Kinetic analysis of Zn(II) removal from water on a strong acid macroreticular resin using a limiting bidisperse pore model

Hydrometallurgy 62 Ž2001. 31–39 www.elsevier.comrlocaterhydromet

Kinetic analysis of Zn žII/ removal from water on a strong acid macroreticular resin using a limiting bidisperse pore model A.M.S. Oancea a,) , E. Pincovschi a , D. Oancea b, M. Cox c a

Department of Inorganic Chemistry, UniÕersity A PolitehnicaB of Bucharest, Str. Polizu nr. 1, Bucharest 78126, Romania b Department of Physical Chemistry, UniÕersity of Bucharest, Bd. Elisabeta nr. 4-12, Bucharest 70346, Romania c Department of Physical Sciences, UniÕersity of Hertfordshire, College Lane, Hatfield AL10 9AB, UK Received 21 March 2001; accepted 15 May 2001

Abstract HqrZn2q integral interdiffusion coefficients on Vionit CS 32 resin Ž –SO 3 H, styrene-divinylbenzene copolymer with a macroporous skeleton. were obtained using the Ruckenstein bidisperse pore model for a two-stage ion exchange process, considering the macropore diffusion to be much faster than microporous diffusion. The HqrZn2q ion exchange rate on the macroporous resin was measured at 298 K using a potentiometric method. The experimental conditions were selected to favour a particle diffusion rate-controlling mechanism, subsequently confirmed by the variation of exchange rate with resin bead size and stirring speed. The HqrZn2q integral interdiffusion coefficients in the macropores were obtained as a function of the ionic composition of the resin for different values of the ratio, bra , of micropore and macropore uptake at equilibrium. Using several discrimination criteria, it was established that the most appropriate value for the system was bra s 0.2. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Ion exchange kinetics; Interdiffusion coefficients; Macroreticular strong acid resin; Bidisperse pore model

1. Introduction Ion exchange is an important technology for the separation and recovery of zinc from wastewater resulting mainly from plating and metal industries. Macroporous ion exchangers are widely used in industrial ion exchange operations as they are much more resistant to osmotic shock, exhibit a smaller loss of volume during drying, and have a higher oxidation resistance when compared with gel resins. It is accepted that a particle of a macroporous resin

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Corresponding author. E-mail addresses: [email protected], [email protected] ŽA.M.S. Oancea..

represents an aggregate of spherical microparticles of normal gel-like porosity Žhighly cross-linked. separated by a continuous non-gel porous structure. The rate of ion exchange is controlled by ion diffusion, either inside the resin particle or in the liquid film surrounding the particle, according to the operating conditions ŽHelfferich, 1995; Helfferich and Hwang, 1991.. The intraparticle diffusion process in a macroporous resin is a more complex problem than in gel resins, due to the bidisperse porous structure. Some studies on the mass transfer in such resins have been reported ŽRuckenstein et al., 1971; Weatherly and Turner, 1976; Yoshida et al., 1985; Yoshida and Kataoka, 1985; McGarvey and Hauser, 1985., giving new data andror proposing mass transfer models.

0304-386Xr01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 6 X Ž 0 1 . 0 0 1 7 7 - 3

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A.M.S. Oancea et al.r Hydrometallurgy 62 (2001) 31–39

The main objectives of the present study are: Ž1. experimental determination of HqrZn2q ion exchange rate using a potentiometric method on Vionit CS 32 resin, which consists of a macroporous styrene-divinylbenzene copolymer matrix with –SO 3 H functional groups; Ž2. determination of HqrZn2q integral interdiffusion coefficients in the macropores and their variation with the ionic composition of the resin using a limiting case of the Ruckenstein et al. Ž1971. bidisperse pore model for a two-stage ion exchange process considering the macropore diffusion much faster than that in the micropores; Ž3. selection of the model parameter a , the ratio of the times necessary to penetrate by diffusion the macropores and the micropores of the resin and bra , the ratio of micropore and macropore uptake at equilibrium using different discrimination criteria namely: the difference in ion exchange rates in macroporous and gel resins; the values of the HqrZn2q integral interdiffusion coefficients on gel resins previously determined ŽOancea et al., 2000a, 2001., and Zn2q and Hq self-diffusion coefficients in dilute aqueous solutions ŽCRC Handbook of Chemistry and Physics, 1998–1999.; Ž4. determining the conditions when the dependence of the fractional attainment of equilibrium, F, in dimensionless time, t , given by the limiting bidisperse pore model ŽRuckenstein et al., 1971., is in agreement with the same function given by the simpler and more popular quasi-homogeneous resin phase model for an infinite solution volume boundary condition ŽHelfferich, 1995; Helfferich and Hwang, 1991..

The mean radius of the swollen resin beads for the size fractions used was determined microscopically. The mean values obtained and their standard deviations, for a Student distribution, were: 0.370 " 0.048, 0.288 " 0.041 and 0.215 " 0.032 mm. The external solution was prepared from Analytical Reagent grade ZnŽNO 3 . 2 P 6H 2 O and the concentration determined by complexometric titration with disodium salt of ethylenediaminetetracetic acid.

2. Experimental

3. Results and discussion

2.1. Resin and solutions

3.1. Ion exchange rate

Macroporous Vionit CS 32 resin ŽVictoria, Romania. was dry sieved, fractionated and purified by consecutive treatment with 1 M HCl, 1 M NaOH and washed with doubly-distilled water. Finally, it was converted to the Hq form and thoroughly washed until free of chloride, dried at room temperature and equilibrated to constant weight by saturation with water vapour in a desiccator with a saturated solution of NaCl.

The rate of ion exchange is reported as the fractional attainment of equilibrium, F, vs. time, t, calculated from the pH variation with time of the external solution

2.2. Ion exchange kinetic measurements Ion exchange kinetics were monitored by measuring the pH variation of the external solution with time at 293.0 " 0.5 K with a Mettler Delta 350 pH-meter, and a combined pH electrode, provided with a temperature probe. The batch reactor was a thermostated glass vessel of cylindrical shape with 55 mm diameter, 80 mm height and the magnetic stirrer ŽTeflon-covered. had 25 mm length and 5 mm outer diameter. The ion exchange process was started by rapid addition of 25 ml of Zn2q solution Ž0.93 mL. to a stirred sample containing a known amount of swollen resin in the Hq form Ž; 1 g. in 25 mL of water. The pH measurement was simultaneously started and continued at suitable time intervals. Under these conditions, the electrolyte desorption interference was negligible as proved by blank experiments. The pH variation during different kinetic runs was 0.32–0.42 pH units. The precision of the pH measurement was "0.001 units for the variation of pH and "0.01 units for the absolute value, the latter being limited by the standard buffer.

Fs

10yp H t y 10yp H 0

. Ž 1. 10yp H` y 10ypH 0 In Eq. Ž1., the ratio of the difference in proton activity is considered to be equal to the ratio of the

A.M.S. Oancea et al.r Hydrometallurgy 62 (2001) 31–39

corresponding differences in concentration. This is a reasonable assumption because the variation of the proton activity coefficient with ionic strength of the external solution can be neglected, the ionic strength being high Ž1.395 M. and varying by only 1.9–2.4% for the various HqrZn2q ion exchange experiments. Fig. 1 shows the kinetics of uptake of Zn2q ions by the macroporous strong acid Vionit CS 32 resin in the Hq form from a 0.465 M ZnŽNO 3 . 2 aqueous solution at 298 K for different size fractions at a stirring speed of 850 miny1 , and for the same fraction at 850 and 1800 miny1 . The experimental results are in agreement with a particle diffusion-control mechanism. This is demonstrated in a situation where the concentration of the external solution is high enough to produce a significant driving force in the liquid film by the rate of ion exchange increasing as the resin bead radius decreases, and within experimental error, no influence on the rate by changes in the stirring speed from 850 to 1800 miny1 . To extend the kinetic analysis of the system at F - 0.20, a kinetic curve was fitted to the experimental points for each run. The best-fit curves, selected for the highest correlation coefficients, and the

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lower and randomly distributed residuals, were used to extrapolate the points for F ™ 0. For example, F vs. t for a swollen bead mean radius of 0.370 and 850 miny1 , from Fig. 1, is represented by the function F s a Ž 1 y exp Ž ybt . . q c Ž 1 y exp Ž ydt . . ,

Ž 2.

where a s 0.1857 " 0.0086, b s 0.0495 " 0.0026, c s 0.7265 " 0.0076 and d s 0.00912 " 0.00012, the correlation coefficient being 0.99994. 3.2. Limiting bidisperse pore kinetic model The mutual ion exchange kinetics on macroporous resins are less well-described than for gel resins because diffusion in the bidispersed porous ion exchanger is more complicated. Results on the kinetics of HqrZn2q ion exchange on the gel resins Purolite C 100 and Vionit CS 3, having –SO 3 H functional groups and a styrene-divinylbenzene matrix, indicated faster ion exchange rate than on the macroporous Vionit CS 32 resin ŽOancea et al., 2000a, 2001.. This behaviour was also observed with other systems with small ions ŽMcGarvey and Hauser, 1985; Oancea et al., 2000b..

Fig. 1. Fractional attainment of equilibrium versus time for HqrZn2q ion exchange on macroporous Vionit CS 32 resin; 0.465 M ZnŽNO 3 . 2 ; 298 K.

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The present kinetic treatment is based on the model of Ruckenstein et al. Ž1971. developed for transient diffusion in solids having a bidisperse pore structure. This model considers a spherical particle of a macroporous ion exchange resin as an agglomerate of highly cross-linked micro-spherical gel particles or micro-beads, between which there are channels, which constitute the macropores. The pores within the gel micro-beads constitute the micropores of the resin. A limiting case of the bidisperse pore model ŽLC-BP. implies a two-stage ion exchange process with macroporous diffusion being much faster than the diffusion within the micropores ŽRuckenstein et al., 1971.. This case is a reasonable hypothesis for the current studies as long as the diffusion path is much more restricted in the micropores than in the macropores, and the mechanical friction due to the interactions with the pore walls is more important in micropores than in macropores. The original model considers the isotope exchange case. In the first stage of the process, ion exchange only occurs in macropores to be followed by a much slower second stage, which takes place only in micropores. The total fractional attainment of equilibrium, F, at any time, t, is a weighed average

Fs

Mt M`

1y s

`

6

p

2

Ý ns1

1 n

2

exp Ž yn2p 2t . q

of both the fractional attainment of equilibrium in macropores, Fa , and micropores, Fi Fs

Ma`

s

M`

Fa q

Mi`

F Ma` q Mi` Ma` q Mi` i 1 b Fa q F 3 a i s , Ž 3. 1 b 1q 3 a where Fa and Fi are the solutions of the isotope diffusion in a quasi-homogeneous resin phase for an infinite solution volume boundary condition ŽHelfferich, 1995; Helfferich and Hwang, 1991. 6 ` 1 F s 1 y 2 Ý 2 exp Ž yn2p 2t X . . Ž 4. p ns1 n In a spherical macroporous resin bead, the macroporous network is considered as one quasi-homogeneous phase and the microporous network of a microsphere as another quasi-homogeneous phase. Thus, the ion exchange process in the macroporous resin occurs in two steps, in two different quasi-homogeneous phases, without interaction between the two steps. It was shown that ŽRuckenstein et al., 1971. 1 b

3 a 1 b 1q 3 a

The parameter a s ŽD i R a2 .rŽDa R i2 . has the physical significance of the ratio between the times required for ion penetration by diffusion in the macrosphere Žmacropores. and in the microsphere Žmicropores.. In the limiting case considered above, a must be equal to or smaller than 10y3 ŽRuckenstein et al., 1971.. The other parameter of the model, bra s 3 Mi` rMa` , is related to the ion exchange capacity of the resin from the functional groups grafted onto the walls of the micropore and macropore, respectively, which are available to the exchanging ion. A computer program was written and used to numerically solve Eq. Ž5.. For a given F and for the selected parameters a and bra , the dimensionless

Mt

1y

`

6

p

2

Ý ns1

1 n2

exp Ž yn2p 2at . .

Ž 5.

time t was computed by an iterative procedure. Knowing the time t necessary to reach the total fractional attainment of equilibrium F, for the macroporous resin bead mean radius, R a , the effective self-diffusion coefficient in the macropores of the species i, Da could be obtained from t s ŽDa t .rŽ R a2 .. The corresponding effective self-diffusion coefficient within the micropores of the species i, D i , could be determined from the parameter a , if the radius of the gel micro-beads, R i , forming the macroporous bead could be evaluated. Eq. Ž5. was used in this work to express the mutual ion exchange in a macroporous resin. The diffusion coefficients obtained are consequently integral interdiffusion coefficients in macropores and

A.M.S. Oancea et al.r Hydrometallurgy 62 (2001) 31–39

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Fig. 2. The total fractional attainment of equilibrium vs. the dimensionless time t calculated with the bidisperse pore model for a two-stage ion exchange process with macropore diffusion much faster than micropore; a s 10y3 .

micropores, respectively. The actual interdiffusion coefficients are variable quantities ŽHelfferich, 1995; Helfferich and Hwang, 1991., while the integral interdiffusion coefficients are constant for each inter-

val 0–F Žor 0–t . ŽOancea et al., 2000b; Tao et al., 1987.. The physical significance of an integral interdiffusion coefficient is the average value of the actual interdiffusion coefficients in the interval 0–F

Fig. 3. Calculated F vs. t curves with the limiting case of the bidisperse pore model, macropore diffusion much faster than micropore diffusion, for different parameters a and bra .

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Table 1 The correspondence between b r a parameter and the ratio of the ion exchange capacity due to micropore and macropore functional groups

b ra Mi` r Ma`

0.01 1:300

0.1 1:30

0.2 1:15

0.3 1:10

0.5 1:6

1.0 1:3

1.5 1:2

3.0 1:1

ŽOancea et al., 2000b; Tao et al., 1987., and varies with the ionic composition of the solution that fills the resin pores. Figs. 2 and 3 give the calculated F vs.t curves, using Eq. Ž5. for 10 values of n in the series and for different values of the parameters a and bra . For F - 0.05, the series in Eq. Ž5. are not convergent when n F 10. Table 1 shows bra values and the ratio Mi` rMa` for the micropore and macropore ion exchange capacities, which are equal to the ratio of the number of functional groups within the micropores and macropores. Fig. 2 shows that the ion exchange process assumed to occur in two steps is notably slower for increasing values bra . From Fig. 3, it can be observed that a decrease of one order of magnitude in a does not affect the F vs. t dependence for t - 0.40. For higher bra ratios and for t ) 1.0, there are differences in F vs. t curves if a decreases, but this represents a region which is outside the practical cases as will be proved subsequently. 3.3. H qr Zn 2 q integral interdiffusion coefficients on the macroporous resin The variation of the HqrZn2q macropore integral interdiffusion coefficient, D Hmqa rZn 2q , with F, was calculated with the LC-BP model for a s 10y3 , n s 10 and different values of parameter bra , and the results are given in Fig. 4a,b. The series in Eq. Ž5. are not convergent for n F 10 if F - 0.05, and this region was not analysed. Fig. 5 compares the above results with HqrZn2q integral interdiffusion coefficients on gel resins with 8% DVB cross-linking and on the macroporous resin obtained from Eq. Ž4. which represents a quasi-homogeneous resin phase-infinite solution volume model ŽQHRP-ISV. ŽOancea et al., 2000a, 2001.. Kinetic measurements using both gel and macroporous resins were carried out for a ratio of the total amount of ions in the resin phase and in the external solution, less than 0.1, thus

ensuring the correct use of the infinite solution boundary conditions. The measured F vs. t curves for the HqrZn2q ion exchange process on the macroporous Vionit CS 32 resin show a slower rate compared with the gel resins ŽOancea et al., 2000a, 2001.. If the mechanism in the macroporous resin is a two-step process, then the parameter a s 10y3 seems to be a reasonable assumption; a smaller value does not change the F Žt . function for t in the usual range of the ion-exchange processes. The selection of a suitable value of the parameter bra can be obtained by inspection of the results given in Figs. 4 and 5. Ž1. If the ion exchange capacity due to functional groups present in micropores is significant Ž bra G 0.3., then the HqrZn2q macropore integral interdiffusion coefficients, D Hmqa rZn 2q , increase by some orders of magnitude with increasing F. The diffusion coefficients of Zn2q and Hq in dilute aqueous solution are 7.03 = 10y1 0 m2 P sy1 and 9.311 = 10y9 m2 P sy1 , respectively ŽCRC Handbook of Chemistry and Physics, 1998–1999.. The maximum value of the HqrZn2q integral interdiffusion coefficient on gel resins with 8% DVB is 4.30 = 10y1 0 m2 sy1 for Purolite C 100 ŽOancea et al., 2000a., and 5.31 = 10y1 0 m2 sy1 for Vionit CS 3 ŽOancea et al., 2001.. Fig. 4b indicates that values of bra ) 0.3 are very improbable because D Hmqa rZn 2q values at increasing F are higher than the Hq diffusion coefficient in dilute aqueous solution. Ž2. In gel resins, the HqrZn2q integral interdiffusion coefficient increases as F increases, attains a maximum value as the resin is progressively converted into the Zn2q form, and decreases near equilibrium, when ion-pairing is a dominant phenomenon. For bra F 0.2, D Hmqa rZn 2q vs. F presents the same behaviour as in gel resins. The increase of D Hmqa rZn 2q with F for very high conversions is less understandable, even in terms of Aa free streaming effectB in the larger pores ŽWeatherly and Turner, 1976.. So bra F 0.2 has a better physical meaning, corresponding also to the fact that sulfonation in the micropores cannot be very extensive as the gel micro-beads are highly cross-linked. Ž3. The overall ion exchange process in the macroporous resin is slower than in gel resins, suggesting a significant contribution of the diffusion in micropores. For bra s 0.2, D Hmqa rZn 2q has the high-

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Fig. 4. Ža. and Žb. HqrZn2q macropore integral interdiffusion coefficients vs. F on macroporous Vionit CS 32 resin, obtained with the bidisperse pore model for a two-stage ion exchange process with macropore diffusion much faster than micropore for different bra parameters; a s 10y3 ; R a s 0.370 mm; 298 K.

est value, very close to gel resins, but at the same time does not increase for F ™ 1. Consequently, this choice seems to be the most suitable. The presence in the macropores of a relatively large amount of co-ions could explain the smaller macropore diffusion coefficients compared with the gel resins. Another possible effect could be the consequence of differences between the ionic radii of completely and

incompletely solvated ions. In macropores, a complete first solvating shell and even a secondary solvating shell are expected, while in some of the pores of the gel resins with 8% DVB, even the first solvating sphere may be perturbed. For bra s 0.01, when a negligible quantity of functional groups is contained in the micropores, D Hmqa rZn 2q has very low values compared with gel

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A.M.S. Oancea et al.r Hydrometallurgy 62 (2001) 31–39

Fig. 5. Comparison of HqrZn2q integral interdiffusion coefficients vs. F on gel-like resins ŽOancea et al., 2000a, 2001. with the HqrZn2q macropore integral interdiffusion coefficients vs. F on macroporous Vionit CS 32 resin obtained with LC-BP and QHRP-ISV models; R a s 0.370 mm; 298 K.

resins. The same results are obtained for the QHRPISV model ŽFig. 5.. The comparison shows that the QHRP-ISV model applied to an ion exchange process in a macroporous resin gives practically the same integral interdiffusion coefficients as those obtained with the LC-BP model with bra s 0.01 for macropores. QHRP-ISV model is therefore a rough approximation for ion exchange kinetics in macroporous resins, and gives lower values for integral interdiffusion coefficients than the bidisperse pore model. 4. Conclusions The HqrZn2q ion exchange kinetics on the macroporous Vionit CS 32 resin Ž –SO 3 H, styrenedivinylbenzene copolymer. were investigated at 298 K using a potentiometric method, under conditions that favoured a particle diffusion mechanism. A limiting case of the Ruckenstein bidisperse pore model, a two-step mechanism with the rate of macroporous diffusion much faster than microporous diffusion, was assumed for the system being investigated. The kinetic curves F vs. t were computed for different a and bra parameters. The HqrZn2q

macroporous integral interdiffusion coefficients vs. the fractional attainment of equilibrium were obtained assuming a s 10y3 and different ratios of the micro and macropore uptakes at equilibrium Ž bra .. The results were compared with HqrZn2q integral interdiffusion coefficients obtained on gel resins and with Hq and Zn2q diffusion coefficients in dilute aqueous solution to select plausible bra values. It was shown that for the system studied bra cannot be greater than 0.3, a more suitable value being bra s 0.2. This means that in Vionit CS 32 resin the ratio of functional groups existing in macropores to those in micropores should be 15:1. The HqrZn2q macropore integral interdiffusion coefficients obtained for a s 10y3 and bra s 0.2 are smaller than the HqrZn2q integral interdiffusion coefficients in similar gel resins with 8% cross-linking. This retardation effect could be explained by the presence of an increased concentration of co-ions and by larger ionic radii of more solvated ions in macropores as compared to gel-like resins. If it is assumed that the micropore ion exchange capacity is negligible Ž bra s 0.01., then the twostep bidisperse pore model with much faster macropore diffusion than micropore diffusion gives F vs. t

A.M.S. Oancea et al.r Hydrometallurgy 62 (2001) 31–39

in very good agreement with the quasi-homogeneous resin phase-infinite solution volume model. Notation D effective intraparticle diffusivity Žm2 sy1 . Da effective macropore diffusivity Žm2 sy1 . Di effective micropore diffusivity Žm2 sy1 . ma D H q rZn 2q protonrzincŽII. macropore integral interdiffusion coefficient Žm2 sy1 . F fractional attainment of equilibrium Fa s MarMa` fractional attainment of equilibrium in macropores Fi s MirMi` fractional attainment of equilibrium in micropores total uptake at time t Mt M` s Ma` q Mi` total uptake at equilibrium macropore uptake at equilibrium Ma` micropore uptake at equilibrium Mi` pH 0 , pH ` , pH t pH of the external solution at time t s 0, at equilibrium and at time t R mean radius of the swollen beads of a gellike resin Žm. macrosphere radius or mean radius of the Ra swollen ion exchanger particles Žm. microsphere radius Žm. Ri t time Žs.

Greek symbols a s ŽD i R a2 .rŽDa R i2 . dimensionless rate parameter bra s 3 Mi` rMa` dimensionless equilibrium parameter t s ŽDa t .rŽ R a2 . dimensionless time t i s ŽDi t .rŽ R i2 . s at dimensionless time t X s ŽDt .rŽ R 2 . dimensionless time

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References CRC Handbook of Chemistry and Physics, 1998–1999, 79th edn. David R. Linde ŽEditor in Chief.. CRC Press, Boca Raton, Table 5-93. Helfferich, F., 1995. Ion Exchange. Dover Publications, New York, pp. 250–322. Helfferich, F.G., Hwang, Y.-L., 1991. Ion exchange kinetics. In: Dorfner, K. ŽEd.., Ion Exchangers. Walter de Gruyter, Berlin, pp. 1277–1309. McGarvey, F.X., Hauser, E.W., 1985. Kinetic studies on gel and macroporous anion exchangers using the uranyl sulfatersulfate exchange. In: Liberti, L., Millar, R.J. ŽEds.., Fundamentals and Applications of Ion Exchange. NATO ASI Series E, 98, Martinus Nijhoff Publishers, Dordrecht, pp. 81–101. Oancea, A.M.S., Pincovschi, E., Oancea, D., 2000a. Quasi-homogeneous resin phase diffusion models in HqrZn2q ion exchange kinetics on Purolite C 100 resin. Proc. Rom. Acad., Ser. B 2, 111–116. Oancea, A.M.S., Pincovschi, E., Oancea, D., 2000b. Protonr nickelŽII. interdiffusion coefficients on strong acid resins. Solvent Extr. Ion Exch. 18 Ž5., 981–1000. Oancea, A.M.S., Pincovschi, E., Barbulea, C., Oancea, D., 2001. HqrZn2q Ion exchange kinetics on Vionit CS 3 resin. Rev. Chim. ŽBucharest. 2 Ž1–2., in press. Ruckenstein, E., Vaidyanathan, A.S., Youngquist, G.R., 1971. Sorption by solids with bidisperse pore structures. Chem. Eng. Sci. 26, 1305–1318. Tao, Z., Zhao, A., Tong, W., Xiao, R., Chen, X., 1987. Studies on ion exchange equilibria and kinetics. I. Naq –Hq cation exchange equilibrium and kinetics. J. Radioanal. Nucl. Chem. 116 Ž1., 35–47. Weatherley, L.R., Turner, J.C.R., 1976. Ion exchange kinetics— comparison between a macroporous and a gel resin. Trans. Inst. Chem. Eng. 54, 89–94. Yoshida, H., Kataoka, T., 1985. Intraparticle mass transfer in bidispersed porous ion exchanger. Part II: Mutual ion exchange. Can. J. Chem. Eng. 63, 430–435. Yoshida, H., Kataoka, T., Ikeda, S., 1985. Intraparticle mass transfer in bidispersed porous ion exchanger. Part I: Isotopic ion exchange. Can. J. Chem. Eng. 63, 422–429.