K+ in Water and

A perfectly mixed batch reactor was used for the determination of the intraparticle diffusivities of Na+ and K+ on commercial Amberlite IR-120 resin i...
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Ind. Eng. Chem. Res. 2002, 41, 3019-3027

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GENERAL RESEARCH Determination of Intraparticle Diffusivities of Na+/K+ in Water and Water/Alcohol Mixed Solvents on a Strong Acid Cation Exchanger Juan F. Rodrı´guez, Antonio de Lucas, Jose R. Leal, and Jose L. Valverde* Department of Chemical Engineering, University of Castilla-La Mancha, Avenida de Camilo Jose´ Cela s/n, 13004 Ciudad Real, Spain

A perfectly mixed batch reactor was used for the determination of the intraparticle diffusivities of Na+ and K+ on commercial Amberlite IR-120 resin in different solvents: water and solvents composed of different percentages of water and methanol, water and ethanol, and water and 2-propanol. A theoretical model was developed to describe the response of the system that includes the resistance to mass transfer inside the particle and the effect of the electric field (NernstPlanck approximation). Equilibria were described using a homogeneous model based on the mass action law in which nonideal behavior for both the solution and the solid phase was taken into account. Simulations were performed to analyze the effect of the model parameters on the response curves. The values of intraparticle diffusivities of Na+ and K+ were obtained by nonlinear regression. Both the regression and the parameters were highly meaningful from the statistical point of view. A simple dimensionless relationship that relates diffusivity (through the Schmidt number) to different physical properties of solvents (namely, viscosity, dipolar moment, and density) and resin swelling was found. Introduction Ion-exchange methods for the separation of both organic and inorganic compounds have been widely used for many years. Because the compounds must show some ionic dissociation for the exchange of ions to occur, it is natural that water is used most commonly as the solvent. In the field of inorganic separations, the incentive to explore the use of nonaqueous or mixed solvents has been enhanced selectivity. Although the use of such media has been described in the literature,1-6 fundamental information about mass transfer and equilibrium in nonaqueous or mixed solvents is scarce. Apart from water, the best-studied solvent is methanol, which is also rather close in its properties to aqueous solutions. Thus, whereas investigations of equilibrium in nonaqueous systems are limited,3,5 kinetics studies are almost nonexistent.3,5-7 To sorb solutes and exchange ions at a practical rate, an ion-exchange resin must swell in the solvent used from its dry form. Water is a useful solvent for the commonly used ion-exchange resins, because even at relatively high levels of cross-linking, these resins swell in water at least several tens of percent beyond their dry volume.3 If, however, a nonaqueous solvent is to be used, ordinary ion-exchange resins might not swell sufficiently. It is well-known that the solvents that cause the greatest swelling are polar solvents. These are the solvents in which exchange occurs most rapidly.2 * To whom correspondence should be addressed. Phone: 34926-295437. Fax: 34-926-295318. E-mail: [email protected]

Chance et al.8 showed that better results in nonaqueous media might be obtained if the resin were not affected by the solvent and were used in the wet condition and regenerated in aqueous medium. In this sense, Lucas et al.4 demonstrated, in the direct purification of crude polyol via ion exchange, that prewetting of the resin with methanol allowed for an increase in the resin capacity and the rate of ion exchange. The models proposed in the literature to predict ionexchange equilibria can be divided into two main groups: those describing ion exchange in terms of the law of mass action and those regarding ion exchange as a phase equilibrium. It was observed that the models based on the law of mass action best represented the ion-exchange equilibrium.10-17 In a more recent paper,18 we reported ion-exchange equilibrium isotherms in pure methanol, ethanol, 2-propanol, 1-butanol, and 1-pentanol and mixtures of these solvents. Experimental equilibrium data were satisfactorily correlated using a homogeneous mass action law model in which the Wilson and Debye-Huckel equations were used to calculate the activity coefficients in the solid and liquid phases, respectively. It was also observed that the dielectric constants of both the solvent and the mixture exerted a strong influence not only on the useful capacity of the resin but also on the equilibrium. Different approaches for modeling the kinetics of ionexchange processes have been used. The resin particles can be described as porous or gel-type particles where only a quasi-homogeneous phase exists. A pore diffusion model can be employed in the first case, whereas a

10.1021/ie010848b CCC: $22.00 © 2002 American Chemical Society Published on Web 05/10/2002

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homogeneous model better represents the physical reality in the second.19 Ion fluxes can be described by either the Fick or the Nernst-Planck approximation. In either case, one must consider the homogeneous or macroporous nature of the particle to obtain a reasonable description of the diffusional process. The Nernst-Planck model accounts for the influence of the electric field caused by the difference in the mobilities of the counterions on the ion-exchange rate, whereas Fick’s law considers only diffusion. Differences between the uptake and elution kinetics for a given system and the dependence of the exchange rate on the concentration of each counterion can be predicted only with the Nernst-Planck model.20 External masstransfer resistance also affects the ion-exchange rate. In general, at low solution concentrations, external mass transfer is dominant, whereas intraparticle diffusional resistance is normally dominant at higher solution concentrations. It is generally accepted that, below a value of 0.1 mol/L, film diffusion is able to be the controlling mechanism.21 Batch, shallow bed, or single-particle methods have been applied for the measurement of intraparticle diffusivities in ion exchangers. In a previous paper,22 we reported the use of the zero-length-column (ZLC) method for measuring intraparticle diffusivities in ion exchangers. A theoretical model that considered the ZLC cell as a continuous stirred tank reactor was developed. The model described ion fluxes with the Nernst-Planck equation and included the resistance to mass transfer both in the particle and in the film. An aqueous binary system was used to test the validity of the model. In this paper, a perfectly mixed batch reactor system was used for the determination of the intraparticle diffusivities of Na+ and K+ using a commercial resin (Amberlite IR-120). The objectives of this work are as follows: (i) to develop a mathematical model to describe the response of a perfectly mixed batch reactor system that includes the resistance to mass transfer inside the particle and the effect of the electric field (Nernst-Planck approximation) and describes equilibria using a homogeneous model based on the mass action law in which nonideal behavior for both the solution and the solid phase is taken into account; (ii) to analyze the effect of the model parameters on the response curves; (iii) to study the ion-exchange kinetics of Na+ and K+ on commercial Amberlite IR-120 resin in different solvents, namely, water and solvents composed of different percentages of water and methanol, water and ethanol, and water and 2-propanol; and (iv) to study the influence of the parameters, such as viscosity, density, dielectric constant, and resin swelling, on the intraparticle diffusivities of Na+ and K+.

exchanger are negligible. (iv) Ion intraparticle diffusivities do not depend on the particle concentration. Considering an ion-exchange isothermal process between an ion i+ (Na+) presaturating the resin and an ion j+ (K+) entering the resin particle

R-i+ + j+ S R-j+ + i+

The general equation corresponding to the flux of ion expressed in terms of the concentration gradient assuming Nernst-Planck equations for the ion flux of both counterions (i and j) and the electroneutrality condition is given by

Ni ) -Dij

∂qi ∂r

(2)

where Dij is the interdiffusion coefficient

[

Dij ) Di 1 +

qi(1 - δ)

]

(3)

Q + qi(δ - 1)

δ ) Di/Djis the ratio between the diffusivities of the species i and j, and Q is the total solid-phase capacity. The mass balance in the resin particle is 2 ∂qi 1 ∂(r Ni) )- 2 ∂t ∂r r

(4)

and by introducing the flux eq 2 together with eq 3, the following equation can be derived22

()

() ( )

2DiQ DiQ(δ - 1) ∂qi ∂qi ∂qi ) ∂t ∂r r[Q + (δ - 1)qi] (Q + (δ - 1)qi)2 ∂r D iQ

∂2qi

[Q + (δ - 1)qi] ∂r2

2

+

(5)

with initial and boundary conditions

t ) 0 qi ) Q, Ci ) Ci0 ) 0 r)0

∂qi )0 ∂r

(6)

and, because the experiments were done with negligible mass-transfer resistance in the solution

r ) Rp Csi ) Ci

(7)

The concentration at the solid resin surface qis is related to the concentration of the liquid Csi by an equilibrium equation based on the mass action law

Mathematical Model The resin used in the present work was a gel-type resin. Thus, the homogeneous model better represents the physical reality inside the particle. Model equations were derived including the following simplifying assumptions: (i) The whole resin is treated as a quasi-homogeneous phase. Resin beads are assumed to be completely spherical in shape and to undergo no appreciable volume change during the ionexchange process. (ii) The effects of pressure gradients are neglected. (iii) Co-ion concentrations in the ion

(1)

K)

(1 - Ysi )γsj Ysi γsi

X iγi (1 - Xi)γj

(8)

where Ysi is the normalized concentration of species i at the external surface of the resin phase and γsi and γi are the activity coefficients of species i at the external surface of the solid and in the solution phase, respectively. In previous papers, the reliability of the Wilson equation in correlating the activity coefficients in the

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solid phase with the composition was demonstrated.9-11,15 Specifically, for a binary mixture, this equation for one of the ions is given by

ln γ j i ) 1 - ln(Yi + YjΛji) -

Yi YjΛij Yi + YjΛji YiΛij + Yj (9)

The activity coefficients of the ions in the solution can be computed by the Debye-Huckel equation

Azi2xI

log γi ) -

(10)

1 + BaixI

where I is the ionic strength of the solution; ai is the ion size parameter (4.0 Å for Na+, 3.5 Å for K+ in aqueous solution); and A and B are calculated from the dielectric constant, density, and temperature of the solvent.23 Taking as reference water as the solvent, the ratio γNa/γK at the experimental conditions of this work (CT ) 0.03 M at 313 K) is equal to 1.0040. Assuming this value to be practically unity for all systems because of the difficulty in estimating specific values of the parameter ai in solvents other than water, eq 8 can be reduced to

K)

(1 - Ysi )γsj

Xi

The mass balance for a perfectly mixed batch reactor is

( )

3WDij ∂qi dCi )dt VFpRp ∂r

(12)

r)Rp

Introducing the following dimensionless variables

Yi )

qi Ci Dit Di r τ) 2 R) δ) Xi ) Q CT Rp Dj R

(13)

p

eq 5 becomes

( ) ( )

∂Yi ∂Yi 2 ) ∂τ R[1 + (δ - 1)Yi] ∂R (δ - 1)

∂Yi 2 ∂R [1 + (δ - 1)Yi]

2

+

( )

∂2Yi 1 (14) [1 + (δ - 1)Yi] ∂R2

The initial and boundary conditions in dimensionless form are, respectively

τ ) 0 Yi ) 1, Xi ) 0 ∂Yi )0 ∂R

(15)

R ) 1 Xsi ) Xi

(16)

R)0

with the equlibrium relation

K)

(1 - Ysi )γsj Ysi γsi

Xi (1 - Xi)

The mass balance for a perfectly mixed batch reactor in dimensionless form is

( )

dXi ∂Yi 3R )dτ [1 + Yi(δ - 1)] ∂R

(17)

R)1

(18)

where

R)

(11)

(1 - Xi)

Ysi γsi

Figure 1. Influence of the number of interior collocation points on two finite elements (Di ) 5 × 10-11 m2/s, δ ) 0.5, R ) 1, K ) 2, Λij ) 1, Λji ) 1).

WQ VCT

(19)

The model parameters are as follows: (i) equilibrium parameters K, ΛNaK, and ΛKNa; (ii) diffusivity ratio δ ) DNa/DK; and (iii) system parameter R ) WQ/VCT. The above system of equations was solved using orthogonal collocation on finite elements with Lagrange polynomials as trial functions.24 By applying this mathematical technique one can reduce the partial differential eqs 14 and 18 to a set of ordinary differential equations. After doing this, one has the values of the solution at each interior collocation point at the new time. Boundary values and the values of the solution at the points between elements were evaluated by solving the corresponding algebraic equations. The Runge-Kutta-Fehlberg method was used in the evaluation of the set of ordinary differential equations, whereas Marquardt’s algorithm was used in the evaluation of the algebraic equations.25 A VBA-Excel application on a Compaq SP750 computer was developed to solve this model. The influence of the number of interior collocation points on two finite elements is depicted in Figure 1. As verified, only with three or more interior points was the necessary tendency to equilibrium achieved. No differences were observed when the number of points was larger than three. We chose this number of interior points because the time needed to attain the solution was the lowest. In this case, less than 3 s (about 200 integration steps) was required to obtain a theoretical desorption curve. Simulation Results When ions with different mobilities are involved in a process, it is necessary to take into account NernstPlanck equations. Figure 2 shows the desorption curves for different values of the relation between the diffusion coefficients, δ ) Di/Dj, at fixed and realistic values of all other parameters. Following the minority rule, the exchange rate in the initial part of the elution process

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Figure 2. Theoretical desorption curves, Xi vs τ, calculated from the model. Influence of the diffusivity ratio of the counterions δ: (a) δ ) 0.5, (b) δ ) 2 (Di ) 5 × 10-11 m2/s, R ) 1, K ) 2, Λij ) 1, Λji ) 1).

Figure 3. Evolution of the dimensionless radial concentration profiles, Yi vs τ, inside the particle. Influence of the diffusivity ratio of the counterions δ: (a) δ ) 0.5, (b) δ ) 2 (Di ) 5 × 10-11 m2/s, R ) 1, K ) 2, Λij ) 1, Λji ) 1).

is limited by the diffusivity of the entering ion. Accordingly, as the diffusivity of the entering ion increases, the flux of the ion initially in the resin grows. One can observe that the concentration of the eluted ion increases more quickly when the diffusivity of the entering ion is larger. As can be seen in Figure 3, for identical operating conditions, the evolution of the dimensionless profiles inside the particle is different for different values of δ. As expected, the concentration profile of the exiting ion inside the particle is more pronounced when the diffusivity of the entering ion is larger.

Figure 4. Theoretical desorption curves, Xi vs τ, calculated from the model. Influence of the equilibrium constant K: (Di ) 5 × 10-11 m2/s, δ ) 0.5, R ) 1, Λij ) 1, Λji ) 1).

Figure 5. Evolution of the dimensionless radial concentration profiles, Yi vs τ, inside the particle. Influence of the equilibrium constant K: (a) K ) 0.1, (b) K ) 10 (Di ) 5 × 10-11 m2/s, δ ) 0.5, R ) 1, Λij ) 1, Λji ) 1).

Figure 4 shows the effect of the equilibrium constant on the response curves for a fixed set of parameters. The process can be equilibrium-controlled when the isotherm is very unfavorable,26 as it is difficult to say anything about the diffusion coefficients under such conditions. As can be seen in Figure 5, for identical operating conditions, the evolution of the dimensionless profiles inside the particle is very different for different K values. When K ) 0.1, the greater affinity of the resin for the sorbed ion produces nearly flat concentration profiles inside the particle; that is, the process is exclusively controlled by the equilibrium relation. On the other hand, the equilibrium constant has a strong influence on the shape of the first part of the

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Figure 7. Theoretical desorption curves, Xi vs τ, calculated from the model. Influence of the system parameter R (Di ) 5 × 10-11 m2/s, δ ) 0.5, K ) 2, Λij ) 1, Λji ) 1). Table 1. Properties of Amberlite IR-120 matrix type functional structure moisture content (%) particle density (kg of dry resin/m3 of wet resin) Total exchange capacity (equiv/kg of dry resin)

Figure 6. Influence of the binary interaction parameters (Λij and Λji) on the prediction of the theoretical desorption curves: (a) equilibrium isotherms, (b) response curves (Di ) 5 × 10-11 m2/s, δ ) 0.5, R ) 1, K ) 2).

desorption curve. For a favorable isotherm, the resin prefers the ion in solution, so the adsorbed ion is easily and rapidly released. In contrast, for a system with an unfavorable isotherm, the ion exchanger prefers the ion initially in the resin, which is more slowly exchanged. In Figure 6 is displayed the effect of the binary interaction parameters (Λij and Λji) on the response curves for a fixed set of parameters. As expected, different values of these parameters lead to different equilibrium isotherms (Figure 6a). This situation forces each of the response curves to tend to different equilibrium values (Figure 6b). Because, in this case, all of the curves were derived using the same values of the intraparticle diffusivities, an important conclusion could be derived. Different values of these diffusivities should be attained depending on the mathematical model used to represent the equilibrium (either an equilibrium model similar to that used here or a simplified one, that is, linear or ideal). The parameters δ and K are related to the physical and chemical properties of the species, and they cannot be varied for a given system. R is the parameter of the system. It includes variables that can be easily modified (W, V, and CT). The trend of the response curves with this parameter is shown in Figure 7. As expected, for lower values of R, higher equilibrium concentrations are reached. Experimental Section Chemical. All of the solvents used in this investigation were PRS grade and were supplied by Panreac

geliform sulfonic 47-62 0.560 5.01

(99%). Sodium and potassium terbutoxide (Alfa, 99%) were used in the preparation of ionic solutions. Demineralized water with a conductivity value lower than 1 µS was used. The cationic resin Amberlite IR-120 supplied by Rohm & Haas was used as the exchanger. As described by Lucas et al.,13 the resin was pretreated and regenerated to convert it to the Na+ form. In the experiments with anhydrous solvents, the commercial solvent was previously dried with calcium oxide and then with sodium under reflux. Prior to experimental runs, the resin was exhaustively washed with the anhydrous solvent to achieve complete moisture removal and swelling of the resin. The physical properties of the resin are summarized in Table 1. In this paper, a unique resin capacity was considered because, as shown elsewhere,18 this magnitude is always determined by the most polar solvent of the mixture (water) and is not dependent on its percentage in the liquid phase. Procedure. The intraparticle diffusion dynamics was studied by measuring the rate at which sodium was released from the resin in a well-mixed tank at 313 K. In a typical experiment, a weighed amount of the ionexchange resin (about 5 g) was mixed with approximately 1 kg of the solvent in a baffled glass mixing vessel equipped with an electric motor driving a standard turbine stirrer with six flat blades. The mixture was then stirred for a period of time. Small aliquots (about 2 × 10-3 g) of the solution were periodically taken, and the potassium content was analyzed by atomic emission spectrometry in a Thermo Jarrel Ash (Smith Hiefje II) spectrometer. Different standard solutions were prepared to account for the matrix effect of the mixed solvents on the spectrometer results. Tables 2 and 3 contain the values of the dielectric constant (), percentage of swelling (H), Debye-Hu¨ckel equation parameters (A and B), average particle radius (Rp), density (F) and viscosity of the solvent (µ), equilibrium constant (K), and binary interaction parameters (ΛNaK and ΛKNa) at 313 K for all of the systems under study. Viscosity measurements were done in a rotating digital viscometer (Brookfield model DV-II). Resin swelling was measured using a microscope.

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Table 2. Dielectric Constant (E), Percentage of Swelling (H), Debye-Hu 1 ckel Equation Parameters (A and B), Average Particle Radius (Rp), and Density (G) and Viscosity of Solvent (µ) for the Binary System Na+/K+ in Pure Solvents and Mixtures on the Resin Amberlite IR-120 at 313 K system (1/2)

% 1:% 2 (by weight)

a (293 K)

H (Na form)

A [(mol/kg)-1/2]

B × 109 [m-1 (mol/kg)-1/2]

Rp × 106 (m)

F (kg/m3)

µ × 103 [kg/(m s)]

H2O CH3OH/H2O CH3OH/H2O CH3OH/H2O C2H5OH/H2O C2H5OH/H2O C2H5OH/H2O i-C3H7OH/H2O i-C3H7OH/H2O i-C3H7OH/H2O

100:0 75:25 50:50 25:75 75:25 50:50 25:75 75:25 50:50 25:75

80.37 44.19 56.32 68.45 37.20 51.35 65.49 29.34 45.65 61.87

89.43 40.72 59.53 75.76 28.71 48.15 68.38 36.22 55.91 73.76

1.05 2.37 1.71 1.32 3.01 1.94 1.31 4.36 2.42 1.53

3.17 3.93 3.60 3.39 4.20 3.73 3.37 4.79 4.14 3.56

409 303 315 328 299 313 327 305 319 330

996.0 818.9 873.8 933.0 856.6 916.3 961.3 795.4 859.0 927.8

6.51 5.14 5.79 6.38 7.88 7.82 7.47 10.9 9.62 8.26

a

Dielectric constant from Wohlfarth.31

Table 3. Equilibrium Constant (K) and Binary Interaction Parameters for the Binary system Na+/K+ (ΛNaK and ΛKNa) in Pure Solvents and Mixtures on the Resin Amberlite IR-120 at 313 K18 system (1/2)

% 1:% 2 (by weight)

K

ΛNaK

ΛKNa

H2O CH3OH/H2O CH3OH/H2O CH3OH/H2O C2H5OH/H2O C2H5OH/H2O C2H5OH/H2O i-C3H7OH/H2O i-C3H7OH/H2O i-C3H7OH/H2O

100:0 75:25 50:50 25:75 75:25 50:50 25:75 75:25 50:50 25:75

1.15 2.37 2.23 2.04 1.56 1.98 1.68 1.39 1.28 1.42

2.08 0.42 0.84 1.50 0.99 1.29 1.24 1.04 1.18 1.06

0.48 3.02 1.33 0.99 0.90 0.80 0.76 1.23 0.69 0.55

Results and Discussion As shown in a previous paper,18 for water and aqueous/organic mixtures, the selectivity of the resin to K+ throughout the range of compositions studied is greater than that to Na+. As mentioned above, the resin capacity is always determined by the most polar solvent. This fact would demonstrate that water seems to be the only solvating agent of the ion-exchange sites and, therefore, that the hydrophilic interaction between the ionic groups of the resin and the polar part of the molecules of the solvent is stronger than the hydrophobic one. The Nernst-Planck model requires knowledge of the resin-phase self-diffusivity of each ion involved in the process. Reliable correlations linking the resin-phase diffusivity to the liquid-phase diffusivity are not available, and the different values proposed in the literature are somewhat contradictory.27 Table 4, in turn, contain the parameters DNa and DK obtained by nonlinear regression.25 In all cases, less than seven iterations were needed. In the same table are shown the results of the tests of statistical significance for all of the systems and their corresponding parameters. A more detailed analysis of this procedure was reported elsewhere.26 A regression is considered to be meaningful whenever the ratio Fc/F (F test) is higher than unity. Likewise, one specific parameter in the model is meaningful whenever the ratio tc/t (t test) is always higher than unity. It can be noted that, for all of the systems studied, both the regression and the parameters are highly meaningful from the statistical point of view. In Figures 8-11, the experimental and predicted concentrations of Na+ in solution are compared. Good agreement is observed.

Figure 8. Plot of Xi vs τ in the perfectly mixed batch reactor system with water as the solvent. Comparison of experimental points ([) and predicted results (s).

Figure 9. Plot of Xi vs τ in the perfectly mixed batch reactor system with methanol (M)/water (W) mixtures as the solvent. Comparison of experimental points for (9) 75% M/25% W by weight, (2) 50% M/50% W by weight, (b) 25% M/75% W by weight and predicted results (solid lines).

It can be verified that, for any given system, DK is always higher than DNa. This result agrees with that reported for a water solution in a previous paper by the authors using the ZLC method.22 In any case, it seems clear that the intraparticle diffusivities of both ions should be affected by the physical properties of the solvents and by resin swelling. Chance et al.8 observed that the rate of solute exchange appears to vary directly as the cube root of the electrical conductivity of the feed solution and inversely as the viscosity of the solution. Wilson and Lapidus7 showed that diffusivities are largely dependent

Ind. Eng. Chem. Res., Vol. 41, No. 12, 2002 3025 Table 4. Intraparticle Diffusivities of Na+ and K+ and Tests of Statistical Significancea for the Binary System Na+/K+ in Pure Solvents and Mixtures on the Resin Amberlite IR-120 at 313 K system (1/2)

% 1:% 2 (by weight)

a

DNa (m2/s)

DK (m2/s)

Fc/F

(tc/t)Na

(tc/t)Na

H2O CH3OH/H2O CH3OH/H2O CH3OH/H2O C2H5OH/H2O C2H5OH/H2O C2H5OH/H2O i-C3H7OH/H2O i-C3H7OH/H2O i-C3H7OH/H2O

100:0 75:25 50:50 25:75 75:25 50:50 25:75 75:25 50:50 25:75

0.942 0.930 0.958 0.983 1.003 0.861 0.825 0.849 0.943 0.463

2.24 × 10-11 8.80 × 10-12 1.36 × 10-11 1.32 × 10-11 9.54 × 10-12 7.63 × 10-12 8.53 × 10-12 3.15 × 10-12 6.32 × 10-12 1.09 × 10-11

5.65 × 10-11 1.75 × 10-11 2.12 × 10-11 3.11 × 10-11 1.45 × 10-11 2.04 × 10-11 2.22 × 10-11 8.97 × 10-12 1.36 × 10-11 2.64 × 10-11

1613 1608 1350 588 506 200 659 3667 607 34.8

6.4 6.3 7.1 4.6 5.3 4.4 7.1 19 5.1 1.3

18 6.2 3.2 7.4 2.3 21 23 310 5.8 2.8

a

Significance level of 0.05.

Figure 10. Plot of Xi vs τ in the perfectly mixed batch reactor system with ethanol (E)/water (W) mixtures as the solvent. Comparison of experimental points for (9) 75% E/25% W by weight, (2) 50% E/50% W by weight, (b) 25% E/75% W by weight and predicted results (solid lines).

Figure 11. Plot of Xi vs τ in the perfectly mixed batch reactor system with 2-propanol (P)/water (W) mixtures. Comparison of experimental points for (9) 75% P/25% W by weight, (2) 50% P/50% W by weight, (b) 25% P/75% W by weight and predicted results (solid lines).

on cross-linking and the solvent composition. On the other hand, Marcus3 observed that the degree of resin swelling and the viscosity of anhydrous and mixed solvents were major factors in determining the diffusion coefficients in the resin. An exponential relationship between resin swelling and the diffusion coefficients was, in turn, proposed by Kataoka et al.29 However, according to Reid et al.,30 a relation also exists between the viscosity and the diffusion coefficients in solution. On the other hand, Lucas et al.5 observed that the diffusion coefficients of potassium in aqueous systems were at least 1 order of magnitude higher than those

Figure 12. Comparison between experimental and predicted dimensionless relation (µ/DF) for (2) Na+ and (9) K+.

in water/methanol/polyol mixtures. Lucas et al.6 likewise noted that the diffusion coefficients for crude polyols were at least 1 order of magnitude smaller than those obtained in the mixtures and found an empirical equation for estimating the diffusion coefficients including viscosity and resin swelling as the main influences. In our case, the different solvents also exhibit different physical properties (namely, viscosity, dipolar moment, and density) that lead to different resin swelling values. In this way, a simple dimensionless relationship that relates diffusivity (through the Schmidt number) to these parameters was found

( ) ( )

( ) ( ) ( ) ( )

µ  ) 4.35 × 105 DNaF water

-4.15

H Hwater

2.08

µ  ) 1.69 × 105 DKF water

-4.15

H Hwater

2.08

(20)

Both the model and the parameters were significant from the statistical point of view: the values of Fc/F and tc/t for all of the parameters were higher than 1. In Figure 12, the experimental and predicted values of the dimensionless quantity µ/DF are compared, and good agreement is observed (average relative error lower than 22%). This empirical expression must not be generalized until more information on different systems has been obtained. Further experiments with solvents would allow us to extend its range of applicability. In this case, it would be interesting to work with pure organic solvents or mixtures of them, because, as indicated in

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a previous paper,18 the polarity of the solvent strongly affects the resin capacity. Conclusions A perfectly mixed batch reactor system was used for the determination of the intraparticle diffusivities of Na+ and K+ on commercial Amberlite IR-120 in different solvents: water and solvents composed of different percentages of water and methanol, water and ethanol, and water and 2-propanol. A mathematical model for analyzing the perfectly mixed batch reactor desorption curve in binary ion-exchange systems was developed. The model included the resistance to mass transfer inside the particle and the effect of the electric field Nernst-Planck approximation). Equilibria were described using a homogeneous model based on the mass action law in which nonideal behavior for both the solution and the solid phase was taken into account. Simulations were performed to analyze the effect of the model parameters on the response curves. The intraparticle diffusivities of Na+ and K+ were obtained by nonlinear regression using this experimental setup. It was observed that the intraparticle diffusivities of both ions should be affected by the physical properties of the solvents and by resin swelling. A simple dimensionless relationship that relates diffusivity to the different physical properties of solvents and resin swelling was found. This empirical expression must not be generalized until more information on different systems has been obtained. Notation A ) Debye-Hu¨ckel equation parameter, (mol/kg)-1/2 ai ) ion-size parameter of species i, m B ) Debye-Hu¨ckel equation parameter, m-1 (mol/kg)-1/2 Ci, Csi ) concentration of species i in the solution phase, mol/m3 Ci0 ) initial concentration of species i, mol/m3 CT ) total ionic concentration in solution, mol/m3 Di ) intraparticle diffusion coefficient of species i, m2/s Dij ) interdiffusion coefficient, eq 3, m2/s Fc/F ) statistical parameter (F test) H ) percentage of swelling I ) ionic strength in molal units K ) thermodynamic equilibrium constant Ni ) flux of ion i, mol/(m2 s) Q ) total solid-phase capacity, mol/m3 qi ) solid-phase concentration of species i, mol/m3 qsi ) concentration of species i at the solid resin surface, mol/m3 R ) dimensionless special coordinate in the particle r ) special coordinate in the particle, m Rp ) particle radius, m t ) time, s tc/t ) statistical parameter (t test) V ) volume of liquid in the vessel, m3 W ) weight of resin, kg Xi, Xsi ) dimensionless concentration of species i in the fluid phase Yi ) dimensionless concentration of species i in the resin phase Ysi ) normalized concentration of species i in the external surface of the resin phase Greek Letters R ) system parameter, eq 19 γi ) activity coefficient of species i in the solution phase

γsi ) activity coefficient of species i in the external surface of the solid δ ) ratio between diffusivities of ions i and j  ) dielectric constant of the fluid Λij ) binary interaction parameter in the Wilson equation µ ) solvent viscosity, kg/m.s F ) solvent density, kg/m3 Fp ) solid density, kg/m3 τ ) dimensionless time

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Received for review October 16, 2001 Revised manuscript received March 1, 2002 Accepted March 9, 2002 IE010848B