Modeling Ion-Exchange Kinetics in Bimetallic Systems - Industrial

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Ind. Eng. Chem. Res. 2002, 41, 1357-1363

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Modeling Ion-Exchange Kinetics in Bimetallic Systems Federico Mijangos,*,† Nikolay Tikhonov,‡ Monika Ortueta,† and Andreei Dautov‡ Chemical Engineering Department, Faculty of Science, University of the Basque Country, Apartado 644, Bilbao 48080, Spain, and Physics Department, Moscow State University, Mathematics Subdivision, Vorobiovy Gory, 119899 Moscow, Russia

The kinetics of simultaneous loading of cobalt and copper onto an iminodiacetic-type resin is investigated here. This process is characterized by the appearance of a maximum of metal load for the cation with less affinity for the functional group. In our case cobalt load shows a maximum over time. When a semireacted bead is analyzed under the microscope, two different colored layers surrounding the central core are observed. The diffusional model proposed here takes into account the mobility of the cations through the two phases of the macroporous ion exchanger. The numerical solution of the system of differential equations yields solid-phase concentration profiles and metal load over time. Model parameters are adjusted to fit the estimated metal load from reacted layer motion. Introduction Ion-exchange processes accompanied by a chemical reaction in multispecies systems are a common phenomenon in many systems which are of both practical and theoretical interest. However, there are few works dealing with basic topics on kinetics which cover this matter.1 There are few kinetic studies of ion exchange with chemical reaction in multispecies systems and even fewer of theoretical aspects, modeling, and kinetic predictions. Multimetallic solutions are usually found in several applied fields, and chelating resins are frequently considered and applied to metal recovery or separation, or even for water purification before disposal.2 These resins are very selective for heavy metals because, besides the main ion-exchange reaction, the functional groups are able to catch metallic cations and form a highly stable chemical structure called a chelate. As a result, “free” and “bound” cation species can be found in the internal structure of these ion exchangers. It is accepted that cations diffuse though pores filled with the external solution and are finally blocked by the solid phase, with cation mobility being almost negligible in this state. Simultaneous loading of several metals on a chelating ion exchanger from an aqueous solution gives rise to “nonnormal” kinetic behavior, where weaker-bound species show a maximum value of solid-phase concentration. As a rule, different concentration layers surrounding an unreacted core are observed when analyzing a semireacted ion-exchanger bead. A sharp boundary between predominance areas appears for each pair of species.3 These layers and boundaries appear as reaction fronts, and they can be observed under certain circumstances, such as when different colors are associated with each chelate: the advance of the reaction can then be evaluated via the motion of these reaction fronts.4 The process can be described adequately in mathematical terms by the use of the unreacted core model (UCM, also called the shell progressive model) developed † ‡

University of the Basque Country. Moscow State University.

under the pseudo-steady-state approach, when only one heavy metal and the counterion initially loaded into the resin are considered. Dealing with only two exchanging species, Kalinitchev5 gives a comprehensive study of the appearance of sharp boundaries and the concentration profile inside the bead. He also discusses the role of the co-ions. After solving the theoretical kinetic model numerically, he concludes that the shape of concentration distributions depends on selectivity, diffusivity, and bulk solution concentration values. Here, we state an approach for loading kinetics of heavy metals onto a chelating resinsthe sodium form of a commercial iminodiacetic-type resinsfrom a bimetallic equimolar solution of cobalt and copper sulfates. In this system, the metal with less affinity for the functional group can achieve a solid-phase concentration of up to 15 times the final equilibrium value for an intermediate reaction time. Here, it is demonstrated that this behavior is in good accordance with measured concentration profiles and can be predicted from basic considerations. The system of nonlinear diffusion equations is used to provide a mathematical description of the process, including the equation in partial derivatives for ion diffusion. We consider the presence in the intraparticle liquid phase of the co-ion, which is not excluded from the exchanger by the Donnan effect because of its association with the counterions and because of counterion association with fixed charges. The estimated effective diffusion coefficients are discussed in terms of the single metal contribution and the co-ion, which is not excluded from the exchanger by the Donnan effect because of its association to the counterion and because of counterion association with fixed charges. Loading Kinetics of Ion Exchange from Bimetallic Solutions Using a batchwise procedure, one particle is taken out of the reactor at a set time. This ion-exchanger bead is analyzed using both optical and scattered electron microscopy.4-6 One important point in this experimental technique is to stop the reaction, which means “freezing” mobile species. Washing the cut bead with alkalized

10.1021/ie010317n CCC: $22.00 © 2002 American Chemical Society Published on Web 02/01/2002

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water suffices to obtain reproducible results. We check the reaction-stopping technique by measuring the width of the reacted layer. If the semireacted bead is washed with ultrapure water, the reaction front continues to move to the center and finally disappears. However, if alkalanized ultrapure water (pH ) 10.2) is used, the reaction front does not move at all, even after several days. The microstructural morphology of the cut surface is examined without using any etching procedures. The microstructure of the unetched specimens is made apparent by using secondary electrons and also backscattered images as a result of an atomic number contrast. Samples for X-ray microanalysis do not require any special preparation, except that the region of interest must be on a flat surface. The conductive coating is made using a thin layer of carbon in order to reduce the absorption effects. Here, specimens are attached to the specimen stub with carbon glue and then coated with a layer of gold or carbon depending on whether the samples are for structural microscopy or microanalysis. A diode sputter coating device, CEA 035, is used for this purpose. This equipment employs a highvacuum evaporative technique for carbon coating at a distance of 35 mm. Microanalysis and examinations are carried out with a JEOL 6400 microscope coupled with energy- and wavelength-dispersive X-ray spectrophotometers, EDX Link EXL and WDX JEOL. Using the microscopic techniques described above, the internal profiles of concentration for copper and cobalt uptake are observed. The optical appearance of these concentration layers is shown in the microphotograph (Figure 1a). Two different colored fronts or layers, surrounding the central core, can be clearly distinguished: an outer blue one, which corresponds to a resin that mainly contains the copper chelate, and an inner pink one, which marks a zone where cobalt chelate is the main species. These reaction fronts are clearly brought out by backscatter electron images (Figure 1b), and the optical colors are associated with three metals with different atomic numbers. In this microphotograph, the softergray tones belong to those elements with higher atomic numbers. It is thus derived from Figure 1b that copper is concentrated in the periphery of the particle. Cobalt fills the intermediate ring. Finally, one element of low atomic number, such as sodium, lies in the central core. When this metallic-ion-exchanger system has been used, no compositional differences have ever been observed within a zone, which should appear as a progressive radial gray tone. Sharp boundaries characterize the metal-metal interfaces, as happens with optic microscopy. X-ray microanalysis and energy-dispersive spectrometry have been used to check for the presence of the majority element in the three zones. The three corresponding spectra are shown together in Figure 2, where the analysis zone is pointed out. Copper, cobalt, sulfur, chlorine, silicon, and sodium appear everywhere tested, but copper and cobalt concentrations change drastically from one zone to another. On the other hand, the other elements appear to be on the same order of magnitude. Chlorine and silicon are homogeneously distributed traces that result from polymerization, synthesis, or preparation stages.

Figure 1. Microphotographs of a partially reacted bead: (a) optical microscope image and (b) scanning electron microscopy (SEM)-backscattered electrons (BSE) technique.

From the above-mentioned results, it can be derived that the layers are characterized by the numbers of heavy metals bounded. That is, the blue outer layer seems to contain mainly copper and small amounts of cobalt, both surrounded by a sodium sulfate ionic medium. In the next layer, which is pink, the only heavy metal is likely to be cobalt. Finally, no heavy metals are found in the core of the bead so no trace of cobalt or copper is likely to be found here. Nevertheless, the spectra collected show that this assertion is only an approach because all of the metals are found in all parts of the bead. Copper and cobalt are the main elements in their respective layers. These metals are minority components within the core, and copper could be classified as a mesocomponent in the cobalt layer. Using optical microscopy, the radial position of the reaction fronts is measured over time, and using these measurements, an estimation of the fractional attainment of equilibrium is obtained from the volumetric ratios of the metallic zones.4 These results are shown in Figure 3, where fractional attainment of equilibrium is represented versus reaction time. The appearance of two different rings inside the bead during the reaction can be explained even if cobalt and copper have similar diffusion coefficients and equal external concentrations, as discussed hereafter, mainly in terms of the selectivity of the ion exchanger for metal ions. Initially, their flux should be similar if the reaction is controlled by pore diffusion. On the other hand, copper is retained in higher amounts than cobalt in the resin (around 50 times higher) by a reversible reaction. Then,

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Figure 2. X-ray spectra collected from the three different layers.

in these circumstances, when copper is retained in the external shell of the particle, cobalt must diffuse inside the resin, forming another reacted ring that contains only cobalt. When the reaction progresses, cobalt is partially displaced from the inner ring by copper so the cobalt ring appears to move to the center. Consequently, initially both metals show an increasing load over time, but the cobalt curve achieves a maximum (after 15 min of reaction, approximately) when its reaction front reaches the center of the particle and the cobalt load then starts to decrease as described in Figure 3. From then on, the solid-phase reaction is merely a cobaltcopper exchange. Mathematical Model A kinetic diffusion model has been developed to describe the behavior in experiments of the ionexchange system shown in Figure 3. In formulating the mathematical model, we proceed from the following assumptions. Let us consider a spherical particle surrounded by an external solution of constant concentration. Resin beads are composed of two phases: a solid porous matrix where ion exchange takes place and pores filled with an inner solution. The components diffuse through the intraparticle solution. Surface or solidphase diffusion is assumed to be negligible.

Figure 3. Copper and cobalt load over time.

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Previous calculations checking the feasibility of the mathematical model have shown that, for the description of experimental data, it suffices to consider a local equilibrium assumption between solution and solid phases inside the grain. The mass action law for local ion exchange gives rise to the set of equations (2) if the general equilibrium equation (1) is considered:

1 1 1 1 R Cu + Mez+ S RzMe + Cu2+ 2 2 z z 2

(1)

where R2Cu means the functional group loaded with copper ions, Me designates any metallic cation with valence z, and RzMe is the chelate formed. It is straightforward to derive that

KNa

( )

CNa CCo ) KCo qNa qCo

1/2

) KCu

( ) CCu qCu

1/2

(2)

Moreover, it is possible to ignore the differences between diffusion coefficients of components in the solution by taking into account a single average value (D).

DCu ) DCo ) DNa ) D

(3)

The continuity equation can be derived under Fick’s law and then stated for spherical geometry as

(

)

hi 1 ∂ 2 ∂C ∂ r (C h i + Fqi) ) D 2 ∂t ∂r ∂r r

0 < r < R (4)

where C h i is the total concentration of species i within the pores, considering both free cations and nondissociated sulfates, as described by eq 5.

C h i ) Ci + Ciso4, i ) Na, Co, Cu

(6)

Moreover, eq 7 needs to considered, that is, the mass balance for the functional group that guarantees charge preservation within ion-exchanger microspheres.

qNa + 2(qCo + qCu) ) Q

(7)

where Q is the acid-base ion-exchange capacity. The condition of electrical neutrality of the solution inside the pore can be expressed by the following equation:

CNa + 2(CCo + CCu) ) 2CSO4

parameter acid-base ion-exchange capacity, Q (mol/kg of Na-form dry resin) effective intraparticle diffusion coefficient, D (m2/s) radius of the ion-exchanger bead, R (m) internal porosity,  density, F (kg of Na-form dry resin/m3 of wet resin) ion-exchange equilibrium constants (pH ) 4) sodium, KNa [(kg/m3)1/2] cobalt, KCo (dimensionless) copper, KCu (dimensionless) metal concentration in the external solution, Ci (mol/m3) copper cobalt dissociation constant, Hi (m3/mol) copper cobalt

value 5.48 1.5 × 10-10 5.7 × 10-4 0.67 730 0.050 0.155 1.000

100 100 0.204 0.316

without discontinuity, so that there is no leap in concentrations at the bead boundary.

When t ) 0, CNa ) Cinit Na , CCo ) CCu ) 0 h Co ) Cext h Cu ) Cext When r ) R, CNa ) 0, C Co , C Cu

(9) (10)

A computer program has been created for simulating metal diffusion inside ion-exchanger grains. The set of equations (2)-(10) are solved by the finite differences method using the implicit scheme of first order of accuracy. The factorization method is applied,7 and the roots of algebraic equations are determined by the Newtonian method. The results of the numerical simulation of the process at different boundary conditions and different values of diffusion coefficients are shown and discussed here.

(5)

Consequently, a correlation needs to be stated between the free ions and nondissociated species. For this purpose the dissociation constants for homogeneous solutions (Hi) can be used, assuming that their values are adequate for the pore solution. Sodium sulfate is assumed to be fully dissociated, HNa ) 0.

Hi ) Ciso4/CiCSO4

Table 1. Values of the Main Equilibrium and Kinetics Parameters of the Lewatit TP207 Ion Exchanger

(8)

Finally, the following initial and boundary conditions are considered (eqs 9 and 10). Here, it is assumed that cations and neutral molecules of CuSO4 and CoSO4 penetrate from the outer to the inner pore solution

Comparing Experimental and Simulated Results: Checking the Model The main parameters of the Lewatit TP 207 ion exchanger are determined experimentally by conventional assays. That is, equilibrium constants, ionexchange capacity, porosity, density, and even effective diffusion coefficients are measured in the laboratory using the conditions under investigation, and the values obtained are reported in Table 1. The values of dissociation constants within the ion-exchanger solution, Hi, are taken from the literature,8 with homogeneous solution data being sought. As a rule, the simulation program is carried out using a boundary value for a total metal concentration of 0.1 M, because the experimental kinetics are done using this value for the concentration of metals in the solution. The first stage of the simulation process is to check the goodness of the mathematical model and the accuracy of the experimentally determined parameters whose values are shown in Table 1, using independent tests. For this purpose, the experimental data set shown in Figure 3 is fitted to the diffusional model using the following procedure. The values of the total amount of metal sorbed by the ion exchanger are first predicted using the parameter values reported in Table 1 and values of MCu(tj) and MCo(tj). If there is not sufficient agreement, other values of the parameters in the model would need to be considered or the hypothesis and

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the “base” values of the parameters. Then the internal concentration profiles for copper and cobalt, qCu(r,tj) and qCo(r,tj), are taken as output data at different reaction times (tj ) 10, 60, and 250 min). These results are plotted in Figure 4 as the dimensionless solid-phase concentration, yi(r,t), along the radial position (r/R).

yi(r,t) )

qi(r,t) Q/zi

(11)

At the early stages of the process, diffusion of copper and cobalt inside the grain takes place and sodium ions are rejected because the affinity of sodium is considerably less than that of cobalt or copper; this can be mathematically described by eq 12. Ion-exchange constants are defined by eq 2 for the equilibrium defined by eq 1 so, in relation to copper exchange, the equilibrium constant should by definition be 1, i.e., KCu ) 1. The ion exchange gives rise to sharp sodium/cobalt and cobalt/copper reaction fronts because

KNa(CNa/qNa)1/2 < KCo < KCu

Figure 4. Simulation of cobalt and copper internal concentration profiles. Values of the physical and chemical parameters are taken from Table 1.

assumptions of the model (eqs 2-10) would need to be reviewed. However, if there is agreement, then the model’s goodness can be improved by defining more exactly the parameter values by fitting the predicted values to the whole set of Mi(t) measurements. Data simulating the total metal uptake over time using the parameter values and conditions shown in Table 1 are depicted in Figure 3 using dotted lines. On the same figure, the continuous lines show the simulated results after fitting the parameters, together with the experimental load of metal. Nevertheless, only the cobalt ion-exchange constant, KCo ) 0.135 kg/m3, needs to be changed: the remaining parameters can be left unchanged to obtain the best fit to the experimental values. The good correspondence of the results is evident. The mathematical model describes the process of metal uptake on the ion-exchange bead with an accuracy level within the margins of experimental error. To describe the experimental system under consideration, there is no need to introduce further complications into the model and its prediction on metal uptake kinetics can be analyzed under different conditions. Simulation of Metal-Ion-Exchange Kinetics Let us call the set parameters reported in Table 1 “base” parameters but change the value of the cobalt ion-exchange constant, KCo, to the fitted value of 0.135 kg/m3. The following cases are considered: (a) Prediction of the Intraparticle Metal Concentration. The set of equations (2)-(10) is solved for

(12)

The front genesis and development of the front are shown in Figure 4. The radial zones or layers are formed because of the prevalence of one component over the others. During the reaction, the central core that is saturated with sodium ions is decreasing in size, as the cobalt layer progresses to the center and enlarges its width. At an intermediate reaction time, the amount of cobalt taken up is much more than the equilibrium value corresponding to the boundary conditions (CCo ) CCu ) 0.1 M). This can be clearly seen in Figure 4 because the local attainment of equilibrium for cobalt at the boundary of the particle, yCo(r,t), is several times smaller than the corresponding value for the cobalt peak. As soon as the cobalt zone reaches the center of the bead, a second stage of the ion-exchange process begins. This process is represented by the bottom profiles (t ) 250 min) in Figure 4. This is characterized by a rejection of cobalt from the resin bead because of diffusion of copper, which is a species with a higher affinity than cobalt for the functional group. During this stage the residual concentration of sodium within the bead is very low; copper and cobalt cations mainly saturate the functional group of the ion exchanger. The ion-exchange process goes on in this way until the final equilibrium state is achieved. From the above description, it is clear that cobalt is accumulated in its own zone, but a flow outward makes the total amount of cobalt in the resin begin to decrease when sodium ions are totally displaced from the resin. As a result of this time dependence, the total amount of cobalt loaded over time, MCo(t), is nonmonotonic: it shows a maximum in the middle of the process. In contrast to the results reported by other authors,9-11 it may be noticed that this behavior is not connected with the values of the diffusion coefficient. In the present work, we take into consideration a sorbent with a twophase structure. It is assumed that metals are accumulated in the solid phase, with the liquid-phase beginning the transport medium. This system behaves like the column ion-exchange process, where the solution components are transferred to the solid grain but by solution flow instead of mass transfer by intraparticle

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profile along the radius (eq 15) and MCo* is the maxi-

MCo(t) )

Figure 5. Effect of cobalt concentration. Simulation of the kinetics of metal uptake from a bimetallic solution of copper and cobalt (other data from Table 1).

diffusion. The nonmonotonic concentration-time curve appears in this case as it usually does in an ionexchange column. So, it is very important to state a twophase diffusional model in simulation because a homogeneous model (one-phase model) could not predict kinetic results qualitatively similar to those from experimental observations. (b) Analysis of the Effect of Parameters on Kinetics. From eqs 2-10, it is easily derived that changes in the diffusion coefficient values (D) or the particle radius (R) do not change the nature of the mathematical solution but only lead to changes in the time scale. Like many other diffusion-controlled systems, here the degree of saturation or reaction is strongly dependent on the dimensionless time (tD/R2). To find out what influence the metal concentration in the external solution has on ion-exchange kinetics, the solution to the differential equations, shown in Figure 5, has been drawn up, as in previous cases, using the “base” parameter values, but the metal concentration in the solution is kept equimolar at 1 order of h ext magnitude lower than before: C h ext Cu ) C Co ) 0.01 M. For convenience in comparing data, a “reduced time” is defined as τ ) t/k, where k is the external concentration ratio in relation to the “base” value; k ) 10 for the case under consideration. The simulation of the bimetallic ion-exchange process is shown in Figure 5 for the case of cobalt uptake. The degree of saturation of the exchanger is calculated as the fractional attainment of equilibrium, that is

X(t) ) MCo(t)/MCo*

(13)

Y(t) ) MCo(t)/Mequil Co

(14)

where the total amount of cobalt loaded in the ion exchanger is calculated by integrating the concentration

∫0Rr2q(r,t) dr

3 R3

(15)

mum value of MCo(t) in the process. Consequently, X(t) must be less than 1. Emphasis must be placed on the difference with Mequil Co , which makes reference to the infinite reaction time value, that is to say, the real equilibrium metal load in resin when that resin is brought into contact with the bimetallic solution. Consequently, X(t) must always be lower than Y(t), and the latter can be higher than 1. Function X(t) is shown in Figure 5a. The dashed line represents the system evolution calculated for the dilute solution. On the same figure, the overplotted solid line corresponds to the results predicted for the “base” conditions. The solid and dashed lines follow an identical tendency, although boundary conditions change by a factor of 10. Of course, changing the concentration in the solution 10 times gives rise to a proportional change in the process rate, but this produces only a slight deformation in the shape of the X(t) curve. This deformation in the kinetics curves is mainly effected by changes in the dissociation ratio, Ciso4/Ci, whose value decreases with the concentration of sulfate, CSO4, in accordance with dissociation equilibrium (eq 6). (c) Effect of the Cobalt Concentration on Intraparticle Metal Concentration Profiles. A cobalt concentration 5 times higher than the copper concentration is considered for the next simulation study. That is, C h ext h ext Cu ) 0.1 M and C Co ) 0.5 M are selected as boundary conditions, replacing the values in eq 10. The remaining parameters are those previously defined as “base” values. The dependence Y(t) obtained by simulation is depicted in Figure 3 using a dashed line. Despite the great influence that the cobalt concentration has on cobalt diffusion, there is no appreciable effect on copper diffusion. The copper uptake curve shown in Figure 3b does not change in any case studied. This can also be observed through the copper concentration profiles shown in Figures 4 and 6. The radial position of the copper reaction front is appreciably the same in both cases, although the cobalt profile has changed because of its external concentration values. The internal concentration profiles are given in Figure 6a for both metals and a reaction time of t ) 10 min. This reaction time is chosen in order to compare these results with those reported in Figure 4a. A considerable increase in Y(10) for CCo ) 0.5 M is reported here in comparison to the “base” case. Finally, the kinetics of metal uptake is simulated, but with the cobalt concentration set at C h ext Co ) 0.02 M. The results are displayed in Figures 5 and 6. The dimensionless local concentration, yi(r,t), is depicted for a reaction time of t ) 60 min. These concentration profiles (Figure 6b) can be compared with those in Figure 4b. No appreciable differences result for copper: the reaction front position is the same in both cases (r/R ) 0.62), but the profile of copper is slightly sharper for low cobalt concentration. However, the concentration peak of cobalt is smaller and sharper and moves more slowly to the center when the external cobalt concentration is low.

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Figure 6. Intraparticle metal concentration profile. Simulation of the kinetics of metal uptake from a bimetallic solution of copper and cobalt (other data from Table 1).

Conclusions Metal diffusion through a macroporous polymeric resin containing an iminodiacetic group is described using a heterogeneous model assuming local equilibrium between the pore solution and the solid phase. The model takes into account a two-phase structure, component diffusion, ion-exchange reaction, and complex formation in the solution. Simulation of experimental results with this mathematical model is in good agreement with the results of metal ion-exchange kinetics from bimetallic and equimolar solutions of copper and cobalt. It has been observed in experiments that during the reaction a layer or ring appears inside the ion-exchanger bead for each diffusing cation and that their fronts move toward the center of the particle as the reaction progresses. It is demonstrated that these layers are zones with prevalence of one component over the others. These layers appear only when there is a high selectivity of the ion exchanger for the metals that are diffusing inside. As a result, the metal with an intermediate affinity for the functional group shows an nonmonotonic curve for the total amount of metal load in resin over time. The simulation study corroborates that this behavior is common in heterogeneous systems such as column separations. Changes in the particle size and diffusion coefficients lead only to changes of the time scale. This is accurately represented by dimensionless number tD/R2. The form of concentration profiles depends essentially upon the ratio between the metal concentrations in the solution. However, the cobalt concentration in the external solution has no appreciable effect on copper diffusion. List of Symbols C(r,t) ) metal concentration as the free cation in the intraparticle liquid phase (mol/m3)

C h ) total concentration of species i within the pores, eq 5 (mol/m3) D ) effective diffusion coefficients in pores (m2/s) H ) dissociation constant, (m3/mol) ext k ) concentration factor, k ) (Cext Co )base/(CCo )test K ) ion-exchange equilibrium constant, defined by eqs 1 and 2 (kg/m3) M(t) ) total metal concentration in the solid phase, eq 15 (mol/kg of dry resin) M* ) maximum value of M(t) in the process (mol/kg of dry resin) Me ) cation of metals in solution Mequil ) metal load at infinite reaction time, equilibrium state (mol/kg of dry resin) Q ) acid-base ion-exchange capacity (mol/kg of Na-form dry resin) q ) local metal concentration in the solid phase (mol/kg of Na-form dry resin) R ) functional group of the ion exchanger R ) ion exchanger bead radius (m) r ) radial position (m) t ) time (s) X(t) ) dimensionless load of cobalt, eq 13 Y(t) ) real fractional attainment to equilibrium, eq 14 y(r,t) ) dimensionless local solid-phase concentration, defined in eq 11 z ) valence Greek Symbols  ) internal porosity F ) dry density (kg of dry resin/m3 of wet resin) τ ) reduced time (s) Subscripts and Superscripts i ) metal init ) initial ext ) external solution j ) iteration

Literature Cited (1) Franzreb, M.; Holl, W. H.; Eberle, S. H. Ind. Eng. Chem. Res. 1995, 34 (8), 2670-2675. (2) LaGrega, M. D.; Buckingham, P. L.; Evans, J. C. Hazardous Waste Management; McGraw-Hill: London, 1998. (3) Mijangos, F. Analysis and modelling of the metal recovery from hydrometallurgical wastewaters by ion exchange. Ph.D. Dissertation, University of the Basque, Bilbao, Spain, July 1989. (4) Mijangos, F.; Dı´az, M. J. Colloid Interface Sci. 1994, 164, 215-222. (5) Kalinitchev, A. I. Solv. Extract. Ion Exchange 1998, 16 (1), 345-379. (6) Mijangos, F.; Bilbao, L. Prog. Ion Exchange 1997, 196, 341348. (7) Samarskii, A. A. Theory of Difference Schemes; Nauka: Moscow, 1989. (8) Allison, J. D.; Brown, D. S. Novo-Gradac, MINTEQA2PRODEFA2. A Geochemical Assessment for Environmental Systems, Version 3.0; EPA/600/3-91/021; U.S. EPA: Athem, GA, 1991. (9) Hwang, Y.-L.; Helfferich, F. React. Polym. 1987, 5, 237. (10) Dolgonosov, A. M.; Khamizov, R. Kh.; Krachak, A. N.; Prudkovsky, A. G. React. Funct. Polym. 1995, 28, 21. (11) Yoshida, H.; Kataoka, T. Intraparticle Ion-exchange Mass Mransfer in Ternary Systems: Theoretical Analysis of Concentration Profiles. Chem. Eng. J. 1988, 5, 55.

Received for review April 10, 2001 Revised manuscript received November 14, 2001 Accepted December 7, 2001 IE010317N