A comprehensive model for copper sulphide heap leaching

Hydrometallurgy 127–128 (2012) 150–161

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A comprehensive model for copper sulphide heap leaching Part 1 Basic formulation and validation through column test simulation C.R. Bennett a, D. McBride a,⁎, M. Cross a, b, J.E. Gebhardt b a b

College of Engineering, Swansea University, Swansea, UK PERI, Salt Lake City, Utah, USA

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 14 March 2012 Received in revised form 19 July 2012 Accepted 7 August 2012 Available online 17 August 2012 Keywords: Heap leach model Copper sulphides Computational fluid dynamics (CFD)

This paper covers the basic formulation of a comprehensive copper heap leach model based on computational fluid dynamics (CFD) technology together with its parameterization and validation against laboratory column test data. For the column test data used here, the model formulation covers an ore with a mixture of chalcocite and pyrite in a column under leach with a ferric raffinate and includes the reaction kinetics of the dissolution of the key minerals within the context of a shrinking core algorithm to accurately model the leach behaviour. Precipitation species, the role of bacteria and the treatment of unsaturated fluid and gas flow in porous media is also described in the context of the model. The current work demonstrates the use of small column test leach data, including particle size distribution, to characterize the ore. The model was then used with the same parameter set to produce a good fit to results from larger columns and different crush sizes. © 2012 Elsevier B.V. All rights reserved.

1. Introduction

which gain energy as a by-product of the reaction (Bailey and Hansford, 1994; Hansford and Bailey, 1992),

Stockpile leaching as a solution mining method to recover metals from primary ore is of increasing importance at a time when demand for metals is continuing to rise, whilst the available mineral grades are degrading; see Bartlett (1998) for an overview of these technologies. Stockpile leaching provides a cost effective technique for the recovery of a range of metals from low grade mineral deposits. In particular it is used for gold and copper, though it is also used for a range of other ores, such as, nickel and uranium. Stockpile leaching typically involves stacking crushed ore in lifts of between 3 and 10 m in height and applying a lixivant (raffinate) to the upper surface that trickles down though the ore, reacting with the minerals present in the ore particles. The resultant pregnant leach solution is collected from the base of the stockpile for further processing to extract the valuable metals. Recovery of metals from sulphide minerals, such as chalcocite, commonly uses bioleaching, where the minerals are oxidized to produce a soluble metal salt. Oxidation normally takes part in two stages, in that the main agent of oxidation is the ferric (Fe 3 +) ion. A reaction with a metal sulphide ore may take the form 3þ

MS þ 2Fe

2þ

→M

þ 2Fe

2þ

þS

ð1Þ

where M is the metal to be leached and S is sulphur. Ferrous (Fe 2 +) ions are then oxidized back to ferric with oxygen dissolved in the raffinate and oxygen in the gas phase catalysed by lithotropic bacteria

⁎ Corresponding author. E-mail address: [email protected] (D. McBride). 0304-386X/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.hydromet.2012.08.004

2þ

Fe

þ

1 1 þ 3þ O þ H →Fe þ H2 O 4 2 2

ð2Þ

The biggest issues in understanding and controlling leaching are scale — both in terms of the physical size of the heaps being leached, which can easily consist of multiple millions of tons, and in time since heaps are leached for many months if not years. The sheer scale also limits what can be done to influence the process. The factors that can be controlled include: • Heap geometry — as in size and shape of lifts • Ore treatment — including particle crush size and any pre-treatment (e.g. agglomeration or acid pre-treatment) • Air injection — the air or gas distribution system and how much is injected. Narrow heaps may draw air in through their sides. Bigger heaps will be built with pipe networks that, if they remain unblocked, can pump air directly into the heap. • Raffinate chemistry — acidity, iron content and bacteria. • Raffinate application — rates, rest periods, method of application (usually drip emitters but sometimes wobblers). Any successful operation depends on extracting the maximum amount of metal for the least cost. This depends upon contacting as much of the ore with solution as possible, generating ideal conditions for reactions to occur and then ensuring that the resultant metal salts are recovered from the pregnant solution. Contacting ore with solution depends upon having good hydraulic properties in the heap, where good means that the flow spreads

C.R. Bennett et al. / Hydrometallurgy 127–128 (2012) 150–161

evenly under the action of gravity and capillary forces and does not channel. Hydraulic properties are closely connected with the available pore sizes in the ore which in turn is a function of the particle size distribution. A fine particle size distribution maximizes surface area and therefore reactivity but can have low hydraulic conductivity preventing effective flow of gas or raffinate. It is also important to be able to extract the soluble salts from the heap. The particle size distribution must also promote physical stability within the heap. The metal dissolution reactions are dependent on the inherent mineral kinetics, diffusion (i.e. the transport of reactants to the mineral surface), temperature and availability of reagents. The best leach process is one whose only limitations are due to factors beyond the control of the operator, as in the inherent kinetics. Reagent availability is driven by consumption and regeneration. Generally, higher temperatures lead to faster reactions but in turn temperature can adversely affect the population and performance of the bacteria whose presence is key to the regeneration of ferric ions. Diffusion is a function of the size of the particles. Due to the time required for reagents to penetrate large particles and for the dissolved metal species to diffuse out, there can often be an effective maximum particle size for a given leach system. Experimental work is fundamental to the exploration of how individual ore bodies can leach. Shake flask tests to determine mineral kinetics are relatively quick but the main experimental work is usually based on column leach tests. Column leach tests replicate many of the main factors present in a full heap but in a controlled environment. Columns can be placed in jackets to allow the temperature within to be kept constant and some effects of lift height and crush size can be explored through the use of differently sized experimental rigs. Columns can vary in size from the laboratory scale (5 cm diameter by 0.5 m high) up to pilot scale (2 m diameter by 10 m high). Individual column tests may take months to complete. A suitably comprehensive mathematical model provides a framework with which to better understand the physics and chemistry involved in the process and provides tools that allow different leach strategies to be explored and optimized cheaply and quickly. The basic process, applying a reactive solution on top of a heap of material and collecting a metal as a dissolved salt in the pregnant leach solution, is simple in concept. However, leach systems are actually very complex physically and chemically, which makes building an effective and reliable mathematical model a difficult challenge. The variability between different ores and between samples (i.e. mineral concentrations, local blockages, impurities) makes it difficult to accurately predict the behavior of a particular system unless the model is calibrated against each specific ore. However, once the model is calibrated for a specific ore type, then it should be possible to determine trends and indicators towards improving leaching at the industrial scale in a fairly reliable manner. Computational models have the advantage of being repeatable and very fast — simulations of hundreds of days of column tests can take place in a few minutes, whilst years of whole heap leach can be delivered within hours. A helpful general description of leach modelling requirements has recently been given by Petersen (2010a, 2010b). To date significant effort has been applied to further the understanding of heap operations by building mathematical models of the heap process. These models have involved studies of fluid flow (Bouffard and Dixon, 2001; Pantelis et al., 2002), chemical dissolution rates (Casas et al., 1998; Dixon and Hendrix, 1993; Madsen and Wadsworth, 1981; Paul et al., 1992a, 1992b), and heat/temperature balances (Cathles and Apps, 1975; Dixon, 2000). Watling (2006) and Dixon (2003) provide recent reviews of the bioleach process and the status of modelling efforts. Dixon and Petersen (2003) present a model for column heap leaching using a model based on raffinate diffusing out to reaction sites from discrete channels through the ore and use comparisons to column test results to generate confidence in

151

the model for predictions of behavior in heaps. Petersen and Dixon (2007) extend the same model to use in zinc leaching. Leahy et al. (2005, 2006, 2007) and Leahy and Schwarz (2009) have also done a good deal of CFD model based work in considering the interactive issues of bacterial effects, heap temperature and the role of gas sparging. More recent work includes the flow model developed by Cariaga et al. (2005, 2007) and the analytical model developed by Mellado and Cisternas (2008) and Mellado et al. (2011). Pyrite leaching has been considered by Bouffard (2008) and Bouffard and Dixon (2009). The aim of the current work is to move towards a practical tool for whole heap simulation based on readily available data from either the laboratory or on the full-scale heap. It builds upon the pioneering work of Petersen and Dixon (2003, 2007) with respect to validation against laboratory column data. Furthermore, this work takes advantage of improvements in computational performance to build a CFD based model that combines true heap geometry with all aspects of the various physical, chemical and biological processes present in a heap as a series of sub models to provide a comprehensive solution (Bennett et al., 2003a, 2003b, 2006, 2008a, 2008b; Cross et al., 2005, 2006; Gebhardt et al., 2007; McBride et al., 2005, 2006). Petersen (2010a, 2010b) provides an excellent overview of the challenges in mathematical modelling the full range of phenomena that play a role in controlling the dynamics of mineral dissolution during heap leaching. From this assessment it is clear that physics and chemistry are fiendishly complex, and the role of the modeller is to find a practical formulation that captures all the key phenomena at an adequate level, so that the resulting model can be reliably parameterized by available measurable data. Rates of mineral dissolution depend upon a balance of diffusion and chemical rate kinetics. Where as many of the factors effecting rate kinetics are well understood, there is still some debate over the most efficient ways of dealing with diffusion. Many large-scale models account for the effect of varying particle sizes by employing average particle sizes and simplified models, such as the shrinking core model. There is much discussion on the validity of the shrinking core method, (Ghorbani et al., 2011), as the assumption of evenly distributed mineral grains and spherical particle geometry is not normally valid. It has been shown that other geotechnical parameters such as, bulk density/stress characteristic, tortuosity, moisture capacity and particle density can effect changes in the diffusion controlled leach rate, (Miller, 2003). However, as such detail is not normally available for a large-scale heap the simplicity of the shrinking core model lends itself well to large-scale operations. The model itself is a very useful tool to analyse commercial heap data (Miller, 2003). The current work uses a simple shrinking core model for multiple representative particles based on the particle size distribution in order to account for diffusion and shows that this approach scales correctly for different particle size distributions using the same ore. The thinking behind this approach was that although the reaction modelling is quite simplistic, inner particle diffusion is dominated by particle size and so long as the full particle size distribution is carefully reflected in the formulation then it should enable the capture of the overall dissolution rate reasonably well. This paper focusses upon the challenge of modelling column leach tests of a chalcocite ore that also contains pyrite. The current work is based on over 10 years of experience in modelling copper heap leach systems with industrial partners who routinely perform column leach tests at a variety of scales to understand how a specific ore body will behave and how best to optimize the full scale heap operation. Column leach tests can typically be modelled as one dimensional systems and are therefore an excellent source of data with which to parameterize and validate aspects of computational models that may then be used to predict the behaviour of larger scale systems. In this work, column tests are used for two related purposes: a) Small column test data is used as a basis for parameterizing the model for a specific ore blend, whilst

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C.R. Bennett et al. / Hydrometallurgy 127–128 (2012) 150–161

b) A larger column test using the same ore blend is used as a basis for enabling a test of the parameterized model to predict its behaviour — that is, to validate the flow and reactive aspects of the mathematical model.

2.2. Modelling requirements Modelling stockpile leaching requires the accurate simulation of a range of physical and chemical phenomena. The resultant multiphase systems include

2. Mathematical modelling 2.1. The system to be modelled The basic system being modelled consists of a small 0.15 m diameter 1.8 m high column of ore containing 0.8% copper in the form of chalcocite (Cu2S) and approximately 2% pyrite (FeS2). The ore is crushed to minus ½-inch (0.0127 m). A raffinate containing 5 g/l iron in the form of ferric sulphate (Fe3 +) at a pH of 1.9 is applied to the top of the column at a rate of 1.3 l/m2/h (3.6 E− 07 m/s) over a period of 100 days, and air is pumped in from the base at 0.32 Nm3/m2/h (8.64 E−05 m/s). About 70% of the copper is extracted over the course of 100 days, the majority of recovery occurring in the initial 50 days after which the rates of reaction tend to fall off. For validation purposes the same set of material parameters used for the small column are also used for a large (6.1 m high, 2 m diameter) column containing the same ore crushed to a P80 of 1½-inch (0.0381 m). In this aspect of the work the principal reactions are the dissolution of chalcocite and pyrite and the regeneration of ferric [as illustrated in Eqs. (1) and (2) above]. In addition an iron precipitation reaction is included as the solubility of ferric is highly dependent on acidity. The chalcocite reaction is split into two stages. This is based on the fundamental work on reaction kinetics of pure chalcocite mineral (Bartlett, 1998; Marcantonio, 1976). The reason for the two different stages is that chalcocite was determined to have two distinct phases during dissolution in ferric sulphate solutions. The first stage involves mixed kinetics, that is, solution diffusion and surface reaction. The second stage is controlled by the surface reaction, mainly electron transfer. The first stage reaction is 3þ

5CU2 S þ 8Fe

2þ

→5CU1:2 S þ 4Cu

þ 8Fe

2þ

ð3Þ

whilst the second stage consists of the breakdown of a covellite like product (Cu1.2S)° generated in the first stage 3þ

5CU1:2 S þ 12Fe

2þ 2þ →5S∘ þ 6Cu þ 12Fe

ð4Þ

The pyrite reaction is 3þ

FeS2 þ 14Fe

2þ

þ 8H2 O→15Fe

2−

þ

þ 2SO4 þ 16H

ð5Þ

The other key reaction is the ferrous oxidation reaction in Eq. (2) which consumes acid as well as oxygen and is controlled or catalyzed by bacterial activity. It can be the main consumer of acid in a column in the absence of gangue minerals. As the solubility of ferric is highly dependent on the acidity of the solution it is also necessary to include an iron precipitation reaction. Insoluble jarosite salts are the most commonly encountered iron precipitates but tend to occur at low pH or when potassium ions are present (KFe3(SO4)2(OH)6). For a pH above 2 the most likely precipitate is soluble ferric hydroxide (Fe(OH)3), 3þ

Fe

þ

þ 3H2 O↔FeðOHÞ3 þ 3H

ð6Þ

Both precipitation events increase acidity in the solution. As the hydroxide reaction is reversible it can have the effect of buffering the acid and iron levels in solution. Although this is a somewhat simplified treatment of the iron cycle such an approach is necessary in order to be able to develop an effective model.

• Transport phenomena, including liquid and gas flows, and mass transfer between liquid, gas and solid phases. • Reaction kinetics for the important mineral species • Bacterial effects on the leach reactions • Heat, energy and acid balances for the overall leach process Using a computational fluid dynamics (CFD) code based on a mass conserving formulation as a starting point, facilitates control over geometry and transport of flow properties and chemical species within the column. It also allows the model to be readily expanded to multiple dimensions. The key physics are captured through source terms in the general transport equations used to govern and conserve the physical properties of the solution flow. A CFD approach depends upon breaking the physical geometry of the column down into a set or mesh of representative elementary volumes, or elements. For the purposes of species transport each element is assumed to have constant properties within its own boundaries. The key to approaching modelling a leach system is in identifying the core physics and in making the best use of what data is available. There are distinct advantages in starting to build a model based on a column test in that the model can be somewhat simplified compared to that required for a full heap. Two key areas of physics in large heaps, the heat balance and bacterial effects, can be simplified in most column tests. Although in some bioleaching column tests the local bacterial environment may have an effect, in the main small column conditions are usually ideal for bugs, especially if they are in the raffinate. The model makes the assumption that the bug environment is ‘ideal’ and it is only the lack of them that retards the process. The heat balance is driven by the heat of reaction, external temperatures and transport through liquid and gas flow. As the ratio of surface area to bulk ore mass is such that columns lose heat more rapidly than they can generate within them, it is the external or ambient temperature that is the dominant factor. Small columns can also be jacketed to maintain a given temperature which can be used to gain insight to the effect of temperature on the reaction kinetics but the key is that the column temperature is unlikely to be significantly influenced by reactions or flows inside the small column. Condensation and evaporation need not be modelled for the small column case, although this can be a mechanism for heat transport in larger systems. Bacteria only limit the leach process when their activity is very low, either due to low population density or to disadvantageous chemical and thermal conditions. When they are present in sufficient quantities and conditions are suitable, the limits on ferrous oxidation will be controlled by the local chemistry. Column tests tend to have ideal conditions and, if using raffinate that is seeded with bacteria, it is quite likely that there will be sufficient bacterial numbers so there will be no noticeable adverse effect on leaching in these circumstances. So, bacteria are therefore ignored in this paper. The remaining important physics can be grouped into three main areas based on their relative timescales. • Liquid phase transport. This covers saturation, leach solution flow and transport of soluble species. It is mainly dependent on the rate of solution application and the hydraulic conductivity of the ore. • Gas phase transport. This covers gas flow and transport of oxygen and to a lesser extent, water vapor. • Process chemistry. This covers dissolution kinetics of the ore, any chemistry in the liquid phase (i.e. iron chemistry and precipitation) and mass transfer between the different phases.

C.R. Bennett et al. / Hydrometallurgy 127–128 (2012) 150–161

The basic formulations for these three areas are described for the heap leach model. 2.2.1. Liquid phase transport Liquid transport is driven primarily by gravity. Matric potential or pore pressure will also act on the moisture although in 1D these effects are small. For unsaturated porous flow, the Darcy flux can be written in terms the pore pressure and moisture content,
 ∂ψ ∂θ q ¼ −K ðθÞ−K ðθÞ − m ∂θ ∂z

ð7Þ

where: θ K(θ) ψm

is the moisture content is the unsaturated hydraulic conductivity is the matric potential or pore pressure.

To ensure mass conservation, the volumetric continuity equation also needs to be satisfied. ∂qy ∂qz ∂θ ∂q ¼− x− − ∂t ∂x ∂y ∂z

ð8Þ

The hydraulic conductivity can be described by the van Genuchten equation (van Genuchten, 1980) 0:5

K ðθÞ ¼ ks SðθÞ
SðθÞ ¼

θ−θr φ−θr

h   i 1=m m 2 1− 1−SðθÞ



 ¼

1 1 þ ðα jψjÞn

m

ð9Þ ð10Þ

=1 − 1/n are empirical constants is the saturated hydraulic conductivity is the porosity is the residual saturation

m α,n ks ϕ θr

2.2.2. Gas phase transport We make the assumption that the influence of the phases on each other is one way. This means that the liquid phase influences gas flow but that the gas flow does not directly influence the liquid phase. The liquid phase is slow moving and over the course of a leach cycle is mostly in a steady state, so the gas flows through a porous medium made up of both liquid and solid phases. Gas flow is primarily driven by boundary conditions such as gas injection through air lines and wind pressure against the sides of heaps. Temperature gradients can also be very important in driving gas flow. Indeed the ideal situation in heaps with large exposed flanks is that a chimney effect is created which draws external air inside. Other possible factors include displacement by liquid flow, oxygen consumption, evaporation and condensation. Gas flow affects the heap through transport of oxygen. It can also spread heat although the low thermal capacity of the gas phase means that this is a minor effect. The gas phase is solved assuming incompressible gas flow with a Boussinesq source term. This approximation is valid as local temperature differences are small and so actual changes in density are very small. The basic continuity equation for the incompressible gas phase transport is

ki gρ μ

ð12Þ

where ρg Sg vg

The saturated conductivity is given by ks ¼

In a 1D unsaturated system, the Van Genuchten equation effectively describes the saturation in the column and the time taken for solution to start to flow from the base of the column. Under constant irrigation rate and assuming there is no change in the hydraulic properties of the column the discharge flow rate will, of course, remain constant. Hydraulic properties of the column could potentially change due to transport of fines, salt precipitation, breakdown of particles under leaching conditions generating more fines and compression. In small columns the most likely cause of any changes will be precipitation but this is highly dependent on the chemistry of the ore and the raffinate.

h i div ρg vg ¼ Sg

where

153

ð11Þ

vg ¼ −

is the gas density is a source term for gas is the gas velocity, is equal to

 kin kg ðSÞ  g ∇p þ ρg g∇z εg μ g

ð13Þ

where ki g ρ μ

is the intrinsic permeability of the media is gravity is the raffinate density is the raffinate viscosity

Permeability is mainly a function of particle size distribution, particularly the amount of fines in the ore which can tend to block the pores through which the liquid flow seeps. The expression for saturation in Eq. (10) provides an easy way to determine pore pressure based on empirical constants and the saturation. Saturation is easy to calculate in the current model as the mass flux of raffinate through the system is conserved. As long as the ore does not become fully saturated the flow can be simply modelled through an explicit scheme evaluating fluxes between individual elements. If parts of the ore approach or become saturated, it is then necessary to implicitly solve for saturation and subsequently derive the fluxes (McBride et al., 2005).

where kin kg(S)

is the intrinsic permeability of the porous media is the unsaturated permeability of the gas at liquid saturation (S), related to the liquid unsaturated permeability kl(S) by kg ðSÞ ¼ 1−kl ðSÞ

εg μg

ð14Þ

is the volume fraction of the gas phase is the gas viscosity

Substitution of Eq. (13) into Eq. (12) gives the following continuity equation for pressure, p. 2 g

∇ p ¼

εg μ g S kin kg ðSÞ g

ð15Þ

154

C.R. Bennett et al. / Hydrometallurgy 127–128 (2012) 150–161

where Sg ¼ Sthermal þ Svol þ Sother

Sthermal

Svol

Sother

ð16Þ

is a Boussinesq source term allowing for natural convection due to the thermal expansion of the gas where there are temperature changes and is therefore not used in an isothermal case. is the source term due to changes in volume available for gas flow due to changes in saturation. As long as the system remains unsaturated, the equations for gas and liquid flows can be solved for separately. The assumption is made that gas flow does not influence liquid flow; therefore, the liquid flow is solved for first. It also contains any changes in gas volume due to consumption of oxygen or evaporation and condensation. contains any other sources of pressure and allows for point injection of gas in multi dimensional models.

2.2.3. Overview of the process chemistry 2.2.3.1. Dissolution kinetics. Minerals are typically present as small grains contained in a matrix of inert material. Reactants diffuse in through pores in the rock matrix of each ore particle, chemical reactions occur and the products diffuse out (Bartlett, 1998; Petersen, 2010a, 2010b). The reaction kinetics of different minerals can vary widely and the overall rate of reaction is also highly dependent on the diffusion of reactants and products. To allow for both these factors the ore is divided into discrete size fractions with characteristic radius and mineral concentrations. The rate of dissolution for each mineral in each characteristic particle size is modelled using a separate shrinking core reaction. The shrinking core reaction normally assumes that there is a homogenous distribution of reactive material. The current work accepts that this is unlikely to be the case for a real ore and in addition the exact distribution of mineral grains and the interaction between different minerals in the same particle can complicate the overall dissolution rates. The model presented here considers the location of a reaction front moving through a representative particle that is the average of the many hundreds or even thousands of particles in the same size class in a particular representative elementary volume, and the average particle will have a homogenous mineral distribution. In this case the shrinking core model appears as a convenient approximation, since the dominant influence of the particle size is captured through explicitly representing the particle size distribution. The equation used to calculate the rate of dissolution of a given mineral is given by (Szekely et al., 1976) drm 3r Mi Deff co Am h   i ¼ − m2 dt 4πr o ρore xi 3Deff r o co þ 2ðr o −r m Þr 2m 1−ε p Am

ð17Þ

where ro rm Am R T Deff εp ρore Mi xi

is the initial particle radius is the current mineral radius comes from the kinetic rate equation for the current mineral. is the gas constant is the temperature in Kelvin is the effective rock diffusion coefficient is the rock voidage is the ore density is the molecular weight of the mineral is the mass fraction of the mineral

The value of Am comes from the general expression for the kinetic rate equations, such as those produced by Paul et al. (1992a), which takes the general form Am ¼

−B dβ ¼ Aeð RT Þ dt

ð18Þ

where β A,B

is the fraction of mineral reacted are functions of the individual kinetic rate equation.

In the modelling below it is convenient to work in terms of the fraction of mineral reacted, β, where β¼


3 rm r0

ð19Þ

This approach allows minerals to react at different rates in an individual particle size, and although it requires the assumption that the minerals can be treated relatively independently, this is not unreasonable given the low concentration of reactive minerals present. This approach allows each mineral in each particle size fraction to be modelled using a single characteristic radius indicating how much has reacted. This in turn allows the model to deal with multiple particle sizes and multiple minerals over large meshes without excessive memory usage in the overall CFD model framework. This approach has an advantage in that it can easily be related to experimental analysis of ore which is commonly given as mineral content by size classification, making validation easier. It also easily allows for different minerals to dominate the reactions at different stages of the leach cycle. Although each reaction in each particle size fraction is considered individually, there can be times, especially early on in the leach process when diffusion is not limiting, where the sum total of reactants consumed can be greater than that available. When this is the case, an optimization routine can be used to share the available reactants between the competing reactions, where the consumption is apportioned proportionally to the relative rates of the individual reactions. As the initial reaction rate can be high with no diffusion, it may also be necessary to use a variable time step for the chemistry. Each of the following kinetic rate equations incorporates a rate constant. This allows the model to be tuned to a given ore using experimental column data. Without setting the rate constants, the model provides information on general trends but better accuracy comes with using these values to fit the model against known data. The rate constants in effect combine to cover factors in the reactions that are not otherwise specified for in any particular ore complex. These can include particular distributions of mineral grains and interactions between different minerals which can be difficult to quantify but may be characteristic of a particular ore body. The rate constants are unrelated to particle size and therefore allow the model to scale from small to large particle size distributions (e.g. from experimental column to heap). Methods to enable the rate constants to be tuned to individual data sets will be discussed in a future paper. 2.2.3.2. Chalcocite reaction rate. The breakdown of chalcocite in a leaching process occurs through many steps and the formation of multiple intermediary copper-deficient phases. Marcantonio (1976) simplified the dissolution or leaching of chalcocite by defining a twostage electrochemical-based reaction mechanism with the second stage reaction initiated after 40% of the copper has been reacted. The second stage (Cu1.2S) mineral has a similar formula to the copper mineral covellite and is often referred to by the same name.

C.R. Bennett et al. / Hydrometallurgy 127–128 (2012) 150–161

The first stage involves mixed kinetics, that is, solution diffusion and surface reaction involving ferric as the reactant. The first stage is strongly influenced by the high mobility of cuprous ion in the solid phases. The second stage for chalcocite dissolution is controlled by the surface reaction, mainly electron transfer. In the shrinking core algorithm, the second stage equation is used if, at the start of the mineral time step, the fraction of copper reacted is greater than 40%. Though the specifics change in each stage, the reaction (involving ferric) considered for the stoichiometric mass balance in the model is 3þ

Cus þ 2Fe

→Cu

2þ

2þ

þ 2Fe

ð20Þ

According to Madsen and Wadsworth (1981), predicted copper recoveries calculated with activities derived from thermodynamic data and the Debye–Huckel theory were consistently higher than experimentally observed copper extractions for chalcocite leach tests. An apparent activity coefficient was determined empirically by fitting to observed leach rate data. Both chalcocite reaction stages use an apparent activity function (γ) with the ferric concentration. The function is as follows: −1:373−239:7⌊Fe3þ ⌋0:5 þ4870⌊Fe3þ ⌋Þ γ ¼ eð

ð21Þ

155

2.2.3.5. Pyrite rate kinetics. The pyrite reaction is 3þ

FeS2 þ 14Fe

2þ

þ 8H 2 O→15Fe

2−

þ 2SO4 þ 16H

þ

ð26Þ

This is governed by the kinetic rate equation which is adapted from Paul et al. (1992a) h i 3þ −10317 dβ RA 601:5 Fe eð T Þ ¼ dt r p ζ ½Fetot ½H þ 0:4

ð27Þ

where β rp ζ T RA

is the fraction of pyrite is the pyrite grain radius (m) is the particle shape function is the temperature (K) is a rate constant based on known data

2.2.3.6. Ferrous oxidation. An equilibrium relationship is used to determine the ferric ion concentration based on the concentrations of ferrous ions, dissolved oxygen and free acid, and an equilibrium constant, K. The relationship is given by, h

2.2.3.3. Stage I — chalcocite (Cu2S). In the first stage, chalcocite is converted to secondary covellite by reaction with ferric according to the following reaction 3þ

5Cu2 S þ 8Fe

2þ

→5Cu1:2 S þ 4Cu

2þ

þ 8Fe

ð22Þ

The rate equation associated with the first stage can be written as h

dβ ¼ dt

i

3þ
 −1404 β RA 8:6γ Fe eð T Þ 1− rp ζ 0:4

ð23Þ

is the fraction of total copper reacted is the mineral grain radius (m) is the particle shape function is the temperature (K). is a rate constant based on a fit to known data

3þ

2þ 2þ →5S∘ þ 6Cu þ 12Fe


 n h io0:54 −9059 1−β 0:5 10 3þ RA 1:26  10 γ Fe eð T Þ 0:6

i h i Fe3þ 1=4 1=4 þ  2þ ¼ K ½O2  H Fe

ð29Þ

þ

þ 3H2 O⇔FeðOHÞ3 þ 3H

ð30Þ

Precipitated ferric hydroxide is generated through relating the maximum soluble ferric concentration to pH using a linear relationship (Garrels and Christ (1965)) and calibrating with experimental data.

ð24Þ

ð25Þ

where β T RA γ [Fe 3 +]

h

3þ

The associated kinetic rate equation is given by (Paul, 1989) dβ ¼ dt

where K = 5.56 × 10 7 (Garrels and Christ (1965)). Concentrations here are in moles/litre. The ratio of ferric ions to ferrous is therefore given by

Fe

2.2.3.4. Stage II — chalcocite (Cu1.2S). The second stage consists of the breakdown of covellite (Cu1.2S) generated in the first stage by a reaction very similar to that for natural covellite: 5Cu1:2 S þ 12Fe

ð28Þ

2.2.3.7. Iron precipitation. Ferric precipitation is controlled by pH, where the maximum ferric solubility is a function of pH. The ferric hydroxide relationship is given by

where β rp ζ T RA

i4 Fe3þ K¼ 4 Fe2þ ½O2  ½Hþ 4

is the fraction of copper reacted is the temperature (K) is a rate constant based on known data is the apparent activity coefficient is the ferric concentration in solution; units in mol/cm 3

All concentrations are in mol/cm 3. This reaction is considerably slower than the first stage reaction.

h i 3þ log Fe

max

¼ 1−pH

ð31Þ

An equilibrium relationship is solved using a Newton–Raphson scheme, and is solved after all other chemical reactions have been completed in a time step. 2.2.3.8. Oxygen mass transfer. The oxygen liquid–gas mass transfer rate can be determined by a number of factors, typically, temperature, liquid and gas composition and liquid–gas interfacial area. Petersen (2010a, 2010b), studied the the gas–liquid mass transfer of oxygen in heap leach scenario's and found that the net mass transfer rate was relatively unaffected by temperature in the range 22–68 °C as an increase due to temperature was offset by a proportional decrease in the solubility of oxygen. The mass transfer tended to increase with materials with higher fines, possibly due to a larger surface area. Due to the large solution time‐steps employed, typically 10 min, and large surface area between the liquid and gas phases the model assumes that the transport rate is effectively instantaneous and an equilibrium state is obtained. Oxygen is partitioned between the liquid and gas phases by using Henry's law.

156

C.R. Bennett et al. / Hydrometallurgy 127–128 (2012) 150–161 Table 2 Summary of particle size data for the small column.

Calculate common element / particle properties

Calculate new core radius Loop over / particles mineral reactions

Loop over liquid reactions

Particles

Mass fraction

Radius (m)

Fraction FeS2

Fraction Cu

1 2 3 4 5 6

0.60% 32.50% 9.40% 21.70% 22.10% 13.70%

0.006250 0.003130 0.002380 0.001000 0.000075 0.000060

1.04% 1.30% 1.46% 1.78% 1.96% 2.00%

0.52% 0.65% 0.73% 0.90% 0.98% 1.00%

Calculate reactants used Loop over elements

MO2 MN2 KH

is the molar weight of oxygen is the molar weight of nitrogen is the Henry's law constant at a given temperature

Build and solve matrix to determine species reacted Henry's law constant is allowed to vary with temperature by using a function based on dissolved oxygen solubility against temperature data under atmospheric conditions. As the partial pressure of oxygen may well be dependent on the level of salts in solution the possible dissolved oxygen levels can also decrease. An equilibrium state is solved using a Newton–Raphson scheme at the end of each time step.

Scale particle Loop over reactions

Loop over particles Calculate revised core radius

Scale reaction

Update species Fig. 1. A Schematic of the algorithm for multiple particles and reactions within each control volume or element.

The equation solved is Og   M ¼ K H Ol O Og þ 1−Og M 2

ð32Þ

N2

where Og Ol

is the mass fraction of oxygen (O2) in the gas phase is the molar concentration of oxygen in the liquid phase

2.2.4. The numerical solution strategy The model is implemented within the context of a CFD software framework which in this case uses the PHYSICA toolkit (Croft et al. (1995)). This uses a finite volume numerical approach which ensures that all spatial variables are transported in a mass conserved manner. Hence, the column volume is set as the solution domain and is divided into a mesh of connected elements. For the column simulations, the flow is essentially one dimensional and so a simple mesh of rectangular elements is used to cover the flow domain. Within each element there are three phases — a static solid phase of the ore body, a liquid phase of solution flowing vertically downwards and a gas phase (of primarily air) flowing vertically upwards. Within each control volume or element the chemical reactions are pursued both for each reaction sequence and also within each particle size fraction simultaneously. Of course, the minerals and pyrite in the solid phase are reacting with various species in the liquid and gaseous phases. A diagram illustrating how the reaction phases are managed within the simulation is shown in Fig. 1 below. It should be clear that there are indeed multiple timescales in the simulation which can be caught by two time steps — one for the overall flow physics of the liquid and gas phases and a second which governs the chemical reactions. The chemical reactions time step is typically 1/10 of the size of the overall flow time step. 3. Parameterization and validation

Table 1 Summary of input data for simulation of column experiments. Property

Small column

Large column

Height 0.15 m Ore crush Ore density Solid fraction Copper mass fraction (chalcocite) Pyrite mass fraction Particle porosity Leach cycle

Diameter 6.1 m P100 0.0127 m (0.5″) 2950 0.59 0.59% 2.15% 5% 97 days on 3 days off 3.6E− 7

1.8 m 2.0 m P80 0.0381 m (1.5″) 2950 0.59 0.59% 2.15% 5% 90 days on 30 off 30 on 10 off 1.576E− 6

4.08 0.10 0.30 0.90 8.64E−5 25

3.50 0.33 0.30 0.85 8.64E−5 25

Raffinate application rate (m/s) Raffinate concentrations Ferric (gpl) Ferrous (gpl) pH Copper (gpl) Air inlet velocity (m/s) Temperature (°C)

Results from simulations of a small and large column and their comparison to experimental data are shown below. The input data is summarized in Tables 1–3. The set of mineral reaction rates, RA in Eqs. (23), (25) and (27) is essentially tuned to enable the best fit

Table 3 Summary of particle size data for the large column. Particle

Mass fraction

Radius (m)

Fraction FeS2

Fraction Cu

1 2 3 4 5 6 7 8 9 10

1.80% 12.50% 21.70% 11.80% 13.70% 12.70% 3.40% 7.40% 8.60% 6.40%

0.037500 0.021900 0.015600 0.010900 0.007810 0.004690 0.002750 0.001690 0.000538 0.000075

2.15% 2.15% 2.15% 2.15% 2.15% 2.15% 2.15% 2.15% 2.15% 2.15%

0.44% 0.53% 0.62% 0.59% 0.63% 0.63% 0.59% 0.74% 0.86% 1.03%

C.R. Bennett et al. / Hydrometallurgy 127–128 (2012) 150–161

4.5

70%

4.0

60%

3.5

Concentration, gpl

80%

50% 40%

Experimental Simulation

30% 20%

157

Experimental Simulation

3.0 2.5 2.0 1.5 1.0

10% 0.5 0% 0

20

40

60

80

100

120

0.0 0

-10%

20

40

60

80

100

120

Leach period, days Fig. 2. Copper recovered to PLS against time for experiment and simulation. Fig. 4. Ferrous ion concentration in the PLS.

possible for the small column and then these data are used to predict the behaviour of the large column. So the data from the small column is used to parameterize the model for a specific ore, whilst the data from the large column provides a validation test for the parameterized model. In the comparisons below we show in detail a number of the key results. It is not too difficult to fix model parameters so that a reliable match can be obtained with the overall rate of copper extraction from the column. It is, however, much more challenging to enable the model to match the behaviour of both the species concentrations and pH within the PLS, and the dissolution in the column by particle size fraction. This requires that the formation of the model capture all the key details of the extraction process. It is worth commenting that most column data recorded in industrial laboratories is to help inform operational decision making or heap design, not to provide the basis for parameterizing a detailed mathematical model. The data reported here is both complete from a simulation perspective and closed – that is it is sufficient to enable a process model to be evaluated, and in that sense, should provide a useful basis for testing and validating other future process models. 3.1. Parameterization against the small scale column In the following figures, a comparison of the simulation is shown against the data from the small column experiments. The operation simulated is based entirely on the conditions as specified in Table 1 and also in Table 2 as the initial conditions for the ore by size fraction. A key parameter to be estimated is the diffusivity by particle size and indeed the values used here are estimated from a wide variety of tests

over an extensive period of multiple comparisons with column experiments. It is the rate parameters for the chalocite and pyrite that are tuned to enable the best fit to the experimental results — both in the PLS and also the residual solid column both overall and by particle size fraction. The simulation of 100 days for the small column takes less than a minute on an ordinary PC. The simulation used 10 min ‘outer’ timesteps and a relatively coarse mesh. Because of the uniform flow, the results did not noticeably alter for a finer mesh. The simulation predicts PLS data for the whole period as well as chemistry and remaining minerals in the ore mix by size fraction, location and time. In Fig. 2, the percentage of the copper in the column recovered to the PLS is shown as a function of time, from which it is clear that the comparison over the 100 days or so is quite good. There is a small discrepancy at around 22 and 38 days, but in each case there were short interruptions to the planned operation which were not recorded in any detail (and so not reflected in the simulation operational conditions). Of course, Table 1 indicates that the solution flow rate is constant into the column for the first 97 days followed by a rest period until the full 100 day period is reached. The simulation cannot effectively capture every aspect of the experiment, such as, variations in overall ore properties from the tested samples through the length of the column. Minor variations in raffinate flow rates and properties can be captured but it may not be time effective or indeed useful to do so, as the key results are the trends. Most models can be tuned to provide a reasonable match to the overall 25

10.0

Concentration, gpl

Concentration, gpl

Experimental Simulation

20

Experimental Simulation

8.0

6.0

4.0

2.0

15

10

5

0.0 0

20

40

60

80

Leach period, days -2.0 Fig. 3. Ferric ion concentration in the PLS.

100

120

0 0

20

40

60

80

Leach period, days Fig. 5. Copper ion concentration in the PLS.

100

120

158

C.R. Bennett et al. / Hydrometallurgy 127–128 (2012) 150–161

7.0

80% 70%

6.0 Experimental Simulation

60%

% Removed

5.0

pH

4.0 3.0

50% Copper Removed Iron Removed

40% 30% 20%

2.0

10% 1.0 0% 0.0

0 0

20

40

60

80

100

20

40

60

80

100

120

Leach period (days)

120

Leach period, days Fig. 8. Percentage of copper and iron (from pyrite) removed from the column against time as predicted by the simulation.

recovery but may not match all of the measured species concentrations. The strength of the model presented here is shown in the simulated concentrations in the PLS and residual solid matrix by size fraction. In Figs. 3 and 4, the comparisons for the ferric and ferrous ion concentration in the PLS are shown. The trends here are well captured, with the initial high ferrous spike and slowly increasing ferric content matching the fast initial reactions with minerals on the surface of the rocks in the column. In Fig. 5, the copper ion concentration in the PLS is shown to compare well with the experimenal data over much of the time range. The early stages of the simulation can lag behind the experiment as the initial conditions in the column may not be well defined. There are also demonstrable differences between the regime within the small and large columns due to the relative particle size distributions. In small particles the shorter diffusion paths for reactants and products lead to relatively fast reactions. In the small column this leads to rapid useage of the available ferric which, assuming a healthy bacteria population, leads to a lot of ferrous oxidation which consumes acid. The ferric salts produced cannot stay in solution and precipitate as hydroxide which releases acid back into the system. This buffering effect ties up iron until the rate of reactions slow and the rate of acid consumption falls, which makes acid available to dissolve the ferric hydroxide. The pH in the PLS is shown in Fig. 6 which compares well with the experimental data except for the early stages where the experimental data is unreliable. The model also caps the PH at 4 because the rates of

Experimental Simulation

2.5

Moles/tonne

90% 80% Radius 6.25E3m

70%

Radius 1.0E-3 m

60%

Radius 7.5E-5 m

50% 40% 30% 20% 10% 0% 0

20

40

60

80

100

Leach Period (days) Fig. 9. Percentage of copper remaining by particle size against time from the simulation.

reaction are effectively zero at this level as there is insufficient acid and ferric solubility is negligable. Another useful way to assess the performance of the model is on the iron balance over the course of the leach cycle as shown in Fig. 7. Again we have a good comparison overall although slighty displaced in time. In Fig. 8, the percentage copper and iron (from iron pyrite) removed from the solid matrix is shown over the time of the experiment. The pyrite reaction rate is significantly slower than for chalcocite but diffusion of reactants and products is the same for all minerals. This means that as the copper

3.5 3.0

100%

% Remaining

Fig. 6. pH level in the PLS.

2.0 1.5 1.0

Table 4 Small column simulated mineral residuals.

0.5 0.0

0

20

40

60

80

100

120

Leach period, days Fig. 7. Iron balance over the course of the leach cycle. This is total iron fed into the column less total iron recovered to PLS.

Particles

Radius (m)

Fraction FeS2 remaining

Fraction Cu remaining

1 2 3 4 5 6

0.006250 0.003130 0.002380 0.001000 0.000075 0.000060

100.00% 99.58% 99.22% 53.92% 50.39% 38.67%

83.89% 57.94% 53.92% 22.21% 0.0% 0.0%

C.R. Bennett et al. / Hydrometallurgy 127–128 (2012) 150–161

4.0 3.5

50%

Concentration, gpl

Fraction Copper Recovered

60%

159

40%

30% Experimental J555 Sim

20%

3.0 2.5 2.0

Experimental

1.5

Simulation

1.0

10% 0.5 0%

0

20

40

60

80

100

120

140

160

0.0

180

0

Leach period, days

50

100

150

200

Leach period, days

Fig. 10. Copper recovered to PLS for the large column.

Fig. 12. Ferrous ion concentration in the PLS for the large column.

3.2. Validation against the large column data Once the chalcocite and pyrite rate data were parameterized against the small column, the model was then set up to simulate the conditions of the large column using the data in Table 1 for the operational conditions and Table 3 for the initial condition of the ore size fractions. This column was run for some 160 days and Fig. 10 shows the predicted recovery over time compared to the measured values. Again, there is good agreement between the model simulation

and the experimental results. There is a gap in the experimental data where date was not recorded during the 30 day rest. The model includes this period in its simulation and predicts the recovery during that time, which is negligible. Again the ferric and ferrous ion concentrations in the PLS (Figs. 11 and 12) are quite well captured by the model — during the rest period the model predicts the ferric concentration will increase, which is not seen in the experimental

9.0 8.0

Experimental Simulation

7.0

Concentration, gpl

reaction rate becomes increasingly dependent on diffusion with dissolution of chalcocite the relative speed of the pyrite reaction increases. Over time these kinds of columns can produce increasing amounts of iron. The copper removal by particle size fraction is shown in Fig. 9 where it is clear that the smaller particle sizes react somewhat faster than the larger size fractions. The final residual copper levels by size fraction is shown in Table 4. This does not take account of how the particle size distribution may change over the course of leaching through particle breakdown. The first part of the leach cycle is dominated by reactions with the most available copper, as in that which is closest to the surface of all particle size fractions. It is thereafter dominated by the two smallest size fractions where diffusion effects are minimized. The leaching of pyrite, which has a much lower kinetic rate than chalcocite, only starts to become significant after the majority of the copper in the smallest size fractions has leached.

6.0 5.0 4.0 3.0 2.0 1.0 0.0 0

50

100

150

200

Leach period, days Fig. 13. Copper ion concentration in the PLS for the large column.

6.0 4.0 5.0 Experimental

3.0

4.0

Simulation

2.5 3.0

pH

Concentration, gpl

3.5

Experimental

2.0

Simulation

1.5

2.0

1.0 1.0 0.5 0.0

0.0 0

50

100

150

Leach period, days Fig. 11. Ferric ion concentration in the PLS for the large column.

200

0

50

100

150

Leach period, days Fig. 14. pH of the PLS for the large column.

200

160

C.R. Bennett et al. / Hydrometallurgy 127–128 (2012) 150–161

References

Table 5 Large column simulated mineral residuals. Particle

Radius (m)

Fraction FeS2 remaining

Fraction Cu remaining

1 2 3 4 5 6 7 8 9 10

0.037500 0.021900 0.015600 0.010900 0.007810 0.004690 0.002750 0.001690 0.000538 0.000075

100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 99.68% 98.64% 78.00% 6.53%

76.32% 86.61% 81.85% 62.35% 51.75% 22.11% 2.43% 0.27% 0.00% 0.00%

data. However, this result can be somewhat misleading as the flow rates fall off dramatically and the actual amount of species in solution is low. It is worth noting that reactions will continue in the column during and after the raffinate flow is turned off. As long as the local acid levels can be maintained through either pyrite dissolution or iron precipitation the local concentration of copper in solution should continue to rise. When the raffinate flow is turned on again the copper that has dissolved during this period will be flushed through, giving an apparent short term boost to the rate of copper recovery. The simulated copper concentration in the PLS is well captured throughout the whole experiment (Fig. 13). Again the pH in the PLS is well captured throughout the experiment (Fig. 14). The residual minerals by particle size is given in Table 5. The predictive match to the experimental results in the large column is generally good especially for copper recovery. All of the main features of the PLS are captured even if exact values vary for reasons of lack of knowledge of the initial conditions present in the column. This model demonstrates the importance of being able to represent multiple particle sizes and also the iron cycle, in particular iron precipitation and its role in buffering iron and acid levels in the column. It is worth noting that the precipitation and buffering effects tend to only occur where reactions take place rapidly and there is a shortage of reagents. In situations where reactions are slower, such as in the large column with a larger particle size distribution, this phenomenon can be either short lived or may not even be not noticeable. The ability of the model to deal with both situations is a major strength. 4. Conclusions The objective of the work presented in this paper has been to summarize a practical computational model for the simulation of well controlled column leach tests. In particular, the results demonstrate how the small column tests can be used to parameterize the model and larger scale column tests to validate the model for scale-up with a particular ore type. It is also worth noting that the simulation results for both the small and large column are broadly correct despite evidence that the behavior of the two systems is different. The small particle size distribution in the small column leads to faster reactions than in the large column, which in turn leads to acid depletion and precipitation of ferric salts, something which does not happen with slower reacting large particles in the large column — as observed both from our model and in column experiments. Modelling heap leaching is a demanding problem with multiple physics and considerable heterogeneity in any experimental data. The primary aim of any successful model is to be able to respond to changes in boundary conditions in the same way as it occurs with heap operations. Careful control of a set of rate parameters can tune the model for a specific ore type to give a close match to experimental data, which will remain relevant over widely varying crush sizes and conditions.

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