idro

dt

ormlogt dar leacoof

t ofdisood

ning mrtancethe avaovervieive techeral deso usedile lea

Hydrometallurgy 127128 (2012) 150161

Contents lists available at SciVerse ScienceDirect

Hydrome

l sethough the ore, reacting with the minerals present in the ore particles.The resultant pregnant leach solution is collected from the base of thestockpile 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 toproduce a soluble metal salt. Oxidation normally takes part in twostages, in that the main agent of oxidation is the ferric (Fe3+) ion. Areaction with a metal sulphide ore may take the form

MS 2Fe3M2 2Fe2 S 1

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 isinjected. Narrow heaps may draw air in through their sides. Biggerheaps will be built with pipe networks that, if they remainunblocked, can pump air directly into the heap.

Rafnate chemistry acidity, iron content and bacteria. Rafnate application rates, rest periods, method of applicationwhere M is the metal to be leached and S isions are then oxidized back to ferric withrafnate and oxygen in the gas phase catalyse

Corresponding author.E-mail address: d.mcbride@swansea.ac.uk (D. McBri

0304-386X/$ see front matter 2012 Elsevier B.V. Alhttp://dx.doi.org/10.1016/j.hydromet.2012.08.004ching typically involvesand 10 m in height andrface that trickles down

heaps are leached for many months if not years. The sheer scale alsolimits what can be done to inuence the process. The factors that canbe controlled include:stacking crushed ore in lifts of between 3applying a lixivant (rafnate) to the upper suStockpile leaching as a solution mifrom primary ore is of increasing impofor metals is continuing to rise, whilstdegrading; see Bartlett (1998) for anStockpile leaching provides a cost effecta range of metals from low grade minused for gold and copper, though it is alsuch as, nickel and uranium. Stockpethod to recover metalsat a time when demandilable mineral grades arew of these technologies.nique for the recovery ofposits. In particular it isfor a range of other ores,

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

Fe2 14O2 HFe3

12H2O 2

The biggest issues in understanding and controlling leaching arescale both in terms of the physical size of the heaps being leached,which can easily consist of multiple millions of tons, and in time since1. IntroductionA comprehensive model for copper sulphPart 1 Basic formulation and validation th

C.R. Bennett a, D. McBride a,, M. Cross a,b, J.E. Gebhara College of Engineering, Swansea University, Swansea, UKb PERI, Salt Lake City, Utah, USA

a b s t r a c ta r t i c l e i n f o

Article history:Received 14 March 2012Received in revised form 19 July 2012Accepted 7 August 2012Available online 17 August 2012

Keywords:Heap leach modelCopper sulphidesComputational uid dynamics (CFD)

This paper covers the basic fuid dynamics (CFD) technotest data. For the column tesand pyrite in a column undethe key minerals within thePrecipitation species, the rolealso described in the contexdata, including particle sizeparameter set to produce a g

j ourna l homepage: www.esulphur. Ferrous (Fe2+)oxygen dissolved in thed by lithotropic bacteria

de).

l rights reserved.e heap leachingugh column test simulationb

ulation of a comprehensive copper heap leach model based on computationaly together with its parameterization and validation against laboratory columnta used here, the model formulation covers an ore with a mixture of chalcocitech with a ferric rafnate and includes the reaction kinetics of the dissolution ofntext of a shrinking core algorithm to accurately model the leach behaviour.bacteria and the treatment of unsaturated uid and gas ow in porous media isthe model. The current work demonstrates the use of small column test leachtribution, to characterize the ore. The model was then used with the samet to results from larger columns and different crush sizes.

2012 Elsevier B.V. All rights reserved.

tallurgy

v ie r .com/ locate /hydromet(usually drip emitters but sometimes wobblers).

Any successful operation depends on extracting the maximumamount of metal for the least cost. This depends upon contacting asmuch of the ore with solution as possible, generating ideal conditionsfor reactions to occur and then ensuring that the resultant metal saltsare recovered from the pregnant solution.

Contacting ore with solution depends upon having good hydraulicproperties in the heap, where good means that the ow spreads

151C.R. Bennett et al. / Hydrometallurgy 127128 (2012) 150161evenly under the action of gravity and capillary forces and doesnot channel. Hydraulic properties are closely connected with theavailable pore sizes in the ore which in turn is a function of theparticle size distribution. A ne particle size distribution maximizessurface area and therefore reactivity but can have low hydraulicconductivity preventing effective ow of gas or rafnate. It is alsoimportant to be able to extract the soluble salts from the heap. Theparticle size distribution must also promote physical stability withinthe heap.

The metal dissolution reactions are dependent on the inherentmineral kinetics, diffusion (i.e. the transport of reactants to the mineralsurface), temperature and availability of reagents. The best leachprocess is one whose only limitations are due to factors beyond thecontrol of the operator, as in the inherent kinetics. Reagent availability isdriven by consumption and regeneration.

Generally, higher temperatures lead to faster reactions but in turntemperature can adversely affect the population and performance ofthe bacteria whose presence is key to the regeneration of ferric ions.

Diffusion is a function of the size of the particles. Due to the timerequired for reagents to penetrate large particles and for the dissolvedmetal species to diffuse out, there can often be an effective maximumparticle size for a given leach system.

Experimental work is fundamental to the exploration of howindividual ore bodies can leach. Shake ask tests to determine mineralkinetics are relatively quick but the main experimental work is usuallybased on column leach tests. Column leach tests replicate many of themain factors present in a full heap but in a controlled environment.Columns can be placed in jackets to allow the temperature within to bekept constant and some effects of lift height and crush size can beexplored through the use of differently sized experimental rigs.Columns can vary in size from the laboratory scale (5 cm diameter by0.5 m high) up to pilot scale (2 m diameter by 10 m high). Individualcolumn tests may take months to complete.

A suitably comprehensive mathematical model provides a frame-work with which to better understand the physics and chemistryinvolved in the process and provides tools that allow different leachstrategies to be explored and optimized cheaply and quickly. The basicprocess, applying a reactive solution on top of a heap of material andcollecting a metal as a dissolved salt in the pregnant leach solution, issimple in concept. However, leach systems are actually very complexphysically and chemically, which makes building an effective andreliable mathematical model a difcult challenge. The variabilitybetween different ores and between samples (i.e. mineral concentra-tions, local blockages, impurities)makes it difcult to accurately predictthe behavior of a particular system unless the model is calibratedagainst each specic ore. However, once the model is calibrated for aspecic ore type, then it should be possible to determine trends andindicators towards improving leaching at the industrial scale in a fairlyreliable manner. Computational models have the advantage of beingrepeatable and very fast simulations of hundreds of days of columntests can take place in a few minutes, whilst years of whole heap leachcan be delivered within hours. A helpful general description of leachmodelling requirements has recently been given by Petersen (2010a,2010b).

To date signicant effort has been applied to further theunderstanding of heap operations by building mathematical modelsof the heap process. These models have involved studies of uid ow(Bouffard and Dixon, 2001; Pantelis et al., 2002), chemical dissolutionrates (Casas et al., 1998; Dixon and Hendrix, 1993; Madsen andWadsworth, 1981; Paul et al., 1992a, 1992b), and heat/temperaturebalances (Cathles and Apps, 1975; Dixon, 2000). Watling (2006) andDixon (2003) provide recent reviews of the bioleach process and thestatus of modelling efforts. Dixon and Petersen (2003) present amodel for column heap leaching using a model based on rafnatediffusing out to reaction sites from discrete channels through the ore

and use comparisons to column test results to generate condence inthe 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 consideringthe interactive issues of bacterial effects, heap temperature and therole of gas sparging. More recent work includes the ow modeldeveloped by Cariaga et al. (2005, 2007) and the analytical modeldeveloped by Mellado and Cisternas (2008) andMellado et al. (2011).Pyrite leaching has been considered by Bouffard (2008) and Bouffardand Dixon (2009).

The aim of the current work is to move towards a practical tool forwhole heap simulation based on readily available data from either thelaboratory or on the full-scale heap. It builds upon the pioneering workof Petersen and Dixon (2003, 2007) with respect to validation againstlaboratory column data. Furthermore, this work takes advantage ofimprovements in computational performance to build a CFD basedmodel that combines true heap geometry with all aspects of the variousphysical, chemical and biological processes present in a heap as a seriesof sub models to provide a comprehensive solution (Bennett et al.,2003a, 2003b, 2006, 2008a, 2008b; Cross et al., 2005, 2006; Gebhardt etal., 2007; McBride et al., 2005, 2006). Petersen (2010a, 2010b) providesan excellent overview of the challenges in mathematical modelling thefull range of phenomena that play a role in controlling the dynamics ofmineral dissolution during heap leaching. From this assessment it isclear that physics and chemistry are endishly complex, and the role ofthe modeller is to nd a practical formulation that captures all the keyphenomena at an adequate level, so that the resulting model can bereliably parameterized by available measurable data.

Rates of mineral dissolution depend upon a balance of diffusion andchemical rate kinetics. Where as many of the factors effecting ratekinetics are well understood, there is still some debate over the mostefcient ways of dealing with diffusion. Many large-scale modelsaccount for the effect of varying particle sizes by employing averageparticle sizes and simplied 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 distributedmineralgrains and spherical particle geometry is not normally valid. It has beenshown that other geotechnical parameters such as, bulk density/stresscharacteristic, tortuosity, moisture capacity and particle density caneffect changes in the diffusion controlled leach rate, (Miller, 2003).However, as such detail is not normally available for a large-scale heapthe simplicity of the shrinking coremodel lends itself well to large-scaleoperations. The model itself is a very useful tool to analyse commercialheap data (Miller, 2003). The current work uses a simple shrinking coremodel for multiple representative particles based on the particle sizedistribution in order to account for diffusion and shows that thisapproach scales correctly for different particle size distributions usingthe same ore. The thinking behind this approach was that althoughthe reaction modelling is quite simplistic, inner particle diffusion isdominated byparticle size and so long as the full particle size distributionis carefully reected in the formulation then it should enable the captureof the overall dissolution rate reasonably well.

This paper focusses upon the challenge of modelling column leachtests of a chalcocite ore that also contains pyrite. The current work isbased on over 10 years of experience in modelling copper heap leachsystems with industrial partners who routinely perform column leachtests at a variety of scales to understand how a specic ore body willbehave and how best to optimize the full scale heap operation.Column leach tests can typically be modelled as one dimensionalsystems and are therefore an excellent source of data with which toparameterize and validate aspects of computational models that maythen be used to predict the behaviour of larger scale systems. In thiswork, column tests are used for two related purposes:

a) Small column test data is used as a basis for parameterizing the

model for a specic ore blend, whilst

152 C.R. Bennett et al. / Hydrometallurgy 127128 (2012) 150161b) A larger column test using the same ore blend is used as a basis forenabling a test of the parameterizedmodel to predict its behaviourthat is, to validate the ow and reactive aspects of the mathematicalmodel.

2. Mathematical modelling

2.1. The system to be modelled

The basic system being modelled consists of a small 0.15 mdiameter 1.8 m high column of ore containing 0.8% copper in the formof chalcocite (Cu2S) and approximately 2% pyrite (FeS2). The ore iscrushed to minus -inch (0.0127 m). A rafnate containing 5 g/l ironin the form of ferric sulphate (Fe3+) at a pH of 1.9 is applied to thetop of the column at a rate of 1.3 l/m2/h (3.6 E07 m/s) over a periodof 100 days, and air is pumped in from the base at 0.32 Nm3/m2/h(8.64 E05 m/s). About 70% of the copper is extracted over thecourse of 100 days, the majority of recovery occurring in the initial50 days after which the rates of reaction tend to fall off.

For validation purposes the same set of material parameters used forthe 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 1-inch (0.0381 m).

In this aspect of the work the principal reactions are the dissolutionof chalcocite and pyrite and the regeneration of ferric [as illustrated inEqs. (1) and (2) above]. In addition an iron precipitation reaction isincluded as the solubility of ferric is highly dependent on acidity.

The chalcocite reaction is split into two stages. This is based on thefundamental work on reaction kinetics of pure chalcocite mineral(Bartlett, 1998; Marcantonio, 1976). The reason for the two differentstages is that chalcocite was determined to have two distinct phasesduring dissolution in ferric sulphate solutions. The rst stage involvesmixed kinetics, that is, solution diffusion and surface reaction. Thesecond stage is controlled by the surface reaction, mainly electrontransfer.

The rst stage reaction is

5CU2S 8Fe35CU1:2S 4Cu2 8Fe2 3whilst the second stage consists of the breakdown of a covellite likeproduct (Cu1.2S) generated in the rst stage

5CU1:2S 12Fe35S 6Cu2 12Fe2 4

The pyrite reaction is

FeS2 14Fe3 8H2O15Fe2 2SO24 16H 5

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

Fe3 3H2OFeOH3 3H 6Both precipitation events increase acidity in the solution. As the

hydroxide reaction is reversible it can have the effect of buffering theacid and iron levels in solution.

Although this is a somewhat simplied treatment of the iron cyclesuch an approach is necessary in order to be able to develop an

effective model.2.2. Modelling requirements

Modelling stockpile leaching requires the accurate simulation of arange of physical and chemical phenomena. The resultant multiphasesystems include

Transport phenomena, including liquid and gas ows, and masstransfer 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 uid dynamics (CFD) code based on a massconserving formulation as a starting point, facilitates control overgeometry and transport of ow properties and chemical specieswithin the column. It also allows the model to be readily expanded tomultiple dimensions. The key physics are captured through sourceterms in the general transport equations used to govern and conservethe physical properties of the solution ow. A CFD approach dependsupon breaking the physical geometry of the column down into a setor mesh of representative elementary volumes, or elements. For thepurposes of species transport each element is assumed to haveconstant properties within its own boundaries.

The key to approaching modelling a leach system is in identifyingthe 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 acolumn test in that the model can be somewhat simplied comparedto that required for a full heap. Two key areas of physics in largeheaps, the heat balance and bacterial effects, can be simplied in mostcolumn tests. Although in some bioleaching column tests the localbacterial environment may have an effect, in the main small columnconditions are usually ideal for bugs, especially if they are in therafnate. The model makes the assumption that the bug environmentis ideal and it is only the lack of them that retards the process.

The heat balance is driven by the heat of reaction, externaltemperatures and transport through liquid and gas ow. As the ratioof surface area to bulk ore mass is such that columns lose heat morerapidly than they can generate within them, it is the external orambient temperature that is the dominant factor. Small columns canalso be jacketed to maintain a given temperature which can be usedto gain insight to the effect of temperature on the reaction kineticsbut the key is that the column temperature is unlikely to besignicantly inuenced by reactions or ows inside the small column.Condensation and evaporation need not be modelled for the smallcolumn case, although this can be a mechanism for heat transport inlarger systems.

Bacteria only limit the leach process when their activity is very low,either due to low population density or to disadvantageous chemicaland thermal conditions. When they are present in sufcient quantitiesand conditions are suitable, the limits on ferrous oxidation will becontrolled by the local chemistry. Column tests tend to have idealconditions and, if using rafnate that is seeded with bacteria, it is quitelikely that there will be sufcient bacterial numbers so there will be nonoticeable adverse effect on leaching in these circumstances. So,bacteria are therefore ignored in this paper.

The remaining important physics can be grouped into three mainareas based on their relative timescales.

Liquid phase transport. This covers saturation, leach solution owand transport of soluble species. It is mainly dependent on the rateof solution application and the hydraulic conductivity of the ore.

Gas phase transport. This covers gas ow and transport of oxygenand to a lesser extent, water vapor.

Process chemistry. This covers dissolution kinetics of the ore, anychemistry in the liquid phase (i.e. iron chemistry and precipitation)

and mass transfer between the different phases.

C.R. Bennett et al. / HydrometallurThe basic formulations for these three areas are described for theheap leach model.

2.2.1. Liquid phase transportLiquid transport is driven primarily by gravity. Matric potential or

pore pressure will also act on the moisture although in 1D these effectsare small.

For unsaturated porous ow, the Darcy ux can be written interms the pore pressure and moisture content,

q K K m

z 7

where:

is the moisture contentK() is the unsaturated hydraulic conductivitym is the matric potential or pore pressure.

To ensure mass conservation, the volumetric continuity equationalso needs to be satised.

t

qxx

qyy

qzz 8

The hydraulic conductivity can be described by the van Genuchtenequation (van Genuchten, 1980)

K ksS 0:5 1 1S 1=m mh i2 9

S rr

1

1 j j n m

10

where

m =11/n,n are empirical constantsks is the saturated hydraulic conductivity is the porosityr is the residual saturation

The saturated conductivity is given by

ks kig

11

where

ki is the intrinsic permeability of the mediag is gravity is the rafnate density is the rafnate viscosity

Permeability is mainly a function of particle size distribution,particularly the amount of nes in the ore which can tend to block thepores through which the liquid ow seeps.

The expression for saturation in Eq. (10) provides an easy way todetermine pore pressure based onempirical constants and the saturation.Saturation is easy to calculate in the current model as the mass ux ofrafnate through the system is conserved.

As long as the ore does not become fully saturated the ow can besimply modelled through an explicit scheme evaluating uxes betweenindividual elements. If parts of the ore approach or become saturated, itis then necessary to implicitly solve for saturation and subsequently

derive the uxes (McBride et al., 2005).In a 1D unsaturated system, the Van Genuchten equation effectivelydescribes the saturation in the columnand the time taken for solution tostart to ow from the base of the column. Under constant irrigation rateand assuming there is no change in the hydraulic properties ofthe column the discharge ow rate will, of course, remain constant.Hydraulic properties of the column could potentially change due totransport of nes, salt precipitation, breakdown of particles underleaching conditions generating more nes and compression. In smallcolumns the most likely cause of any changes will be precipitation butthis is highly dependent on the chemistry of the ore and the rafnate.

2.2.2. Gas phase transportWe make the assumption that the inuence of the phases on each

other is one way. This means that the liquid phase inuences gas owbut that the gas ow does not directly inuence the liquid phase. Theliquid phase is slow moving and over the course of a leach cycle ismostly in a steady state, so the gas ows through a porous mediummade up of both liquid and solid phases.

Gas ow is primarily driven by boundary conditions such as gasinjection through air lines and wind pressure against the sides ofheaps. Temperature gradients can also be very important in drivinggas ow. Indeed the ideal situation in heaps with large exposed anksis that a chimney effect is created which draws external air inside.Other possible factors include displacement by liquid ow, oxygenconsumption, evaporation and condensation.

Gas ow affects the heap through transport of oxygen. It can alsospread heat although the low thermal capacity of the gas phasemeans that this is a minor effect.

The gas phase is solved assuming incompressible gas ow with aBoussinesq source term. This approximation is valid as local tempera-ture differences are small and so actual changes in density are verysmall. The basic continuity equation for the incompressible gas phasetransport is

div gvgh i

Sg 12

where

g is the gas densitySg is a source term for gasvg is the gas velocity, is equal to

vg kinkg S gg

pg ggz

13

where

kin is the intrinsic permeability of the porous mediakg(S) is the unsaturated permeability of the gas at liquid saturation

(S), related to the liquid unsaturated permeability kl(S) by

kg S 1kl S 14

g is the volume fraction of the gas phaseg is the gas viscosity

Substitution of Eq. (13) into Eq. (12) gives the following continuityequation for pressure, p.

2pg ggkinkg S

Sg 15

153gy 127128 (2012) 150161

154 C.R. Bennett et al. / Hydrometallurgy 127128 (2012) 150161where

Sg Sthermal Svol Sother 16

Sthermal is a Boussinesq source term allowing for natural convectiondue to the thermal expansion of the gas where there aretemperature changes and is therefore not used in anisothermal case.

Svol is the source term due to changes in volume available forgas ow due to changes in saturation. As long as the systemremains unsaturated, the equations for gas and liquid owscan be solved for separately. The assumption is made thatgas ow does not inuence liquid ow; therefore, the liquidow is solved for rst. It also contains any changes in gasvolume due to consumption of oxygen or evaporation andcondensation.

Sother contains any other sources of pressure and allows for pointinjection 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 smallgrains contained in a matrix of inert material. Reactants diffuse inthrough pores in the rock matrix of each ore particle, chemicalreactions occur and the products diffuse out (Bartlett, 1998; Petersen,2010a, 2010b). The reaction kinetics of different minerals can varywidely and the overall rate of reaction is also highly dependent on thediffusion of reactants and products. To allow for both these factors theore is divided into discrete size fractions with characteristic radiusand mineral concentrations. The rate of dissolution for each mineralin each characteristic particle size is modelled using a separateshrinking core reaction.

The shrinking core reaction normally assumes that there is ahomogenous distribution of reactivematerial. The currentwork acceptsthat this is unlikely to be the case for a real ore and in addition the exactdistribution of mineral grains and the interaction between differentminerals in the same particle can complicate the overall dissolutionrates. The model presented here considers the location of a reactionfrontmoving through a representative particle that is the average of themany hundreds or even thousands of particles in the same size class in aparticular representative elementary volume, and the average particlewill have a homogenous mineral distribution. In this case the shrinkingcoremodel appears as a convenient approximation, since the dominantinuence of the particle size is captured through explicitly representingthe particle size distribution.

The equation used to calculate the rate of dissolution of a givenmineral is given by (Szekely et al., 1976)

drmdt

3rm4r2o

Miorexi

DeffcoAm3Deff roco 2 rorm r2m 1p

Am

h i 17

where

ro is the initial particle radiusrm is the current mineral radiusAm comes from the kinetic rate equation for the current

mineral.R is the gas constantT is the temperature in KelvinDeff is the effective rock diffusion coefcientp is the rock voidageore is the ore densityMi is the molecular weight of the mineral

xi is the mass fraction of the mineralThe value of Am comes from the general expression for the kineticrate equations, such as those produced by Paul et al. (1992a), whichtakes the general form

Am ddt

Ae BRT 18

where

is the fraction of mineral reactedA,B are functions of the individual kinetic rate equation.

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

rmr0

319

This approach allows minerals to react at different rates in anindividual particle size, and although it requires the assumption that theminerals can be treated relatively independently, this is not unreason-able given the low concentration of reactive minerals present. Thisapproach allows each mineral in each particle size fraction to bemodelled using a single characteristic radius indicating how much hasreacted. This in turn allows the model to deal with multiple particlesizes and multiple minerals over large meshes without excessivememory usage in the overall CFD model framework.

This approach has an advantage in that it can easily be related toexperimental analysis of ore which is commonly given as mineralcontent by size classication, making validation easier. It also easilyallows for different minerals to dominate the reactions at differentstages of the leach cycle.

Although each reaction in each particle size fraction is consideredindividually, there can be times, especially early on in the leachprocess when diffusion is not limiting, where the sum total ofreactants consumed can be greater than that available. When this isthe case, an optimization routine can be used to share the availablereactants between the competing reactions, where the consumptionis apportioned proportionally to the relative rates of the individualreactions.

As the initial reaction rate can be high with no diffusion, it mayalso be necessary to use a variable time step for the chemistry.

Each of the following kinetic rate equations incorporates a rateconstant. This allows the model to be tuned to a given ore usingexperimental column data. Without setting the rate constants, themodel provides information on general trends but better accuracycomes with using these values to t the model against known data.The rate constants in effect combine to cover factors in the reactionsthat are not otherwise specied for in any particular ore complex.These can include particular distributions of mineral grains andinteractions between different minerals which can be difcult toquantify but may be characteristic of a particular ore body. The rateconstants are unrelated to particle size and therefore allow the modelto scale from small to large particle size distributions (e.g. fromexperimental column to heap). Methods to enable the rate constantsto 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 aleaching process occurs through many steps and the formation ofmultiple intermediary copper-decient phases. Marcantonio (1976)simplied the dissolution or leaching of chalcocite by dening a two-stage electrochemical-based reaction mechanism with the secondstage reaction initiated after 40% of the copper has been reacted. Thesecond stage (Cu1.2S) mineral has a similar formula to the copper

mineral covellite and is often referred to by the same name.

155C.R. Bennett et al. / Hydrometallurgy 127128 (2012) 150161The rst stage involves mixed kinetics, that is, solution diffusionand surface reaction involving ferric as the reactant. The rst stage isstrongly inuenced by the high mobility of cuprous ion in the solidphases. The second stage for chalcocite dissolution is controlled bythe surface reaction, mainly electron transfer.

In the shrinking core algorithm, the second stage equation is usedif, at the start of the mineral time step, the fraction of copper reactedis greater than 40%.

Though the specics change in each stage, the reaction (involvingferric) considered for the stoichiometric mass balance in the model is

Cus 2Fe3Cu2 2Fe2 20According to Madsen and Wadsworth (1981), predicted copper

recoveries calculated with activities derived from thermodynamicdata and the DebyeHuckel theory were consistently higher thanexperimentally observed copper extractions for chalcocite leach tests.An apparent activity coefcient was determined empirically by ttingto observed leach rate data. Both chalcocite reaction stages use anapparent activity function () with the ferric concentration. Thefunction is as follows:

e 1:373239:7Fe30:54870Fe3 21

2.2.3.3. Stage I chalcocite (Cu2S). In the rst stage, chalcocite isconverted to secondary covellite by reaction with ferric according tothe following reaction

5Cu2S 8Fe35Cu1:2S 4Cu2 8Fe2 22The rate equation associated with the rst stage can be written as

ddt

1 0:4

RA8:6 Fe3h i

rpe

1404T 23

where

is the fraction of total copper reactedrp is the mineral grain radius (m) is the particle shape functionT is the temperature (K).RA is a rate constant based on a t to known data

2.2.3.4. Stage II chalcocite (Cu1.2S). The second stage consists of thebreakdown of covellite (Cu1.2S) generated in the rst stage by areaction very similar to that for natural covellite:

5Cu1:2S 12Fe35S 6Cu2 12Fe2 24The associated kinetic rate equation is given by (Paul, 1989)

ddt

10:6

0:5RA1:26 1010 Fe3

h in o0:54e

9059T 25

where

is the fraction of copper reactedT is the temperature (K)RA is a rate constant based on known data is the apparent activity coefcient[Fe3+] is the ferric concentration in solution; units in mol/cm3

All concentrations are in mol/cm3. This reaction is considerably

slower than the rst stage reaction.2.2.3.5. Pyrite rate kinetics. The pyrite reaction is

FeS2 14Fe3 8H2O15Fe2 2SO24 16H 26

This is governed by the kinetic rate equation which is adaptedfrom Paul et al. (1992a)

ddt

RA601:5 Fe

3h irp Fetot H 0:4

e10317

T 27

where

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

2.2.3.6. Ferrous oxidation. An equilibrium relationship is used todetermine the ferric ion concentration based on the concentrations offerrous ions, dissolved oxygen and free acid, and an equilibriumconstant, K. The relationship is given by,

K Fe3h i4

Fe2 4 O2 H 4 28

where K=5.56107 (Garrels and Christ (1965)).Concentrations here are in moles/litre.The ratio of ferric ions to ferrous is therefore given by

Fe3h iFe2 K1=4 O2 1=4 H

h i29

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

Fe3 3H2OFeOH3 3H 30Precipitated 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.

log Fe3h i

max 1pH 31

An equilibrium relationship is solved using a NewtonRaphsonscheme, and is solved after all other chemical reactions have beencompleted in a time step.

2.2.3.8. Oxygen mass transfer. The oxygen liquidgas mass transfer ratecan be determined by a number of factors, typically, temperature,liquid and gas composition and liquidgas interfacial area. Petersen(2010a, 2010b), studied the the gasliquid mass transfer of oxygen inheap leach scenario's and found that the net mass transfer rate wasrelatively unaffected by temperature in the range 2268 C as anincrease due to temperature was offset by a proportional decrease inthe solubility of oxygen. The mass transfer tended to increase withmaterials with higher nes, possibly due to a larger surface area.

Due to the large solution timesteps employed, typically 10 min,and large surface area between the liquid and gas phases the modelassumes that the transport rate is effectively instantaneous and anequilibrium state is obtained. Oxygen is partitioned between the

liquid and gas phases by using Henry's law.

MO2 is the molar weight of oxygenMN2 is the molar weight of nitrogen

Calculate reactants used

Calculate new core radius

Build and solve matrix to determine species reacted

Loop over liquid reactions

Loop over / particlesmineral reactions

Scale particle

Calculate common element / particle properties

Loop over elements

Table 2Summary of particle size data for the small column.

Particles Mass fraction Radius (m) Fraction FeS2 Fraction Cu

1 0.60% 0.006250 1.04% 0.52%2 32.50% 0.003130 1.30% 0.65%3 9.40% 0.002380 1.46% 0.73%4 21.70% 0.001000 1.78% 0.90%5 22.10% 0.000075 1.96% 0.98%6 13.70% 0.000060 2.00% 1.00%

156 C.R. Bennett et al. / Hydrometallurgy 127128 (2012) 150161Calculate revised core radius

Loop over reactions

Loop over particles

Scale reactionThe equation solved is

Og

Og 1Og

MO2MN2

KHOl 32

where

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

Update species

Fig. 1. A Schematic of the algorithm for multiple particles and reactions within eachcontrol volume or element.

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

Property Small column Large column

Height Diameter 1.8 m0.15 m 6.1 m 2.0 mOre crush P100 0.0127 m (0.5) P80 0.0381 m (1.5)Ore density 2950 2950Solid fraction 0.59 0.59Copper mass fraction (chalcocite) 0.59% 0.59%Pyrite mass fraction 2.15% 2.15%Particle porosity 5% 5%Leach cycle 97 days on

3 days off90 days on 30 off 30on 10 off

Rafnate application rate (m/s) 3.6E7 1.576E6Rafnate concentrationsFerric (gpl) 4.08 3.50Ferrous (gpl) 0.10 0.33pH 0.30 0.30Copper (gpl) 0.90 0.85Air inlet velocity (m/s) 8.64E5 8.64E5Temperature (C) 25 25KH is the Henry's law constant at a given temperature

Henry's law constant is allowed to vary with temperature by usinga function based on dissolved oxygen solubility against temperaturedata under atmospheric conditions. As the partial pressure of oxygenmay well be dependent on the level of salts in solution the possibledissolved oxygen levels can also decrease. An equilibrium state issolved using a NewtonRaphson scheme at the end of each time step.

2.2.4. The numerical solution strategyThe 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 nite volume numerical approach which ensuresthat all spatial variables are transported in a mass conserved manner.Hence, the column volume is set as the solution domain and is dividedinto a mesh of connected elements. For the column simulations, theow is essentially one dimensional and so a simple mesh of rectangularelements is used to cover the ow domain. Within each element thereare three phases a static solid phase of the ore body, a liquid phase ofsolution owing vertically downwards and a gas phase (of primarilyair) owing vertically upwards. Within each control volume or elementthe chemical reactions are pursued both for each reaction sequence andalso within each particle size fraction simultaneously. Of course, theminerals and pyrite in the solid phase are reacting with various speciesin the liquid and gaseous phases. A diagram illustrating how thereaction phases are managed within the simulation is shown in Fig. 1below. It should be clear that there are indeedmultiple timescales in thesimulation which can be caught by two time steps one for the overallow physics of the liquid and gas phases and a second which governsthe chemical reactions. The chemical reactions time step is typically1/10 of the size of the overall ow time step.

3. Parameterization and validation

Results from simulations of a small and large column and theircomparison to experimental data are shown below. The input data issummarized in Tables 13. The set of mineral reaction rates, RA inEqs. (23), (25) and (27) is essentially tuned to enable the best t

Table 3

Summary of particle size data for the large column.

Particle Mass fraction Radius (m) Fraction FeS2 Fraction Cu

1 1.80% 0.037500 2.15% 0.44%2 12.50% 0.021900 2.15% 0.53%3 21.70% 0.015600 2.15% 0.62%4 11.80% 0.010900 2.15% 0.59%5 13.70% 0.007810 2.15% 0.63%6 12.70% 0.004690 2.15% 0.63%7 3.40% 0.002750 2.15% 0.59%8 7.40% 0.001690 2.15% 0.74%9 8.60% 0.000538 2.15% 0.86%10 6.40% 0.000075 2.15% 1.03%

models can be tuned to provide a reasonable match to the overall

-10%

0%

10%

20%

30%

40%

50%

60%

70%

80%

0 20 40 60 80 100 120

ExperimentalSimulation

Fig. 2. Copper recovered to PLS against time for experiment and simulation.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 20 40 60 80 100 120Leach period, days

Conc

entra

tion,

gpl

ExperimentalSimulation

Fig. 4. Ferrous ion concentration in the PLS.

157C.R. Bennett et al. / Hydrometallurgy 127128 (2012) 150161possible for the small column and then these data are used to predictthe behaviour of the large column. So the data from the small columnis used to parameterize the model for a specic ore, whilst the datafrom the large column provides a validation test for the parameter-ized model. In the comparisons below we show in detail a number ofthe key results. It is not too difcult to x model parameters so that areliable match can be obtained with the overall rate of copperextraction from the column. It is, however, much more challenging toenable the model to match the behaviour of both the speciesconcentrations and pH within the PLS, and the dissolution in thecolumn by particle size fraction. This requires that the formation ofthe model capture all the key details of the extraction process.

It is worth commenting that most column data recorded inindustrial laboratories is to help inform operational decision makingor heap design, not to provide the basis for parameterizing a detailedmathematical model. The data reported here is both complete from asimulation perspective and closed that is it is sufcient to enable aprocess model to be evaluated, and in that sense, should provide auseful basis for testing and validating other future process models.

3.1. Parameterization against the small scale column

In the following gures, a comparison of the simulation is shownagainst the data from the small column experiments. The operationsimulated is based entirely on the conditions as specied in Table 1and 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 andindeed the values used here are estimated from a wide variety of tests-2.0

0.0

2.0

4.0

6.0

8.0

10.0

0 20 40 60 80 100 120Leach period, days

Conc

entra

tion,

gpl

ExperimentalSimulation

Fig. 3. Ferric ion concentration in the PLS.over an extensive period of multiple comparisons with columnexperiments. It is the rate parameters for the chalocite and pyrite thatare tuned to enable the best t to the experimental results both inthe PLS and also the residual solid column both overall and by particlesize fraction.

The simulation of 100 days for the small column takes less than aminute on an ordinary PC. The simulation used 10min outer time-steps and a relatively coarse mesh. Because of the uniform ow, theresults did not noticeably alter for a ner mesh. The simulation predictsPLS data for the whole period as well as chemistry and remainingminerals in the ore mix by size fraction, location and time. In Fig. 2, thepercentage of the copper in the column recovered to the PLS is shown asa function of time, from which it is clear that the comparison over the100 days or so is quite good. There is a small discrepancy at around 22and 38 days, but in each case there were short interruptions to theplanned operation which were not recorded in any detail (and so notreected in the simulation operational conditions). Of course, Table 1indicates that the solution ow rate is constant into the column for therst 97 days followed by a rest period until the full 100 day period isreached.

The simulation cannot effectively capture every aspect of the experi-ment, such as, variations in overall ore properties from the testedsamples through the length of the column.Minor variations in rafnateow rates and properties can be captured but it may not be timeeffective or indeed useful to do so, as the key results are the trends.Most0

5

10

15

20

25

0 20 40 60 80 100 120Leach period, days

Conc

entra

tion,

gpl

ExperimentalSimulation

Fig. 5. Copper ion concentration in the PLS.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 20 40 60 80 100 120Leach period, days

pH

ExperimentalSimulation

Fig. 6. pH level in the PLS.

30%

40%

50%

60%

70%

80%

% R

emov

ed Copper RemovedIron Removed

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 20 40 60 80 100

Leach Period (days)

% R

emai

ning

Radius 6.25E-3 mRadius 1.0E-3m

Radius 7.5E-5m

Fig. 9. Percentage of copper remaining by particle size against time from thesimulation.

158 C.R. Bennett et al. / Hydrometallurgy 127128 (2012) 150161recovery butmay notmatch all of themeasured species concentrations.The strength of the model presented here is shown in the simulatedconcentrations in the PLS and residual solid matrix by size fraction. InFigs. 3 and 4, the comparisons for the ferric and ferrous ion concen-tration in the PLS are shown. The trends here arewell captured,with theinitial high ferrous spike and slowly increasing ferric content matchingthe fast initial reactions with minerals on the surface of the rocks in thecolumn. In Fig. 5, the copper ion concentration in the PLS is shown tocompare well with the experimenal data over much of the time range.The early stages of the simulation can lag behind the experiment as theinitial conditions in the column may not be well dened.

There are also demonstrable differences between the regimewithin the small and large columns due to the relative particle sizedistributions. In small particles the shorter diffusion paths for reactantsand products lead to relatively fast reactions. In the small column thisleads to rapid useage of the available ferric which, assuming a healthybacteria population, leads to a lot of ferrous oxidation which consumesacid. The ferric salts produced cannot stay in solution and precipitate ashydroxide which releases acid back into the system. This bufferingeffect ties up iron until the rate of reactions slow and the rate of acidconsumption falls, which makes acid available to dissolve the ferrichydroxide.

The pH in the PLS is shown in Fig. 6 which compares well with theexperimental data except for the early stages where the experimentaldata is unreliable. The model also caps the PH at 4 because the rates of0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 20 40 60 80 100 120Leach period, days

Mol

es/to

nne

ExperimentalSimulation

Fig. 7. Iron balance over the course of the leach cycle. This is total iron fed into thecolumn less total iron recovered to PLS.0%

10%

20%

0 20 40 60 80 100 120Leach period (days)

Fig. 8. Percentage of copper and iron (from pyrite) removed from the column againsttime as predicted by the simulation.reaction are effectively zero at this level as there is insufcient acidand ferric solubility is negligable. Another useful way to assess theperformance of the model is on the iron balance over the course of theleach cycle as shown in Fig. 7. Again we have a good comparisonoverall although slighty displaced in time. In Fig. 8, the percentagecopper and iron (from iron pyrite) removed from the solid matrix isshown over the time of the experiment. The pyrite reaction rate issignicantly slower than for chalcocite but diffusion of reactants andproducts is the same for all minerals. This means that as the copper

Table 4Small column simulated mineral residuals.

Particles Radius (m) Fraction FeS2 remaining Fraction Cu remaining

1 0.006250 100.00% 83.89%2 0.003130 99.58% 57.94%3 0.002380 99.22% 53.92%4 0.001000 53.92% 22.21%5 0.000075 50.39% 0.0%6 0.000060 38.67% 0.0%

and the experimental results. There is a gap in the experimental datawhere date was not recorded during the 30 day rest. The modelincludes this period in its simulation and predicts the recovery duringthat time, which is negligible. Again the ferric and ferrous ionconcentrations in the PLS (Figs. 11 and 12) are quite well captured bythe model during the rest period the model predicts the ferricconcentration will increase, which is not seen in the experimental

0%

10%

20%

30%

40%

50%

60%

0 20 40 60 80 100 120 140 160 180Leach period, days

Frac

tion

Copp

er R

ecov

ered

ExperimentalJ555 Sim

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

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 50 100 150 200Leach period, days

Conc

entra

tion,

gpl

Experimental

Simulation

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

4.0

5.0

6.0

7.0

8.0

9.0

cent

ratio

n, g

pl

ExperimentalSimulation

159C.R. Bennett et al. / Hydrometallurgy 127128 (2012) 150161reaction rate becomes increasingly dependent on diffusion withdissolution of chalcocite the relative speed of the pyrite reactionincreases. Over time these kinds of columns can produce increasingamounts of iron. The copper removal by particle size fraction isshown in Fig. 9 where it is clear that the smaller particle sizes reactsomewhat faster than the larger size fractions. The nal residualcopper levels by size fraction is shown in Table 4. This does not takeaccount of how the particle size distribution may change over thecourse of leaching through particle breakdown.

The rst part of the leach cycle is dominated by reactions with themost available copper, as in that which is closest to the surface of allparticle size fractions. It is thereafter dominated by the two smallestsize fractions where diffusion effects are minimized. The leaching ofpyrite, which has a much lower kinetic rate than chalcocite, onlystarts to become signicant after the majority of the copper in thesmallest size fractions has leached.

3.2. Validation against the large column data

Once the chalcocite and pyrite rate data were parameterizedagainst the small column, the model was then set up to simulate theconditions of the large column using the data in Table 1 for theoperational conditions and Table 3 for the initial condition of the oresize fractions. This column was run for some 160 days and Fig. 10shows the predicted recovery over time compared to the measured

values. Again, there is good agreement between the model simulation

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 50 100 150 200Leach period, days

Conc

entra

tion,

gpl

ExperimentalSimulation

Fig. 11. Ferric ion concentration in the PLS for the large column.0.0

1.0

2.0

3.0

0 50 100 150 200Leach period, days

ConFig. 13. Copper ion concentration in the PLS for the large column.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 50 100 150 200Leach period, days

pH

Experimental

Simulation

Fig. 14. pH of the PLS for the large column.

160 C.R. Bennett et al. / Hydrometallurgy 127128 (2012) 150161data. However, this result can be somewhat misleading as the owrates fall off dramatically and the actual amount of species in solutionis low. It is worth noting that reactions will continue in the columnduring and after the rafnate ow is turned off. As long as the localacid levels can be maintained through either pyrite dissolution or ironprecipitation the local concentration of copper in solution shouldcontinue to rise. When the rafnate ow is turned on again thecopper that has dissolved during this period will be ushed through,giving an apparent short term boost to the rate of copper recovery.

The simulated copper concentration in the PLS is well capturedthroughout the whole experiment (Fig. 13). Again the pH in the PLS iswell captured throughout the experiment (Fig. 14). The residualminerals by particle size is given in Table 5.

The predictive match to the experimental results in the largecolumn is generally good especially for copper recovery. All of themain features of the PLS are captured even if exact values vary forreasons of lack of knowledge of the initial conditions present in thecolumn. This model demonstrates the importance of being able torepresent multiple particle sizes and also the iron cycle, in particulariron precipitation and its role in buffering iron and acid levels in thecolumn. It is worth noting that the precipitation and buffering effectstend to only occur where reactions take place rapidly and there is ashortage of reagents. In situations where reactions are slower, suchas in the large column with a larger particle size distribution, thisphenomenon can be either short lived or may not even be notnoticeable. The ability of the model to deal with both situations is amajor strength.

4. Conclusions

The objective of the work presented in this paper has been tosummarize a practical computational model for the simulation of wellcontrolled column leach tests. In particular, the results demonstratehow the small column tests can be used to parameterize the modeland larger scale column tests to validate the model for scale-up with aparticular ore type.

It is also worth noting that the simulation results for both the

Table 5Large column simulated mineral residuals.

Particle Radius (m) Fraction FeS2 remaining Fraction Cu remaining

1 0.037500 100.00% 76.32%2 0.021900 100.00% 86.61%3 0.015600 100.00% 81.85%4 0.010900 100.00% 62.35%5 0.007810 100.00% 51.75%6 0.004690 100.00% 22.11%7 0.002750 99.68% 2.43%8 0.001690 98.64% 0.27%9 0.000538 78.00% 0.00%10 0.000075 6.53% 0.00%small and large column are broadly correct despite evidence that thebehavior of the two systems is different. The small particle sizedistribution in the small column leads to faster reactions than in thelarge column, which in turn leads to acid depletion and precipitationof ferric salts, something which does not happen with slower reactinglarge particles in the large column as observed both from our modeland in column experiments.

Modelling heap leaching is a demanding problem with multiplephysics and considerable heterogeneity in any experimental data. Theprimary aim of any successful model is to be able to respond tochanges in boundary conditions in the same way as it occurs withheap operations. Careful control of a set of rate parameters can tunethe model for a specic ore type to give a close match to experimentaldata, which will remain relevant over widely varying crush sizes andconditions.References

Bailey, A., Hansford, G., 1994. Oxygen mass transfer limitation of batch bio-oxidation athigh solids concentration. Miner. Eng. 7 (23), 293303.

Bartlett, R.W., 1998. Solution Mining, 2nd edition. Gordon & Breach Science Publishers,Amsterdam, The Netherlands. 325 pp.

Bennett, C.R., Cross, M., Croft, T.N., Uhrie, J.L., Green, C.R., Gebhardt, J.E., 2003a. Acomprehensive copper stockpile leach model: background and model formulation.In: Young, C.A., et al. (Ed.), Hydrometallurgy. TMS, Warrendale, PA, pp. 315319.

Bennett, C.R., Cross, M., Croft, T.N., Uhrie, J.L., Green, C.R., Gebhardt, J.E., 2003b. Acomprehensive copper stockpile leach model: background and model sensitivity.In: Riveros, P.A., Dixon, D., Dreisinger, D., Menacho, J. (Eds.), Copper 2003 Cobre2003 Hydrometallurgy of Copper, VI. MetSoc, Quebec, pp. 563579.

Bennett, C.R., McBride, D., Cross, M., Gebhardt, J.E., Taylor, D.A., 2006. Simulationtechnology to support base metal ore heap leaching. Trans. Inst. Min. Metall. Sect. C115 (1), 4148.

Bennett, C.R., McBride, D., Cross, M., Croft, T.N., Gebhardt, J.E., 2008a. Computationalmodelling of reactive porous media in hydrometallurgy. In: Topping, B.H.V.,Papadrakakis, M. (Eds.), Trends in Engineering Computational Technology. Saxe-Coburg Publications, pp. 151171.

Bennett, C.R., McBride, D., Cross, M., Gebhardt, J.E., 2008b. Computational modelling ofreactive multiphase ows in hydrometallurgical processes. Conference Proceed-ings, WCCM8 ECCOMAS.

Bouffard, S., 2008. Application of the HeapSim model to the heap bioleaching of thePueblo Viejo ore deposit. Hydrometallurgy 93 (34), 116123.

Bouffard, S.C., Dixon, D.G., 2001. Investigative study into the hydrodynamics of heapleaching processes. Metall. Mater. Trans. B 32B, 763776.

Bouffard, S., Dixon, D., 2009. Modeling pyrite bioleaching in isothermal test columnswith the HeapSim model. Hydrometallurgy 95 (34), 215226.

Cariaga, E., Concha, F., Sepulveda, M., 2005. Flow through porous media withapplications to heap leaching of copper ores. Chem. Eng. J. 111 (23), 151165.

Cariaga, E., Concha, F., Sepulveda, M., 2007. Convergence of a MFEFV method for twophase ow with applications to heap leaching of copper ores. Comput. MethodsAppl. Mech. Eng. 196 (2528), 25412554.

Casas, J.M., Martinez, J., Moreno, L., Vargas, T., 1998. Bioleaching model of a copper-sulphideore bed in heap and dump congurations. Metall. Mater. Trans. B 29B, 899909.

Cathles, L.M., Apps, J.A., 1975. A model of the dump leaching process that incorporatesoxygen balance, heat balance, and air convection. Metall. Trans. B 6B, 617624.

Croft, T.N., Pericleous, K., Cross, M., 1995. PHYSICA: a multi-physics environment forcomplex ow processes. In: Taylor, C., et al. (Ed.), Numerical Methods Laminar andTurbulent Flow'95, IX, pp. 12691280.

Cross, M., Bennett, C.R., McBride, D., Gebhardt, J.E., Taylor, D.A., 2005. Computationalmodelling of primary metal ore heap leaching. CD Proceedings of the Intl Conf onComputational Modelling in Minerals Processing, Cape Town.

Cross, M., Bennett, C.R., Croft, T.N., McBride, D., Gebhardt, J.E., 2006. Computationalmodeling of reactive multi-phase ows in porous media: applications to metalsextraction and environmental recovery processes. Miner. Eng. 19, 10981108.

Dixon, D.G., 2000. Analysis of heat conservation during copper sulphide heap leaching.Hydrometallurgy 58, 2741.

Dixon, D.G., 2003. In: Young, C.A., et al. (Ed.), Heap leach modeling the current stateof the art, hydrometallurgy, vol. 1. The Minerals, Metals & Materials Society,Warrendale, PA, pp. 289314.

Dixon, D.G., Hendrix, J.L., 1993. A general model for leaching one or more solidreactants from porous ore particles. Metall. Trans. B 24B, 157169.

Dixon, D.G., Petersen, J., 2003. Comprehensive modelling study of chalcocite columnand heap bioleaching. In: Riveros, P.A., Dixon, D., Dreisinger, D., Menacho, J. (Eds.),Copper 2003-Volume VI: Hydrometallurgy of Copper (Book 2). CIM, Montreal,Canada, pp. 493516.

Garrels, R.M., Christ, C.L., 1965. Solutions, Minerals, and Equilibria. Freeman, Cooper &Company, San Francisco, CA.

Gebhardt, J.E., Hernandez, A., Cross, M., Bennett, C.R., McBride, D., 2007. Analysis ofheap leach processes with modeling and simulation tools. In: Riveros, P.A., Dixon,D.G., Dreisinger, D.B., Collins, M.J. (Eds.), Copper 2007-Cobre 2007 The John DutrizacInternational Symposium on Copper Hydrometallurgy, pp. 231244. IV(Book 1).

Ghorbani, Y., Becker, M., Mainza, A., Franzidis, J.-P., Petersen, J., 2011. Large particleeffects in chemical/biochemical heap leach processes a review. Miner. Eng. 24,11721184.

Hansford, G., Bailey, A., 1992. The logistic equation for modelling bacterial oxidationkinetics. Miner. Eng. 5 (1012), 13551364.

Leahy, M.J., Schwarz, M.P., 2009. Modelling jarosite precipitation in isothermalchalcopyrite bioleaching columns. Hydrometallurgy 98, 181191.

Leahy, M.J., Davidson, M.R., Schwarz, M.P., 2005. A model for heap bioleaching ofchalcocite with heat balance: bacterial temperature dependence. Miner. Eng. 18,12391252.

Leahy, M.J., Schwarz, M.P., Davidson, M.R., 2006. An air sparging CFD model of heapbioleaching of chalocite. Appl. Math. Modell. 30, 14281444.

Leahy, M.J., Davidson, M.R., Schwarz, M.P., 2007. Amodel for heap bioleaching of chalcocitewith heat balance: mesophiles and moderate thermophiles. Hydrometallurgy 85,2441.

Madsen, B.W., Wadsworth, M.E., 1981. A mixed kinetics dump leaching model of orescontaining a variety of copper sulde minerals. USBM Report of Investigations,8547, pp. 144.

Marcantonio, P.J. 1976. Chalcocite dissolution in acidic ferric sulfate solutions, Ph.D.Thesis, University of Utah.

McBride, D., Cross, M., Croft, T.N., Bennett, C.R., Gebhardt, J.E., 2005. Modelling variably

saturated ow in porous media for heap leach analysis. In: Dixon, D.G., Dry, M.J.

(Eds.), Computational Analysis in Hydrometallurgy. MetSoc, Montreal, Quebec,Canada, pp. 4559.

McBride, D., Cross, M., Croft, T.N., Bennett, C.R., Gebhardt, J.E., 2006. Computationalmodelling of variably saturated ow in porous media with complex three-dimensional geometries. Int. J. Numer. Methods Fluids 50, 10851117.

Mellado, M.E., Cisternas, L.A., 2008. An analyticalnumerical method for solving a heapleaching problem of one or more solid reactants from porous pellets. Comput.Chem. Eng. 32, 23952402.

Mellado, M.E., Casanova, M.P., Cisternas, L.A., Glvez, E.D., 2011. On scalable analyticalmodels for heap leaching. Comput. Chem. Eng. 35, 220225.

Miller, G., 2003. Ore Geotechnical Effects on Copper Heap Leach Kinetics. TMS, TheMinerals, Metals and Materials Society, pp. 329342.

Pantelis, G., Ritchie, A.I.M., Stepanyants, Y.A., 2002. A conceptual model for thedescription of oxidation and transport processes in sulphidic waste rock dumps.Appl. Math. Modell. 26, 751770.

Paul, B.C., 1989. Economic and technical feasibility of modied in-situ recovery ofcopper, Ph.D. Thesis, University of Utah.

Paul, B.C., Sohn, H.Y., McCarter, M.K., 1992a. Model for ferric sulfate leaching of copperores containing a variety of sulde minerals: part I: modeling uniform size orefragments. Metall. Trans. B 23B, 537548.

Paul, B.C., Sohn, H.Y., McCarter, M.K., 1992b. Model for ferric sulfate leaching of copperores containing a variety of sulde minerals: part II. Process modeling of in-situoperations. Metall. Trans. B 23B, 549555.

Petersen, J., 2010a. Modelling of bioleach processes: connection between science andengineering. Hydrometallurgy 104, 404409.

Petersen, J., 2010b. Determination of oxygen gasliquid mass transfer rates in heapbioleach reactors. Miner. Eng. 23, 504510.

Petersen, J., Dixon, D., 2003. The dynamics of chalcocite heap bioleaching. In: Young, C.,et al. (Eds.), Hydrometallurgy 2003 5th Int. Conf. In Honour of Prof. Ian Ritchie,Leaching and Solution Purication, vol. 1. TMS, pp. 351364.

Petersen, J., Dixon, D.G., 2007. Modelling zinc heap bioleaching. Hydrometallurgy 85(24), 127143.

Szekely, J., Evans, J.W., Sohn, H.Y., 1976. Gas-Solid Reactions. Academic Press.van Genuchten, M.T., 1980. A closed form of equation for predicting the hydraulic

conductivity of unsaturated soils. Soil Sci. Am. J. 44, 892898.Watling, H.R., 2006. The bioleaching of sulphide minerals with emphasis on copper

sulphides a review. Hydrometallurgy 84, 81108.

161C.R. Bennett et al. / Hydrometallurgy 127128 (2012) 150161

A comprehensive model for copper sulphide heap leaching. Part 1 Basic formulation and validation through column test simulation1. Introduction2. Mathematical modelling2.1. The system to be modelled2.2. Modelling requirements2.2.1. Liquid phase transport2.2.2. Gas phase transport2.2.3. Overview of the process chemistry2.2.3.1. Dissolution kinetics2.2.3.2. Chalcocite reaction rate2.2.3.3. Stage I chalcocite (Cu2S)2.2.3.4. Stage II chalcocite (Cu1.2S)2.2.3.5. Pyrite rate kinetics2.2.3.6. Ferrous oxidation2.2.3.7. Iron precipitation2.2.3.8. Oxygen mass transfer

2.2.4. The numerical solution strategy

3. Parameterization and validation3.1. Parameterization against the small scale column3.2. Validation against the large column data

4. ConclusionsReferences