2618
Ind. Eng. Chem. Res. 1995,34, 2618-2624
Chloride/Sulfate Ion Exchange Kinetics on Fibrous Resins. Two Independent Models for Film Diffusion Control Domenico Petruzzelli,*”Anatolyi Kalinitchev? Vladimir S. Soldatov? and Giovanni Tiravantit Istituto di Ricerca sulle Acque, National Research Council, 5, Via De Blasio, 70123 Bari, Italy, Institute of Physical Chemistry, Russian Academy of Science, 31 Lenin Avenue, Moscow GSP 117915, Russia, and Institute of Physic0 Organic Chemistry, Belarus Academy of Science, 13 Surganova Ul., 220603 Minsk, Belarus
Fibrous ion exchangers appear t o be very promising for full scale applications in water conditioning (demineralization, softening), wastewater treatment for pollutant control, as well as for applications in new areas promoted by the physical structure of the materials. The kinetic performances of these exchangers appear to be faster than traditional bead-shaped resins. In a previous paper, the kinetic behavior of chloride/sulfate ion exchange system on fibrous resins was investigated in a wide range of liquid-phase concentrations (i.e., 0.006-1.8 N). The NernstPlanck model was adopted for data correlation, and the rate-determining step was associated with the mass transfer resistance to the interdiffusion of ions in the liquid film around the microfibers. In the present paper, the same data are revisited on the basis of an alternative model for film diffusion control, based on the numerical solution of a phenomenological equation accounting for the selectivity of the resin toward the exchanging counterions. In this context, equilibrium studies were also carried out and criticized to get relevant data for model implementation. Both kinetic models gave equivalent results in experimental data correlation, thus confirming the exposed surface area of the fibrous exchangers, a more important factor with respect to selectivity in determining the peculiar kinetic performance of these materials.
Introduction Ion exchange technology based on the use of fibrous polymers is starting to be applied for full scale applications in traditional areas such as water conditioning (demineralization, softening, and polishing) (Chatelin, 1992a),wastewater treatment for pollutant control and recovery (Soldatov, 1991; Bytsan et al., 19931, as well as in other areas h e . , medicine, material science), by taking advantage of the physical structure of the reference materials (Masaru et al., 1992; Chatelin, 1992b). Interest is focused on the favorable kinetic performances of fibrous materials which, on the average, are at least 1order of magnitude faster than the equivalent bead-shaped resins (Petruzzelli et al., 1993; Liquan et al., 1994). In a previous paper (Petruzzelli et al., 1993) was discussed the kinetic behavior of a set of fibrous anion resins, differing in the basicity of the functional groups, and compared to those of equivalent bead-shaped (spherular) resins. Results confirmed the remarkably good performances of the fibrous materials and the NernstPlanck model was applied for data correlation. The rate-determining step of the overall kinetic process was associated with the mass transfer resistance to the ionic interdiffusion in the stationary liquid film around the constituting microfibers of the exchanger material (felts, cloths, yarns). In the present paper, the same kinetic data on fibrous resins are revisited in light of an alternative phenomenological model, with respect to the Nernst-Planck, which is based on the numerical solution of the dif-
* To whom correspondence should be addressed. E-mail:
[email protected]. National Research Council. Russian Academy of Science. E-mail: Kalinitchea 1mm.phyche.msk.su. 8 Belarus Academy of Science.
ferential phenomenological equation describing liquidphase mass transfer in charged systems. Thermodynamic data are also reported for convenience of correlation of the phenomenological model, and experimental exchange isotherms are used for the numerical solution of the kinetic equations. Thermodynamic implications are discussed on the basis of a critical evaluation of the equilibrium data.
Theory Nernst-Planck Film Diffusion Control (FDC) Model. The reference physical model assumes that overall exchange phenomena are controlled by the liquid-phase mass transfer resistance to ionic interdiffusion in the stationary liquid film around the exchanger particle, on the basis of the “Nernst film concept” (Glueckauf and Coates, 1947). The model assumes the following as fundamental premises: the exchanger phase containing ion A as the only counterion, in contact with the solution of constant concentration of counterion B; linear driving force for the ion concentration profiles through the stationary liquid film; constant separation factor, a B A , for exchanging ions at a given solution concentration; no concentration gradients of ions and no uptake or release of water by the solid phase; quasi-stationary state of liquid-phase mass transfer. The first assumption corresponds to what is commonly termed the “infinite solution volume” (ISV) condition (Helfferich, 1962d):
+
0888-5885/95/2634-2618$09.o0/0
(For symbols see the Nomenclature section at the end of the paper.). 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2619 The basic flux equation for ionic interdiffusion under the Nernst film concept is (Glueckauf and Coates, 1947)
J = DACId
(2)
which, integrated under conditions 1,leads to (Helfferich, 1962a)
+
hl(1 - F) (1- a B A ) F = - 3 D C a ~ ~ t / d d c (3) According to eq 3 conversion kinetics is inversely proportiorgl to the fiber diameter, d, to the ion exchange capacity, C, and to the thickness of the stationary liquid film around the fibers, 6, and directly proportional to the separation factor, a B A , the liquid-phase concentration, C, and to the ionic diffusivities in liquid film, D . The dependence on the fiber size reflects the fact that the surface area per unit volume of material is inversely proportional to the fiber radius or diameter. The dependence on ion exchange capacity depends on the fact that, at a given rate, the time t o fill a reservoir is dependent on its capacity. The dependence on the thickness of the Nernst liquid film reflects the length of the ionic diffusional patterns offering resistance t o mass transfer. Of special interest for the discussions to follow is the limiting case with large values of the separation factor, a B A , corresponding to those very selective systems with a strong preference for the ion initially in the liquid phase. For a B A >> 1 and not too small values of 1 - F (initial stages of the kinetic process), eq 3 reduces to
F M 3DCtlddE
b, = 3D,/d6 and i = 1,2, ..., m.
dY,/dt = b,'[l = X,(Y,)l; b,' = b,C/Q (s-')
(5)
(6)
and numerical integration, the following expression form for the kinetic fractional conversion, F(t), is obtained: F(t)
(4)
This reflects a linear dependence of the fractional conversion,F, vs time and thus an almost constant rate of exchange. Physically, this is explained in terms of the almost constant concentration gradients in the liquid film, depending on the high selectivityof the resin for the ion initially in the liquid phase, whose concentration a t the interface remains negligible till almost complete conversion,whereas the bulk concentration of the same ion is constant (ISV conditions). The concentration gradients in the liquid film remain constant, thus leading to constant rate of exchange (Helfferich et al., 1985). Moreover, in the limiting case of eq 4 kinetics are independent of the separation factor (selectivity), due to the fact that an even higher affinity of the resin €or the ion being taken up would merely further reduce an already negligible concentration a t the liquid-solid interface, responsible for the mentioned constant concentration gradient. At later stages of the kinetic process, when the concentration profiles in the liquid phase tend ineluctably to level off, the separation factor tends to regain its influence according to eq 3. In the case at hand, the mass tranfer parameter, Le., DI6, values may be evaluated by the slope of the linear experimental F(t) kinetic curves. Phenomenological Equation for Film Diffusion Controlled Ion Exchange Kinetics. The alternative model for description of film diffusion controlled ion exchange kinetics is discussed as follows. The model is based on the well-known general differential equation €or mass transfer control of ions in the liquid film around the particles (Helfferich, 1962d): dC,"ldt = bJCio- @JC*)]
where Q1(C,*,C2*,...,&,* ) are multicomponent isotherms of a mixture of m components. Based on eq 5 the model assumes linear concentration gradients in the liquid film and postulates that the rate of the kinetic process is proportional to the distance of the system from the equilibrium condition (i.e., the leveling off of the concentration gradients). In addition to the above assumptions the model assumes also the following conditions: no concentration gradients in the solid phase; no uptake or release of water by the solid phase; quasi stationary state of the liquid-phase mass transfer; infinite solution volume kinetic conditions. Differently from the_previous model the concentration of the invading ion (C,*), at the liquid-solid interface does not remain negligible during the kinetic process and is related to the solid-phase concentration of the same ion by the condition of equilibrium at the liquidsolid interface, i.e., th_e multicomponent equilibrium isotherm equation, @(C,*). This is particularly important in the second part of the kinetic process (t > t o 5 ) when the concentration gradients are no more constant (i.e., constant rate, see eq 4 ) and begin to level off. After conversion of eq 5 in dimensionless form
JFdY,/[l
- X,(Y,)] = b,'t
(7)
The following polynomial approximation for the experimental equilibrium isotherms is adopted for numerical iteration of eq 7. [l
- X,(Y,)] = k,Y, + k2Y12+ ... + k7YZ7
(8)
kl, ..., k7 are arbitrary constants and X and Y the ionic fractions. The kinetic coefficient b,' and thus the related mass transfer parameter, D/6, may be calculated from experimental data by evaluating the initial slope of the F vs t kinetic curve:
b,' = tar@) where
(9)
is the angle a t point 0,O of the experimental
F(t) curve. Comparison of the Kinetic Models By comparing respective formulations, specifically eqs 3 and 5, the phenomenological model results to be more general than the Nernst-Planck and main differences and advantages of one over the other can be summarized as follows: In the phenomenological model the separation factor, a", accounting for the selectivity of the resin toward the exchanging counterions, is not assumed constant but is variable according t o the experimental equilibrium isotherm formulation, Xi(YJ (see eq 7). As a consequence, the concentration gradients in the liquid film are not constant because the concentration of the ionic species at the film-particle interface varies according to the isotherm formulation. A similar evaluation of the liquid-solid interface concentrations in ion exchange kinetic systems was
2620 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 Table 1. Main Physicochemical Properties of the Fibrous Resins Investigated resin matrix functional group capacity (mequidg) Fiban A1 polypropylene-polystyrene quaternary ammonium 2.87 secondary-tertiary amine 5.03 Fiban AK22 polyacrylic Fiban AK22-1 polyacrylic secondary-tertiary amine 3.28
made by the authors for the case of a solution for simultaneous film and particle diffusion control, but it referred to binary systems (Petruzzelli et al., 1987). Generally speaking, the phenomenological model gives a more accurate description of the actual phenomena determining mass transfer of ions in the stationary liquid film around the particle. Last but not least, the phenomenological model allows for correlation of multicomponent systems, thus rendering this latter model more general.
A -
t Ll
Materials and Methods Fibrous Resin. Fibrous exchangers were manufactured by chemical modification of commercial polypropylene fibers(So1datov et al., 1987, 1988). Resin Fiban A1 (type I strong-base quaternary ammonium) was prepared by grafting polypropylene fibers with divinylbenzene followed by functionalization (chloromethylation and amination) of this latter by using trimethylamine. Resins AK22 and AK22-1 (weak base secondary and tertiary amine), differing in their exchange capacity, were functionalized by amination of polyacrylic fibers with diethyltriamine pendant groups (Soldatov et al., 1987, 1988). The main physico-chemical properties of the resins are reported in Table 1. Conditioning of the Resins. After three preliminary acid-base reconversions of the resins in chloride and free base form (1 M HC1 and NaOH) to remove solvents and other preparation chemicals, the samples were converted in chloride or sulfate form by exhaustive elution of NaCI/HCl or Na2SO&I2S04 mixtures of 0.006, 0.06, and 1.8 N, pH 3, respectively. Conditioned resin were then dried under vacuum (35 "C, 1,333 Pa) and stored in dry atmosphere. Exchange Capacities and Equilibrium Isotherms. The exchange capacities of the weak base resins Fiban AK22 and AK22-1 were determined from both chloride and sulfate form resins by adopting the batch procedure proposed by Helfferich for weak electrolyte resins (Helfferich, 1962131, accounting for hydrolytic phenomena. For strong base anion resin Fiban A1 the column technique was preferred (Helfferich, 196213). Equilibrium experiments were run by using jacketed glass columns (25 cm length, 1 cm i.d.1. For this purpose, weighed amounts of conditioned C1-form resin samples were loaded into columns and eluted exhaustively with the equilibrating solutions at different total concentration (0.006, 0.06, and 1.8 N, pH 3) and different C1-/S042- ionic ratios (i.e., Xcl = 0.25, 0.50, 0.75). After equilibration, columns were quickly rinsed with 1 bed volume of methanol-water mixture and resin samples were then regenerated with 1 M NaOH. Chloride and sulfate ions in the spent regeneration eluates, corresponding to the amounts of ions retained by the resin, were determined by ion chromatography with a Dionex Model 4000i apparatus equipped with a AS9 SC general column for anions (AG9 SC precolumn), both from Dionex Sunnyvale, CA. All chemicals were analytical grade from Carlo Erba, Milan, Italy.
fiber diameter @m) 30-60 20-40 20-40
0 -
X
I
P
I
Figure 1. Experimental setup for kinetic experiments. (A) The Kressman-Kitchener stirrer-reactor. (B)RE, reactor; b, reaction batch; M, S, stirrers; EL, electrodes; UT, ultrathermostat; PA, pneumatic lift; E, potentiometer; R, recorder; DAS, data acquisition system; C, computer; P, plotter.
Batch Kinetic Experiments. Kinetic experiments were run in triplicate by means of the KressmannKitchener stirrer-reactor technique, and were monitored automatically through the experimental setup reported in Figure 1 (Petruzzelli and Palmisano, 1981). For this purpose, weighed amounts (10 mg) of C1-form conditioned resin were loaded in the reactor which was immersed while rotating into the reaction batch containing 1 L of Na2S04 solution, pH 3, at predetermined concentration (i.e., 0.006,0.06,1.8N). Infinite solution volume (ISV)kinetic conditions (i.e., negligible concentration buildup in solution for the counterion released by the resin in solution) were adopted throughout the experiments. Chloride ion released by the resin was continuously monitored in the batch by means of an ion-selective electrode (Model 39604 from Beckman Instruments, USA). Potential vs time kinetic data were acquired by means of a data acquisition system (Hewlett Packard 3478A Multimeter and Hewlett Packard 372038 Extender) interfaced to an IBM System 2 Personal Computer (4 MB), for real-time plotting of the fractional conversion vs time kinetic curves. Before kinetic experiments the electronic chain (potentiometer, data
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2621 acquisition systems, computer) was dynamically calibrated by dosing known amounts of chloride ions to the kinetic batch by injecting solutions of chloride ion at known concentration and strictly controlled flow rates, thus simulating the kinetic experiment. Correlations of experimental data to credited theoretical model models was run automatically through the same computer system (Liberti et al., 1984).
b
a
C
Results and Discussion Thermodynamics of the CUS04 System on Fibrous Resins. From the thermodynamic point of view ion exchange phenomena account for two types of interactions a t the liquid-solid interface: the breakup of the hydrated structure of the free ions and the electrostatic interaction at the functional groups within the solid phase. The preferred ion is the one that shows the most favorable energetic balance (Eisenman, 1982). In reference t o the the ion exchange reaction: 2RC1+
S O : -
--L
R,SO,
+ 2C1-
X
(10)
the overall variation of the standard free energy is therefore: (AGex)CUS04= (1/2XAGex)so4- (AGex)Cl= (AGCi - (1/2)AGS0,)hydr- (AGc1 - (1/2)AGso4),1 (11) The predominance of the hydrostatic (hydr) or the electrostatic (el) term depends on the charge density, i.e., on the strength of the electric field associated with the interacting species. The influence of the electrostatic term will gradually increase passing from strongbase anion resins, with quaternary ammonium groups (regarded as low field strength due to the large size of the functional groups, thus low charge density), to the weak-base anion resins with primary and secondary amino groups (small-sizefunctional groups, high charge density, high field strength). The contribution of the hydrostatic term can be calculated by applying Born's theory (Glueckauf, 19641, whereas the electrostatic interaction within the resin phase is calculated by a Taylor's series including Coulombic interaction, induced polarization and multipolar moments of ions (Boari et al., 1974). Based on the Eisenman theory, and in reference to the chloride/sulfate ionic system a t hand, Liberti (Boari et al., 1974) evaluated the selectivity sequence of different amino group functionalities on a set of spherular anion resins of different basicity. According to his conclusions, the sulfate ion is preferred over chloride ion with a selectivity sequence decreasing in the following order: primary amino > secondary amino tertiary amino > quaternary ammonium (12) In reference to the fibrous resins, Figure 2 shows the equilibrium isotherms, and Table 2 summarizes the separation factors, acuso4, determined at the three indicated concentrations (i.e., 0.006, 0.06, and 1.8 N). In the table are also reported the experimental values of the selectivity coefficients, Kcyso4,averaged for a solidphase equivalent fraction Y = 0.5. The experimental isotherms shown in Figure 2 confirm theoretical expectations: The weak-base resins A K 2 2 and AK22-1 show a stronger affinity for the sulfate ion in a large concentration range, whereas less
Figure 2. Chloride/sulfate equilibrium isotherms for the fibrous resin Fiban A1 (a),Fiban AK22 (b), and Fiban AK22-1 (c) at three different concentrations: 0.006 N (d); 0.06 N (e); 1.8 N (0. (2' = 298 K, (M) chlorides; (0) sulfates.) Table 2. Average Separation Factors' and Selectivity Coefficients for the Chloride/Sulfate System on Fibrous Resins 0.006 N 0.06 N 1.8 N resin W V S O ~ KCIISO~ ~ C V S O ~K C V S O ~aClsOl KCIISO? 7.80 10.66 0.55 23.83 0.50 242 Fiban A1 FibanAK22 0.35 29.28 0.10 8.16 0.45-2.5 1.22-6.8c 1.13 FibanAK22-1 0.12 65.6 0.12 6.98 0.60 a
Averaged as linear regression of the analytical formulation.
* Selectivity coefficients, Kcyso4,calculated a t solid-phase equiva-
lent fraction Y = 0.5. Sigmoid equilibrium isotherm (see Figure 2): the first value refers to the average figure in the range of Y = 0 -0.4 and the second to the range of Y = 0.4 -1.
pronounced is the affinity for the sulfate ion for the quaternary ammonium resin (Fiban Al). At high solution concentration (1.8 N), when the electroselectivity effect is reduced (Helfferich, 1962e1, quaternary ammonium resin (Fiban A11 prefers chloride ion over sulfate. Resin Fiban A22 deserves a special mention: The isotherm a t high solution concentration (1.8 N, Figure 2) shows a peculiar S-shaped conformation with selectivity reversal at fractional conversion Y = 0.4. This behavior could be interpreted in terms of different functionalities present simultaneously on the resin matrix, i.e., tertiary and secondary amino groups: At low sulfate ion fractions in the liquid phase, the secondary amino groups have a stronger affinity for this latter ion with respect to tertiary functionalities (see selectivity sequence 12). When the secondary amino functionalities are almost saturated by sulfates, the selectivity of the resin is reversed toward chloride ion, only tertiary amino groups now being available for the exchange reaction. In other words, the exchanger prefers the counterion which is present in the liquid phase at lower concentration (Helfferich, 1962~). Possibly another interpretation of the selectivity reversal phenomena could be associated to inhomogeneities of the polymeric matrix in correspondence with areas of different cross-linking and porosity. It follows a sieving effect associated with the respective hydrated
2622 Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995 .
Table 3. Half-ExchangeTimes, t0.5, for the Fibrous Resins at Different Concentrations and Stirring Speed (rpm)
-..
.
.
.-
.
2,000 r p i
3,000 r p i
t n . 5 (SI
resin Fiban A1 FibanAK22 FibanAK22-1 A. IRA 458O K. A102* b
2000 rpm 10 13 11
0.006 N 2500 3000 rpm rpm 8 7 9 7 11
0.06N 3000rpm 5 3 4
7
42 197
1.8 N 3000rpm 3 2 6 29 67
a Quaternary ammonium spherular equivalent of Fiban Al. Secondary amine spherular equivalent of Fiban AK22 and AK22-
2,000 r p 3,000 r p
0.06N
0.006N
1.
a
b
C
1,000
F
Figure 3. Experimental kinetic curves and theoretical correlations according to the two models: (- - -) Phenomenological model; (-) Nernst-Planck FDC. (a) 0.006 N; (b) 0.06 N; (c) 1.8 N; (d) resin Fiban Al; (e) resin Fiban AK22; (0 resin Fiban AK22-1.
ionic radii of the competing counterions excluding one or the other ion (Helfferich, 1962~). Kinetics of the CVSO, System on Fibrous Resins. Table 3 reports the half exchange times, t0.5, for the kinetic experiments run on fibrous resins at different liquid-phase concentration and stirring speed. In the table are also reported data for equivalent bead-shaped commercial resins for comparison. Data clearly show faster kinetics for the fibers with respect to the spherular bead equivalent resins, with differences ranging in 1 order of magnitude in favor of the fibers (Petruzzelli et al., 1993,Liquan et al., 1994). Figure 3 shows the experimental kinetic trends for the three resins in reference to experiments run at different liquid-phase concentrations. The figure also shows the theoretical correlations according to both theoretical models, i.e., Nernst-Planck and the phenomenological equation models. Both models gave good correlation of data at almost all liquid-phase concentrations investigated, with the only exception the phenomenological model for resin AK22 at 1.8 N. It has to be noted that for this latter resin the equilibrium isotherm is not a monotonic function (see AK22 isotherm a t 1.8 N in Figure 2), and
3,000
r Pm
Figure 4. Linear correlation of half-exchange times vs concentration, C, and,vice versa, constant stirring speed at different liquidphase concentrations, for the two resins Fiban A1 (upper boxes) and Fiban AK22 (lower boxes).
possibly the selectivity reversal is responsible for the poor correlations of the phenomenological model equation. Contrary to spherular resins, when a switching from film to particle diffusion control was observed for liquidphase concentrations higher than ~ 1 . M 5 (Liberti et al., 19871, in the case of fibrous resins the kinetics are film controlled at all the concentration ranges investigated. The larger surface area exposed, associated with the smaller size of the particles (average diameter of the fibers 20-50 pm vs 100-500 pm of the bead particles), the reduced solid-phase diffusional patterns, and resin functionalities mostly distributed on the surface of polymer matrix, favors the formation of ionic gradients in the stationary liquid film around the fibers. Accordingly, the exposed surface area of the fibers plays a major role in determining the overall kinetic performances of these exchangers. In this context, by reducing 10-20-fold the average geometrical dimensions of the particles, the concentration limit for particle diffusion control is not reached, at least for the substantially high solution concentration (1.8 N) tested in the study. As expected (see eq 4 and related comments), the selectivity of the resin toward different ionic species results a minor factor influencing kinetic performances. The sensible decrease of the exchange rates with stirring speed (rpm) and liquid-phase concentration, C (see Table 3), and the linear correlations of the mentioned parameters vs the half-times of exchange,t0.5 (see Figure 4) are additional evidence in favor of the film diffusion control. In fact, smoother concentration gradients in the liquid film with reducing liquid-phase concentrations leads to lower driving forces for the ionic interdiffusion and corresponding slower kinetics. On the other hand, slower kinetics at low stirring speed can be associated with the formation of a thicker liquid film around the particle, thus to longer diffusional patterns for the ionic interdiffusion in the stationary liquid film determining the overall ion exchange kinetic control. Moreover, the peculiar linearity of the F ( t ) kinetic
Ind. Eng. Chem. Res., Vol. 34, No. 8,1995 2623
curves is symptomatic of constant concentration gradients in the liquid film, thus constant ionic fluxes and corresponding constant rates of exchange (see eq 4 and related comments). On the basis of eqs 4, 5 , and 9, almost equivalent figures for the film thickness were evaluated with both models, which resulted in the range of 0.4 x cm. The figure, which is consistent with that obtained for spherular resins (Liberti et al., 1983),is quite reasonable by considering that the experimental parameters influencing hydrodynamics of the system (i.e., rotation speed of the reactor, viscosity and temperature of the liquid p h a s e ) were the same for the experiments run on fibrous resins and on corresponding bead exchangers. Conclusion Fibrous resins could represent the future of ion exchange technology in the near future. Quite a number of applications, based on innovative concepts of unit operations, are increasingly appearing in the literature. In this context, unit operations, traditionally based on fxed bed systems, could be revisited in the light of the peculiar structure of reference materials, possibly by adopting more practical moving bed systems based, for example, on belt units recirculating between the exhaustion and regeneration batches, or other innovative and more profitable ways in order to take the maximum advantage from the favorable kinetic performances of the exchanging fibers. Insufficient basic knowledge on kinetics and thermodynamics of fibrous exchangers is available in the literature, and accordingly, studies in this direction are needed to better characterize reference materials for improved performances in different applicative areas, Together with classical thermodynamic considerations, we have discussed two models for film diffusion controlled ion exchange kinetics, leading t o equivalent conclusions. In the case at hand, the specific rates being controlled by almost constant concentration gradients in the liquid film, the selectivity of the resin toward the exchanging ionic species is not an important ratedetermining factor. As expected, the exposed surface area of fibrous materials is the key factor determining the overall kinetic behavior of the system involved. Generally speaking, particular care and attention has to be devoted in the correlation of experimental kinetic data to models. In this case we have coincident conclusions but this, of course, cannot be generalized. Model correlation of data is a very versatile tool if used properly, on the basis of a deep knowledge of the system involved, or it turns to be a mere computer exercise if used with insufficient experimental data or lacking basic knowledge of the physico-chemical system.
Nomenclature t = time (s) t 0 . 5 = half-exchange time of the kinetic process C = total solution concentration of ions (mmol ~ m - ~ ) C = total counterion concentration in the resin phase (mmol ~ m - ~ ) C; = bulk solution concentration of counterion i (mmol ~ m - ~ ) Ci* = concentration of the i t h counterion at the liquidsolid interface (mmol ~ m - ~ ) AC = concentration difference of counterions across the liquid film (mmoYcm3) D,= effective liquid-phase diffusivity of ith ion (cm2 s-l) F = fractional attainment of equilibrium
J = flux of counterions (mmol cm-2 s-l) d = diameter of the fibers (cm) 6 = liquid-film thickness (c-m) am = separation factor = cAC~CBCA T = temperature (K) Xi = ionic fraction of ith ion in the liquid phase Yi = ionic fraction of the ith ion in the resin phase Q = ion exchange capacity of the resin (mmol ~ m - ~ ) KBA= selectivity coefficient for ions A and B G,,= free energy of exchange ( k J mol-l) bi = 3Dild @
