Kinetic Modeling of High-Pressure Pyrite Oxidation with Parameter

Article pubs.acs.org/IECR

Kinetic Modeling of High-Pressure Pyrite Oxidation with Parameter Estimation and Reliability Analysis Using the Markov Chain Monte Carlo Method Vladimir V. Zhukov,* Arto Laari, and Tuomas Koiranen LUT School of Engineering Science, Lappeenranta University of Technology, Skinnarilankatu 34, P.O. Box 20, 53851 Lappeenranta, Finland ABSTRACT: The aim of this work was to develop a comprehensive kinetic model for aqueous-phase oxidation of pyrite that can explain all the relevant phenomena that govern pyrite oxidation, including the effects of temperature, oxygen partial pressure, particle size, and multiple surface and bulk phase reactions at different slurry concentrations. The developed model is based on experimental data presented in the literature. For the current case the reaction process formulation of the shrinking particle model in the usual integrated form is challenging, and the model is therefore given as differential equations that are solved numerically. A surface passivation model was implemented to explain experimentally observed surface passivation. The kinetic model includes multiple experimental parameters, which were estimated by comparison with experimental data. The reliability of the estimated parameters and model predictions was studied using novel Markov chain Monte Carlo (MCMC) methods. The MCMC analysis indicated that all model parameters were well identified without any cross correlation. The results also show that pyrite surface reactions with both molecular oxygen and ferric iron are important at the studied experimental conditions, and their relative importance depends on the pyrite slurry concentration.

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and iron can be dissolved at 210 °C with high conversion (>95%) in 240 min, while Pb remains in the residue in the form of lead. The chemistry of oxidative leaching is complicated, and there has been much debate about the various leaching mechanisms involved. Recently, Chandra and Gerson4 published a review article about the mechanisms of pyrite oxidation and leaching. After thorough study of the available literature, they concluded that due to the heterogeneous nature of pyrite, it is very difficult to prove that any specific mechanism is valid for all pyrite oxidation. However, there is a general agreement that the dissolution reaction proceeds via an electrochemical reaction mechanism. It is generally held that at high temperatures in acidic conditions pyrite can react directly with dissolved oxygen or with ferric iron according to the reactions1,5,6

INTRODUCTION Gold is often present in sulfidic ores, such as pyrite, arsenopyrite, or chalcopyrite, in the form of very small particles embedded in the sulfidic structure. This embedded structure prevents gold separation by the usual gravimetric means. In order to enhance the release of gold particles and to make sulfidic ores amenable for gold extraction, oxidation of the sulfidic structure is often carried out as an essential pretreatment step in gold processing. Industrially, the oxidation process takes place in acidic conditions at high temperature (above 170 °C) and high oxygen partial pressure (above 300 kPa) in large autoclaves with multicompartmental sections. The main oxidizing agents in the process are dissolved oxygen and ferric iron, which oxidize the concentrate or ore particles. The oxidation transfers pyrite, arsenopyrite, iron, sulfur, and arsenic into the liquid phase. After oxidation, the pyrite and arsenopyrite are almost completely oxidized and the solution is ready for the gold leaching process. Aqueous-phase oxidation of pyrite, arsenopyrite, or other pyrite-containing minerals has been studied extensively in recent decades. However, only a few studies have been carried out at the higher temperatures typically used in industrial pressure oxidation processes. Long and Dixon1 conducted experimental work on high-pressure pyrite oxidation in a laboratory-scale autoclave under acidic conditions. The dependence of pyrite conversion on temperature, oxygen partial pressure, and particle size was studied. Rusanen et al.2 studied pressure oxidation of pyrite−arsenopyrite at different temperatures, oxygen partial pressures, and slurry densities. It was found in their study that all three studied factors influence pyrite conversion. The oxygen partial pressure was found to be the most important factor. Matkovic et al.3 studied pressure oxidative leaching of a complex Pb−Zn−Cu−Fe sulfide concentrate under elevated temperature and pressure in acidic conditions. They found that copper, zinc, © XXXX American Chemical Society

FeS2 + 7/2O2 + H 2O → Fe2 + + 2SO24 − + 2H+

(R1)

FeS2 + 14Fe3 + + 8H 2O → 15Fe2 + + 2SO24 − + 16H+ (R2)

The ferrous iron produced in reaction R1 is oxidized back to ferric iron according to the bulk liquid-phase reaction Fe2 + + 1/4O2 + H+ → Fe3 + + 1/2H 2O

(R3)

Comparing the relative rates of reactions R1 and R2, Moses et al.7 showed that at ambient-temperature reaction R2 with Fe3+ as the oxidant is much faster than the direct oxidation with molecular oxygen. Therefore, reaction R1 is important only in cases where Received: June 30, 2015 Revised: September 28, 2015 Accepted: September 29, 2015

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DOI: 10.1021/acs.iecr.5b02374 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

oxidation rate is first order with respect to the ferrous ion and oxygen concentrations at pH values lower than 3.5.

the dissolved iron concentration remains low. The main importance of oxygen is the back oxidation of Fe2+ to Fe3+ by reaction R3, which together with reaction R2 forms a redox reaction cycle. In many cases, when the reaction R2 is relatively fast, reaction R3 may be the limiting factor that controls the overall leaching rate. Reactions R1 and R2 take place at the surface of the pyrite. However, it has been earlier shown8 that the oxidation reactions are mostly confined to regions of high surface energy. Therefore, the reactive or effective surface area may be more important than the total surface area. Such reactive regions are related to grain edges and corners, defects, solid and fluid inclusion pits, cleavages, and fractures. Moreover, different pyrite minerals typically have small deviations from the theoretical molecular Fe:S ratio of 1:2. These deviations are caused by lattice substitutions of the Fe2+ and S2− ions with atoms of similar radius and charge or net polarity. According to Abraitis et al.,9 these trace elements can alter the semiconducting bulk properties of the pyrite and significantly affect the reactivity of the pyrite surface. Long and Dixon1 found in their pressure oxidation experiments that surface passivation takes place at low slurry concentrations and is most likely by elemental sulfur. They proposed that instead of reaction R1, the oxidation reaction produces thiosulfate ions that rapidly disproportionate to form elemental sulfur

−

rFe2+ =

(R5)

It can be expected that sulfur, most likely in the form of polysulfides, causes surface passivation. The resulting sulfite is further oxidized by ferric iron to sulfate (R6)

The kinetics of pyrite oxidation has been investigated in several studies.1,4−6 According to Holmes and Crundwell6 pyrite dissolution is governed by an electrochemical process where the rate-determining step is the charge transfer at the particle surface−solution interface. On the basis of this electrochemical mechanism, the following rate equation was derived for pyrite dissolution by both ferric iron and dissolved oxygen. + ⎞ kFeS2c H−+1/2 ⎛ kFe3+c Fe3+ + k O2cO2c H0.14 ⎜ ⎟ = 14F ⎜⎝ kFeS2c H−+1/2 + kFe2+c Fe2+ ⎟⎠

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1 + (a1/a 2)c Fe2+

e−(E / RT )

(3)

EXPERIMENTAL DATA

The experimental data used for kinetic modeling is based on Long and Dixon’s study1 on pyrite high-pressure oxidation in a laboratory-scale autoclave under acidic conditions. In their study, high-grade massive pyrite specimens originating from Zacateces (Mexico) were used as the raw material for the experiments. The purity of the pyrite samples was 97 ± 1%, and the molar ratio of sulfur to iron was 1.92. Oxygen was fed into the reactor to maintain constant oxygen partial pressure. Temperature was controlled by removing the reaction heat using a cooling coil. The effects of temperature (170−230 °C), particle size (49−125 μm), oxygen partial pressure (345−1035 kPa), and pulp density (1−20 kg/m3) were evaluated in order to investigate the pyrite oxidation kinetics. Sulfuric acid concentration in the experiments was 0.5 mol/dm3 to prevent iron precipitation as ferric hydroxide. The experimental data can be organized in 10 data sets, the conditions of which are shown in Table 1. In the experiments, an agitation speed of 800 rpm was found to be high enough not to show any effect on the reaction rate. Therefore, this agitation speed can be assumed to guarantee oxygen saturation in the

1/2

rFeS2

2 c a1c Fe 2 + L,O 2

According to eq 3 the reaction is first order with respect to oxygen and second order with respect to ferrous iron at low ferrous iron concentrations. At high ferrous iron concentration, the rate is first order with respect to both oxygen and ferrous iron concentrations. As shown above, the pyrite oxidation process is complicated and includes several surface and bulk phase reactions. The overall leaching rate depends on factors such as temperature, oxygen partial pressure, particle size, and reactant concentrations. In order to cast new light on the oxidation mechanism, it is important to develop kinetic models that are capable of predicting the oxidation outcome at different conditions. In this work, the objective is to develop a comprehensive kinetic model for pyrite oxidation that can explain the effects of the multiple surface and liquid-phase reactions and describe the influence on pyrite oxidation kinetics of temperature, oxygen partial pressure, particle size, and reactant and slurry concentrations. It is recognized, however, that this is a challenging task since the pyrite oxidation process also involves phenomena such as iron precipitation that might be difficult to take into account in model development. The developed model is based on pyrite oxidation experimental data given in the literature by Long and Dixon.1 The study extends the work of Long and Dixon1 by including multiple surface and bulk phase reactions. Up-to-date mathematical methods are used to estimate the model parameters in this complicated multiparameter model. The reliability of the estimated parameters and model predictions is studied using Markov chain Monte Carlo (MCMC) methods.

(R4)

S2 O32 − + 2Fe3 + + H 2O → SO24 − + 2Fe 2 + + 2H+

(2)

Rönnholm et al.12 derived a rate equation for ferrous ion oxidation for concentrated H2SO4−FeSO4 solutions based on mechanism studies. They give the rate equation in the form

FeS2 + 7/4O2 + H+ → Fe3 + + S2 O32 − + 1/2H 2O

S2 O32 − → S0 + SO32 −

dc Fe2+ = kc Fe2+c L,O2 dt

(1)

According to eq 1 the electrochemical mechanism gives a reaction order close to 0.5 with respect to dissolved oxygen at low iron concentration. There is evidence that at higher ion concentration, anions such as Cl− and SO42− have a detrimental effect on the pyrite oxidation rate.4 This effect is attributed to adsorption of Cl− and SO42− on the pyrite surface, which might inhibit the access of oxidants. In addition, other ions, such as Fe2+, may adsorb on the pyrite surface, competing for vacant sites with the oxidants. The effect of Fe2+ is shown as a negative reaction order or as a term in the denominator in rate equation. Several kinetic models for ferrous iron oxidation (reaction R3) have been presented in the literature, and according to Garrels and Thompson10 and Singer and Stumm11 the ferrous iron B


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As experimental results, Long and Dixon1 give pyrite conversion and the ratio of ferric iron concentration to total iron concentration at different reaction times. The experimental data are shown in Figures 1 and 2. Model Development. The rate of the surface reactions at the pyrite surface depends on the concentration of pyrite in the solids, concentration of dissolved oxygen and ferric ions in the solution, temperature, and available surface area. Pyrite concentration at the solid surface of the particles is characteristic for the concentrate and can be assumed to remain constant during oxidation. As noted by Long and Dixon,1 experimental evidence shows that the oxidation process starts by production of ferric iron (see Figure 2). Therefore, it may be postulated that the oxidation proceeds following reactions R4 and R2. Reaction R6 may cause some problems for kinetic modeling since no sulfite data is

Table 1. Conditions in Oxidation Experimental Runs run

T (°C)

di (μm)

PO2 (kPa)

mFeS2 (kg/m3)

1 2 3 4 5 6 7 8 9 10

170 190 210 210 210 210 210 210 230 210

62.6 62.6 48.2 62.6 62.6 62.6 88.1 125.1 62.6 62.6

690 690 690 345 690 1035 690 690 690 690

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 20.0

liquid phase. Details of the experimental equipment, procedures, and analytical methods are given in Long and Dixon.1

Figure 1. Experimental data and model prediction for pyrite conversion at various conditions. C


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Figure 2. Experimental data and model predictions for ferric to total iron concentration ratio at various conditions (experimental data for run 10 are missing).

passivation term (1 − XFeS2)n1. This term takes into account the negative effects of passivation on the pyrite oxidation rate as the reaction proceeds. Parameter n1 is an experimental parameter that determines the effect of passivation. Since it is assumed that the observed passivation depends on the surface coverage of the passivating species and the leaching reaction rate is given as a surface reaction, it can be expected that the passivation effect of sulfur is the same for both oxygen and ferric iron oxidation reactions. The reactant concentrations change during the reaction process due to dissolution and the reactions of ferric and ferrous iron. Therefore, it is difficult to present the shrinking particle model in the usual implicit integral form. Instead, the kinetic model is given in the form of differential equations where the pyrite oxidation reaction rates are presented by the surface reaction rate between pyrite and oxygen (eq 4)

available and the reaction consumes ferric iron and produces ferrous iron. For this reason, it is probably difficult to separate the effects of reactions R2 and R6. However, since this reaction is not directly involved in the pyrite surface reactions, it can be considered to be of secondary importance and may be omitted from the kinetic model. In the model, the reaction rate of pyrite is specified as a surface reaction between oxygen and pyrite and ferric ions and pyrite (reactions R4 and R2). It is assumed that the oxidation process closely follows a shrinking particle mechanism, and the active area, therefore, continuously decreases as the reactions proceeds. As mentioned earlier, surface passivation, most likely by elemental sulfur, was detected at low slurry concentrations during the experiments.1 It is necessary to take this passivation effect into account in model development. In this work, a simple passivation model is implemented to calculate pyrite conversion using a D


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where the equilibrium constant Keq can be easily calculated from the final ferric to total iron ratio, given in Figure 2, and dissolved oxygen concentration. For a batch reactor, differential mass balances for pyrite, ferrous iron, and ferric iron can be written

and the surface reaction rate of the reaction of pyrite with Fe3+ (eq 5) ⎛ 1 − XFeS ⎞n1⎛ c L,O ⎞n2 2 2 ⎟⎟ ⎜⎜ ⎟⎟ r1 = k1c FeS2,0⎜⎜ X c ⎝ FeS2,mean ⎠ ⎝ L,O2,mean ⎠

(4)

dXFeS2

⎛ 1 − XFeS ⎞n1⎛ c 3+ ⎞n3 2 ⎟⎟ ⎜⎜ Fe ⎟⎟ r2 = k 2c FeS2,0⎜⎜ X ⎝ FeS2,mean ⎠ ⎝ c Fe3+,mean ⎠

dt (5)

value

dc Fe3+ An An = r1 i − 14r2 i + r3 − r −3 ε ε dt

−5

cL,O2,mean (mol m )

1.0 × 10

cFe2+,mean (mol m−3) cFe3+,mean (mol m−3) Tmean (°C)

1.0 × 10−6 2.5 × 10−6 210

Ai = π (1 − XFeS2)2/3 d02

(12)

where d0 is the initial particle size. One additional complication for model development is that ferric iron tends to be hydrolyzed and precipitates out from the solution. The extent of precipitation depends on temperature, acidity, and the concentration of ferric iron and other species. Long and Dixon1 found experimentally that at 210 °C and at slurry concentration 20 kg/m3 90% of iron is in the precipitate. However, at low slurry concentrations (1 kg/m3) precipitation was not found to be important. Precipitation was also found to take place rapidly. Consequently, the amount of precipitated iron can be taken into account in the model by multiplying the ferric iron production rate due to the surface reaction (r1 in eq 11) by the fraction of nonprecipitated iron in the solution. In the reaction rate in eqs 4 and 6, dissolved oxygen concentration is needed. The mixing rate that was used in the laboratory experiments by Long and Dixon1 was intensive. Therefore, it can be assumed that the reaction rate is controlled by the chemical reactions and that the dissolved oxygen concentration equals the saturation concentration at the reaction conditions. The dependence of the molar saturation concentration of dissolved oxygen on the oxygen partial pressure is evaluated by the Tromans13 model

(6)

For each reaction, the dependency of the reaction rates on temperature is taken into account by the reparametrized Arrhenius equation ⎛ E ⎛1 1 ⎞⎞ k i = k i,mean exp⎜⎜ − i ⎜ − ⎟⎟⎟ Tmean ⎠⎠ ⎝ R ⎝T

(7)

Although redox reactions such as reaction R3 are not usually considered as equilibrium reactions, Long and Dixon1 found in practice that an equilibrium is in fact reached between ferrous and ferric iron (see Figure 2). In order to be able to use the measured ferric to total iron concentration data in parameter estimation, reaction R3 was given as an equilibrium reaction. The backward reaction r−3 is then given by r r −3 = 3 Keq (8)

2 ⎛ ⎞ * = 101.325 Ψ exp⎜ −0.046T + 203.357T ln(T /298) − (299.378 + 0.092T )(T − 298) − 20591 ⎟p c L,O 2 8.3143T ⎝ ⎠ O2

(13)

eqs 9, 10, and 11. These equations together with the reaction rates in eqs 4, 5, and 6 and the particle surface area in eq 12 were numerically solved from the initial state to the time of the data points shown in Figure 1. The effect of iron precipitation at high temperature and slurry concentration was taken into account for run 10 in Table 1 by applying a nonprecipitated iron fraction in eq 11 for ferric iron production by reaction rate r1. It was found, however, that by using a value of 10% for the nonprecipitated fraction, as suggested by Long and Dixon,1 the model gives

where the oxygen partial pressure is in atm and Ψ is a term that depends on sulfuric acid molal concentration. ⎛ ⎞0.168954 1 ⎜ ⎟⎟ Ψ=⎜ ⎝ 1 + 2.01628c H2SO4 ⎠

(11)

According to the shrinking particle model, the particle size and surface area reduce continuously as the reaction proceeds. The surface area of the particles can be calculated from the pyrite conversion

Since it is expected that ferrous iron concentration remains at relatively low levels, eq 3 can be implemented and the oxidation reaction of ferrous ions back to ferric ions in the liquid phase can be expressed in the form n ⎛ c 2+ ⎞n4 ⎛ c L,O ⎞ 5 2 ⎟⎟ ⎟⎟ ⎜⎜ r3 = k 3⎜⎜ Fe ⎝ c Fe2+,mean ⎠ ⎝ c L,O2,mean ⎠

(10)

where ε is the volume fraction of liquid in the slurry. The production rate of Fe3+ is calculated from

0.5 −3

(9)

dc Fe2+ An = 15r2 i − r3 + r −3 ε dt

Table 2. Mean Values of Parameters XFeS2,mean (%)

(r1 + r2)Ai n c FeS2,0

where Ai is the surface area of the particle and n is the number of particles per slurry volume. Production of Fe2+ is obtained from

The pyrite conversion and the reagent concentrations are divided by their mean values to improve parameter identification in the parameter estimation. The used mean values are given in Table 2.

parameter, dimension

=

(14)

Parameter Estimation and Model Predictions. The pyrite oxidation model consists of three differential equations, E


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a considerable underprediction for pyrite conversion for run 10 in Table 1. To overcome the underprediction, the best practice is to include the nonprecipitated iron fraction as an estimable parameter in the model. Taking this into account, the model has altogether 12 parameters: reaction rates at mean temperature k1,mean, k2,mean, and k3,mean, activation energies E1, E2, and E3, reaction orders n1, n2, n3, n4, and n5, and the nonprecipitated iron fraction f pre. The model parameters were first estimated with standard leastsquares fitting by minimizing the squared difference between the measured and the calculated pyrite conversions and the ferric to the total iron concentration ratios. The goodness of the fit was determined by using the R2 value and the standard errors from the estimation. However, to thoroughly evaluate the accuracy and reliability of the estimated parameters in a nonlinear multiparameter

Table 3. Model Parameter Values Obtained from Estimation parameter

status

value

est rel. std error, %

k1,mean (mol/m2 s) k2,mean (mol/m2 s) k3,mean (mol/m3 s) E1 (J/mol) E2 (J/mol) E3 (J/mol) n1 n2 n3 n4 n5 f pre

estimated estimated estimated estimated estimated estimated estimated fixed6,8 fixed12 fixed12 fixed6 estimated

2.95 × 10−2 2.77 × 10−3 7.09 × 10−5 46.1 × 10+3 50.0 × 10+3 109 × 10+3 0.45 0.5 1.0 2.0 0.5 0.486

2.5 8.5 14.0 6.2 12.8 9.0 14.0

19.4

Figure 3. Model prediction of the importance of the direct oxidation rate at various conditions. F


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Figure 4. Ratio of particle size to initial particle size.

According to a Bayesian paradigm all the parametrizations of the model that statistically fit the data equally well are determined. A distribution of the unknown parameters is generated using available prior information (e.g., results obtained from previous studies or bound constraints for the parameters) and statistical knowledge of the observation noise. Computationally, the distribution is generated using the MCMC sampling approach. The length of the calculated chain was 50 000 samples. Simple flat, uninformative priors with minimum and maximum bounds set for each parameter were used in the calculation of the chain. Up-to-date adaptive computational schemes are employed in order to make the simulations as effective as possible.17,18 In this study, a FORTRAN 90 software package, MODEST 15,19 was used for both the least-squares and the MCMC estimation. The two methods are also implemented in a MATLAB package.18,20

model, it is important also to consider possible cross-correlation and identifiability of the parameters. Classical statistical analysis that gives the optimal parameter values, their error estimates, and correlations between them is based on linearization of the model and is therefore approximate. Furthermore, it may sometimes even be quite misleading, especially if the available data are limited and the parameters are poorly identified. The reliability of the models and their parameters was investigated in this study using Markov chain Monte Carlo (MCMC) methods. The MCMC method is based on Bayesian inference and gives a probability distribution of solutions. MCMC methods have recently been successfully applied in chemical engineering to study parameter reliability.14−16 Moreover, the question of the reliability of the model predictions remains unaddressed, i.e., how the uncertainty in the model parameters is reflected in the model response. G


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Figure 5. Results of the MCMC analysis (two-dimensional posterior distributions for the parameters. The density of the dots in each subfigure represents probability, and the two circles represent 95% and 90% probability regions of the parameter).

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RESULTS AND DISCUSSION In the parameter estimation, the reaction orders n2, n3, n4, and n5 were fixed to values proposed in the literature and the remaining 8 parameters were fitted to experimental data. The estimated parameters are given in Table 3. Obtained values of activation energy E1 and E2 are close to values obtained by Long and Dixon1 (33.2 kJ/mol). The obtained results show that the model predicts the pyrite oxidation data at least with reasonable agreement, as shown in Figures 1 and 2. The coefficient of determination for the fit is 0.981. The model also predicts the ratio of ferric to total iron reasonably well. The largest deviation can be seen in the ferric to total iron concentration ratio for some data sets (runs 3, 6, and 8). This deviation is not surprising in view of the lack of reliability of

the chemical analysis of ferrous and ferric iron concentrations and the detected equilibrium concentration. It was also found that the reaction orders taken directly from the literature1 fit the data well, and there is no need to fit them separately. The simple passivation term used in the model is capable of predicting the effect of passivation on leaching reasonably well. As can be seen from Figure 1, the model slightly overestimates the achieved conversion at low temperatures at the later stages of leaching. Contrary to the passivation model given by Long and Dixon,1 this model does not predict complete surface passivation. An interesting conclusion can be drawn from the results. The reaction starts with direct molecular oxygen, and during the initial stages the reaction seems to produce ferric iron instead of ferrous iron. This seems to support the hypothesis that the H


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Figure 6. Uncertainty distributions for model predictions (run 1) for pyrite conversion (left) and ferric to total iron ions ratio (right). The dark gray area represents the 95% confidence region for model prediction. The light gray area represents the 95% confidence region for measurements (model uncertainty combined with experimental error).

represents probability, and the two circles represent the 95% and 90% probability regions of the parameter. It can be seen from Figure 5 that all parameters are well identified and are not correlated. This can be seen from the well-centered round probability distributions and the sharp peaks in the projected 1-dimensional distributions. Using MCMC analysis enables study of how the parameter uncertainty is translated into the model response and model predictions, which cannot be easily done with classical regression analysis. As an example, the calculated uncertainties for model predictions are shown in Figure 6 for run 1. The dark gray area represents the 95% confidence region for model predictions. As can be seen from Figure 6, the uncertainty region is quite narrow for pyrite conversion but wider for the predicted ferric to total iron ratio. The light gray area represents the 95% confidence region for measurements (model uncertainty combined with experimental error).

molecular oxygen reaction proceeds according to reaction R4. Reaction R2 is fast; therefore, the ferric iron concentration initially drops rapidly. However, after 5−10 min from the start of experiments the surface reactions have slowed down and the bulk liquid reaction reaction R3, starts to produce more ferric iron than is consumed in the surface reaction, reaction R2. Therefore, in later stages the ferric iron concentration gradually increases, finally approaching the equilibrium concentration. Another interesting conclusion can be drawn from the relative magnitudes of the surface reaction rates r1 and r2, shown in Figure 3, where the ratio of the reaction rate r1 to the total reaction rate r1 + r2 gradually drops during the process. In the initial stages, the oxidation proceeds entirely by molecular oxygen (reaction R1) as there is no ferric or ferrous iron present in the solution. However, as the reaction proceeds, the ferric iron concentration gradually increases and reaction R2 becomes more important. Because of the low slurry concentration (1 kg/m3) used in most of the experiments, the level of ferric iron concentration remains at a relatively low level, and therefore, the direct oxygen reaction always corresponds to at least 75% of the overall oxidation rate. At high slurry concentration (20 kg/m3) the situation is completely different and the ferric iron reaction r2 dominates. Despite this and despite the fact that only one data set was available at high slurry concentration, the model predicts the leaching behavior also at high slurry concentration reasonably well (see Figure 1, run 10). Indeed, it was discovered that for higher slurry concentration the direct oxygen reaction corresponds to only 20% of the total reaction rate. Figure 4 shows how the calculated particle size decreases over time. As discussed by Long and Dixon,1 the decrease should be linear over time, based on the shrinking particle model. The nonlinear particle size shrinkage means that some surface passivation takes place. The implemented passivation model predicts the particle shrinkage reasonably well. Some deviation from experimental data can be seen at later stages as the model does not predict complete passivation. The results of the MCMC analysis for each model are shown in Figure 5, which presents two-dimensional posterior distributions for the parameters. The density of the dots in each subfigure

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CONCLUSIONS A comprehensive kinetic model for pyrite oxidation in the aqueous phase is presented that includes the effects of temperature, oxygen partial pressure, particle size, and multiple surface and bulk phase reactions at different slurry concentrations. However, the leaching rate probably also depends on pyrite properties, such as pyrite semiconducting properties or catalytic effects of other minerals or metals which are difficult to take into account in modeling. It is believed, however, that since the model is mechanistic, the form of the model can be used for other types of pyrite minerals by just conducting a new parameter fitting for available data. The model can be considered comprehensive because it includes the most important influencing factors that govern pyrite aqueous-phase oxidation at different conditions. The model is based on a shrinking core model given in the form of differential equations and has seven experimental parameters that were estimated by comparison to experimental data. Very good agreement was obtained between the predicted and the experimental data for pyrite conversion and ferric to total iron concentration ratio (coefficient of determination > 0.981). I


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Industrial & Engineering Chemistry Research The reliability of the kinetic parameters was studied using MCMC analysis. Although the model has multiple parameters, the MCMC analysis shows that all parameters are well identified and do not correlate to each other. The uncertainties in the model prediction, i.e., the simulated pyrite conversion and ferric to total iron concentration ratio, were also studied by MCMC analysis. It was shown that the uncertainty in model prediction is low, which can be seen by the narrow 95% confidence region for pyrite conversion. A simple surface passivation model was implemented in which the passivation is linked to the pyrite conversion. This simple passivation model explains the surface passivation reasonably well. Unlike the passivation model given by Long and Dixon,1 it does not predict complete surface passivation at low temperatures. The results also show that pyrite surface reactions with both molecular oxygen and ferric iron are important in these conditions. At low slurry concentration, when the total iron concentration in the solution is low, the reaction with molecular oxygen dominates. However, at high slurry concentration the opposite is true and the pyrite reaction with ferric iron dominates. Although only one data set was available for high slurry concentrations, the model still explained reasonably well the oxidation process in these conditions. The good results despite the limited data suggest that it would be beneficial to further verify the model behavior at different slurry concentrations. The presented pyrite oxidation model gives new information about the pyrite oxidation process, the importance of different surface reactions, and the conditions in which each reaction dominates. The study also demonstrates how novel MCMC analysis methods can be used to study the reliability of model parameters and model predictions. MCMC analysis is of value in the development of more accurate and reliable models for chemical processes.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: vladimir.zhukov@lut.fi. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Funding from the Green Mining Programme of the Finnish Funding Agency for Innovation (Tekes) and Outotec Oyj is gratefully acknowledged.

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NOMENCLATURE a1, a2 = experimental parameters in eq 3, Ai = inner surface area of particle (reactive area), m2 cH2SO4 = molal concentration of H2SO4, mol kg−1 cO2 = dissolved oxygen concentration, mol m−3 cH+ = hydrogen concentration, mol m−3 cL,O2* = dissolved oxygen concentration at the saturation point, mol m−3 cL,O2 = dissolved oxygen concentration, mol m−3 cL,O2,mean = mean dissolved oxygen concentration, mol m−3 cFeS2,0 = initial pyrite concentration, mol m−3 cFetot = concentration of total dissolved iron, mol m−3 cFe3+ = concentration of ferric ions, mol m−3 cFe2+ = concentration of ferrous ions, mol m−3 cFe2+, mean = mean concentration of ferrous ions, mol m−3 cFe3+, mean = mean concentration of ferric ions, mol m−3

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di = inner diameter of particle, m dmean = mean particle size, m d0 = outer diameter of particle, m E1 = activation energy, kJ mol−1 E2 = activation energy, kJ mol−1 E3 = activation energy, kJ mol−1 F = Faraday constant, C mol−1 f pre = fraction of nonprecipitated iron, Keq = equilibrium constant, KH = Henry’s constant, L Pa mol−1 kO2 = oxygen mass transfer coefficient, s−1 k1 = reaction rate constant for reaction of pyrite with oxygen k2 = reaction rate constant for reaction of pyrite with Fe3+ k3 = reaction rate constant for reaction of ferrous ion oxidation k1,mean = reaction rate constant at mean temperature, mol m−2 s−1 k2,mean = reaction rate constant at mean temperature, mol m−2 s−1 k3,mean = reaction rate constant at mean temperature, mol m−3 s−1 kFeS2 = rate constant for reaction of pyrite oxidation, s−1 kFe3+ = ferric reduction rate constant, s−1 kFe2+ = ferrous oxidation rate constant, s−1 MFeS2 = molar mass of pyrite, kg mol−1 mFeS2 = pulp density, kg m−3 n = number of solid particles in reactor per liter slurry, m−3 n1 = reaction order for pyrite conversion, n2 = reaction order for concentration of dissolved oxygen in reaction R1, n3 = reaction order for concentration of ferric ions, n4 = reaction order for concentration of ferrous ions, n5 = reaction order for concentration of dissolved oxygen in reaction R3, PO2 = oxygen partial pressure, kPa R = gas constant, R = 0.008314, kJ mol−1 K−1 r1 = surface reaction rate for reaction of pyrite with oxygen, mol m−2 min−1 r2 = surface reaction rate for reaction of pyrite with ferric ions, mol m−2 min−1 r3 = reaction rate for reaction of ferrous iron with oxygen, mol m−3 s−1 r−3 = reaction rate of ferric iron to ferrous iron, mol m−3 s−1 rFeS2 = reaction rate for pyrite oxidation, mol m−3 s−1 rFe2+ = reaction rate for ferrous ions, mol m−3 s−1 rO2 = oxygen mass transfer rate, mol m−3 s−1 T = temperature, K Tmean = mean temperature, K VFeS2 = volume of pyrite, m3 Vslurry = volume of slurry, m3 XFeS2 = pyrite conversion, % XFeS2, mean = mean pyrite conversion, % ρFeS2 = density of pyrite, kg m−3 ε = volume fraction of liquid in the slurry, m3 m−3slurry ψ = shape factor MCMC = Markov chain Monte Carlo

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Kinetic Modeling of High-Pressure Pyrite Oxidation with Parameter

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