Development of an integrated heap leach solution flow and mineral leaching model

Hydrometallurgy 169 (2017) 79–88

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Development of an integrated heap leach solution flow and mineral leaching model Stefan Robertson Mintek, Randburg, South Africa

a r t i c l e

i n f o

Article history: Received 15 March 2016 Received in revised form 15 December 2016 Accepted 19 December 2016 Available online 21 December 2016 Keywords: Heap leach model Richard's equation Shrinking core

a b s t r a c t A 1-dimensional solution flow and mineral leaching model was developed to simulate data from leach columns and to demonstrate a dual porosity approach whereby an ore bed is divided into mobile (advective) flow and stagnant (diffusional control) regimes. Hydrodynamic properties were modelled with Richard's and van Genuchten's equations, and hydrodynamic column tests were performed to measure the parameters needed to solve these equations. The solute balance was performed using the standard advection-dispersion equation used in soil dispersion models. However, the term describing the desorption of solute (copper) from the solid into the liquid phase was replaced with a shrinking core reaction model rate term. The model shows that the proportion of the ore bed governed by diffusion increases as the diameter and height of the bed increase. The hydrodynamic properties therefore appear to have a significant effect on the copper extraction profiles and, if so, this could provide an explanation as to why large scale heaps leach slower than columns. The mass transfer coefficients for solute transfer between the mobile and stagnant regimes were of a similar order of magnitude as reported in tracer studies in literature. © 2016 Mintek. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction A heap leach pad under irrigation is governed by various sub-processes ranging from bulk transport of solution and reagents by advection, inter-particle diffusion in stagnant clusters, intra-particle diffusion in reaction pores and chemical reaction at the mineral surface (Dixon and Petersen, 2003). The heterogeneity of hydrodynamic conditions in an ore bed has been reported (e.g. Bouffard and Dixon, 2001 and de Andrade Lima, 2006) and dual porosity models describing flow through soils have been developed in 2D and 3D by authors such as Simunek and van Genuchten (2008). The application of techniques for the physical and hydraulic characterisation of ores for percolation processes have been described by a number of authors including Guzman et al. (2008), Milczarek et al. (2012) and Robertson et al. (2013). Notwithstanding this, fully published dual porosity models are limited to hydrodynamics or soil dispersion, whereas most leach models found in literature treat the ore bed as a single ore phase governed by either advection or diffusion (e.g. Bartlett, 1998; Miller, 2003; Bennet et al., 2012 and Cariaga et al., 2015). A number of models describing both advective and diffusional control are summarised in Table 1. Bartlett (1998) and Bennet et al. (2012) model the reaction rate as a shrinking core within a single liquid phase governed by advection. Bartlett (1998) models the flow of E-mail address: [email protected].

solution as a vertical front moving through the ore bed at constant velocity and constant liquid hold-up, whereas Bennet et al. (2012) and Cariaga et al. (2015) use Richard's equation to describe the change in liquid hold-up with flowrate. Simunek and van Genuchten (2008) and Robertson et al. (2013) use soil dispersion models which describe the hydraulic properties with Richard's equation and the solute transport with the advection-dispersion equation. The ore bed is divided into advective and stagnant flow regimes (dual porosity). The transfer of metal species from solids to solution is proportional to the concentration of metal in the solids (linear desorption), but this approach is not an accurate representation of heap leaching since the reaction rate is also a function of the concentration of reagent(s) at the mineral surface. For the shrinking core reaction model where the chemical rate is limiting, the concentration of acid at the mineral surface is equal to the concentration in the bulk solution phase. The bulk solution acid concentration is, in turn, a function of the supply of acid through the ore bed by either advective transport (mobile phase) or diffusion (stagnant phase). Bouffard and Dixon (2001) and de Andrade Lima (2006) model the ore bed as comprising both mobile and immobile phases. They quantify the mass transfer coefficient of solute between the stagnant and mobile phases, as well as the ore bed porosity and the stagnant and flowing liquid hold-up. However, the work is limited to tracer tests and do not incorporate leaching studies. Miller (2003) proposes that a heap may be represented by stagnant “macro-particles” governed entirely by diffusion and characterised by a

http://dx.doi.org/10.1016/j.hydromet.2016.12.010 0304-386X/© 2016 Mintek. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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S. Robertson / Hydrometallurgy 169 (2017) 79–88

Nomenclature c concentration of solute (kg/m3) concentration of acid in bulk (g/L) Cb cmo and cim concentrations of solute in the mobile and immobile regions (kg/m3) d particle diameter (mm) dispersion coefficient of mobile phase (m2/s) D, Dmo incremental mass of acid reacted per unit volume (kg/ dMacid m3) incremental mass of copper species reacted per unit dMCu volume (kg/m3) dF incremental overall conversion of species in ore bed incremental conversion of species in size fraction i dFi F mass fraction of species dissolved (conversion) fraction of sorption sites in contact with the mobile fmo phase (kg/kg) h hydraulic pressure head (cm) K hydraulic conductivity (cm/s) kChrysocolla mineral rate constant (mm/(g/L)/h) copper rate constant (mm/(g/L)/h) kCu kGangue gangue rate constant (mm/(g/L)/h) kMalachite mineral rate constant (mm/(g/L)/h) kPseudo-malachite mineral rate constant (mm/(g/L)/h rate constant (mm/(g/L)/h) kRate saturated hydraulic conductivity (cm/s) Ksat empirical constant (1/cm) k1 m, n empirical constants mass of copper species in contact with the immobile MCuim phase per unit volume (kg/m3) initial mass of copper species in the ore bed per unit volMCu0 ume (kg/m3) mass of copper species in contact with the mobile phase MCumo per unit volume (kg/m3) N number of size fractions Pe Peclet number q linear velocity in downwards direction (cm/s) volumetric fluid flux density in mobile phase (m/s) qmo Re Reynolds number S relative degree of saturation s sorbed concentration of species (kg/kg) smo and sim sorbed concentrations in contact with the mobile and immobile regions (kg/kg) u linear velocity (m/s) mass fraction of copper in each size fraction (g/g) Wi z length of ore bed in downwards direction (cm) mass transfer rate term for solutes between the mobile гs and immobile regions (kg/m3/s) θ moisture content (m3/m3) moisture content in immobile phase (m3/m3) θim moisture content in mobile phase (m3/m3) θmo residual moisture content (m3/m3) θr saturation moisture content (m3/m3) θs μ dynamic viscosity (kg/m/s) ρ bulk density (kg/m3) ѱ suction pressure (cm) ω, ωACID, ωCu, ωmim mass transfer coefficient for solutes between the mobile and immobile phases (1/s)

total effective diffusional path length. Dixon and Petersen (2003) divide the heap into discrete advection flow channels separated by stagnant macro-particles within which diffusion is limiting. Mineral dissolution

takes place within the stagnant pores and the intrinsic reaction rate is governed by the generalised equation: dF ¼ kðTÞf ðCÞð1−FÞϕ dt

ð1Þ

The particle size distribution is represented by a single topological exponent ϕ, where F is the mineral conversion, k(T) is a temperaturedependent rate constant and f(C) is a function of the solution composition. Whereas scale-up from columns to larger diameter columns or heaps may be represented by an increase in the diffusional path length in the above models, a more appropriate representation will be a dual porosity model with an increase in the proportion of the ore bed governed by diffusion. In order to demonstrate this effect, a dual porosity model similar to the soil dispersion model was developed. However, the intrinsic reaction rate was described by a shrinking core reaction model solved over a number of size fractions, rather than the linear desorption expression used previously. Experimental testing was performed to measure the hydraulic conductivity and pore pressure relationships needed to solve Richard's equation and to validate the flow properties in the ore bed. The model was applied to the leaching of a copper oxide ore, where the intrinsic reaction rate is easier to model and the hydrodynamic effects on the overall copper leach kinetics can be better illustrated. Whereas copper oxide orebodies are becoming depleted in traditional heap leach regions such as Chile, new African projects often entail the processing of copper oxide ores with high clay content. Therefore, an understanding of the physical and hydraulic properties which govern the flow of solution through the ore bed is important. The model will be extended to include the leaching of mineral sulphides at a later stage. 2. Theory The continuity equation for conservation of solution in a control volume can be expressed by Richard's equation for dual porosity, where q (cm/s) is the linear velocity, θmo (m3/m3) is the mobile moisture content, θim (m3/m3) is the immobile moisture content, K (cm/s) is the hydraulic conductivity, hmo (cm) is the hydraulic pressure head and z (cm) is the height of the ore bed (Simunek and van Genuchten, 2008). 
 
 ∂θmo ∂q ∂ ∂hmo ∂θ ∂ ∂hmo ∂θ − im ¼ − im ð2Þ ¼− −K K ¼− ∂z ∂z ∂z ∂t ∂z ∂t ∂z ∂t The immobile moisture in the ore bed is essentially filled up during agglomeration, and any further moisture increase during irrigation can therefore be assumed to be mobile moisture, hence ∂θ∂tim ¼ 0 . Then, substituting h = z + ѱ, where ψ (cm) is the pore suction pressure, and changing the convention of flow so that z increases in the downwards direction, Eq. (2) can be reduced to: ∂θ ¼ ∂t

! !!
2 2 2 ∂θ ∂K ∂ψ ∂ ψ ∂ψ ∂ θ ∂K ∂θ − − −K 2 −K 2 ∂z ∂θ ∂θ ∂θ ∂z ∂θ ∂z ∂θ

ð3Þ

The hydraulic conductivity and the pore pressure relationships can be described by van Genuchten's Eqs. (4), (5) and (6), where S is the relative degree of saturation, θr (m3/m3) is the residual moisture content, θs (m3/m3) is the moisture content at saturation, Ksat (cm/s) is the saturated hydraulic conductivity, and m, n and k1 are constants. Taking the derivative with respect to S yields Eqs. (7), (8) and (9), which may be substituted into Eq. (3). The change in moisture content with respect ) may be solved with the Crank-Nicolson to ore bed height ( ∂θ ∂z

S. Robertson / Hydrometallurgy 169 (2017) 79–88

81

Table 1 Model comparison. Author

Bouffard and Dixon, 2001

Dixon and Petersen, 2003

Fluid flow model

Plug flow channels with stagnant regions Plug flow with solute transfer at stagnant/flowing interface Tracer test

Plug flow channels with stagnant Soil dispersion model macro-particles with Richard's equation Advection in flow channels; Advection-dispersion diffusion in pore channels equation with dual porosity

Solute transport model Mineral reaction rate model Effect of temperature Measurement of van Genuchten parameters Number of phases considered Rate parameters

Simunek and van Genuchten, 2008

Bennet et al., 2012; Cariaga et al., 2015

This model

Soil dispersion model with Richard's and van Genuchten's equations Advection-dispersion equation with dual porosity

Richard's and van Genuchten's equations Advection

Soil dispersion model with Richard's and van Genuchten's equations Advection-dispersion equation with dual porosity Shrinking core with summation over particle size distribution No

Instantaneous and Instantaneous and first first order desorption order desorption

No

Generalised rate equation with particle size distribution represented by topological exponent Yes

No

No

Shrinking core with summation over particle size distribution No

No

No

No

No

No

Yes

Two phases; diffusion only in stagnant phase ϴmo = 0.02 m3/m3 ϴim = 0.11 m3/m3 ω = 5.8 × 10−6 s−1

Two phases; mineral reaction only in stagnant phase

Two phases

Two phases

Single phase

KGangue = 0.1–1 (g/L)−1.h−1

ω = 1 × 10−5 s−1

Two phases; reaction in advective and stagnant phases ϴmo = 0.065 m3/m3 ϴim = 0.12 m3/m3 ω = 3.3 × 10−6 s−1 kRate = 0.001–0.06 (mm)(g/L)−1.h−1

discretisation method (Cariaga et al., 2015).

S¼

Robertson et al., 2013

θ−θr θs −θr

h   i 1 1 m 2 K ¼ Ksat S2 1− 1−Sm

ð4Þ

ð5Þ

h   i2     1 1 m 1 1 m ∂K 1 þ Ksat S2 2 1− 1−Sm ¼ Ksat S−2 1− 1−Sm ∂S 2
   1 m−1 1 1 − Sm−1  ð−mÞ 1−Sm m  
 1−1  1 ∂ψ 1 1 1 −m1 −1 n − ¼ S−m −1 S k1 m ∂S n 2

∂ ψ 2

 n1  1 1 ψ ¼ S−m −1 k1

∂S ð6Þ

ϴs = 0.33 m3/m3 ϴr = 0 m3/m3 Ksat = 7 cm/s n = 2.3 k1 = 0.23 cm−1

¼


 n1−2 
1  1
1  1 1 1 1 − −1 S−m −1 S−m−1 − S−m−1 nk1 n m  m

1 n−1 1 1  −m1 1 1 þ S −1 − − −1 S−m−2 nk1 m m

Fig. 1. Mathematical model for solution of solute concentrations.

ð7Þ ð8Þ

ð9Þ

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S. Robertson / Hydrometallurgy 169 (2017) 79–88

Fig. 2. Hydraulic conductivity versus degree of saturation.

The solute transport in soil dispersion models is described by the advection-dispersion equation, adapted for dual porosity in Eqs. (10) to (12), where fmo (kg/kg) is the fraction of reaction sites in contact with the mobile phase, cmo (kg/m3) and cim (kg/m3) are the concentrations of solute in the mobile and immobile phases, smo (kg/kg) and sim (kg/ kg) are the sorbed concentrations in contact with mobile and immobile phases, ᴦs (kg/m3/s) is the mass transfer rate term for solutes between the mobile and immobile regions, D (m2/s) is the dispersion coefficient and ωmim (1/s) is the mass transfer coefficient for solutes between the mobile and immobile phases (Simunek and van Genuchten, 2008).
 ∂θmo cmo ∂smo ∂ ∂cmo ∂q cmo − mo θmo Dmo þ f mo ρ ¼ −Γs ∂z ∂t ∂t ∂z ∂z

ð10Þ

∂θim cim ∂s þ ð1−f mo Þρ im ¼ Γs ∂t ∂t

ð11Þ

Γs ¼ ωmim ðcmo −cim Þ

ð12Þ

The advection-dispersion equation is now modified by replacing the sorption term with a term representing the change in the mass of copper species, where MCumo (kg/m3) and MCuim (kg/m3) represent the mass of copper per unit volume in contact with the mobile and immobile phases, respectively. ∂θmo cmo ∂MCumo þ ∂t
∂t  ∂ ∂cmo ∂q cmo − mo θmo Dmo −ωmim ðcmo −cim Þ ¼ ∂z ∂z ∂z ∂θim cim ∂MCuim þ ¼ ωmim ðCmo −C im Þ ∂t ∂t

ð14Þ

ð15Þ

The axial dispersion coefficient (Dmo) was estimated from the relationship between the Peclet number (Eq. (16)) and the Reynolds number (Eq. (17)) derived for packed beds by Wilhelm (1962), where d (m) is the particle diameter, u (m/s) is the linear velocity, μ (kg/m/s) is the dynamic viscosity and ρ (kg/m3) is the density of the solution.

Furthermore, the overall moisture content is divided into mobile and immobile regions as indicated by the equation:

Pe ¼

du D

ð16Þ

θ ¼ θmo þ θim

Re ¼

d uρ μ

ð17Þ

ð13Þ

Fig. 3. Degree of saturation versus pore pressure.

S. Robertson / Hydrometallurgy 169 (2017) 79–88

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Table 2 Solid head and residue size distributions and assays-by-size. Size fraction mm

−25 + 12.5 −12.5 + 4.75 −4.75 + 2.36 −2.36 + 1.18 −1.18 + 0.425 −0.425 Average

Head

Residue (6 m column, C5)

Mass fraction kg/kg

Acid-soluble Cu %

Cyanide-soluble Cu ppm

Residual Cu %

Mass fraction kg/kg

Acid-soluble Cu %

Cyanide-soluble Cu ppm

Residual Cu %

0.31 0.19 0.10 0.09 0.10 0.20 1.00

2.54 2.77 2.76 2.76 2.73 2.90 2.71

88 97 112 92 76 178 109

0.13 0.16 0.15 0.17 0.18 0.19 0.16

0.14 0.24 0.12 0.10 0.14 0.25 1.00

0.082 0.025 0.016 0.019 0.018 0.012 0.027

60 60 64 79 77 72 68

0.10 0.11 0.10 0.12 0.12 0.16 0.12

The conversion of copper species by the shrinking core reaction model is represented by Eq. (18) (Gibor and Jia, 2004), where F is the mass fraction of mineral dissolved (conversion), Cb (kg/m3) is the bulk acid concentration, d (mm) is the particle diameter and kCu (mm/(g/ L)/h) is the rate constant. The shrinking core reaction model was used to solve the degree of extraction for each size interval separately, and the overall mass reacted was calculated from a weighted average of the individual fractions, as indicated in Eq. (19). Wi is the mass fraction of copper in size fraction i, di (mm) is the particle diameter in size fraction i and Fi is the extent of conversion of copper in size fraction i. MCu0 (kg/m3) is the initial mass of copper per unit volume in the ore bed. dF ¼

2 kCu ðTÞ Cb ð1− FÞ3 dt d

and a 1-dimensional spreadsheet model was applied to each interval using discrete time intervals. The model was then adapted to scale by summation of incremental ore bed heights.

3. Experimental 3.1. Column leach tests Column leach tests were performed on a copper oxide ore sample, crushed to 80% passing 12 mm, in water-jacketed 1 m (160 mm ID) and 6 m (200 mm ID) columns at 20 °C. The ore was agglomerated with synthetic raffinate and concentrated sulphuric acid (5 kg/t). Columns were irrigated at 10 L/m2/h with synthetic raffinate containing 10 g/L H2SO4 (pH 1.1), 1.6 g/L Al, 1.5 g/L Mg, 0.5 g/L Ca, 0.0145 g/L Mn, 1.5 g/L Fe3+, 0.23 g/L Co, 0.6 g/L Zn, 0.6 g/L K and 0.15 g/L Na. The mass, volume, SG, pH and redox potential of the irrigation and drainage solutions were recorded daily. Copper and iron concentrations in the drainage solutions were measured daily by atomic absorption spectroscopy (AAS), and the ferrous (Fe2+) and H2SO4 concentrations by titration. Columns were operated in closed circuit with batch solvent

ð18Þ N

dMCu ¼ MCu0 dF ¼ MCu0 ∑ Wi i¼1

2 kCu ðTÞ Cb ð1− FÞ3 dt di

ð19Þ

The concentrations of copper and acid in the ore bed are solved incrementally from the above equations, in accordance with the mass balance illustrated in Fig. 1. The column was divided into vertical sections,

Table 3 Model parameters. Parameter

Unit

C3

C5

C5

C5

C6

Fraction of bed in contact with mobile phase fmo Column height Column ID Irrigation rate [H2SO4] in irrigation solution Dry loaded bulk density Curing acid dosage k Malachite (20 °C) k Pseudo-malachite (20 °C) k Chrysocolla (20 °C) k Gangue (20 °C) Solute transfer coefficient ωCu Solute transfer coefficient ωACID Curing acid reacted Cu reacted during curing Saturated hydraulic conductivity Ksat Saturation moisture Өs Residual moisture Өr Moisture content Ө Moisture content of mobile phase Өmo Moisture content of immobile phase Өim Degree of saturation S Dispersion coefficient Dmo van Genuchten constant m van Genuchten constant k1 Moisture content ϴ, boundary condition z = 0 Moisture content ϴ, boundary condition t = 0 Temperature Mass percentage of leachable gangue mineral Mass of acid reacted per mass of gangue

kg/kg m mm L/m2/h g/L t/m3 kg/t mm/(g/L)/h mm/(g/L)/h mm/(g/L)/h mm/(g/L)/h 1/s 1/s % % cm/s m3/m3 m3/m3 m3/m3 m3/m3 m3/m3

0.8 1 160 10 12 1.4 5 0.06 0.02 0.01 0.001 3.3E-06 3.3E-06 100 10 0.16 0.45 0.12 0.28 0.16 0.12 0.45 2.2E-08 0.95 0.13 0.28 0.12 20 6 0.9

0 6 200 10 10 1.5 5 0.11 0.02 0.01 0.001 4.2E-06 4.2E-06 100 10 0.16 0.45 0.12 0.19 0.07 0.12 0.23 2.2E-08 0.95 0.13 0.19 0.12 20 6 0.9

0.5 6 200 10 10 1.5 5 0.06 0.02 0.01 0.001 3.3E-06 3.3E-06 100 10 0.16 0.45 0.12 0.19 0.07 0.12 0.23 2.2E-08 0.95 0.13 0.19 0.12 20 6 0.9

1 6 200 10 10 1.5 5 0.002 0.001 0.001 0.001 3.3E-06 3.3E-06 100 10 0.16 0.45 0.12 0.19 0.07 0.12 0.23 2.2E-08 0.95 0.13 0.19 0.12 20 6 0.9

0.5 6 200 15 12 1.5 5 0.06 0.02 0.01 0.001 3.3E-06 3.3E-06 100 10 0.16 0.45 0.12 0.22 0.10 0.12 0.30 3.3E-08 0.95 0.13 0.22 0.12 20 6 0.9

m2/s

m3/m3 m3/m3 °C % kg/kg

84

S. Robertson / Hydrometallurgy 169 (2017) 79–88

Fig. 4. Moisture content profiles (6 m column, 10 L/m2/h).

extraction to reduce the copper in the raffinate to below 0.2 g/L. Solid head and residues were wet screened on a vibratory screen at 9.5 mm, 4.75 mm, 2.36 mm, 1 mm and 0.5 mm. Assay-by-size was performed by Inductively Coupled Plasma-Optical Emission Spectroscopy (ICPOES) and by a copper diagnostic leach (acid-soluble, cyanide-soluble and residual copper). Mineralogy on the ore sample was performed by Quantitative Evaluation of Minerals by Scanning Electron Microscope (QEMSCAN) to determine the relative proportions of minerals present. 3.2. Hydrodynamic column tests Hydrodynamic column tests were performed in 150 mm ID, 76 cm tall columns. A column was loaded with agglomerated ore at a uniform target bulk density and was irrigated at incremental solution application rates until localised ponding occurred. The irrigation rate was then reduced and the column slowly filled to saturate all the void spaces and to measure the saturated hydraulic conductivity. The column was then allowed to drain and a drain-down profile was generated. The pore pressures were measured at each flowrate using a tensiometer connected to a porous plug. A solution balance was performed from the load cell masses of the column and feed reservoir to calculate the degree of saturation and moisture content relationships at each incremental irrigation rate. The column was discharged and the solids were dried to determine the final moisture content.

4. Results and discussion The hydraulic conductivity and pore pressure relationships measured for a copper ore at uniform bulk density in the hydrodynamic column tests are presented in Figs. 2 and 3, from which the van Genuchten constants (m, k1) were derived. Since hydrodynamic columns are loaded at uniform bulk density, the suction pressure is assumed to be uniform throughout the depth of the ore bed, such that q = −K. Hence a measure of q through the ore bed is also a measure of K. Solid head and residue particle size distributions and copper grades of a column leach test is summarised in Table 2. Leach columns were loaded at uniform bulk density and moisture content with moderate change in bulk density and particle size distribution during the leach cycle due to wall effects. Hence it is reasonable to assume that the van Genuchten parameters do not change during the leach cycle. The change in hydraulic parameters as a function of leach cycle on account of ore decrepitation and fines generation will be addressed in subsequent versions of the model. The ore head sample contained 4.9% malachite, 0.8% chrysocolla, 0.4% pseudo-malachite and 0.1% chalcocite. Copper deportments were 81.9% from malachite, 7.5% from chrysocolla, 7% from pseudo-malachite and 2.3% from chalcocite. Major reactive isosilicate gangue minerals were amphibole (1.1%) and pyroxene (0.1%), whereas reactive phyllosilicates were mica (biotite or muscovite; 18.4%), chlorite (5.4%) and

Fig. 5. Solution flowrate profiles (6 m column, 10 L/m2/h).

S. Robertson / Hydrometallurgy 169 (2017) 79–88

85

Fig. 6. Copper extraction profiles.

feldspar (4.3%). Other gangue minerals present were quartz (50.8%), goethite (6.2%), kaolinite (4%) and talc (0.5%). Non-carbonate acid consumers (Baum, 1999) typically consume in the order of 0.48 to 1.4 kg H2SO4/kg mineral, e.g. Ca-feldspar (1.4 kg/kg), biotite (1.1 kg/kg), montmorillonite (0.75 kg/kg) and chlorite (0.49 kg/kg). Gangue minerals were grouped together as a single species since it was not possible to calculate extents of reaction of individual gangue minerals from the experimental data (this would require “sacrificial” columns with mineralogy at various times in the leach cycle). The percentage total reactive gangue mineral in the head sample and the ratio of acid to gangue mineral reacted was made a variable and the reaction was assumed to take place according to the shrinking core reaction model. A summary of model parameters is provided in Table 3. Leach columns were operated at constant temperature with isothermal cooling jackets, so it is reasonable to ignore temperature and heat transfer effects, with all rate constants derived at 20 °C. Chemical rate constants for first order gangue and acid-soluble copper minerals (malachite, chrysocolla and pseudo-malachite) were assumed to be constant during scale-up.

Since the contribution from chalcocite to the total copper content is small, and refractory copper sulphide minerals are expected to remain insoluble, the current model did not consider such minerals. However, this will be a focus of subsequent development. The solution chemistry was limited to copper and acid, as minerals were acid-soluble and therefore the effect of iron concentrations are limited. Acid generation and consumption from iron precipitation and oxidation was ignored as columns were not aerated. Iron precipitation occurred primarily during the first 40 days of irrigation in the 6 m columns, generating an acid equivalent of approximately 5% of the total acid consumption, if precipitated as ferric hydroxide. Change in particle size distribution from iron precipitation was ignored for simplicity. The moisture content (Fig. 4) increased from approximately 0.12 m3/m3 after agglomeration to approximately 0.19 m3/m3 at steady state after 2 days at the “start-up” irrigation rate of 10 L/m2/h. The initial moisture content profile compares well with the experimental result. The model predicts a constant moisture content of approximately 0.19 m3/m3 and does not take into account further increase in moisture content with time as a result of decrepitation and fines generation.

Fig. 7. Acid consumption profiles.

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S. Robertson / Hydrometallurgy 169 (2017) 79–88

Fig. 8. Copper conversion profiles (6 m column, 10 L/m2/h).

The irrigation and drainage flowrates predicted by the model are plotted with the leach column data (Fig. 5). The copper extraction profiles, based on mass of copper drained from the base of the column (Fig. 6), and acid consumption profiles (Fig. 7) closely resemble the data, as indicted by the correlation coefficients in brackets. The copper extraction data for the 6 m column (C5) was modelled with fmo = 0, 0.5 and 1, respectively (Fig. 6). It was not possible to obtain fits for fmo = 0 (ore bed governed solely by diffusion) or fmo = 1 (ore bed governed solely by advection). A combination of advection and diffusion (fmo = 0.5) was required to fit the experimental data. Rate constants were iterated to fit the experimental copper extraction and acid consumption profiles. The copper extraction profiles are based on total mass of copper drained from the columns, which is different from the overall copper conversion (mass fraction of copper dissolved; F), since it takes time for the leached copper to drain. Separate conversions of copper are calculated for mobile and immobile zones, respectively (Fig. 8). The immobile moisture content, θim, is determined from the residual moisture in the hydrodynamic column after completion of the drain-down test, and is assumed to remain constant during the column leach tests. Since it is not possible to measure pore pressures and hydraulic conductivities in both immobile and mobile phases, the measurements in the hydrodynamic column is used to calculate the overall moisture content, and the mobile moisture is then calculated by subtracting the residual moisture from the overall moisture content. The acid consumption profile takes into account an initial curing acid addition. It is assumed that 100% of acid reacts during curing. From the experimental data, all of the 5 kg/t curing acid reacted (i.e. no acid drained from the columns initially). Hence the initial copper conversion

equivalent to the 5 kg/t acid was set to 11% for the first time interval. The acid concentration profile is presented in Fig. 9. Fig. 10 indicates the contribution of the three copper minerals to the total copper conversion. The shrinking core reaction model describes a reaction front moving through a representative particle with homogeneous mineral distribution and minerals react at different rates in an individual particle size, each mineral (malachite, chrysocolla and pseudomalachite) modelled with a single characteristic radius (Bennet et al., 2012). As indicated in Table 3, the solute mass transfer coefficients for acid and copper (ωACID and ωCu), the intrinsic rate constants (kMalachite, kPseudo-malachite, kChrysocolla and kGangue), and the proportion of the ore bed in contact with the mobile phase (fmo) was constant when scaling between the 6 m column irrigated at 10 L/m2/h with 10 g/L acid (C5) and the 6 m column irrigated at 15 L/m2/h with 12 g/L acid (C6). Boundary conditions for θ are provided in Table 3. At time = 0, θ = θim = θr and at z = 0, θ is the same as the moisture content at steady state in the column (0.19–0.28 m3/m3). Values for the axial dispersion coefficient (Dmo) are provided in Table 3 and are 2.2 × 10−8 m2/s at 10 L/m2/h and 3.3 × 10−8 m2/s at 15 L/m2/h, respectively. The dispersion term in Eq. (10) is small compared with the advection term (Pe = 0.5) and therefore the flow in the mobile phase approaches plug flow. As indicated in Table 1, the solute mass transfer coefficients (ω) were of the same order as reported by Bouffard and Dixon (2001) and Simunek and van Genuchten (2008). The concentration of solute and acid in the immobile phase was also assumed to be uniform in a unit block of volume and time to avoid additional complexity. The solute mass transfer coefficients were also kept

Fig. 9. Acid concentration profiles (6 m column, 10 L/m2/h).

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Fig. 10. Contribution to total copper conversion (6 m column, 10 L/m2/h).

constant when scaling from the 1 m (160 mm ID) column (C3) to the 6 m (200 mm ID) column (C5). However, it was necessary to increase the fraction of the solids in contact with the mobile solution (fmo) from 0.5 in the 6 m columns to 0.8 in the 1 m column to fit the experimental data. This suggests that a shrinking core reaction model with solution transport almost entirely by advection accurately fits the experimental result of the 1 m column. However, in the 6 m column, a larger proportion of the ore bed is governed by diffusion. This appears to be consistent with a higher moisture content and degree of saturation observed in the 1 m column, as indicated in Table 3. From Figs. 6 and 11 it is clear that the 6 m column cannot be described by advection only, but that a portion of flow governed by diffusion is necessary to accurately fit the leaching profile. The results therefore suggest that the scale-up from column to large-scale heaps cannot be done without an understanding of the proportion of the ore bed that is governed by advection and diffusion. This is not currently addressed by flow models such as Bennet et al. (2012), where the flow in the ore bed is represented by advection only, and authors such as Dixon and Petersen (2003), where the flow in the bed is represented entirely by diffusion. As the column diameter (or dripper spacing) and the ore bed height increase, the proportion of immobile moisture increases, resulting in longer leach cycles. Various rules of thumb for scaling up from laboratory scale to heaps have been proposed, with large scale heaps always leaching slower than columns (Miller and Newton, 1999; Jansen and Taylor, 2002 and John, 2011).

5. Conclusions A 1-dimensional model closely predicts the copper extraction and acid consumption profiles in a 160 mm ID 1 m column and a 200 mm ID 6 m column when applied to an acid-soluble copper oxide ore. The current model combines a dual porosity modelling of the hydrodynamics with a shrinking core reaction model. This is an improvement on classical soil dispersion models where the reaction rate is modelled by linear desorption, which does not apply to copper leaching where the chemical rate is also dependent on the concentration of acid at the mineral surface. It was clearly demonstrated that a shrinking core reaction approach with solution flow solely by advection, accurately fits the rate data in a short, narrow (160 mm ID) column. However, as the column diameter and ore bed height increase, the proportion of flow governed by diffusion cannot be ignored, and becomes significant. The author therefore contends that an understanding of the ratio of the ore bed governed by diffusion versus advection is essential in scaling up heap leach data. Hence it is necessary to quantify the change in the dual porosity ratio with increase in column diameter. Such a model will be the focus of further test work, and may be performed with tracer tests and hydrodynamic column tests at various column diameters. Acknowledgements The author wishes to thank Societé d'Exploitation de Kipoi for permission to publish the experimental data. Amado Guzman is acknowledged

Fig. 11. Effect of fmo on copper extraction (6 m column, 10 L/m2/h).

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for his geomechanical experimental procedures, equipment and assistance with model development. The useful suggestions by Petrus Basson and Petrus van Staden are also acknowledged. This paper is published with the permission of Mintek. References Bartlett, R.W., 1998. Solution Mining: Leaching and Fluid Recovery of Materials. Gordon and Breach Science Publishers. Baum, W., 1999. The use of a mineralogical data base for production forecasting and troubleshooting in copper leach operations. In: Young, S.K., Dreisinger, D.B., Hackl, R.P., Dixon, D.G. (Eds.), Proceedings of Copper 99 International Conference, Volume IV Hydrometallurgy of Copper. The Minerals, Metals and Materials Society. Bennet, C.R., McBride, D., Cross, M., Gebhardt, J.E., 2012. A comprehensive model for copper sulphide heap leaching. Hydrometallurgy 127–128, 150–161. Bouffard, S., Dixon, D.G., 2001. Investigative study into the hydrodynamics of heap leaching processes. Metall. Mater. Trans. 32B, 763–775. Cariaga, E., Martinez, R., Sepulveda, M., 2015. Estimation of hydraulic parameters under unsaturated flow conditions in heap leaching. Math. Comput. Simul. 109C, 20–31. De Andrade Lima, L.R.P., 2006. Liquid axial dispersion and holdup in column leaching. Miner. Eng. 19, 37–47. Dixon, D.G., Petersen, J., 2003. Comprehensive Modelling Study of Chalcocite Column and Heap Bioleaching. Copper 2003, Volume VI-Hydrometallurgy of Copper (Book 2), Santiago. pp. 493–516.

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