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Modeling and Process Design of Intraparticle Adsorption in Single-Stage and Multistage Continuous Stirred Reactors: An Analytical Kinetics Approach Mohammad Outokesh and Ali Naderi Department of Energy Engineering, Sharif University of Technology, Azadi Ave., P.O. Box 113658639, Tehran, Iran

Ali Reza Khanchi* and Javad Karimi Sabet Jaber Ebne Hayyan National Research Laboratory, NSTRI, Tehran, Iran S Supporting Information *

ABSTRACT: Continuous adsorption in stirred reactors in the form of carbon in pulp (CIP) and resin in pulp (RIP) is an established process for the extraction of gold and uranium. Under the circumstance of intraparticle diffusion resistance, CIP and RIP have been accurately modeled by the Boyd’s series (reversible adsorption) and shrinking core model (irreversible adsorption). The present study, in its first part, introduces an analytical formula that most closely approximates both models. Using such formula, the study addresses a basic algorithm for optimization of single-stage continuous adsorption systems through linking of the major process variables. Furthermore, this study is devoted to developing an “analytical kinetics approach” for the design of multistage CIP and RIP processes via application of Glauekauf’s multiple series. Advantages of the new approach over the McCabe−Thiele “Equilibrium Approach” are (1) the incorporation of the kinetics and equilibrium into one unified model, and (2) accurate determination of the number of stages, reactor size, and optimum operational conditions.

1. INTRODUCTION The adsorption process is usually carried out in fixed-bed contactors in industry.1,2 Nevertheless, there are important cases such as resin-in-pulp (RIP) and carbon-in-pulp (CIP) processes of gold and uranium extractions that favor the application of stirred reactors.3−10 Adsorption in the stirred contactors can be performed in any of the three possible modes of batch, liquid continuous, and continuous (Figure 1), depending upon the volume and concentration of the processed liquid. In the RIP and CIP, where concentration of the gold or uranium is in the ppm range, and volume of the solutions tremendous,7,11 attaining a reasonable throughput involves using of the multistage continuous stirred tank reactor (CSTR). Prevailing of the “perfect mixing” and “steady state” conditions in CSTRs12 causes the concentration of the entering liquid to decrease from the initial value of C0 to the uniform value of Cf; despite the fact that, for the adsorbent, uptake proceeds gradually (Figure 2). During the uptake, constancy of the Cf concentration at the outer surface of resin particles greatly simplifies the mathematics of the adsorption. The current approach in the gold extraction industry is to develop highly selective resins3 for the uptake of the Au(CN)2− species. Selective resins are normally characterized by a slow rate of uptake that obeys the intraparticle kinetics.13,14 In the instance of uranium, the RIP process uses anion-exchange resins with particles larger than 1 or even 3 mm.7,15,16 Such large resin beads are again subject to intraparticle resistance kinetics. The discussion of the last paragraphs shows that modeling of the intraparticle diffusion under constant boundary conditions is of great importance in the economically significant sectors of © 2013 American Chemical Society

industry. One of the most famous equations for modeling of this problem is the Boyd’s infinite series:17 U (t ) = 1 −

6 π2

∞

∑ n=1

⎛ Dtπ 2n2 ⎞ 1 ⎜− ⎟ exp n2 r02 ⎠ ⎝

(1)

where U(t) stands for the fractional approach to equilibrium, D is the diffusion coefficient, and r0 denotes the radius of the particles. Equation 1 is the solution of Fick’s equation (eq 2) for a spherical bead under boundary conditions and initial conditions (eqs 3).18 ∂qA ∂t ∂q ∂r

=

1 ∂ ⎛ 2 ∂qA ⎞ ⎜r D ⎟ ∂r ⎠ r 2 ∂r ⎝ =0

(2)

for t > 0 (3-1)

r=0

q|r = r0 = q1 = constant q = q0 = constant

for t > 0 for 0 < r < r0

(3-2)

at t = 0

(3-3)

where q indicates the concentration of the adsorbed ions (mol/cm3) in the sorbent matrix. Glauekauf worked out a more-general case, in which the resin is subjected to the successive boundary conditions shown in eqs 4:19 Received: Revised: Accepted: Published: 305

July 6, 2013 November 20, 2013 November 27, 2013 November 27, 2013 dx.doi.org/10.1021/ie402142z | Ind. Eng. Chem. Res. 2014, 53, 305−315

Industrial & Engineering Chemistry Research

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Figure 1. Three possible modes of adsorption in a stirred reactor: (a) in batch mode, both liquid and solid are initially filled in the reactor and after elapsing residence time (ts) discharged together; (b) in liquid-continuous process, liquid continuously flows through the reactor, while the loaded resin remains for a definite period; and (c) in fully continuous mode, both solid and liquid are continuously feeding (to) and discharging (from) reactor.

Figure 2. Temporal variation of liquid concentration (C(t)) and resin loading (q(t)), in (a) batch reactors and (b) fully continuous reactors.

q|r = r0 = q1 = constant for 0 ≤ t ≤ t1

(4‐1)

q|r = r0 = q2 = constant for t1 ≤ t ≤ t 2

(4‐2)

q|r = r0 = q3 = constant for t 2 ≤ t ≤ t3

(4‐3)

⋮

⋮

ions uptake by chelating resins, but later, it was also applied to the reversible uptakes.20 The SCM equation in the case of constant boundary conditions is written as 1 − 3(1 − U )2/3 + 2(1 − U ) =

⋮

where C0 (mmol/cm ) indicates the concentration of the solution, D (cm2/s) is the diffusion coefficient of the ions in the resin, r0(cm) is the radius of the resin particles, q* is the adsorption capacity of the resin, t (s) represents the adsorption time, and U is an abbreviated version of the term U(t). The first part of the current investigation was aimed at finding a new analytical formula that could provide a close approximation of both of the Boyd’s series and the SCM equation (i.e., reversible and irreversible adsorption). Using such a formula, the second part of study was devoted on developing an algorithm for the optimum design of a singlestage continuous adsorption reactor. At present, the most established method for estimating the number of stages in a multistage countercurrent adsorption process is the McCabe−Thiele method. This method, despite its simplicity, suffers from the following drawbacks: (1) It assumes equilibrium between the liquid and resin leaving the stages, while, in reality, the equilibrium is approached only at an infinite residence time. Hence, the McCabe−Thiele method often underestimates the number of adsorption stages.

Solution of Fick’s equation (eq 2) to initial conditions (eq 3-3) and boundary conditions (eq 3-1) and eqs 4 is given as the following multibracketed series:19 ∞

∑ n=1

⎛ Dπ 2n2t ⎞⎤ 1 ⎟⎥ exp⎜ − 2 n r02 ⎠⎥⎦ ⎝

⎡ 6 + (q2 − q1)⎢1 − 2 ⎢⎣ π

∑

⎡ 6 + (q3 − q2)⎢1 − 2 ⎢⎣ π

∑

⎛ Dtπ 2n2(t − t ) ⎞⎤ 1 1 ⎥ ⎟ exp⎜ − 2 n r02 ⎝ ⎠⎥⎦

∞ n=1 ∞ n=1

⎡ 6 + (qk − qk − 1)⎢1 − 2 ⎢⎣ π

⎛ Dtπ 2n2(t − t ) ⎞⎤ 1 2 ⎥ ⎟ + ··· exp⎜ − 2 n r02 ⎝ ⎠⎦⎥

∞

∑ n=1

(6)

3

q|r = r0 = qk = constant for tk − 1 ≤ t ≤ tk (4‐4)

⎡ 6 U (t ) = q1⎢1 − 2 ⎢⎣ π

C0D r0 2q*

⎛ Dtπ 2n2(t − t ) ⎞⎤ 1 k−1 ⎥ ⎟ exp⎜ − 2 n r02 ⎝ ⎠⎥⎦

(5)

Another equation that is widely used to model intraparticle diffusion is the shrinking core model (SCM).20,21 SCM was originally developed for irreversible adsorptions such as metal 306

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the results were tabulated in Table 2. The error percentage in Table 2 is defined as follows (here, τ′ = Dπ2t/r02):

(2) The last stage’s loading of the resin (qN), estimated from the McCabe−Thiele method, often exceeds the output loading (qout), given by an overall mass balance; i.e., the McCabe−Thiele method violates the law of mass conservation. In order to overcome the above drawbacks, the third part of this study worked out an “analytical kinetics approach” for the design of multistage CIP and RIP processes. The new approach, which was based on Glauekauf’s multiple series (described by eq 5), is able to accurately determine the number of stages, reactor size, and optimum operational conditions.

error (%) =

square root modified square root Vermeulen a

6 Dt 1/2 r0 π 6 Dt 1/2

( ) U (t ) = r ( π ) U (t ) =

0

U (t ) =

0 < U(t) < 0.3 3Dt

(7)

−

⎛ ψD2π 4t 2 ⎞ 3Dt ⎟ ⎜− exp r0 4 ⎠ r0 2 ⎝

(8)

⎤ ⎛ βt ⎞ 6⎡ πβt − ⎜ ⎟ exp( −ψβ 2t 2)⎥ 2⎢ ⎝ ⎠ ⎣ ⎦ 2 π

(9)

In contrast to MSRF, whose errors at proximity of the equilibrium are rapidly growing, the new eq 8 keeps a maximum error of 1.26%, even at the U(t) = 0.999446 level (see Table 2). Taking the entire range of adsorption (i.e., 0 ≤ U(t) ≤ 1) into consideration, eq 8 is even more effective than Vermeulen’s formula (see Table 2). The slightly better behavior of Vermeulen’s formula in the vicinity of equilibrium is of little practical importance, because the uptake kinetics at this level is so slow that most of industrial processes are intentionally terminated in it.22 To further elucidate this point, a brief look at Table 2 is useful; as for the function U(t), it takes equal time to ascend from 0 to 0.92, or from 0.92 to 0.99. In the context of

0 < U(t) < 0.5

r02

1/2 ⎡ ⎛ −Dtπ 2 ⎞⎤ U (t ) = ⎢1 − exp⎜ 2 ⎟⎥ ⎝ r0 ⎠⎦ ⎣

× 100

where ψ is an adjustable parameter whose precise setting results in the accurate fitting of the Boyd’s series. It was found that exact fitting is provided by ψ = 0.0027. By introducing a new parameter, β = Dπ2/r02, eq 8 can be rewritten as

application range (according to ref 18)

−

6 ⎛⎜ Dt ⎞⎟ r0 ⎝ π ⎠

1/2

U (t ) =

Table 1. Various Approximations of Boyd’s Equationa formula

U (τ′)Boyd

As Table 2 indicates, in the main course of the adsorption (i.e., 0.0001 ≤ U(t) ≤0.95), the modified square root formula (MSRF) presents a better approximation than the square root formula and Vermeulen’s formula. This result is in contrast with the statement of ref 18, as that reference considers the range of applicability of MSRF as U(t) ≤ 0.5.18 The accuracy of MSRF at the vicinity of equilibrium (U(t) ≥ 0.95), can be enhanced by multiplying its second term in an exponential factor as follows:

2. THEORY 2.1. Search for the Best Closed Approximation of Boyd’S Equation. Convergence of the Boyd’s series, particularly in the region of small t, is quite slow, such that, often, summation over 200 terms is necessary. Thus, some closed formulas were developed as its alternatives18 (see Table (1).

name

U (τ′)Formula − U (τ′)Boyd

0 < U(t) < 1

In all models, τ denotes dimensionless time (τ = Dt/r02).

In search for finding an appropriate closed alternative for the Boyd’s series, we first examined the closeness of approximation of it with the formulas of Table 1. The U(t) values for this purpose were obtained by summing over the first 200 terms of the Boyd’s series, as well as the aforementioned formulas, and

Table 2. Comparison of Accuracy of the Suggested Formula with the Older Formulas in Approximation of the Boyd’s Equationa U(τ′)

Relative Error (%)

τ′

Boyd’s equation (eq 1)

square root

modified square root

Vermeulen

new (eq 8)

square root

modified square root

Vermeulen

new (eq 8)

0.0001 0.001 0.01 0.10 0.20 0.40 0.70 0.90 1.20 1.50 2.06 2.40 2.50 2.73 3.01 3.41 4.12 7.00

0.01075 0.03376 0.10437 0.31010 0.42087 0.55973 0.68862 0.748567 0.815574 0.863952 0.922477 0.94484 0.950091 0.960349 0.970033 0.980013 0.990124 0.999446

0.01078 0.03408 0.10778 0.34084 0.48202 0.68168 0.90177 1.022517 1.180701 1.320064 1.546974 1.669764 1.704195 1.780864 1.869961 1.991795 2.187751 2.851664

0.01076 0.03378 0.10474 0.31043 0.42120 0.56005 0.68892 0.748847 0.815807 0.863947 0.920573 0.939976 0.9440 0.95073 0.954686 0.953368 0.934949 0.723116

0.01 0.03161 0.09975 0.30848 0.42576 0.57418 0.70952 0.770344 0.835946 0.881402 0.934102 0.953563 0.958079 0.966841 0.975043 0.983424 0.991844 0.999544

0.010747 0.033779 0.104741 0.310432 0.42121 0.560099 0.689201 0.749444 0.817223 0.866709 0.927708 0.951237 0.956720 0.967267 0.976803 0.985556 0.991070 0.986889

0.326 0.933 3.267 9.913 14.529 21.787 30.953 36.596 44.769 52.793 67.697 76.724 79.371 85.439 92.772 103.241 120.957 185.324

0.043 0.033 0.355 0.106 0.078 0.057 0.044 0.0373 0.0285 −0.0006 −0.206 −0.514 −0.641 −1.001 −1.582 −2.718 −5.572 −27.648

−6.919 −6.368 −4.427 −0.522 1.162 2.582 3.035 2.909 2.497 2.019 1.260 0.923 0.8407 0.676 0.516 0.348 0.173 0.0098

0.043 0.033 0.351 0.111 0.08 0.065 0.083 0.117 0.202 0.319 0.567 0.677 0.697 0.720 0.697 0.565 0.0954 −1.256

In calculation of the U(τ) by Boyd’s equation, 500 terms were considered in the summation. Parameter ψ in eq 8 was taken as 0.0027. In this table, τ′ = βt = π2τ.

a

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It should be pointed out that the value of ψ that provides the best fitting of the SCM by eq 8 is 0.01, which is different from that determined from the Boyd’s series (ψ = 0.0027). This can differentiate the reversible adsorption (modeling by the Boyd’s series) from irreversible uptake, which is presented by SCM.20,23 Figure 3 shows a significant correlation between the SCM and new model. From Table S1 in the Supporting Information, we can also deduce the following important points: • In almost all cases, a fairly close correlation (i.e., R2 = 0.999) is observed between the SCM and new model. • D SCM and β (or, in other words, D Boyd ) vary proportionally. • For a given value of β, DSCM varies with the initial concentration of the solution (i.e., C0). This behavior is not astonishing, because the SCM model assumes an irreversible reaction between the diffusing ions and the resins, whose rate of progress, in addition to diffusivity (DSCM), depend on the concentration of the solution20,23 (eq 6). 2.3. Procedure for Design of a Single-Stage RIP or CIP Adsorption Reactor. Referring to Figure 1, the mass balance of solute about a continuous adsorption reactor can be written as12

the chemical reactor theory, increasing the reaction time corresponds to increasing the reactor volume.21 However, while increasing the reaction time proportionally increases the volume (i.e., capital cost) of the system, the economic gain due to enhancement of the adsorption level in the region of U(t) > 0.92 is almost negligible. 2.2. Equivalence between the Suggested Formula and the SCM Model. As was mentioned in the Introduction, the SCM model has been successfully employed for modeling intraparticle adsorption kinetics. Since the suggested formula (eq 9) is working in the same area, it is useful to determine how closely the two equations match each other. This anticipation is based on numerous reports in which the same intraparticle adsorptions were modeled fairly well by both the SCM and distributed models. Unfortunately, most of the published data are not useful for our purposes, because they were obtained via batch experiments, in which the concentration of the solution (C), instead of eq 6, obeys the following SCM equation:23 1 − 3(1 − U )2/3 + 2(1 − U ) =

D r0 2q*

∫0

t

C dt (10)

Our procedure for comparing the suggested formula with the SCM model was to select an arbitrary value for β in eq 9 and then change the parameter D in eq 6 until the closest regression is obtained. The closeness of fitting was evaluated by the coefficient of determination (i.e., R2), which is defined as SS R = 1 − err SStot

Q (C0 − Cf ) =

(11-1)

∑ (yi − fi )2

(11-2)

SStot =

∑ (yi − yi ̅ )2

(11-3)

(12)

where Q and ṁ denote the volumetric flow rate of the liquid and the mass flow rate of the resin, respectively, and ρS stands for the density of the resin particles. In eq 12, q has units of mg/cm3 or mol/cm3, where cm3 denotes the unit volume of resin matrix. In most of the adsorption processes, the initial loading (q0) is zero and the final loading (qf) can be expressed in terms of equilibrium loading (q∞) and the previously introduced function, U(t)18:

2

SSerr =

ṁ (q − q 0 ) ρS f

⎛ ṁ ⎞ Q (C0 − Cf ) = ⎜⎜ ⎟⎟ q∞U (tS) ⎝ ρS ⎠

where yi and f i indicate the values of SCM data and the suggested model, respectively, and y ̅ stands for the average of yis”. Figure 3 illustrates a typical case of regression of the SCM

(13)

where tS signifies the residence time of the resin. The function U(t) in the above equation can be either the Boyd’s series, or any of its closed approximations, such as MSRF or the new eq 9. Here, a question may arise as: Is it possible to separate the fully loaded resin from the partially loaded one in the outlet of the reactor, and only allow the former to leave the reactor? This inquiry is in fact related to the residence time distribution of the solid adsorbent in the RIP−CIP reactors, and is addressed in some detail in Appendix A. In a CSTR reactor, the equilibrium loading (q∞) is related to the final concentration (Cf) by the adsorption isotherm (f(Cf)); furthermore, the average residence time of the resin is given by tS =

φSρS V (14)

ṁ

where V shows the volume of the reactor and φS is the holdup (i.e., volume fraction) of the resin inside the reactor. Incorporation of the above expressions into a single relation will result in Figure 3. Regression of SCM model (marked points), with the suggested formula (solid line) for the case: C0 = 3 mmol/L, r0 = 0.01 cm, β = 10−5 s−1, and DSCM = 11 × 10−7 cm2/s.

⎛ ṁ ⎞ Q (C0 − Cf ) = ⎜⎜ ⎟⎟ f (Cf )U (tS) ⎝ ρS ⎠ ⎛ ṁ ⎞ ⎛φ ρ V ⎞ = ⎜⎜ ⎟⎟ f (Cf )U ⎜ S S ⎟ ⎝ ṁ ⎠ ⎝ ρS ⎠

model by the suggested formula, and Table S1 in the Supporting Information lists the results of such regression for a wider range of D and β. 308

(15)



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The parameters on the left-hand side of eq 15 are specified by the process requirement and, thus, are fixed. For instance, for a plant that is designed to process 5000 m3 uranium solution/day from an initial concentration of 300 ppm to a final concentration of 5 ppm, all values of Q, C0, and Cf are known. On the right-hand side of eq 15, however, there are three adjustable parametersV, m, and φSthat can be specified by the design engineers to optimize the adsorption process. Note that only two of these variables are independent. If, for instance, the values of φS and ṁ are specified, the volume of the reactor (V) will be set by eq 15. Referring to the last paragraph, and by drawing an analogy to the other chemical engineering processes, it follows that there is only one ordered pair (ṁ ,φS) that minimizes the total cost of the plant. Discussion about the details of such cost optimization is beyond the scope of this study; however, the effects of the independent variables on the total cost can be qualitatively assessed as follows: • By increasing ṁ , on one hand, the energy cost for transferring the resin, and the attrition rate of the resin, are increased; however, on the other hand, the lower amount of the resin loading (q∞), which is arisen from a higher value of ṁ , gives rise to a higher rate of uptake and, in turn, reduces the required residence time of the resin. Lowering the residence time according to eq 14 decreases size of the reactor, and brings about an appreciable savings of the capital cost. • Increasing the holdup also can have some contradictory effects, because, according to eq 14, for a defined residence time tS, the larger values of φS correspond to the smaller sizes of the reactor (i.e., lower capital costs); while at the same time, the rising of the holdup increases the inventory of the resin, and its rate of attrition, and thus adds to the total costs of the plant. Overall, the preliminary cost analysis showed that, except at very large values of φS, increasing the holdup decreases the total cost. Note that, because of the hydrodynamic limitations, neither φS nor ṁ can increase limitlessly. In fact, in the RIP process of uranium and nickel extraction, the upper limit of holdups has been reported as φS = 0.15 and φS = 0.3, respectively.24 For the gold RIP, lower values of holdup as low as φS = 0.01 are used, but in gold CIP, φS can be chosen to be 0.05.10 The above explanations address the ewffect of the major process variable, as well as the influential factors, on the total cost of the plant. Based on such discussions, a simplified algorithm for attaining the optimum design condition was devised in Figure 4. As can be seen from Figure 4, the application of eq 15 in designing a RIP system needs a good estimation of the diffusion coefficient D or its alternative β factor. Two widely employed methods for this purpose include:13,14 (1) the shallow bed technique, in which a continuous stream of the fresh solution is percolated through a thin layer of the resin, and (2) the conventional batch experiment. In the shallow bed method, because of the presence of constant boundary conditions, one can use the Boyd series, the Vermeulen equation, or the new eq 9 (with ψ = 0.0027) to obtain the coefficient D, provided that intraparticle diffusion is the rate-controlling mechanism. When adsorption proceeds via an irreversible reaction (e.g., ion adsorption by chelating resins), one can either use the SCM (eq 6) or the new eq 9 with a parameter ψ = 0.01 to estimate the diffusion coefficient.

Figure 4. Basic flowchart for the optimization of a single-stage adsorption process.

Since, in the shallow bed technique, the kinetics behavior must be obtained by laborious analysis of the resin granules, researchers often prefer to work with the more-convenient batch method. In the batch experiments, concentration of the solution is variable and, thus, a more-complex kinetic model is needed to estimate the coefficient D. A famous distributed (i.e., reversible) model for this condition is the Paterson equation:18 U (t ) =

w ⎧ 1 ⎨1 − [γ exp(γ 2τ )(1 + erf(γτ1/2) γ−λ w + 1⎩ ⎫ − λ exp(λ 2τ )(1 + erf(λτ1/2)]⎬ ⎭

(16)

In this equation, τ = and w = qV ̅ ̅ /C0V, where q̅ and V̅ are the saturation capacity (mmol/cm3) and total volume (cm3) of the resin in the solution, respectively, and C0 and V are the initial concentration and volume of the solution, respectively, with the same units as q̅ and V̅ , respectively. Parameters γ and λ are roots of the equation x2 + 3xw − 3w = 0. All of the parameters of eq 16 except D are known beforehand, and this parameter should be obtained by fitting the experimental uptake data with the above equation. For the case of irreversible adsorption in a batch experiment, one can use the SCM (eq 10) to estimate the diffusion coefficient. However, after DSCM has been determined, it must be converted to β, using Table S1 in the Supporting Information. This β value then can be used in eq 15 to design a continuous reactor, provided that U(t) in eq 15 is given by eq 9. 2.4. Procedure for Designing Multistage Countercurrent RIP and CIP Adsorption Processes. Figure 5 exhibits the basic flowsheet of a continuous countercurrent adsorption process that includes the nomenclature for flow rates, concentrations, and stage numbers. In our convention, solution and sorbent leaving a stage get the number of that stage. A material balance between the left end of the above flowsheet and the point immediately before stage k gives Dt/r02

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equal to its internal concentration. Using f(C) to represent the adsorption isotherm, one can write qi ∗ = f (Ci) Figure 5. Basic flowchart of a countercurrent continuous adsorption process.

Q (C in − Ck) =

ṁ (q − qk − 1) ρS out

For the sake of simplicity, the following abbreviation will be used for the bracketed series of eq 19: ⎡ 6 U ′(itS) = ⎢1 − 2 ⎢⎣ π

(17)

∞

∑ n=1

∑

⎡ 6 + (q3* − q2*)⎢1 − 2 ⎢⎣ π

∑

∞ n=1 ∞

(22) (18)

Three first examples of application of this formula are

n=1

∞ n=1

⎛ Dtπ 2n2t ⎞⎤ 1 S ⎥ ⎟ exp⎜ − 2 n r02 ⎠⎥⎦ ⎝

(23-1)

q2 = q1*U ′(2tS) + (q2* − q1*)U ′(tS)

(23-2)

(23-3)

Now, the procedure for designing a multistage adsorption process, using eq 14 and eqs 17−20 will be as follows: • Specify the resin holdup (φs); the φs values must be in the technically permitted range for the intended process (see section 2.3 for more details). • Specify the mass flow rate of resin (ṁ ). For any given set of (φs, ṁ ), the number of stages and the reactor volume can be figured by the following method: (1) Set the number of stages (N) in Figure 5. The starting point in selecting N is the minimum number of stages that are obtained from the McCabe−Thiele method for the specified value of ṁ . Note that Q is process-invariant, and the slope of the operating line in the McCabe− Thiele diagram is dependent only on ṁ . (2) Select a reactor volume V. (3) Calculate the residence time tS using eq 14 (4) Use eq 20 and C1 = Cout to obtain q*1 (5) Use eq 23-1 to estimate q1 (6) Calculate C2 by using eq 17. (7) Repeat the above steps (4)−(6) to obtain q2*, q3* (by eq 23-2), and C3. In the same way, repeat those steps to determine all values of q*i , qi, and Ci+1, for 3 ≤ i ≤ N. (8) If qN, obtained from step (7), is equal to qout calculating from eq 18, the values of N and V are correct; otherwise, the aforementioned trial-and-error process must be repeated by first changing the volume, and then the number of stages, to arrive at an acceptable answer. For the sake of simplicity, the procedure described by the above steps (1)−(8) is hereafter referenced by the expression “O−K−N”.

⎛ Dtπ 2n2(k − 2)t ) ⎞⎤ 1 S ⎥ ⎟ exp⎜ − 2 n r0 2 ⎝ ⎠⎥⎦

∑

q1 = q1*U ′(tS)

q3 = q1*U ′(3tS) + (q2* − q1*)U ′(2tS) + (q3* − q2*)U ′(tS)

⎛ Dtπ 2n2(k − 1)t ⎞⎤ 1 S ⎥ ⎜− ⎟ exp n2 r0 2 ⎝ ⎠⎥⎦

⎡ 6 + ··· + (qk* − qk*− 1)⎢1 − 2 ⎢⎣ π

(21)

+ (q3* − q2*)U ′((k − 2)tS)+ ··· + (qk* − qk*− 1)U ′(tS)

⎛ Dπ 2n2kt ⎞⎤ 1 S ⎥ ⎟ exp⎜ − 2 n r0 2 ⎠⎥⎦ ⎝

⎡ 6 + (q2* − q1*)⎢1 − 2 ⎢⎣ π

n=1

⎛ Dπ 2n2it ⎞⎤ 1 S ⎥ ⎜− ⎟ exp n2 r0 2 ⎠⎥⎦ ⎝

qk = q1*U ′(ktS) + (q2* − q1*)U ′((k − 1)tS)

In industrial applications, the quantities on the left side of eq 18 are specified by the process requirements and, thus, are invariant; however, on the right-hand side, by changing the adsorbent flow rate ṁ , loading of the solute on the outgoing resins (qout) is changed. Note that, normally, qin = 0. In the conventional (i.e., McCabe−Thiele) method of design of the countercurrent uptake processes, the solution and sorbent leaving a stage are assumed to be in equilibrium.25 Thus, by drawing the regular staircase construction between the equilibrium curve and the operating line, the number of stages is obtained. Figure 6 shows an example of application of this method whose calculation details will be given in the Results and Discussion section. It is known that attaining equilibrium in a process requires an infinite amount of time. This means that, in the McCabe− Thiele method, the residence time of each adsorption stage and, in turn, the volume of each adsorption reactoris infinitely long (large). In addition, the loading of the last stage of the process (qn) that is obtained by the projection method from the McCabe−Thiele diagram almost always surpasses qout or the outlet loading of resin obtained from the overall mass balance in eq 18. This happens because the number of stages in the McCabe−Thiele diagram is an integer. To overcome this type of problem, we exploited a modified form of the Glauekauf equation (eq 5) as follows. Assume that the residence time of resin in all successive reactors of Figure 5 is equal to tS. For the resin that has passed k stages of the adsorption cascade; the Glauekauf equation (eq 5) is written as ⎡ 6 qk = q1*⎢1 − 2 ⎢⎣ π

∞

∑

Using the above convention, eq 19 can be rewritten as

Equation 17 represents the operating line. The unknown quantity, qout, in this equation can be obtained by writing a mass balance around the entire cascade: ṁ Q (C in − Ck) = (qout − qin) ρS

(20)

3. RESULTS AND DISCUSSION In order to demonstrate the applicability of the developed O− K−N method for designing a multistage continuous adsorption process, we worked out the two following practical examples: an RIP process for the recovery of uranium from an acidic ore leachate and CIP for the recovery of gold from a cyanidation leachate.

(19)

where q*i shows the equilibrium loading of the resin in equilibrium with the output concentration of stage i (Ci), and tS is given by eq 14. Note that, because of the “perfect mixing” condition, the output concentration of a stirred reactor (Ci) is 310



Article

uranium and gold uptake was controlled by intraparticle masstransfer resistance.7,10 The second worthy point for consideration pertains to the reversibility of uranium and gold adsorption by the abovementioned sorbents. The normal procedure to identify whether an uptake reaction is reversible or not is to perform Fourier transform infrared (FTIR) analysis of the adsorbent before and after the adsorption. If an ion is sequestered on an adsorbent irreversibly, the formation of a covalent bond will shift the peaks of the FTIR spectrum remarkably. References 7 and 10 did not provide the FTIR data of their samples. However, for the following reasons, it seems that the intended adsorption reactions are reversible: (1) Strongly basic anion exchanger resins with a structure similar to that of AMn adsorb uranium selectively and reversibly. Because of this property, they have been utilized in the RIP process of uranium extraction for more than 60 years.11 Regeneration of these resins can be readily performed by nitric acid, sulfuric acid, or other inexpensive reagents.11 (2) Adsorbent-grade active carbon (e.g., G210) adsorbs and desorbs Au ions in a slightly different fashion. In adsorption, it adsorbs gold as the Au(CN)2− complex; however, in commercial desorption media, the speciation of gold changes and the formed new species no longer tends to remain on the carbon surface.30 Identifying this property has led to extensive application of carbon in the hydrometallurgical extraction of gold for several decades.3 3.1. Uranium RIP Process. Figure 6 depicts the McCabe− Thiele diagram for the uranium RIP process with the conditions cited in ref 7. Two different mass flow rates of resins were considered, for which the calculated values of the loadings are given as follows:

RIP Process for Recovery of Uranium from Acidic Ore Leachate (ref 7): • resin type: strongly basic anion exchanger • resin name: AMn, Synthesized in All the Russian Institute of Chemical Technology • diffusion coefficient: D = 1 × 10−8 cm2/s • particle radius: r = 0.55 mm • Cin = 200 ppm • Cout = 1 ppm • adsorption isotherm: q = 0.75C0.69, where q is given in units of mg U/cm3 resin and C is given in units of mg/L • liquid flow rate: Q = 0.018 m3/s • resin holdup: φs = 0.05 • density of resin: ρ = 1.1 g/cm3 (estimated from the data of ref 12) CIP for Recovery of Gold from Cyanidation Leachate (ref 10): • adsorbent: activated carbon with tradename G210, made from coconut shell by PICA SA France • diffusion coefficient: D = 1 × 10−8 cm2/s • particle radius: r = 0.5 mm • Cin = 50 ppm • Cout = 1 ppm • adsorption isotherm: q = 34.08C0.393, where q is given in units of mg Au/cm3 resin and C is given in units of mg/L • liquid flow rate: Q = 0.014 m3/s • resin holdup: φs = 0.05 • density of resin: ρ = 1.2 g/cm3 Before we detail the calculations that have been undertaken, it is worthwhile to draw attention to two important points about the employed data. The first point is about the kinetics regime that prevails in the adsorption of uranium and gold by the aforementioned adsorbents. Generally speaking, an adsorption reaction, depending on the concentration of the solution, the size of the adsorbent particles, and intensity of the agitation, can proceed via one of the following cases: intraparticle resistance, liquid film controlled, or the mixed intraparticle−liquid film regime. As for the adsorption of gold and uranium by active carbon and resins, all of the aforementioned cases have been reported in the literature;10,26−29 however, since our analytical approach is only applicable to intraparticle resistance systems, we used the data of refs 7 and 10, in which the kinetics of

ṁ = 330 g/s → qout = 11.94 mg U/cm 3 resin and qN = 14.92 mg U/cm 3 resin ṁ = 220 g/s → qout = 17.91 mg U/cm 3 resin and qN = 18.86 mg/(mg U/cm 3 resin)

The number of stages in the aforementioned cases was found to be 3 and 4 by the graphical method (see Figure 6); however, with such numbers of stages, the last stage loading (i.e., qN) would exceed qout or the output loading that is obtained from the overall mass balance of the solute. If, instead of the McCabe−Thiele scheme, the new O−K−N method is applied to the design problem; it can result in the optimum number of stages, as well as the optimum reactor volume. Table S2 in the Supporting Information illustrates application of the new method on such a problem (see the end of the manuscript). The data from this table were obtained by multiple running of a computer program that worked with the O−K−N algorithm for a fixed set of (φS, ṁ ). An acceptable result in Table S2 in the Supporting Information, denoted by the symbol “A”, is the one whose qN value differs from qout by only ±5%. The last column of Table S2 in the Supporting Information indicates the construction cost of the adsorption reactors, obtained by multiplication of the number of stages by the price of a reactor unit. According to ref 31, the price of a reactor is the summation of the price of its vessel and that of its driving

Figure 6. McCabe−Thiele diagram for the recovery of uranium via the RIP process. In this figure, ṁ = 220 g/s; other conditions are given in the text. 311



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3.2. Gold CIP Process. Analogous to the uranium RIP process, the study first attempted to determine the equilibrium number of stages for the gold CIP process. The results that were obtained by the McCabe−Thiele method are depicted in Figure 7. The calculated values of the loading for the two considered cases are

system (electromotor + gearbox). The reported formulas for estimation of these price components are31

C Drive = 2590P 0.54

(24)

⎛ 1050V ⎞0.72 ⎟ C Vessel = 14 ⎜ ⎝ 3.7854 ⎠

(25)

C Total = (C Vessel + CDrive)n

(26)

where n denotes the number of stages, P the power of the electromotor (in units of horsepower, hp), and V the volume of the liquid in the reactor. The parameters CDrive, CVessel, and CTotal are all reported in U.S. dollars. Details of derivation of eqs 24 and 25, and method of estimation of power of an agitator, are given in Appendix B. Even a brief look at Table S2 in the Supporting Information reveals the existence of a local minimum in the total construction cost of the uranium adsorption process. For ṁ = 330 g/s, this optimum occurs at V = 48 m3 and n = 5 stages; however, for ṁ = 220 g/s, the minimum is attained at V = 30 m3 and n = 9 stages. The optimum numbers of stages obtained via this method are evidently different from those achieved using the McCabe−Thiele method. The discussion of the last paragraph demonstrates the necessity of undertaking a global (i.e., wider) optimization for the adsorption process. This optimization can be accomplished through the algorithm of section 2.4 by selecting different ordered pairs (φS, ṁ ), and minimizing the total cost of the plant, including all of its capital and operational components. The results of such optimization will be the optimum number of stages, optimum reactor volumes, and eventually optimum resin flow rate and resin holdup (φS, ṁ ). Undertaking global optimization requires access to a comprehensive set of the cost components whose finding is evidently outside the scope of the present study. In the following, we instead have briefly tried to address a qualitative description of the cost components. Capital cost includes: • construction costs of the reactors, which are dependent on the number of stages and volume of each reactor unit; • the cost of the liquid and solid transferring systems, including air lift pumps (if present), liquid pumps, screens, etc. (the transferring cost is principally determined by the magnitude of Q and ṁ and the number of stages); • indirect capital costs including land price and building, electrical, utility, instrumental, installation, commissioning, etc.; and • the total cost of the resin (CResin) is determined from its inventory as C Resin = C R nVφS ρS

Figure 7. McCabe−Thiele diagram for the recovery of gold via the CIP process. In this figure, ṁ = 7.2 g/s; other conditions are given in the text.

ṁ = 6 g/s → qN = 114.23 mg Au/cm 3 resin and qout = 137.19 mg Au/ cm 3 resin ṁ = 7.2 g/s → qout = 114.33 mg Au/cm 3 resin and qN = 125.40 mg Au/cm 3 resin

In both of the above cases, the number of stages is three (3); however, in the second case, the last-stage loading of the resin (i.e., qN) exceeds the mass balance loading (qout). Application of the new O−K−N method on the above CIP process resulted in Table S3 in the Supporting Information (see the end of the manuscript). The acceptable data (denoted by the symbol “A”) in Table S3 in the Supporting Information are those whose last-stage loading satisfies the following relation: qN ∈ [0.95qout , 1.05qout]

(28)

N ≥ NMcCabe−Thiele

(29)

Similar to the case of uranium, the gold CIP process indicates some local optimum points in the last column of Table S3 in the Supporting Information. Therefore, it shall similarly render an absolute minimum in its total costs. This absolute minimum point shall be obtained by undertaking the same global optimization procedures that was described in section 3.2.

(27)

where CR indicates the unit price of the resin (in $/kg resin) and ρS denotes its density. Operational cost includes: • the cost of electrical energy of the reactor drivers, which depends on the number of stages, and the power of each unit; • the cost of electrical energy of the liquid and the solid transferring systems; • other utility costs; • the resin replacement cost, which depends on the rate of its attrition and breakage; • personnel and labor costs; and • maintenance and overhaul costs.

4. CONCLUSIONS Adsorption in continuous stirred tank reactors (CSTRs) under the names of RIP and CIP has become an established process for extraction of gold and uranium,4,14 and, recently, zinc, copper, nickel, and cobalt.32−34 One of the most applied adsorption model in resin-in-pulp (RIP) and carbon-in-pulp (CIP) processes when intraparticle mass-transfer resistance prevails is the Boyd’s series model. The current study in its first part introduced an analytical formula (eq 8), which could approximate the Boyd’s series 312



Article

(i.e., desorb) it in another solution, and in this respect, only the average loading of the resin is significant. Equations A1−A4 indicate that the average residence times and residence time distributions of resin and liquid in a CSTR are different. The liquid entering these reactors is mixed with the liquid content of the vessels almost immediately (see Figure 2), and because of such a rapid equilibration, the residence time of the liquid plays almost no role in the overall adsorption kinetics. The process engineers, in order to keep resins inside the reactor for a longer period, and make a sharper residence time distribution for it, so far, have devised the two following methods: (1) Use of a cord baffle, which prevents short-circuiting of the suspended resin and allows only a small fraction of it to exit (see Figure A1a)

more closely than the Vermeulen formula and other known formulas. The new formula with a slight change of its ψ parameter could also model the irreversible uptake processes as closely as the shrinking core model (SCM).20,23 The second part of this study was devoted on developing a stepwise procedure for design of the single-stage CIP and RIP processes. In this procedure, first, the major design variables, including the reactor volume (V), the mass flow rate of the resin (ṁ ), and its holdup (φS) were linked through eqs 14 and 15; then, a basic optimization algorithm was introduced that could minimize the total cost of the process (recall Figure 5). The third part of the current study devised a new approach for design of the multistage countercurrent adsorption processes based on the Glauekauf multiple series. The developed “O−K−N” method can overcome disadvantages of the McCabe−Thiele method. It can also support the following optimization processes: • Local optimization: If the (φS, ṁ ) ordered pair is fixed, the O−K−N method can precisely predict the optimum reactor size and the number of stages. • Global optimization: By changing the two parameters φS and ṁ and undertaking the algorithm of section 2.4, one can arrive at a global optimization in which all process condition are optimized. Undertaking such global optimization requires a comprehensive knowledge of all cost components of the process.

■

APPENDIX A. RESIDENCE TIME DISTRIBUTION IN RIP AND CIP REACTORS AND ITS INFLUENCE ON KINETICS OF ADSORPTION The residence time distribution of the liquid “E(t)” in a CSTR is given by the relation E L (t ) =

e −t / t L tL

(A1)

where tL represents the average residence time of the liquid obtained from eq A2: tL =

(1 − φS)V Q

(A2)

Parameters of eq A2 have similar meanings with those given in the main text. For the resin particles, analogous equations to eq A1 can be used, with a slight difference in the definition of the average residence time (tS): ES(t ) = tS =

e −t / t S tS

(A3)

φS ρS V

Figure A1. Two ways to achieve an adequate residence time of resin: (a) mounting a cord baffle prevents short-circuiting of the entered resin and keeps its particles inside the reactor for a longer period; (b) use of an air lift pump discharges both liquid and resin together, but a part of the resin is recycled, which prolongs the residence time of the resin.

(A4) ṁ Because of the statistical nature of the residence time, some resin particles may remain in the reactor longer than tS and the others shorter than this period. The longer is the residence time of a resin, the higher is the extent of its loading. Consequently, in the outlet of the adsorption reactor, a mixture of the resin particles is formed whose loading distribution varies in a wide range. There is no mean by which one can separate the resin of high loading from those of lower loadings at the exit of the reactor. Nevertheless, this issue is of little practical importance, because resin itself is a medium whose function is to uptake a special target ion from an aqueous mixture of ions and transfer

(2) Utilization of an air lift pump along with a separating screen (see Figure A1b). At any moment, the air lift pump withdraws only a small fraction of the solution and resin, and in addition, a part of the withdrawn resin may be recycled to reactor to further complete its adsorption (Figure A1b). An air lift pump can be installed behind 313



Article

dimensionless number NQ approaches 0.74 at high Re.1 Therefore,

the cord baffle, and, in that case, it is possible to take advantage of both systems in a combined scheme.

■

vs = 0.183 m/s

APPENDIX B. ESTIMATION OF AGITATION POWER OF A STIRRED REACTOR, AND DERIVATION OF EQS 24 AND 25 Our approach for estimation of the power of an agitator is the “Chemineer Co.” approach that has been presented by that company in a series of publications and summarized in ref 1. According to this approach, any of the services offered by a stirred reactor (i.e., mixer) is characterized with a specif ic superf icial velocity. The superficial velocity (vs) is defined by the following relation: vs =

QP =

NP =

3

0.74(+/3)

7/3

V 2/3

35

All quantities in the above relation have SI units (e.g., the volume V has units of m3). In horsepower units (hp), eq B10 becomes P (hp) = 1.121V 2/3

(B11)

The power calculated by eq B11 is the power required for the liquid agitation. To estimate the power consumed by the electromotor, this power should be multiplied by three correction factors: • A factor of 1/cos φ, to account for the electrical inefficiency of the electromotor • An approximate factor of 1/0.95, for mechanical frictions • A service factor of 1.5, for continuous loading of the electromotor By taking a regular value of cos φ = 0.85, we will have

(B4)

P (hp) =

where 2

1.121 × 1.5V 2/3 = 2.082V 2/3 0.95 × 0.85

(B12)

The power determined by eq B12 then can be inserted in eq 24 to yield the price of the driver unit. On the other hand, the construction cost of the vessel, according to ref 31, is given as

(B5)

In the current study, d/+ = 1/3 and the density (ρ) and viscosity (μ) of the solution were assumed to be equal to those of water. Similar to most dimensionless numbers, in the baffled vessels, the flow number NQ becomes independent of the Reynolds number (Re) in the developed turbulent region:1 Re > 10000

3

27 ( 0.74 ) × ( π4 )

(B10)

where d and N are the diameter and rotation speed of the impeller, respectively. The flow number NQ is a function of the Reynolds number of the impeller, as well as the ratio of impeller diameter to the vessel diameter:

d Nρ μ

0.74+ 3

= 836V 2/3

(B3)

⎛ d⎞ ⎟ NQ = f ⎜Re , ⎝ +⎠

27Q P

(B9)

P (W) =

(B2)

Nd3

=

P N3d5ρ

1.2 × 1000 × (0.183)3 ×

QP

⎛d⎞ NQ = f ⎜ ⎟ ⎝+⎠

QP

Similar to NQ, the power number NP approaches a constant value in the turbulent region.1 For the “Pitched blade turbine 45°” impeller, NP tends toward 1.2. Combining eqs B7−B9 yields a relation for computing power:

where V is the reactor volume that is obtained from the residence time of resin using eq 14. According to ref 1, the specific superficial velocity for a uniform suspension of resin particles is vs = 0.6 ft/s = 0.183 m/ s. The unknown quantity of eq B1 is the pumping flow rate (QP), which can be obtained by making use of the aforementioned values of vs and + . After the value of QP was found, the dimensionless flow number (NQ) can be determined using the following relation:

Re =

N=

The required rotation speed of the impeller N that is calculated from the above relations can be used for estimation of the consumed power P by using the dimensionless power number (NP), which is defined as follows:1

where + is the diameter of the reactor vessel and QP is the pumping flow rate of the impeller. Similar to many other industrial processes, the adsorption process normally uses a square stirring vessel, whose height and diameter are equal, such that

NQ =

→

(B7)

(B8)

(B1)

⎛ 4V ⎞1/3 +=⎜ ⎟ ⎝ π ⎠

0.183π (4V /π )2/3 0.183π + 2 = 4 4

NQ = 0.74 m/s

QP π (+ 2/4)

→

C Vessel = 14V ′0.72

(B13)

where V′ in the above equation is the volume of liquid (in the reactor) (given in gallons). Here, we considered a correction factor of 1.05, to convert the liquid volume to the reactor volume. Thereafter, by converting from cubic meters (m3) to gallons, we will have

(B6)

Our selected impeller was “Pitched blade turbine 45°”, which is a recommended impeller for the suspension of solids (i.e., resins) in liquids. For such a turbine, for d/+ = 1/3, the

V′ =

1.05 × 1000V 3.78541

(B14)

Combining eqs B13 and B14 will eventually result in eq 25. 314



Article

(16) Yahorava, V.; Scheepers, J.; Kotze, M. H.; Auerswald, D. Impact of Silica on Hydrometallurgical and Mechanical Properties of RIP Grade Resins for Uranium Recovery. J. South. Afr. Inst. Min. Metall. 2009, 109, 609. (17) Boyd, G. E.; Adamson, A. W.; Myers, L. S. The Exchange Adsorption of Ions from Aqueous Solutions by Organic Zeolite. J. Am. Chem. Soc. 1947, 69, 2836. (18) Helfferich, F. Ion Exchange; McGraw−Hill: New York, 1962. (19) Glaueckauf, E. Theory of Chromatography. Part 9. The Theoretical Plate Concept in Column Separations. Trans. Faraday Soc. 1955, 51, 1540. (20) Helfferich, F. G. Models and Reality in Ion-Exchange Kinetics. React. Funct. Polym. 1990, 13, 191. (21) Levenspiel, O. Chemical Reaction Engineering; John Wiley and Sons: New York, 1999. (22) McDougall, G. J.; Hancockt, R. D.; Nicol, M. J.; Wellington, O. L.; Copperthwaite, R. G. The Mechanism of the Adsorption of Gold Cyanide on Activated Carbon. J. South. Afr. Inst. Min. Metall. 1980, 80, 340. (23) Gopala Rao, M.; Gupta, A. K. Ion Exchange Processes Accompanied by Ionic Reactions. Chem. Eng. J. 1982, 24, 181. (24) Carr, J.; Zontov, N.; Yamin, S. Meeting the Future Challenges of the Uranium Industry. ALTA 2008 Nickel, Cobalt, Copper and Uranium Conference, Perth, Australia. 2008, (25) Treybal, R. E. Mass-Transfer Operations; McGraw−Hill: Auckland, New Zealand, 1981. (26) Fleming, C. A.; Nicol, M. J. A Comparative Study of Kinetic Models for the Extraction of Uranium by Strong-Base Anion-Exchange Resins. J. South. Afr. Inst. Min. Metall. 1980, 80, 89. (27) Streat, M.; Takel, G. N. J. Anion Exchange Kinetics of Uranium in Sulphate Media. J. Inorg. Nucl. Chem. 1981, 43, 807. (28) Gonzalez-Luque, S.; Streat, M. Uranium Sorption from Phosphoric Acid Solutions Using Selective Ion Exchange Resins: Part II. Kinetic Studies. Hydrometallurgy 1983, 11, 227. (29) Woollacott, L. C.; Afewu, K. I. Scale-up Procedures for GoldAdsorption Systems. Part 1: Adsorption Kinetics. J. South. Afr. Inst. Min. Metall. 1995, 95, 167. (30) Fleming, C. A.; Nicol, M. J. Adsorption of Gold Cyanide onto Activated Carbon. III. Factors Influencing the Rate of Loading and the Equilibrium Capacity. J. South. Afr. Inst. Min. Metall. 1984, 84, 85. (31) Seider, W. D.; Seader, J. D.; Lewine, D. R. Process Design Principles: Synthesis, Analysis, and Evaluation; John Wiley and Sons: New York, 1999. (32) Zainol, Z. The Development of a Resin-in-pulp Process for the Recovery of Nickel and Cobalt from Laterite Leach Slurries. Ph.D. Dissertation, Murdoch, Western Australia, Australia, 2005. (33) Yahorava, V.; Kotze, M. H. RIP Pilot Plant for the Recovery of Copper and Cobalt from Tailings. In Proceedings of the 6th Southern African Base Metals Conference, Phalaborwa, South Africa, 2011; p 149. (34) Makhubela, T. R. Base Metal Recovery by Resin-In-Pulp (RIP) Technology. M.Sc. Dissertation, Tshwane University of Technology, Pretoria, South Africa. 2006.

Note that the prices used in eqs 24 and 25 are those of the year 1999 (the publication date of ref 31) and were not corrected by the price index of the current year.

■

ASSOCIATED CONTENT

S Supporting Information *

Data showing the correlation between the suggested models and the shrinking core model (SCM) (Table S1), the results of calculation of a resin-in-pulp (RIP) process of uranium adsorption (Table S2), and the results of calculation of a carbon-in-pulp (RIP) process of gold adsorption (Table S3) are included as Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Walas, S. M. Chemical Process Equipment Selection and Design; Butterworth−Heinemann: Boston, MA, 1990. (2) Dahlke, T.; Chen, Y. H.; Franzreb, M.; Höll, W. H. Continuous Removal of Copper Ions from Dilute Feed Streams Using Magnetic Weak-Base Anion Exchangers in a Continuous Stirred Tank Reactor (CSTR). React. Funct. Polym. 2006, 66, 1062. (3) De Andrade Lima, L. R. P. Dynamic Simulation of the Carbon-inPulp and Carbon-in-Leach Process. Braz. J. Chem. Eng. 2007, 24, 623. (4) Lan, X.; Liang, S.; Song, Y. Recovery of Rhenium from MolybdeniteCalcine by a Resin-in-Pulp Process. Hydrometallurgy 2006, 82, 133. (5) Udayar, T.; Kotze, M. H.; Yahorava, V. Recovery of Uranium from Dense Slurries via Resin-in-Pulp. In Proceedings of the 6th Southern African Base Metals Conference, Phalaborwa, South Africa, 2011; p 49. (6) Gray, D. E. A Quantitative Study into Carbon-in-pulp Adsorption Operations. M.Sc. Dissertation, Cape Peninsula University of Technology, Cape Town, South Africa, 1999. (7) Mirjalili, K.; Roshani, M. Resin-in-Pulp Method for Uranium Recovery from Leached Pulp of Low Grade Uranium Ore. Hydrometallurgy 2007, 85, 103. (8) Fleming, C. A.; Cromberge, G. Small-scale Pilot-Plant Tests on the Resin-in-Pulp Extraction of Gold from Cyanide Media. J. South. Afr. Inst. Min. Metall. 1984, 84, 369. (9) Griffiths, R. A. Experiments with a Resin-in-pulp Process for Treating Lead-Contaminated Soil. J. Environ. Eng. 2002, 128, 416. (10) Le Roux, J. O.; Brysont, A. W.; Young, B. O. A Comparison of Several Kinetic Models for the Adsorption of Gold Cyanide onto Activated Carbon. J. South. Afr. Inst. Min. Metall. 1991, 91, 95. (11) Benedict, M.; Pigford, T.; Levi, H. W. Nuclear Chemical Engineering; McGraw−Hill: New York, 1981. (12) Seader, J. D.; Henley, E. J. Separation Process Principles; John Wiley and Sons: New York, 2006. (13) Cortina, J. L.; Warshawsky, A.; Kahana, N.; Kampel, V.; Sampaio, C. H.; Kautzmann, R. M. Kinetics of Goldcyanide Extraction Using Ion-exchange Resins Containing Piperazine Functionality. React. Funct. Polym. 2003, 54, 25. (14) Dicinoski, W. G.; Gahan, R. L.; Lawson, J. P.; Rideout, John A. Application of the Shrinking Core Model to the Kinetics of Extraction of Gold (I), Silver (I) and Nickel (II) Cyanide Complexes by Novel Anion Exchange Resins. Hydrometallurgy 2000, 56, 323. (15) McKevitt, B.; Abbasi, P.; Dreisinger, D. A Comparison of Large Bead Ion Exchange Resins for the Recovery of Base Metals in a Resinin-Pulp (RIP) Circuit. In Proceedings of the 6th Southern African Base Metals Conference, Phalaborwa, South Africa, 2011; p 337. 315


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