Ind. Eng. C h e m . Res. 1994,33, 623-630
623
Quaternized Poly(4-vinylpyridine) Gel-Coated on Silica. Fast Kinetics of Diffusion-Controlled Sorption of Organic Sulfonates M a n a s Chandat and G a r r y L. Rempel' Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3Gl
A new anionic sorbent has been prepared by gel-coating highly quaternized poly(4-vinylpyridine) as a thin layer on high surface area silica. The resulting granular sorbent, designated as SiOz..[QPVP(Cl-)], affords nearly 80 7% attainment of theoretical capacity and remarkably faster kinetics with 1-2 orders of magnitude higher rate of sorption and elution, as compared to conventional bead resins HPQ(C1-) and IRA-400(Cl-), in the removal of 2-naphthalenesulfonate from dilute aqueous solutions. Shell progressive batch loading models with changing bulk concentration for both beadform and gel-coated ion-exchange resins have been derived analytically. These models fit the 1.88 X and 7.47 X experimental loading data well, yielding average values of 0.88 X cm2/s for the product of distribution coefficient and effective diffusivity in the sorption of 2-naphthalenesulfonate on HPQ(Cl-), IRA-400(Cl-), and SiOz-[QPVP(Cl-)I, respectively. Introduction Applications of ion-exchangeprocesses are often limited by slow kinetics of loading and elution. In most cases, the rate-determining step in the overall kinetics has been established to be diffusion through either the resin bead or externally adherent liquid film (Helfferich,1991). While film diffusional resistance can be minimized by effective agitation of the liquid, intraparticle diffusion is largely influenced by the nature of the resin matrix. Not only is part of the cross-sectional area of the resin bead blocked by the presence of the matrix, but diffusion around the matrix strands has to follow a tortuous and thus longer path. The rate of sorption may therefore decrease significantly with progressive resin conversion, necessitating prolonged contact with sorbate to reach equilibrium. A practical example of this is the sorption of plutonium nitrate onto a weak-base anion-exchange resin which requires in excess of 500 h to reach equilibrium (Streat, 1984). Since a very slow rate of attainment of equilibrium sorption is related to the greater inaccessibility of sorption sites in the interior of resin beads and longer diffusion path associated with progressive resin conversion, a significant improvement in rate would be expected to result from preparation of the sorbent as a thin gel layer on a high surface area substrate. The sorption rate may also be significantly affected by the water content of the sorbent. Thus according to the model of Mackie and Meares (1955)which relates diffusion cofficientof a species in ion exchanger (D) to the diffusion coefficient in solution (D)by D = D [ d ( 2 - ell2, where t is the fractional intraparticle void volume (satisfactorily approximated by weight fraction of imbibed solvent),larger amount of water in the resin contributes to greater diffusivity. The advantages of surface-coated adsorbents have led to their use in several practical applications. Mention may be made of the products Spherosil and Spherodex of Sepracor which are ion exchangers coated on silica. These adsorbents are used not only in high-performance liquid chromatography (HPLC) but also on a large scale, an example being the purification of human albumin (Tayot et al., 1978; Van der Wiel and Wesselingh, 1989). Unlike an ordinary gel resin with a large volume of imbibed solvent which tends to deform and agglomerate, +
On leave from Indian Institute of Science, Bangalore, India.
deposition of gel as a thin layer on a granular high surface area solid is found to impart excellent stability and abrasion resistance facilitating its use in the same way as a conventional granular resin. Using a process developed by us for gel-coating poly(4-vinylpyridine) on high surface area silica (Chanda and Rempel, 19931, we thus prepared a granular sorbent which exhibited an order of magnitude higher rate of sorption and elution in the resin layer as compared to conventional weak- and strong-base resins in the sorption of UOzS04 from dilute aqueous solutions at a relatively high pH (14). Encouraged by the above performance of a gel-coated weak-base chelating sorbent, we further prepared a strong base type sorbent by gel-coating highly quaternized poly(4-vinylpyridine) on high surface area silica. For comparison with conventional bead type anionic sorbents, rate measurements were made with aromatic sulfonates as these are likely to undergo diffusion-controlled sorption in the resin phase in view of their relatively large molecular size. Preliminary measurements revealed a remarkably faster kinetic behavior of the gel-coated strong-base resin as compared to similar strong-base resins in bead form. Detailed sorption studies were then conducted with 2-naphthalenesulfonate for comparative kinetic evaluation of the sorbents. Since the organic sulfonate showed no sorption on the silica support used for the gel-coated sorbent, the sorption kinetic data on the gel-coated sorbent could be directly applied to a theoretical model for such sorbents. The results of the study are presented below. Experimental Section Preparation of Sorbent. The process of gel-coating highly quaternized poly(4-vinylpyridine) (QPVP) on silica is shown schematically in Figure 1. It is similar to the process described previously (Chanda and Rempel, 1993) for gel-coating poly(4-vinylpyridine) (PVP) but modified so as to achieve quaternization of most of the pyridine units of the gel-coated PVP. The polymer used had an average molecular weight of 1.4 X 106 and was prepared by bulk polymerization of 4-vinylpyridine (Aldrich, 95 % , purified by distillation) using cumenyl hydroperoxide (Aldrich, 73%) initiator to the extent of 0.5% (w/v) at 55 "C. In a typical procedure, 100 g of silica gel (Aldrich, surface area 650 m2/g, pore volume 0.65 cm3/g, particle size 0.30-0.96 mm) was soaked in 200 mL of 1%(w/v) CuSOp5Hz0 solution and evaporated to dryness on a water
o a s s - ~ s a ~ ~ ~ ~ / ~ ~ ~ ~0 -1994 o ~ American ~ ~ $ o ~Chemical . ~ o / oSociety
624 Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994
/
PVP in
CU2+
Figure 1. Schematic representation of the process of gel-coating highly quaternized poly(4-vinylpyridine) on silica.
-
1680
1640
1600
I
1660
I
1
1620
Wavenumber,
I
1
1580
1
1
'
1640
1
I
1600
cm-'
Figure 2. Infrared spectra (Nicol510 FT-IR)of silica gel supported PVP resins: (a) after complexing with Cu(II), followed by (b) crosslinking quaternization with l,4-dibromobutane and (c) treatment successively with NHdOH and NaCl followed by reaction with n-C4H9Br and treatment with NaCl.
bath with continuous mixing (I). PVP turnings (10 g) were dissolved in 1000mL of chloroform (11). I was added to I1 and agitated with a paddle stirrer for 6 h. Chloroform was decanted, and the resin-coated silica (111)was washed several times with methanol. To perform cross-linking quaternization, I11 was again dispersed in 250 mL of methanol to which the cross-linking reagent 1,4-dibromobutane was added (in steps) to the extent of 200 % in excess of the theoretical amount required for cross-linking the pyridine units. The mixture was stirred at 65 "C for 5 days in a slow current of nitrogen. The quaternized product (IV) was filtered and washed with methanol. It had an elemental composition of C 2.21%, H 0.4196, N 0.36%, Br 0.5276, and Cu 0.32% (w/ w), the remaining elements being Si, S, and 0 of Si02 and sod2-.Assuming that all Br is accounted for by bromide counterions of quaternized N in IV, the maximum percentage of pyridine units quaternized is about 25%, suggesting that less than 25% of all pyridine units are cross-linked by quaternization. The IR spectra of this product, shown in Figure 2b, exhibit prominent absorptions at 1600 em-' (due to free pyridine units), at 1620 cm-' (due to complexed pyridine units), and at 1640 cm-' (due to quaternized pyridine units). To achieve a high degree of quaternization, IV was treated successively with 2 M NH40H and 2 M NaCl and then subjected to further quaternization with n-butyl bromide. In the latter process, the sorbent was dispersed
Table 1. Properties of Sorbents Used for Comparison of Naphthalenesulfonate Sorption sorbent properties Si02-[QPVP(Cl-)I HPQ(C1-)a IRA-4OO(C1-) water content 0.45 0.61 0.47 (g/g wet sorbent) particle diameter (mm) 0.14-0.69 0.15475 wet 0.37-1.15 0.13-0.65 0.14-0.73 dry 0.34-1.05 surface area 408 19 0.7 (m2/gdry) pore volume 0.60 1.2 0.68 (cm3/gdry) 3.1 3.3 3.6 capacityb (mequiv/g dry)
*
a About 70% of pyridine rings are quaternized. Strong-base capacity.
in 250 mL of methanol to which n-butyl bromide was added to the extent of 200 76 in excess of the theoretical amount required for cross-linking all the residual pyridine units and the mixture was agitated at 65 "C for 3 days in a slow current of nitrogen. The product was filtered and washed successively with methanol, 2 M NaC1, and water. The gel-coated sorbent prepared in this way was designated SiO2.4QPVP(Cl-)I in accordance with the system of notation introduced by Warshawsky and Upson (1989). The IR spectra of this product, shown in Figure 2c, exhibit a major absorption peak at 1640 cm-I indicating a high degree of quaternization. The product had an elemental compositionofC 2.90%,H0.65%,N0.31%,andC10.62% (the remaining elements being Si and 0 of SiOz) and a proton uptake capacity of 0.06 mequiv/g (dry) sorbent. Therefore about 70% of the pyridine units in the final product is quaternized. For the purpose of comparison, a quaternized PVP resin Reillex HPQ supplied by Reilly Industries, Inc., Indianapolis, IN, and a strong-base resin Amberlite IRA-400, both in chloride form, were activated by cyclic wash with 2 N NH40H and 1 N HC1. The properties of these two sorbents are summarized in Table 1 along with those of SiOr-[QPVP(Cl-)]. Since silica itself is a good sorbent, the possibility of sorption of 2-naphthalenesulfonate (used in the study) by the silica support of the gel-coated resin cannot be ignored. In order to determine the extent of sorption that is attributable to the silica support, a control material was prepared by separately subjecting the same silica to the entire sequence of treatments used in the preparation of SiO~-[QPVP(Cl-)],except that in place of poly(4-vinylpy-
Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 625 ridine) only pyridine was used. This specially treated silica was used in control experiments. Sorption Experiments. For measurement of equilibrium sorption of 2-naphthalenesulfonate, small-scale dynamic contacts between the sorbent and sodium 2-naphthalenesulfonate aqueous solutions of specified compositions were effected in tightly stoppered flasks for 20 h on a mechanical shaker. The extent of sorption was calculated from the residual concentration of the sorbate in the equilibrated solution. Parallel experiments performed with control silica showed that it did not adsorb the sulfonate at all, and consequently the extent of sorption on SiOy.[QPVP(Cl-)I calculated from the measurement of residual sorbate concentration could be attributed solely to the gel-coated resin. For determination of sorption kinetics wet-sieved sorbents of narrow particle size ranges were used. Dynamic contacts between sorbent and solution were effect in small batches, separately for each time period. For relatively long contact periods (21min), the suspension of sorbent in solution was shaken on a wrist-action mechanical shaker and the system was frozen after a specified period by draining the solution quickly through a screen with opening small enough to retain the resin particles. The ratio of drainage time to the time of shaking was much less than 0.1 for dynamic contact periods greater than 1 min. For experiments involving relatively short dynamic contacts (4min), a rectangular basket made of polypropylene screen (0.45-mm opening) was used to hold the granular sorbent. The basket was fixed to the shaft of a motor and rotated while the sorbate solution was brought into contact for a specified period. In this way, the sorbent could be instantaneously separated from the sorbate solution at any specified time. Dynamic contacts between sorbent and solution were effected at different shaking and stirring speeds to determine the minimum speed above which the kinetics are independent of the degree of agitation and hence are not influenced by film diffusion. Sorption rates were always measured at shaking and stirring speeds much above the respective minimum. Analysis. For analysis of 2-naphthalenesulfonate (NS) in aqueous solution, visible spectroscopy (Shimadzu UV2100) was used. A simple and sensitive method (Sargent and Rieman, 1956) based on oxidation of the organic compound by dichromate in sulfuric acid medium and spectrophotometric measurement of the resulting Cr(II1) was calibrated using standard solutions of NS in the concentration range employed in the work. Surface area and pore volume of sorbents were measured with a Micromeritics Flowsorb 2300 instrument using helium-nitrogen mixtures.
Results and Discussion Sorption Isotherm. The equilibrium sorption of sodium 2-naphthalenesulfonate by the gel-coated QPVP(Cl-) resin is compared in Figure 3 with sorptions by HPQ(C1-) and IRA-400(Cl-) measured under similar conditions. The significantly higher sorption by the gelcoated QPVP(C1-) resin as compared to bead-form HPQ(C1-), both of which depend on quaternized pyridine-N sites for anionic sorption and have about 70% of all N-sites quaternized, is a clear indication of much greater accessibility of sorption sites in the former contributing significantly to higher capacity. The equilibrium sorption data in Figure 3 fitted well to the Langmuir isotherm, yielding correlation coefficients in the range 0.988-0.999. The parameters A, and Kb,
5 1 QPVP
3
(cr)
(gel-coat) HPQ (CI-)
2 1
Equilibrium concent rat ion , m mol I L
Figure 3. Sorption isotherms of sodium 2-naphthalenesulfonate on resins. Loading 1.0 g (wet)/L; pH 5-6; temperature 20 OC. Table 2. Langmuir Isotherm Parameters for Sorption of Sodium 2-Naphthalenesulfonateon Different Sorbents sorbent QPVP(C1-) (gel coat) HPQ(C1-) IRA-400(C1-)
Lanmuir isotherm, ea 1 A. (mmol/g dry) Kb &/mol) correln coeff 3.06 2.80 3.47
8255 4381 11887
0.995 0.988 0.999
representing, respectively,the saturation sorption capacity (mmolof NS/g of dry resin) and sorption binding constant (L/mol), the Langmuir isotherm is written as
where x* is the equilibrium sorption (mmol of NS/g of dry resin) and C* is the equilibrium sorbate concentration (mmol/L). The values of A, and Kb determined by linear regression on the equilibrium sorption data are presented in Table 2. It may be noted that the saturation capacity of the gel-coated QPVP(C1-) resin, 3.1 mmol of NS/g of dry resin, accounts for nearly 80% of its theoretical capacity, while for the bead-form polymeric sorbent HPQ(C1-) and IRA-400(Cl-),the saturation capacities are 64% and 70% of the respective theoretical values. Because hydrophobic bonding should play a more extensive role in the case of polystyrene-based resin, it is not surprising that IRA-400(Cl-) exhibits the highest binding constant of the three sorbents under comparison. Similarly, the higher binding constant of the gel-coated QPVP(C1-) resin as compared to HPQ(Cl-) would be expected in view of the greater organic content of the former. Effect of pH. The equilibrium sorption of naphthalenesulfonate from aqueous solution was measured at different pH values of the substrate on SiOy.[&PVP(Cl-)] and also on control Si02 in parallel experiments. Since the latter showed no sorption of naphthalenesulfonate, the sorption measured on the sorbent was attributed solely to the resin. For comparison, the equilibrium sorption was also measured under similar conditions on the conventional bead-form resins HPQ and IRA-400, both in chloride form. From the results plotted in Figure 4, it is seen that the sorption is unaffected over a wide pH range 5-11, indicating relatively strong binding of the organic anion on the sorbents. However, while the sorption on both SiOy[QPVP(Cl-)] and HPQ(C1-) decreases markedly at higher pH (>ll), that on IRA-4OO(C1-) is only slightly affected, indicating that the alkali stripping of the organic anion from the latter would be more difficult. Kinetic Considerations. For kinetic measurements the three sorbents used for comparative study were wet-
626 Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 i. n" . QPVP (Cl-1 (Gel-coat)
0.8 0.6
C,,mM Sodium 2 - naphthalene Sulfonate
0.4 HPQ (CL-1
0
0.25 0.50 1.00
-0
c
Y
\
5
8
7
6
9
1 0 1 1
12
PH
0
Figure 4. Effect of pH on equilibrium sorption of sodium 2-naphthalenesulfonate on sorbents. Initial concentration 1.5 mmol/L; sorbent loading 1.0 g (wet)/L; pH 5-6; temperature 20 OC.
/ A
c
0
0.2
A
L
U
0
0
-1
0
80
100
120
I
I
0
60
40
C,, mM Sodium 2-naphthalene sulfonate
I
,
20
Figure 6. Rate of sorption on SiOr[QPVP(Cl-)] of bead size (diameter)0.60-0.82 mm in sodium 2-naphthalenesulfonatesolutions of different concentrations, CO;pH 5-6; loading 0.8-8.0 g (wet)/L; temperature 20 OC; vigorous agitation.
/
Ad
nLL
X
Time,s
I
-
V
I
,
-0 6
1
-0 2
,
j
log (T,mm)
Figure 5. Logarithmic plot of tO.5 of sorption vs average radius of resin bead? (mm);to.6 in secondsfor S~OT.[QPVP(CI-)] and in minutes for HPQ(C1-) and IRA-400(Cl-). Initial concentration of sodium 2-naphthalenesulfonate in solution 0.1-2.0 mmol/L; pH 5-6; sorbent loading 0.8-8.0 g (wet)/L; temperature 20 OC; vigorous agitation.
sieved to obtain four size fractions with average particle diameters (a) 0.41, 0.57, 0.78, and 1.10 mm for SiOy [QPVP(Cl-)I, 0.18,0.30,0.44, and 0.61 mm for HPQ(C1-), and 0.20, 0.31, 0.48 and 0.66 mm for IRA-400(Cl-). Sorption rates were measured for different particle size fractions employing in each case a high degree of agitation to obviate film diffusional resistance. A plot of sorption half-time versus bead radius given in Figure 5 shows that the rate of sorption is inversely proportional to the square of the bead radius. An inverse square relationship is predicted by models assuming particle-diffusion control (pdc) of the exchange rate (Helfferich, 1962), including shell-core theories, in contrast with predictions from the chemical reaction control (Liberti et al., 1978), according to which the rate should be independent of particle size. The dependence of sorption kinetics on external concentration of sorbate may also help in finding the controlling mechanism. The effect of external solution concentration on the rate of sorption was measured under a high degree of agitation on the three sorbents using in each case one particle size which constituted the bulk of the sorbent. The data reported in Figures 6-8 show that the external solution concentration in each case has a pronounced effect on the rate of sorption. The data also show a remarkable (1-2 orders of magnitude) superiority of the gel-coated sorbent over conventional bead-form sorbents in rate behavior. Thus the sorption half-time is typically 5-17 s for gel-coated QPVP(C1-) compared to 5-15 min for HPQ(C1-) and IRA-400(Cl-). The concen-
5
I
I
10
15
x
I
0.5 1.0 2.0 4.0
1
25
20
30
Time,min
Figure 7. Rate of sorption on HPQ(C1-) of bead size (diameter) 0.3H.53 mm in sodium 2-naphthalenesulfonatesolutionsof different concentrations, CO;resin loading 0.3-2.4 g (wet)/L; pH 5-6; temperature 20 OC; vigorous agitation.
-1
r:'
C,,mM
Sodium 2-naphthalene sulfonate 0 0.5
L
2 0
0
U
U
5
10
15
20
x
, 4.0
25
,
30
I
Time,min
Figure 8. Rate of sorption on IRA-400(C1-)of bead size (diameter) 0.38-0.57 mm in sodium 2-naphthalenesulfonatesolutionsof different concentrations, CO; resin loading 0.20-1.6 g (wet)/L; pH 5-6; temperature 20 OC; vigorous agitation.
tration effects as shown by data in Figures 6-8 are not consistent with predictions from the ordinary pdc model (Helfferich, 19621, but are in accord with the shell-core or reacted-layer diffusion control model. An analytical solution is available for shell-core model of sorption kinetics with constant bulk concentration (Schmuckler and Goldstein, 1974),which is referred to as infinite solution volume condition. The model, however, does not apply to batch experiments with limited solution volume or changing bulk concentration, which are more convenient to perform for comparative evaluation of sorbents. In the present work, an analytical solution has been obtained for a shell-core diffusion control model which assumes quasi-stationary-state diffusion but allows change of bulk solution concentration with progressive
Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 627 conversion of bead-form resin. An analytical expression has also been derived for a similar shell-progressive model which is, however, applicable to gel-coat sorbents with sorption confined to gel-coated resin layer, as in the present case. Finite-Base Model for Bead-Form Sorbent. The possibility of a shell-progressive mechanism or moving boundary process in ion-exchange beads arises when conversion of the resin involves consumption (or fixation) of the entering ion by a chemical reaction which is fast compared to diffusion. In order to describe the ionexchange kinetics in such a moving boundary process, in the present work, the following assumptions and simplifications are made: 1. The sorbent consists of isotropic spherical particles of equal constant diameter. 2. The sorbent is considered as quasi-homogeneous phase for mathematical treatment. 3. The sorbent particles undergo negligible swelling or shrinking during reaction. 4. The sorption reaction is fast, and the overall rate of the process is controlled by intraparticle diffusion for which Fick's law with a constant diffusivity in the resin phase is applicable. It is assumed that the effects of electric transference and electric coupling are not important. 5. The volume of the substrate solution is very large compared to the pore volume of the resin and is therefore considered to be constant. 6. Departure from isothermal behavior is negligible. According to the assumptions under 1-6 the concentration field of the sorbate in the reacted layer of the resin bead can be formulated as follows:
term of the sum on the right-hand side represents the quantity of the sorbate in the bulk solution at a certain time t and the second term denotes the quantity of sorbate which has reacted with fixed sorption sites in the shell. After integration and rearrangement of eq 4 the following equation is derived C(t)
Co[l - ( ~ ( 1R*3)] -
(5)
where
(7)
Equations 5-7 relate the concentration of the finite volume bulk solution at any time t to the radial position of the moving boundary, R ( t ) . Employing pseudo-steady-state approximation (Levenspiel, 1961) to the above problem allows considerable simplification in the mathematics such that an analytical solution becomes possible. By means of this simplification eq 2 reduces to
Noting that C = XC at r = ro and of eq 8 yields
c = 0 at r = R, integration
( R ( t )Ir I r,, t > 0) where
is the concentration of the sorbate in resin bead,
D is the diffusion coefficient, R ( t ) is the radial position of the moving boundary, and ro is the radius of the resin bead. The initial and boundary conditions of eq 2 are
c =c,
( t = 0)
Differentiating eq 9 with respect to r and substituting for (a€/ar) at r = R from eq 3d, we obtain the following expression for the rate of movement of moving boundary:
dR - -dt
(3a)
DXC
CP(1- R/ro)
(10)
Substituting for C from eq 5 and for R from eq 7, eq 10 then yields
C = 0 (r = R ( t ) ,t > 0 ) aC = -e, dR Ddr dt (r = R ( t ) ,t > 0)
(34 (3d) Equation 11 can be integrated to yield
where C is the sorbate concentration in external solution, Xis the molar distribution coefficient,and Cris the sorption capacity per unit volume of the unreacted resin bead. It is assumed that an equilibrium condition exists across the surface of the bead, as represented by eq 3b. For any given run, X is assumed to be constant. For a shell-core system with finite volume of the bulk solution the following mass balance holds:
VLCo= V,C(t)
+ nJ:47rR2C,
dR (t 1 0 )
(2)
+ constant = a ln[03 + R*31 -
where
(4)
(with n as the number of sorbent particles in each test). While the term on the left-hand side of eq 4 represents the initial quantity of sorbate in the bulk solution, the first
The constant is evaluated at t = to, R* = R*o, where to is the initial time at which a measurable sorption on the
628 Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 0.6
2
0.5
-
0.4
-
Table 3. Values of AD for Sorption of 2-Naphthalenesulfonateon HPQ(Cl-) and IRA-400(C1-)at Different Concentrations of External Solution
Co,mM Sodium 2 - n a p h thalenc rulfs 0 0.5 a 1.0 0 2.0
x
concn of 2-naphthalenesulfonate in soln (mmol)/L
4.0
E 0 3 -
A D (cm2/s)calcd from eq 13 HPQ(Cl-) 1.11x 0.98 X 0.77 X 0.69 X
0.5 1.0 2.0 4.0
IRA-4W(C1-)
106 106 106 106
2.57 X 2.02 x 1.69 X 1.25 X
:
106 106 106 1od
denoted by r8,the mass balance equation (cf. eq 5) takes the form
O .0 ’ 0
8
4
12
16
20
24
(t-t o), min
Figure 9. Test of eq 13for sorption of sodium 2-naphthalenesulfonate on HPQ(C1-) of bead size (diameter) 0.36-0.53 mm; pH 5-6; temperature 20 “C; vigorous agitation.
where r8* = r8/roand (16)
C,,.
mM Sodium 2-naphthalene sulfonate
Substituting for C from eq 15 and for R from eq 7 in eq 10, we obtain the following expression for the rate of movement of the sorption front:
0.5
R*3)- (1r t C p * ( 1 - R*)(I - rs*3)
m* - ADC,[cu(l-dt
(17)
The position of the sorption front R* is related to the fractional conversion of the gel-coated resin layer X by 0
I
0
4
8
t
12
1
1
1
16
I
20
I
I
24
I
,
28
,
X=1-
32
R*3-
*3 r8
(18)
1- r8*3
( t - t o ) , min
Figure 10. Test of eq 13 for sorption of sodium 2-naphthalenesulfonate on IRA-400(C1-)of bead size (diameter) 0.38-0.57 mm; pH 5-6; temperature 20 “C; vigorous agitation.
Equation 17 can thus be written in terms of resin conversion as
resin is obtained. Equation 12 then becomes
dX -dt
3ADC0(1- h X ) [ l - X(1- r,*3)11/3 (19) r,2Cr(1- r8*3)[1- 11- X(1- r8*3))’/3~
Using the approximation
since X(1 - rs*3)
