the microscopic oxidation rates and the global oxidation rate (GOR) of the heap as modified by the macroscopic transport processes. In the “shrinking core model” the oxidation of a spherical particle occurs by a diffusing reaction front originating from the outer surface and directed into the center. The oxidation rate of a particle is therefore highly dependent on the particle size and, as is indicated above, is a simple way to change the oxidation rate at the microscopic level. Figure 1 demonstrates this for two particle sizes, a = 0.005 m and a = 0.002 m, the curves of which were obtained by numerically solving (10) and assuming that the oxygen concentration at the surface of the particle was fixed at c = cO_ The heaps were chosen to have vertical sides of height 20 m and radius 20 m. This is small for a typical commercial copper leach dump but about the size of dumps for preoxidation of auriferous pyrite ores. Cathles and Schlitt’O restricted their examples to permeabilities of 10P9 m2, but measurements of the waste rock dumps at Rum JungleIs suggest permeabilities typically on the

order of IO-” m2. Our simulations indicated that for dumps of the dimensions considered here, convection could not really be considered significant within the first 3-4 years for permeabilities less than about IO “’ m2. Therefore as a first example a permeability of K = 10 ‘Om2 is used with a particle size a = 0.005 m, resulting in the time scale t,, = 0.56 y and t, = 4.3 y. Figure 2 shows two stages in the evolution of oxygen concentration and temperature in the heap of permeability K = IO- “’ m2 and particle size a = 0.005 m. An ambient temperature of 30°C was used in all the examples considered here. At year 2, oxygen transport is largely by diffusion from the atmosphere/heap interface, which includes the top surface. The inward bending of oxygen contours at the side near the base indicates the onset of convection. At year 4, heatinduced aitflow has increased, and convection in from the sides is clearly evident; however, air flowing vertically upward works against diffusion directed downward. The temperature peak that follows the reacting region moving in from the side has reached the ceiling of T,, = lOO”C, resulting in the local switching off of the oxidation reaction. The air velocity field shown in Figure 2 indicates that air flows predominantly from the side of the heap through the hot region and then up through the top surface. In doing so, it penetrates, in some places, the cooler interior regions. The air velocity field also indicates some circulation at the bottom center of the heap. This circulation is more apparent in the air velocity field shown in Figure 3. Higher permeabilities lead to the earlier establishment of convection and of high temperatures. Figure 3 shows that with K = low9 m2, convection is well

Appl. Math. Modelling,

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Rate-limiting

factor in dump leaching of pyritic ores: G. Pantelis and A. I. M. Ritchie

Figure 2. Oxygen content and temperature contours and air velocity fields in a cylindrical heap at years 2 and 4. The permeability K = lo-” m2, and the particle size a = 0.005 m. The top two figures correspond to the percentage of oxygen relative to atmospheric oxygen content within the heap at years 2 and 4. The middle two figures show temperature contours for years 2 and 4 corresponding to an ambient atmospheric temperature of 30°C. The bottom two figures show the corresponding air velocity fields. A vector length of 0.1 in axis units corresponds to 9.68 x 10 -’ m/s for year 2 and 1.05 x 10 m-zm/s for year 4

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Rate-limiting

factor in dump leaching of pyritic ores: G. Pantelis and A. 1. M. Ritchie

established by 1 year and that a large region of the heap is at 100°C. The oxidation reaction is switched off in this region, which means that oxygen, carried by the air flow, can penetrate through it and fuel oxidation in the cooler parts of the heap. This has a profound effect on both the oxygen concentration and solid reactant profiles, as can be seen in Figure 3. Since the computer run time was a concern for all examples considered, tests on the accuracy of the time stepping and mesh convergence were somewhat limited. Short runs, representing l/2-year simulation time, were made for selected cases using spatial grid systems ranging from 11 x 11 to 41 x 41 and different maximum time step sizes. The results of the trial runs suggested that a spatial grid system of 21 x 21 with a maximum time step size corresponding to 10 P5 year was adequate for all examples. The fraction of pyrite left in the dump as a function of time in heaps of differing particle size and permeability are shown in Figure 4. These curves should be examined in relation to Figure I. For each permeabil-

increasing the microscopic oxidation rate does not lead to a comensurate increase in the GOR. This clearly indicates that macroscopic transport is an important factor in heap leaching operations. For the heap geometry considered here, increasing the permeability increases the GOR, indicating further the effectiveness of air convection. Curves (c) and (d), associated with permeabilities K = 10 ~ lo m2, do not indicate significantly improved rates over the simulation in which convection is removed all together (curves (a) and (b)). For permeabilities K 2 lo-” m2, air convection becomes an important gas transport process within the first 2-3 years of heap life and leads to significantly greater GOR. However, overheating, resulting in a switching off of the reaction, becomes important and negates the effectiveness of the induced air flow. ity,

Conclusion In industrial heaps containing pyritic ores that measure some tens to hundreds of meters in height, convection

Figure 3. Oxygen content, solid reactant content, temperature, and air velocities at year 1 in a heap with permeability K = 10m9 m* and particle size a = 0.005 m. The oxygen content is given as a percentage of atmospheric oxygen content in air, and the temperature corresponds to an ambient atmospheric temperature of 30°C. The solid reactant content is given as a percentage of its initial concentration. An air velocity vector length of 0.1 of the axis units corresponds to 1.70 x 10m4 m/s

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convection must also depend highly on heap geometry. To investigate these and other effects, it is necessary to design more generalized numerical schemes that are robust enough to handle the changing dominance of the various macroscopic transport mechanisms. The strong dependence of the variables on the many heap parameters, especially permeability, indicates the need to develop reliable techniques and equipment to obtain field data that define the individual heap under consideration. Nomenclature

I 0.4

0.8

I.2

1.5

2.0

Time (years)

Figure 4. Fraction of solid reactant left as a function of time in a heap of height 20 m and radius 20 m. The six curves correspond to (a) diffusion only and particle size a = 0.005 m, (b) diffusion only and particle size a = 0.002 m, (c) permeability K = IO-” m* and particle size a = 0.005 m, (d) permeability K = IO-” m2 and particle size a = 0.002 m, (e) permeability K = 10 ’ m2 and particle size a = 0.005 m, and (f 1permeability K = 10 ~’ m* and particle size a = 0.002 m

can significantly increase the oxidation rate. Initially, oxygen will enter the heap from the atmosphere/heap interface, and oxidation of the pyritic ore will be confined to a thin region around the surface. As the heat of reaction increases, a convective air current will be induced within the heap and increase the rate of oxygen input into the heap. The degree to which convection takes off is highly dependent on the permeability of the heap material. While convection can significantly increase the oxidation of the heap, it is limited by the process of overheating, which results in the switching off of the pyritic oxidation reaction. The time interval or startup period for the onset of convection varies for each individual dump but is sufficiently long to necessitate the need to include both transport mechanisms of diffusion and convection in the model. Our simulations have shown that an increase in the oxidation rate at the microscopic level by many factors translates to only about a l-5% increase in the GOR of the heap studied. This suggests that parameters describing the reaction kinetics, as measured, say, from shake flask or column experiments, are important but cannot be translated readily to describe the oxidation rates of large heaps. The model presented in this paper included only the solid and air phases. However, any attempt at a more quantitative model of real systems of heap bioleaching must include the water phase. Water content can significantly affect the air permeability of the ore material, and water draining at the base of the heap can also act as an effective heat remover from the heap. As well as a strong dependence on permeability, induction of

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particle radius (0.002-0.005 m) oxygen concentration (kg/m3) oxygen concentration in air (0.265 kg/m3) density of reactant in the solid phase (kg/m3) specific heat of the LYphase (c, = 866 J kg-’ KP’, c, = IO3 J kg-’ K-‘) coefficient of heat diffusion (0.65 J rn-’ K-’ s-1) diffusion coefficient of oxygen in air in the heap (1.5067 x lops m*/s) oxygen diffusion coefficient in particles (2.6 x 10 -9 m2/s) acceleration due to gravity (9.8 m/s2) permeability of the porous matrix (m2) characteristic length scale of heap (m) pressure of the air phase (kg rn-’ s -2) temperature relative to atmosphere (K) time (s) macroscopic velocity of the air phase (m/s) height (m) coefficient of thermal expansion (3.4 x lop3 KP ‘) a proportionality constant encompassing both Henry’s law and gas law (0.03) mass of oxygen used per mass of solid reactant in oxidation reaction (1.746) volume fraction of the (Yphase (E, = 0.3, E, = 0.7) density of the (Yphase (kg/m*) (p., = 1500 kg/m3) intrinsic density of the (Yphase (kg/m’) density of air (1.2 kg/m3) initial density of reactant in solid phase (15 kg/m3) viscosity of the air phase (1.9 x 10 -5 kg m ’ s-1)

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