A population balance approach to predict the performance of continuous leaching reactors: Model validation in a pilot plant using a roasted zinc concentrate

Hydrometallurgy 194 (2020) 105301

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A population balance approach to predict the performance of continuous leaching reactors: Model validation in a pilot plant using a roasted zinc concentrate

T

Fabrício Eduardo Bortot Coelhoa, Julio Cézar Balarinia, Estêvão Magno Rodrigues Araújoa, Tânia Lúcia Santos Mirandaa, Antônio Eduardo Clark Peresb, Afonso Henriques Martinsb, ⁎ Adriane Saluma, a b

Chemical Engineering Department, Federal University of Minas Gerais, Belo Horizonte, MG, Brazil Metallurgical and Materials Engineering Department, Federal University of Minas Gerais, Belo Horizonte, MG, Brazil

ARTICLE INFO

ABSTRACT

Keywords: Population balance Maximum Mixedness Segregated flow Residence time distribution Roasted zinc concentrate

A Population Balance model for the continuous leaching of a roasted zinc concentrate was developed. The Maximum Mixedness and the Segregated Flow approaches were chosen to model the flow patterns and mixing conditions in the reactors. In order to compare these approaches and validate the proposed model, leaching experiments were carried out in a pilot plant. For that, it was also determined the residence time distributions in this plant reactors. As a result, it was observed that the Segregated Flow did not represent the actual pilot plant operation, since the conditions in which the Segregated Flow is recommended (i.e. segregation in the micromixing of the slurry phase, low diffusivity, and high slurry viscosity) were not present. On the other hand, using the Maximum Mixedness resulted in an excellent agreement between the experimental and modeled data (mean absolute error of 0.5% for the zincite conversion). This indicates that the complex particulate phenomena that occur during the leaching, as well as the flow patterns in the continuous reactors were adequately accounted in the model. Therefore, a robust tool for the design and control of leaching plants is delivered, since it is able to predict the effect of operational parameters (e.g. particle size distribution, solid/liquid ratio, feed flow rate) on the reactors performance. This can lead to better decision-making and costs reduction, contributing also to the prediction of problems faced in the operation of these plants. In a direct application of the developed model, it was predicted that the pilot plant could operate with the doubled capacity.

1. Introduction Leaching can contribute up to 30% of the total capital expenditure in a hydrometallurgy plant (Crundwell, 2005). The absence of accurate design methods resulted in excessive capital spending, poor control, and erroneous engineering decision-making. Historically, the determination of kinetic mechanisms has received great attention and it has provided important parameters for leaching design (Elgersma et al., 1992; Lampinen et al., 2015; Lapidus, 1992; Salmi et al., 2010). However, traditional kinetic models, such as the shrinking core model, were developed for monodispersed particles; thus, changes in operational parameters (e.g. the feed particle size distribution) could lead to large errors in plant design (Crundwell and Bryson, 1992). In addition, some of these traditional models have a limited industrial application since they are valid only for low solid/liquid ratios and constant acid

⁎

concentrations (Doyle et al., 1987; Larba et al., 2013). Therefore, predicting a leaching plant performance has been a challenging task (Crundwell, 1995; Crundwell et al., 2013; Giona et al., 2002; LeBlanc and Fogler, 1987). In order to overcome these limitations, several mathematical models have been proposed for leaching systems. The Population Balance (PB) has proved to be the most appropriate approach to model polydisperse systems (Crundwell et al., 2013; Dixon, 1996; ElduayenEchave et al., 2019; Sepulveda and Herbst, 1978). Since it considers that the solid particles have a distribution of properties (e.g. size and composition), it can predicts the influence of these polydisperse particles in the reactors performance (Dixon, 1996, Dixon, 1995; Dorfling et al., 2013; Giona et al., 2002; Rubisov and Papangelakis, 1997). The PB approach has been applied with success to model several leaching systems, such as the continuous pressure leaching of sphalerite

Corresponding author. E-mail address: [email protected] (A. Salum).

https://doi.org/10.1016/j.hydromet.2020.105301 Received 24 November 2019; Received in revised form 19 February 2020; Accepted 29 February 2020 Available online 12 March 2020 0304-386X/ © 2020 Published by Elsevier B.V.

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(Crundwell and Bryson, 1992) and the bacterial leaching (Crundwell, 2000, Crundwell, 1994a), which have led to excellent agreement between the predictions of the PB models and the data from pilot plants. Nevertheless, in order to model continuous reactors, the flow patterns and the mixing conditions in the reactors should be accounted in the equations. To achieve this, several approaches are available, such as the Maximum Mixedness and the Segregated Flow (Crundwell, 1994b), which are associated to the two limiting regimes of micromixing. In the first one, all molecules of the same age remain together as they travel through the reactor and are not mixed with molecules of other ages. In other words, the inlet stream (or slurry) is broken into small finite volume elements that remain intact and behave as individual batch reactors during their passage through the reactor (Crundwell, 1994b). This regime corresponds to the Segregated Flow approach. In the second regime, molecules of different ages are completely mixed, on a molecular scale, as they enter the reactor, so all the particles leaving the reactor have the same age distribution and were completely mixed for the duration of their stay in the reactor, which correspond to the Maximum Mixedness approach (Crundwell, 2005; Fogler, 2016). A theoretically comparison of these approaches has been done by Crundwell (1994b), who has demonstrated that the Segregated Flow and the Maximum Mixedness led to quite different results. Other authors (Crundwell, 2000; Rubisov and Papangelakis, 2000) validated experimentally their PB models, considering only the Maximum Mixedness condition. However, no previous work has compared the leaching data obtained from a pilot plant with the results generated by a PB model using both approaches (Segregated Flow and Maximum Mixedness), which is one of the objectives of this paper. In order to achieve this, the sulfuric leaching of a roasted zinc sulfide concentrate (RZC), in a pilot plant, was investigated and modeled. This system attracts attention since more than 85% of the world's zinc production is through the hydrometallurgical process (Habashi, 1997; Herrero et al., 2010; Hewitt and Wall, 2000), but it has not been comprehensively modeled. Although Doyle et al. (1987) modeled the leaching of zinc oxide, it was in diluted acid solutions and without using of the Population Balance. Balarini (2009) modeled the RZC leaching with the PB model, but no experimental verification was carried out. Other authors developed models for zinc sulfides, such the ones developed by Corriou et al. (1988) and Lampinen et al. (2015), but these models may not be adequate for the RZC leaching, since they work with pressure leaching in autoclaves, which involves other phenomena (e.g. gas-liquid equilibriums); then, quite different kinetics of that studied in present work. Therefore, in a previous work (Coelho et al., 2018), the first BP model for the RZC leaching resulted in an excellent agreement between the experimental data and the developed model, but only the batch leaching was studied. In this context, the purpose of this paper is to apply the Population Balance to model the continuous leaching of a roasted zinc concentrate in sulfuric acid solutions, using the kinetic data and parameters determined for this system in the latter work (Coelho et al., 2018). Moreover, it is intended to validate this model with experimental data obtained in a pilot plant, and to investigate the flow patterns and mixing conditions in the reactors of this pilot plant by determining its residence time distributions (RTD). This allows comparing experimentally the Maximum Mixedness and the Segregated Flow approaches in the context of the PB modeling. Therefore, the validation of such model will contribute for further understanding of zinc leaching processes providing better techniques for plant design and control; since the PB model is capable to simulate industrial conditions, predicting the effect of several parameters in the reactor performance (e.g. the solid particles polydispersity, the acid consumption, and the feeding flow rate).

2. Materials and methods 2.1. Population balance modeling In order to model the continuous leaching of the roasted zinc concentrate (RZC), it was used a population balance (PB) for solid particles and one mass balance for the leaching agent (sulfuric acid). The kinetic parameters were determined in a previous work (Coelho et al., 2018), in which a PB model for the batch RZC leaching was experimentally validated. Therefore, it is considered, initially, the general Population Balance for particles in a continuous system, according to Eq. (1) which detailed derivation can be found in work of Herbst (1979):

t

(V ) = Qin

in

Qout

out

+ V (B

j

D)

j=1

(vj ) j

(1)

In this equation, ψ is the population density function; Qin and Qout are the inlet and outlet volumetric flows, respectively; and υ(D) is the time rate of change in the particles diameter, which is given by the reaction intrinsic kinetics. This equation can be simplified considering the following: the leaching reactors have the same constant volume (V) and they are operating in a steady state condition (the accumulation term is null, V / t = 0 ); there are no particle birth (B ) or death (D ) events; and the only studied property is the particle diameter (ζj = D). Then, the macroscopic PB equation for the continuous leaching is given by Eq. (2) which is a non-linear first order differential equation that can be solved by analytical and numerical methods (Crundwell, 2005).

Qout

out

(D) = Qin

in (D )

V

d [v (D) (D)] dD

(2)

The particle size distribution in the inlet stream, ψin, is given in a number basis, while the empirical models (e.g. log-normal, gamma, GGS (Gates-Gaudin-Schumann), and RRB (Rosin-Rammler-Bennett), which represents the particle size distribution measured experimentally, are usually determined in a mass basis ( fmass(D)). However, it is possible to transform a size distribution in a mass basis to a number basis and vice versa (Herbst, 1979). For the RZC it was used the RRB (Rosin-RammlerBennett) model (Eq. (3)), which has two adjustable parameters, m and D63.2 (represents the aperture of a sieve in witch 63.2% of the material would not be retained).

fmass (D) =

m D . D63.2 D63.2

m 1

. exp

D D63.2

m

(3)

Then, according to Eq. (4), the conversion (X) in the leaching reactor is calculated from the inlet and outlet mass flows of solids in the reactor, Min and Mout, respectively. These values can be obtained from the change in the particles size distributions (Herbst, 1979).

X=1

Mout =1 Min

0 0

D3 D3

out

(D ) dD

in (D )

dD

(4)

Concerning the leaching intrinsic kinetics, Balarini (2009) observed that the RZC leaching could be described by the Shrinking-Particle Model (also known as Progressive-Conversion Model) under the surface reaction control. However, in that work low slurry concentrations and large surpluses of sulfuric acid were considered. Since these conditions are not usually practiced industrially, it was investigated in a previous work (Coelho et al., 2018) the leaching under high slurry concentrations and different molar ratios between the RZC and the sulfuric acid. As expected, these conditions led to deviations from the Shrinking-

2

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in the reactors (Eq. (4)), it is required to determine the size distribution of the particles in the outlet stream. That can done by solving the population balance (Eq. (2)) and the leaching agent mass balance (Eq. (7)). Therefore, the flow patterns and mixing conditions inside the reactor must be accounted in the model. In order to achieve that, the Segregated Flow and the Maximum Mixedness approaches can be evaluated.

Table 1 Parameters used in the model for the roasted zinc concentrate (RZC) leaching. (Source: Coelho et al., 2018). Origin

Parameter

Value

Reaction kinetics (Balarini, 2009)

Model Controlling stage ks α ZnO percentage (TZnO) Molar density (ρB) Model D63.2 m σ/μ

Progressive conversion Chemical reaction 18,000 μm.min−1 5496 μm.min−1 76.1%

Experimental data RZC properties RZC size analysis

69.2 mol.L RRB 41.65 μm 1.022 0.98

2.1.1. Maximum mixedness In this approach, molecules of different ages are completely mixed as they enter the reactor, so all the particles leaving the reactor have the same age distribution. This means that all particles stayed inside the reactor for the same period, which is called residence time (τ = V/Q). Under these conditions, the particle size distribution inside the reactor is equal to the one in the outlet stream ( = out ). In addition, it is also considered that the continuous and dispersed phases are homogenous, and that the slurry inlet and outlet flow rates are equal (Qin = Qout = Q). Therefore, applying these conditions to Eq. (2), ψout can be obtained from Eq. ((10).

−1

Particle Model. These deviations were also caused by the fact that several minerals are present in the RZC, which affects the leaching mechanism. Nevertheless, in that work it was obtained an equation for the velocity, υ(D), derived from the Shrinking-Particle Model under the surface reaction control, as shown in Eq. (5).

dD 2 v (D ) = = [ks CAf dt s

(CA0

CAf )]

out

out

=

nBf

=

nZn, f

XZn (t ) (7)

nFerrite, f nB 0

=

(8)

v0 (CZn, f

out

(10)

D 0

D

in (D)

F dD

(11)

dD (D )

(12)

=

batch (t ) E (t ) dt

0

(13)

The Segregated flow approach is usually given in terms of the conversion (X), according to Eq. (14), after the transformation of size distribution from a number to a mass basis (Crundwell, 2005, Crundwell, 1994b; Dixon, 1996; Levenspiel, 2012).

X=1

0

0

[1

Xbatch (t )]

in, mass

dDE (t ) dt

(14)

The conversion considering a batch reactor (Xbatch) was obtained from the solution of the population balance model developed for the batch leaching of the RZC, in a previous work (Coelho et al., 2018). From the same work, the values of parameters required to solve the model (Table 1) were obtained. The RTD were measured (Section 2.3) under the same conditions utilized for the leaching experiments in the pilot plant. Finally, the model was numerically solved through an algorithm implemented in the software Mathematica 11® (Wolfram Research).

CFe, f /2) nB0

in (D )]

2.1.2. Segregated flow In the Segregated Flow approach, it is assumed that all molecules of the same age remain together as they travel through the reactor and are not mixed with molecules of other ages (Levenspiel, 2012). In other words, the inlet stream is broken into volume elements of fluid that remain intact and behave as individual batch reactors during their passage through the reactor (Crundwell, 1994b). Therefore, the size distribution of particles leaving the reactor is given by Eq. (13), in which E(t) is the residence time distribution (RTD) and ψbatch(t) is the size distribution of particles in a batch reactor as a function of the reaction time.

In Eq. (8) v0 is the volumetric flow rate of leaching solution, n B0 is calculated by the mass flow of RZC fed to the reactor (m B0) times the percentage of zincite in the RZC (TZnO) divided by the zincite molar mass (MMZnO = 81.38 g.mol−1). Only zincite (ZnO) was accounted for the calculations, but part of zinc ferrite (ZnFe2O4) is also solubilized during the leaching (Coelho et al., 2018); therefore, in order to calculate the correct zincite conversion (XZn), it should considered only the amount of zincite that was solubilized in the liquor (nBf ). Thus, as shown in Eq. (9), from the concentration of zinc measured in the liquor (CZn,f) it was subtracted half of the iron concentration measured in the liquor (CFe,f), which represents the concentration of zinc originated from the dissolution of zinc ferrite. The concentrations of zinc and iron in the liquor were measured by AAS, as described in Section 2.2.

nB0

1 d [v (D) dD

(6)

ZnSO4(aq) + H2 O(l)

nA0 v C v0 CA0 = 0 A0 = nB0 nB 0 mB0 TZnO / MMZnO

XZn =

+

1 (D ) F

(D ) =

F = exp

Therefore, the mass balance for the leaching agent (sulfuric acid) can be given as a function of the zincite conversion (XZn(t)), according to Eq. (7). In this equation, η is the molar ratio between the molar flows of sulfuric acid (n A0) and zincite present in the RZC (B0) fed to the reactor, as shown in Eq. (8).

CAf (t ) = CA0 1

in (D )

For the cases in which υ(D) do not depends on the particle size, as in the present work, this equation becomes a 1st order quasi-linear differential equation that can be solved by the Method of Characteristics (Crundwell, 2013). Therefore, the size distribution of the particles leaving the reactor can be obtained solving Eqs. (11) and (12). Then, the conversion can be calculated from Eq. (4). In this work, the model was solved numerically using an algorithm implemented in MATLAB® R2015b (MathWorks).

(5)

In this equation, α [μm.min−1] is an adjustable parameter, which value was obtained through a parameter estimation procedure using the experimental data (Coelho et al., 2018); ks [μm.min−1] is a rate constant for the RZC leaching determined by Balarini (2009) for a process under the surface reaction control; ρs [mol.L−1] is the molar density of zinc in the RZC; CA0 and CAf [mol.L−1] are the concentrations of the leaching agent (sulfuric acid) in the inlet and outlet streams, respectively. The values of these parameters are shown in Table 1. Since the adjustable parameter (α) was introduced in the model, it can be considered that the only reaction consuming the sulfuric acid is the leaching of zincite (ZnO), according to the Eq. (6).

ZnO(s) + H2 SO4(aq)

(D ) =

(9)

In order to solve the model and determine the conversion achieved 3

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F.E. Bortot Coelho, et al.

Element

Zn

Fe

Pb

Cu

Ca

% (by mass)

65.0

4.7

1.1

0.6

1.4

Cumulative mass fraction

Table 2 Chemical composition of the roasted zinc concentrate. (Source: Coelho et al., 2018). Method: samples were digested in a mixture of hydrochloric and nitric acid (3/1 v/v) in boiling temperature and then analyzed by AAS in the spectrophotometer XplorAA (GBC).

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1

1

10 100 Diameter (µm)

1000

Fig. 2. RZC particle size distribution: (♦) experimental data, (e) RRB model. (Source: Coelho et al., 2018). Method: Wet sieving using ASTM standard sieves in the range between 38 and 297 μm. A fraction of the RZC, which passed through a 74 μm sieve, was analyzed by laser diffraction, using the Helos12LA (Sympatec) particle-size analyzer. Table 3 Parameters investigated in the RTD measurement experiments.

Fig. 1. X-ray diffractrogram of the roasted zinc concentrate. (Source: Coelho et al., 2018). Method: samples were crushed below 74 μm and analyzed in the diffractometer Empyrean (PANanalitical) with a Xe detector and a Cu anode.

Parameter

Level

Number of reactors in series υ0 feed flow rate (L.min−1) Stirring velocity (rpm)

1, 2, 3 0.21, 0.41 250, 500, 750, 1000

plant in order to determine the residence time distributions in an attempt to elucidate the flow patterns inside these reactors. The operational parameters investigated (feed volumetric flow rate (υ0), stirring velocity, and number of reactors) are shown in Table 3. These RTD experiments and the leaching ones were carried out in a pilot plant (Fig. 3) with three baffled reactors in series. The capacity of each reactor was 6.0 L (measured without stirring), the temperature in each one was controlled using thermostatic baths, and each reactor had a mechanical stirring system, in which the impeller had two pairs of angled blades. The solid was fed using a hopper with a rotating plate, and the liquid by using the peristaltic pump Qdos 30 (Watson-Marlow). Both solid and liquid feeds were fed to the first reactor. For the RTD measurements, the injected tracer was hydrochloric acid and the fluid was pre-acidified water (in order to remove dissolved carbonates, as pointed by Choi et al. (2004)). Initially, this pre-acidified water was used for the start-up of pilot plant until the hydrodynamic regime was reached. Next, at time t = 0, an amount of the tracer, sufficient to reach concentration C0, was injected in the feed fluid reservoir and mixed. Then, the tracer concentration as a function of the time (C(t)) was measured in the effluent stream using a pH meter (M220, Denver Instrument), which recorded the pH in intervals of 20 s. Next, the residence time distribution function (E(t)) and the mean residence time ( M ) were calculated using Eqs. (15) and (16), respectively (Fogler, 2016). Then, the obtained RTD functions were presented in their dimensionless form using the correlation θ = t/τ.

2.2. Roasted zinc concentrate characterization The roasted zinc concentrate (RZC) was provided by a metallurgical plant located in the state of Minas Gerais, Brazil. In this plant, a zinc sphalerite concentrate is roasted and then leached for metallic zinc production by electrowinning. A detailed characterization of this material was done previously (Balarini et al., 2008; Coelho et al., 2018). The chemical analysis of the RZC, determined by atomic absorption spectrometry (AAS), is shown in Table 2. The X-ray diffraction spectrometry analysis (Fig. 1) indicated the majority presence of zincite (ZnO) and zinc ferrite (ZnFe2O4) in the RZC sample. Sphalerite (ZnS) and pyrite (FeS2) were also detected, which evidences that the roasting of the zinc sulfide concentrate was incomplete. The zincite content in the RZC (76.1% w/w) was determined by AAS after the digestion of the sample in a mixture of NH3 (25%) and NH4Cl, as done by Elgersma et al. (1992). The zinc ferrite amount of 10.2% w/w was estimated using a mass balance with the total amounts of Zn and Fe present in the RZC (Coelho et al., 2018). The size analysis of the RZC, presented in Fig. 2, demonstrated a high percentage of small particles in this material, since half of the particles have a diameter smaller than 30 μm. Using the least squares curve fitting method, among models tested (GGS, RRB, Log-normal and Sigmoid), the RRB model (D63.2 = 41.65 and m = 1.022) presented the smallest error and it was used to represent the RZC particle size distribution. The leaching solution was an aqueous solution of sulfuric acid (98% w/w - Vetec) and anhydrous lithium sulfate (99% w/w - SigmaAldrich). Li2SO4 was added as a tracer for correcting errors of sampling and dilution, since Li is chemical inert in the process and easily quantified by AAS.

d C (t ) dt C0

E (t ) =

M

=

0

tE (t ) dt

Step

(15) (16)

In a perfectly mixed tank, upon entering the reactor, the fluid mixes instantaneously and uniformly with the contents of the reactor, so that the composition of the effluent stream is identical to the composition of the entire reactor (Fogler, 2016; Levenspiel, 2012). Thus, in an

2.3. Residence time distribution (RTD) measurements Positive step experiments were performed in the reactors of the pilot 4

Hydrometallurgy 194 (2020) 105301

F.E. Bortot Coelho, et al.

Fig. 3. Leaching pilot plant used in the experiments.

operation without bypassing or dead volumes, the RDT of this system can be described by the perfect mixed cell model, according to Eq. (17).

E (t ) =

(Table 4) were chosen based on typical operating conditions of industrial processes. In such processes, the initial acid concentration (CA0) varies between 0.3 and 1.5 mol.L−1 (Gupta and Mukherjee, 1990; Habashi, 1997; Sancho et al., 2000). A value of 0.5 mol.L−1 was chosen for safety reasons, since large volumes of this leaching solution had to be manipulated for the experiments. A value of 1.0 was chosen for the molar ratio between the acid and the zincite in the RZC added to the reactor (η), since values slight lower or higher than one are applied industrially, depending on the process design (Habashi, 1997; Sancho et al., 2000). Two volumetric flow rates (0.21 and 0.41 L.min−1) were tested for the feeding of leaching solution, considering the limits of the peristaltic pump available and that these two values were the ones used for the determination of the residence time distributions (RTD). The mass flow rates of RZC (mB0 ) used in the experiments were calculated from Eq.(8). A moderate temperature value (40 °C) was chosen for all tests in order to guarantee the safety during the operation of the pilot plant. It is worth emphasizing that the chosen value is not so distant from the industrial one (around 60 °C) (Sancho et al., 2000). In addition, at higher temperatures, the RZC leaching rate is quite fast (Balarini, 2009; Napo et al., 2011), which would make the kinetic study much harder. The experimental procedure started with the preparation of the leaching solution and calibration of the RZC feeding mass flow rate. Then, with the empty reactors, the feedings of leaching solution and RZC to the first reactor were started, as well as the reactors stirring systems. At this moment the timer was started. When the slurry started flowing out of the last reactor, samples of slurry were collected at specific times and filtered under vacuum for the solid/liquid separation. The resulting liquid samples were diluted and stored for Zn, Li, and Fe analysis by AAS. Each experiment was done in triplicate and a Student's t-distribution with a confidence level of 95% was used to calculate confidence intervals.

t

e

(17)

For a series of N-perfectly mixed tanks of equal volume and mean residence times, the RTD function is given by the Tanks-in-series model (TIS), as shown in Eq.(18).

E (t ) =

t

N 1

NN e (N 1)!

tN

1 (18)

2.4. Leaching experiments A series of continuous leaching experiments were carried out in a pilot plant, in order to verify if the proposed Population Balance model was able to describe the RZC continuous leaching. It was also intended to compare the Segregated Flow and the Maximum Mixedness approaches for the flow patterns and mixing conditions in the reactors. Initially, using one reactor of the pilot plant, batch-leaching experiments were done in order to determine the minimum speed sufficient to homogenize completely the pulp and to reduce the effects of the mass transfer during the reaction. For these tests, several stirring speeds (250, 500, and 750 rpm) were tested for an initial acid concentration of 1.0 mol.L−1 and a molar ratio (η) equal to one, which implied in a solid/liquid ratio of 100 g.L−1. After selecting the optimum stirring speed (section 3.1), further experiments were done using 500 rpm. The parameters used in the continuous RZC leaching experiments Table 4 Fixed and investigated parameters in the leaching experiments. Parameter CA0 η mB0 υ0 Q T

Set 1 −1

Acid concentration (mol.L ) Molar ratio RZC mass flow rate (g.min−1) Leaching solution flow rate (L.min−1) Slurry flow rate (L.min−1) Temperature Stirring velocity (rpm)

0.5 1.0 11.1 0.21 0.213 40 °C 500

Set 2

3. Results and discussion 3.1. Residence time distributions (RTD)

21.6 0.41 0.415

The residence time distribution curves obtained for one single reactor of the pilot plant, for the feeding flow rate (υ0) of 0.21 and 0.41 L.min−1, are shown in Fig. 4. In this figure the RTD are presented 5

Hydrometallurgy 194 (2020) 105301

F.E. Bortot Coelho, et al.

acid as tracer has been as validated by other authors (Choi et al., 2004). Second, it could be caused by errors in the measures of the reactor volume (V) or in the feeding flow rate (υ0). The feeding flow rate was accurately measured but the actual reactor capacity without stirring was different from the capacity under stirring, since the rotating impeller promotes the formation of a vortex (Pant et al., 2015). Therefore, this could be a plausible reason to explain the deviation. Third, it could be caused by the existence of multiple mixing zones in the reactor (Stegowski et al., 2010). In this case, the reactor is composed by two regions, an active zone and a stagnant one, with parallel exchange of flow between them (Dagadu et al., 2012; IAEA, 2008; Stegowski et al., 2010). In order to verify this hypothesis, the mixing cells with exchange model (TISEx: Tanks-in-Series with Exchange), which describes a system with an active and a stagnant regions, was fitted to the experimental data through a least squares method and compared with the perfect mixer cell model (Table 5). The TISEx model equations are presented at the Appendix A.1. From Table 5, it can be seen that both models were in a good agreement with the experimental data. As expected, the TISEx model has presented slightly smaller values for the sums of squared residuals (SSR) than the perfect mixer cell model, since it was has three adjustable parameters while perfect mixer cell model has two. Using the TISEx model, it was obtained an average of only 2% for the fraction of the inlet flow that is exchanged with the stagnant zone (a). Therefore, considering this result and that the SSR values for both models were quite similar, the perfect mixer cell model was considered suitable to describe the flow in one reactor of the pilot plant. In the same table, it can also be seen that increasing the stirring speed there is a reduction of the mean residence time ( ), which becomes closer to the theoretical value ( M ), since higher stirring can improve the mixing and eliminate possible stagnant zones (Pant et al., 2015). Moreover, comparing the two inlet flow rates (0.21 and 0.41 L.min−1), the lower value implies in a longer residence time and in a longer time of mixing inside the reactor, which favors the outlet stream homogeneity and makes the reactor operation closer to an ideal CSTR (Choi et al., 2004). In Fig. 5, are shown the RTD for the systems with two and three reactors in series. Once it was determined that the flow pattern in one reactor of the pilot plant could be described by the perfect mixer cell model and that the three reactors of the pilot plant are identical, it was expected that the reactors in series also behave as perfect mixers in series. Therefore, the Tanks-in-Series (TIS) model was fitted to the residence time distribution obtained experimentally. As shown in Fig. 5, the TIS model and the experimental data are in a good agreement. It should be noticed that the residence time distribution was determined for the liquid phase. In order to obtain RTDs of the solid particles in the reactors, more complex methods such as the use of radiotracers (IAEA, 2008) would be necessary. Since there was no sign of slurry segregation in the present work, it was considered that the solid and liquid phases have the same RTD.

Fig. 4. RTD curves obtained for one single reactor and the simulation for the perfect mixer cell model. Operating conditions: temperature = 25 °C, feed flow rate of (a) 0.21 L.min−1 and (b) 0.41 L.min−1.

as the normalized residence time distribution function (E(θ)) and it was also plotted the simulated RTD curves using the perfect mixer cell model (Ideal CSTR). From Fig. 4, it can be seen that no bypassing (short-circuiting) occurred in the system, since no sharp early peaks or long tails were observed in the RTD curves (Levenspiel, 2012). However, it can also be noticed that the deviation between the experimental curve and the ideal CSTR curve is grater in the beginning of the test. This initial time lag can be caused by individual factors or a combination of them. First, the injection may not be a perfect spike. Second, the agitation may not be sufficiently high. Third, the path between the fluid reservoir and the inlet of the first reactor may be too long, which makes this piece of tubing act as a small reactor (Levenspiel, 2012). For these cases, the Fractional Tanks-in-series model could be applied. This model accepts fractional values for the number of reactors (N). Mathematically, it would led to better fits between the data and the model but fractional values of N would have no physical meaning, as stated by Rao et al. (2012). Therefore, this model was not addressed in this paper. It is worth emphasizing that all the values obtained for the mean residence time ( M ) were higher than the theoretical ones (τ=V/υ0), as shown in Table 5. This result disregards the presence of dead volumes in the reactor, but it may indicate the occurrence of other phenomena (Dagadu et al., 2012; Stegowski et al., 2010). First, it could be caused by a noninert tracer, which is not probable since the use of hydrochloric

3.2. Effect of stirring velocity on the leaching rate In Fig. 6, are shown the zincite conversion rates (dXZn/dt) obtained in batch leaching experiments for the different stirring speed tested using a single reactor of the pilot plant. From this figure, it can be noticed that there is no statistical difference in the zinc leaching rates for the stirring speeds tested. However, it was observed sedimentation of solids at 250 rpm and pulp spillage at 750 rpm. Therefore, considering these facts, the stirring speed of 500 rpm was chosen for use in the continuous RZC leaching experiments in the pilot plant. 3.3. The maximum mixedness and the segregated flow approaches In the PB model developed in this work, the flow patterns and mixing conditions in the reactors were accounted in the model by two 6

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Table 5 Fitting of the RTD models to the experimental data. Feed flow rate, υ0 (L.min−1)

Stirring velocity (rpm)

0.41

250 500 750 1000 250 500 750 1000

Normalized tracer concentration, E( )

0.21

Theoretical mean residence time, (min)

Mean residence time, M (min)

14.6

Perfect mixer cell (TIS)

16.3 16.1 15.6 15.0 29.5 29.4 29.3 28.9

28.6

1 Reactor 2 Reactors in series 3 Reactors in series TIS model, N=1 TIS model, N=2 TIS model, N=3

0.05 0.04 0.03 0.02 0.01 0.00 0.0

0.5

1.0

1.5

2.0

2.5

Dimensionless time, Fig. 5. Experimental and simulated RTD curves for one, two and three reactors in series. Operating conditions: temperature = 25 °C, feed flow rate = 0.41 L.min−1, stirring speed = 500 rpm.

Sum of squared residuals

Model parameters

16.6 16.2 15.8 15.2 29.3 29.2 28.9 28.3

0.11 0.05 0.01 0.12 0.02 0.04 0.08 0.03

15.8 15.5 15.8 15.2 29.4 29.3 28.9 28.6

(

0.20 500 rpm 750 rpm 0.10 0.05 0.00 0

5

10

τ1 (min) τ2 (min) a 35.7 16.3 17.8 17.1 0.27 0.24 0.23 0.29

0.04 0.06 0.02 0.01 0.02 0.02 0.01 0.01

Sum of squared residuals 0.09 0.04 0.01 0.06 0.02 0.04 0.08 0.02

100 n

y exp y y

), as presented in Table 6.

According to the values obtained for the sum of squared residuals and mean percentage error (Table 6), the Population Balance model, using the Maximum Mixedness approach to describe the flow patterns and mixing condition, is the most suitable for modeling the continuous leaching of the roasted zinc concentrate in the pilot plant. Similarly, good correlations between PB models, using the Maximum Mixedness approach, with experimental data have been observed for other systems, such as the pressure leaching of sphalerite in a autoclave with four tanks (Crundwell and Bryson, 1992) and the bacterial leaching of an iron sulphide ore (Crundwell, 2000). Furthermore, although the residence time distributions (RTD) obtained for the pilot plant reactors (Section 3.1) could be well described by the perfectly mixed tanks equations (Tanks-in-series model) the Maximum Mixedness and the Segregated Flow approaches led to different results. For heterogeneous reactions, these two approaches lead to the same numerical results when reaction is of zero order or pseudozero-order. In these cases, the rate of the particles dissolution does not depends on the leaching agent concentration or the leaching agent was added in a large surplus. Thus, each particle behaves like a batch reactor, in which the micro-mixing has no effect on the reactor performance (Crundwell, 1994b). The RZC leaching reaction depends on the sulfuric acid concentration; thus, it is not a zero-order reaction. Therefore, it was already expected that the Maximum Mixedness and Segregated Flow models would not led to identical results. Indeed, as shown in Fig. 9, the Segregated Flow led to higher conversion values (XZn) than the ones obtained with the Maximum Mixedness for a same

250 rpm 0.15 dXZn/dt

Fitted mean residence time, τ (min)

slurry was fed to the first reactor and then transported to the next reactors by overflow. Since all three reactor of the pilot plant have the same volume (6.0 L), the theoretical residence time in each reactor is given by τ = 6/Q. A similar procedure was used by other authors (Crundwell, 2000; Crundwell and Bryson, 1992) to assess the suitability of leaching models. From Fig. 7, it can be noticed that the PB model with the Segregated Flow approach did not provided a good representation of the pilot plant data. All the calculated values of zincite conversion (XZn) were higher than the experimental ones for each reactor and slurry flow rate. Consequentially, as the liquor acid concentration (CAf) is calculated from the zincite conversion values (Eq. (7)), all the model values for CAf were lower than the values obtained in the pilot plant. On the other hand, as it can be seen in Fig. 8, the results of PB model with the Maximum Mixedness approach were in an excellent agreement with the pilot plant data, for both zincite conversion and liquor acid concentration. In order to validate statistically the proposed PB model and to verify it with one of the two approaches for the flow pattern and mixing conditions is the most suitable, it was calculated the sum of squared (y exp y )2 ), between the model values ( y ) and the residuals (SSR= experimental data (yexp), and the mean absolute percentage error

0.07 0.06

Mixing cells with exchange (TISEx)

15

Time (min) Fig. 6. Effect of stirring speed on zinc conversion rate (dXZn/dt). Operating conditions: temperature = 40 °C, molar ratio = 1.0, solid/liquid ratio of 100 g.L−1.

different approaches, the Maximum Mixedness and the Segregated Flow. In order to compare these approaches and validate the model, simulations were done for the continuous RZC leaching under the operational conditions used in the pilot plant experiments (Table 4). The residence time distributions (RTD) in the pilot plant, required in the Segregated Flow approach, were determined in Section 3.1. The experimental and simulated values for the zincite conversion (XZn) and liquor acid concentration (CAf) as functions of the slurry flow rate (Q) are shown in Figs. 7 and 8. It is worth emphasizing that the 7

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0.16

(b) Acid concentration in liquor CAf (mol.L-1)

Zincite conversion, XZn

(a) 0.90 0.86 0.82 0.78

Segregated Flow Reactor 1 Reactor 2 Reactor 3 Pilot plant Q=0.213 L/min Pilot Plant Q=0.415 L/min

0.74 0.70

0.1

0.2 0.3 0.4 0.5 0.6 Slurry flow rate, Q (L.min-1)

Segregated Flow Reactor 1 Reactor 2 Reactor 3 Pilot Plant Q=0.213 L/min Pilot Plant Q=0.415 L/min

0.14 0.12 0.10 0.08 0.06

0.7

0.1

0.2 0.3 0.4 0.5 0.6 Slurry flow rate, Q (L.min-1)

0.7

Fig. 7. Model results using the Segregated Flow approach and the experimental data for the effect of the inlet slurry flow rate on the (a) zincite conversion and (b) acid concentration in liquor obtained in the pilot plant. Operating conditions: temperature = 40 °C, molar ratio = 1.0, and initial acid concentration = 0.5 mol.L−1.

reactor and a same residence time. It can also be seen in this figure that the difference between the zincite conversions obtained using these two approaches were smaller for higher residence times. The reason for this fact is that for higher residence times, the reactor behavior approaches the one of a batch reactor, in which the flow patterns and micromixing conditions have no effect on this reactor performance (Crundwell, 1994b). As stated by Crundwell (2005), the Segregated Flow is appropriate for operational conditions of low diffusivity and high viscosity. However, these conditions are unusual in leaching tanks, in which normally high stirring is applied to ensure adequate mixing and to reduce the effects of mass transfer in the liquid layer that surrounds the solid particles. Therefore, since the Segregated Flow approach did not present a good correlation with the experimental data from the RZC leaching in the pilot plant, it can be said that there was no segregation in the micromixing of the slurry phase. Crundwell (1994b) demonstrated theoretically that the best approach to the Population Balance modeling of leaching systems is considering the Maximum Mixedness conditions over the Segregated Flow ones. In the present work, this statement was experimentally verified by comparing the leaching pilot plant data with model simulations. With this procedure, it was also possible to validate the proposed model. In a direct application of the developed model, it can be predicted that the pilot plant could operate with the doubled capacity. Considering a slurry flow rate of 0.41 L.min−1, the residence time in

one reactor is around 15 min, which leads to 86% of zinc conversion at the third reactor (Fig. 9). If the slurry flow rate was doubled to 0.81 L.min−1, the residence time would be around 8 min and almost the same zinc conversion (85%) would be achieved in third reactor with this new operational condition. 4. Conclusion In the present work, the Population Balance was applied to develop a model for the continuous leaching of a roasted zinc concentrate. The flow patterns and mixing conditions in the reactors were incorporated in the model using the Maximum Mixedness and the Segregated Flow approaches. Experiments using a tracer in the liquid phase demonstrated that the flow patterns in the pilot plant reactors are similar to the ones observed in ideal continuous stirred-tank reactors (CSTR). Therefore, the Tanksin-Series model was adequate to describe the residence time distributions in the pilot plant reactors. Since it was not observed signs of slurry segregation, it was considered that the solid and the leaching fluid have the same residence time distributions. For the experimental leaching data of this work, the model results considering the Segregated Flow did not provide a good representation of the pilot plant data, resulting in higher conversion values than the experimental ones. This indicates that the conditions in which the Segregated Flow model is recommended (i.e. segregation in the micromixing of the slurry phase, low diffusivity, and high slurry viscosity)

(b)

0.14

0.86

0.12

0.82 0.78 0.74 0.70

0.16

Acid concentration in liquor CAf (mol.L-1)

Zincite conversion, XZn

(a) 0.90

Maximum Mixedness Reactor 1 Reactor 2 Reactor 3 Pilot Plant Q=0.213 L/min Pilot Plant Q=0.415 L/min

0.10

Maximum Mixedness Reactor 1 Reactor 2 Reactor 3 Pilot Plant Q=0.213 L/min Pilot Plant Q=0.415 L/min 0.1

0.2 0.3 0.4 0.5 0.6 Slurry flow rate, Q (L.min-1)

0.08 0.06

0.7

0.1

0.2 0.3 0.4 0.5 0.6 Slurry flow rate, Q (L.min-1)

0.7

Fig. 8. Model results using the Maximum Mixedness approach and experimental data for the effect of the inlet slurry flow rate on the (a) zincite conversion and (b) acid concentration in liquor obtained in the pilot plant. Operating conditions: temperature = 40 °C, molar ratio = 1.0, and initial acid concentration = 0.5 mol.L−1. 8

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Table 6 Comparison of the pilot plant data with the PB model results obtained using the Maximum Mixedness or the Segregate Flow approaches. Reactor

0.41

1 2 3 0.21 1 2 3 Sum of squared residuals Mean absolute percentage error (%)

Pilot plant

Maximum mixedness

Segregated flow

Pilot plant

Maximum mixedness

Segregated flow

0.782 0.835 0.857 0.808 0.849 0.862 – –

0.787 0.843 0.858 0.812 0.853 0.863 1 × 10−4 0.5

0.846 0.866 0.868 0.856 0.868 0.868 8 × 10−3 3.7

0.114 0.087 0.075 0.101 0.079 0.073 – –

0.117 0.079 0.071 0.094 0.073 0.069 2 × 10−4 6.2

0.077 0.067 0.066 0.072 0.066 0.066 3 × 10−3 20.4

One tank

(b) 0.9

(c) 0.9

Two tanks

XZn

XZn

(a) 0.9

Acid concentration in liquor CAf (mol.L−1)

Zincite conversion XZn

0.8

XZn

Slurry flow rate, Q (L.min−1)

0.8

Segregated Flow

0

10 20 30 40 Residence time - (min)

0.8

Maximum Mixedness

Maximum Mixedness

0.7

Three tanks

Segregated Flow

0.7 0

10 20 30 40 Residence time- (min)

Maximum Mixedness Segregated Flow

0.7 0

10 20 30 40 Residence time - (min)

Fig. 9. Simulations using the proposed PB model with both Maximum Mixedness and Segregated Flow approaches for the continuous RZC leaching in the pilot plant.

were not present at the RZC leaching in the pilot plant. On the other hand, excellent agreement between the experimental data and the PB model was obtained when the Maximum Mixedness approach was considered. This approach resulted in small mean absolute errors, 0.5 and 6.2% for the zincite conversion and acid concentration in the liquor, respectively. Therefore, it can be said that the developed model adequately accounts for the complex particulate phenomena that occur during leaching, the flow patterns, and the mixing conditions. In this perspective, the proposed model has potential applications, such as predicting the effect of operational parameters (e.g. particle size distribution, acid concentration, solid/liquid ratio, feed flow rate) on the reactor performance, which can lead to better decision-making and costs reduction. In conclusion, the present findings enlighten the incorporation of the flow patterns in real reactors into Population Balance models, with important implications in the modeling of continuous leaching process. In addition, a robust method is delivered for the design and control of leaching plants, contributing also to the prediction of problems faced in the operation of these plants.

D D63.2 E(t) fmass ks m M mB0 MMZnO nA0 nB0 nBf nFerrite, f

nZn, f PB Qin Qout RRB RZC SSR t TZnO V v0 X Xbatch XRD

Nomenclature a AAS B C(t) C0 CA0 CAf CFe,f CZn,f D

fraction of the inlet flow is exchanged with the stagnant zone (TISEx model) atomic absorption spectroscopy average rate of particles birth tracer concentration in the outlet stream tracer concentration in the inlet stream molar concentration of the leaching agent (sulfuric acid) in the feed stream molar concentration of the leaching agent (sulfuric acid) in the outlet stream (liquor) molar concentration of iron in the outlet stream (liquor) molar concentration of zinc in the outlet stream (liquor) particle diameter

average rate of particles death aperture of a sieve in with 63.2% of the material would not be retained residence time distribution function particle size distribution in a mass basis reaction rate constant RRB model parameter mass flow rate of solids in the reactor mass flow of RZC fed the reactor zincite molar mass molar flow of leaching agent (sulfuric acid) fed to the reactor molar flow of zincite fed to the reactor molar flow of zincite that was solubilized in the liquor molar flow of zinc from the zinc ferrite solubilized in the liquor molar flow of zinc in the liquor Population Balance inlet volumetric flow rate outlet volumetric flow rate Rosin-Rammler-Bennett model for size distribution roasted zinc concentrate sum of squared residuals time percentage of zincite (ZnO) in the RZC reactor volume volumetric flow rate of leaching solution fed to the reactor conversion of zinc or iron conversion achieved in a batch reactor X-ray diffraction spectroscopy

Greek symbols α η θ 9

adjustable parameter molar ratio between the acid and the zincite in the RZC added to the reactor dimensionless time

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time rate of change in particle diameter solid molar density covariance theoretical residence time mean residence time M residence time in the active zone (TISEx model) τ1 τ2 residence time in the stagnant zone (TISEx model) ψ population density function ψbatch(t) population density function in a batch reactor in a given reaction time

interests or personal relationships that could have appeared to influence the work reported in this paper.

ν(D) ρs σ/μ τ

Acknowledgments This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) Finance Code 001. The authors also acknowledge the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and the Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) for the support.

Declaration of Competing Interest The authors declare that they have no known competing financial Appendix A. Appendix

Tanks-in-Series with Exchange Model (TISEx). In the Tanks-in-Series with Exchange model (or mixing cells with exchange), each stirred tank is composed of two zone, as shown in Fig. A.1. The active zone, of volume V1, exchanges flow with the stagnant zone, of volume V2 (Dagadu et al., 2012; Stegowski et al., 2010).

Fig. A.1. TISEx model scheme.

A fraction aQ of the inlet flow rate (Q) is exchanged between the active and stagnant zones. The mean residence time in the active zone is given by τ1 = V1/Q, while in the stagnant zone it is given by τ2 = V2/aQ. Therefore, the total mean residence time of the tank is M = τ1 + aτ2 (IAEA, 2008; Stegowski et al., 2010). Considering the concentrations of tracer in the first zone (C1(t)) and in the second zone (C2(t)), the mass balance for tracer through the tank, gives the TISEx model equations:

dC1 (t ) 1 = [Cin (t ) dt 1

(1 + a) C1 (t )

dC2 (t ) 1 = [C1 (t ) dt 2

C2 (t )]

aC2 (t )]

(A.1) (A.2)

Therefore, with tracer injection experiments, knowing the concentration of the tracer in the inlet stream (C1(t)), it is obtained the tracer concentration profile in the outlet stream (C1(t)). Then, Eqs. (A.1) and (A.2) can be fitted to the experimental data, and the TISEx model parameters (τ1, τ2, a) determined.

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