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Gel-coated ion-exchange resin: A new kinetic model
Chemical Engineering Science 54 (1999) 3723}3733
Gel-coated ion-exchange resin: A new kinetic model Manas Chanda , Garry L. Rempel * Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 Received 16 June 1998; accepted 17 October 1998
Abstract Regenerable &gel-coated' cationic resins with fast sorption kinetics and high sorption capacity have application potential for removal of trace metal ions even in large-scale operations. Poly(acrylic acid) has been gel-coated on high-surface area silica (pre-coated with ethylene}vinyl acetate copolymer providing a thin barrier layer) and insolubilized by crosslinking with a lowmolecular-weight diepoxide (epoxy equivalent 180 g) in the presence of benzyl dimethylamine catalyst at 703C. In experiments performed for Ca> sorption from dilute aqueous solutions of Ca(NO ) , the gel-coated acrylic resin is found to have nearly 40% higher sorption capacity than the bead-form commercial methacrylic resin Amberlite IRC-50 and also several times higher rate of sorption. The sorption on the gel-coated sorbent under vigorous agitation has the characteristics of particle di!usion control with homogeneous (gel) di!usion in resin phase. A new mathematical model is proposed for such sorption on gel-coated ion-exchange resin in "nite bath and solved by applying operator-theoretic methods. The analytical solution so obtained shows good agreement with experimental sorption kinetics at relatively low levels ((70%) of resin conversion. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Gel-coated resin; Poly(acrylic acid); Weakly acidic resin; Sorption kinetic model
1. Introduction While the sorption selectivity of a resin is basically a property of the functional group it contains, the sorption rate behavior is determined largely by the physical characteristics of the sorbent. Moreover, the sorption capacity of bead-form resins is often found to be much less than the theoretical capacity based on the content of functional groups and this is usually due to the inaccessibility of many sorption sites buried inside the resin matrix. This is especially true for resins synthesized from functional group containing monomers. For example, poly(4-vinyl pyridine) has a measured proton capacity of 5.7 meq/g dry resin (Chanda et al., 1983), as compared to the theoretical proton capacity of ca. 8.5 meq/g dry resin. Similarly, commercial polybenzimidazole has a measured proton capacity of 4.5 meq/g dry resin (Chanda et al., 1985), as compared to the theoretical capacity of 6.5 meq/g dry resin. The discrepancy between measured
*Corresponding author. Tel.: #1 519 888 4567; fax: #1 519 746 4979; e-mail: [email protected].
and theoretical capacities becomes more pronounced for larger size sorbents. Practical applications of ion-exchange separation processes are often limited by slow kinetics of sorption and elution. The rate determining step in most cases has been established to be either external "lm di!usion or intraparticle di!usion. The chemical reaction on the "xed charge is usually too fast to control the sorption rate unless chemical modi"cations occur during the sorption (Hel!erich and Hwang, 1991). While "lm di!usional resistance can be eliminated or minimized by e!ective agitation of the external liquid, intraparticle di!usional resistance is largely in#uenced by the physical characteristics of the resin matrix. As the di!usion path becomes longer and more tortuous with increasing conversion of the resin bead, depending on the nature of the matrix, the rate of sorption may decrease with progressive resin conversion, necessitating prolonged contact with the sorbent to reach equilibrium. As notable examples may be cited the sorption of uranium on Duolite ES467 which requires 10}12 h and the sorption of plutonium nitrate onto a weak-base anion-exchange resin which requires in excess of 500 h to reach equilibrium (Streat, 1984).
0009-2509/99/$ } see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 2 8 7 - 5
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M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
Since both the lower attainable capacity and slower rate of sorption of bead-form sorbents stem from relative inaccessibility of sorption sites in the interior of resin beads, a signi"cant improvement in both these respects would be expected to result from the preparation of the sorbent as a thin gel layer on a high surface area substrate. The sorption rate may also be enhanced by preparing the gel with a relatively high water content. Thus, according to the model of Mackie and Meares (1955), which relates di!usion coe$cient of a species in ion exchanger (DM ) to the di!usion coe$cient in solution (D) by DM "D[e/(2!e)], where e is the fractional intraparticle void volume (satisfactorily approximated by weight fraction of the imbibed solvent), larger amount of water in the resin contributes to greater di!usivity. Whereas an ordinary gel resin with a large volume of imbibed water is not stable and tends to deform and agglomerate, a gel-coated sorbent made by depositing a thin layer of resin on a granular high surface area solid is, however, found to have enhanced stability and abrasion resistance, facilitating its use in the same way as a conventional granular sorbent (Chanda and Rempel, 1993, 1994, 1997). The advantages of surface-coated adsorbents have led to their commercial development. Mention may be made of the products Spherosil and Spherodex of Sepracor which are ion exchangers coated on silica. These adsorbents "nd use not only in high performance liquid chromatography but also on a large scale, an example being the puri"cation of human albumin (Tayot et al., 1978; Van der Wiel and Wesselingh, 1989). A drawback of gel-coated sorbents is the need to employ relatively large quantities in view of the reduced overall capacity which limits their application in largescale commercial separation processes. However, the signi"cantly faster kinetics of these sorbents often become invaluable when sorbates present in trace concentrations in large volumes of substrate are to be separated. Earlier we have developed processes for making gelcoated sorbents from chelating resins such as poly(4vinyl pyridine) (Chanda and Rempel, 1993) and polyethyleneimine (Chanda and Rempel, 1995). In the present paper, we describe a process for making a gel-coated ion-exchange sorbent by partially crosslinking of polyacrylic acid on high-surface area silica. In ion-exchange operations with a high degree of agitation of the external solution, the process is often found to be particle-di!usion controlled (pdc). In the ordinary pdc model, di!usion of the sorbate is assumed to occur throughout the resin bead, whereas in the shell-core model the di!usion is con"ned to the reacted layer as the boundary between the reacted and unreacted layers moves toward the center of the bead. For spherical beadtype sorbents, "nite bath sorption kinetics have been formulated both for the ordinary pdc model with homogeneous di!usion in the resin (Hel!erich, 1962) and for
the pdc model with shell-core mechanism (Hel!erich, 1965; Harris et al., 1986). However, kinetic models for gel-coated sorbents are not readily available. Earlier we reported (Chanda and Rempel, 1994) an analytical solution of the "nite bath model for gel-coated sorbents when pdc with shell-core mechanism is applicable. Recently, we also reported (Chanda and Rempel, 1997) a new mathematical model for gel-coated chelating sorbent in "nite bath when sorption is characterized by ordinary pdc with homogeneous di!usion behavior. Operator theoretic methods (Ramkrishna, 1984) were applied yielding an analytical solution which was tested with the experimental rate data of Cr(III) sorption on gel-coated polyethyleneimine. In the present paper, the method has been further extended to gel-coated ion-exchange sorbent and the resulting analytical solution has been tested with experimental kinetic data of ion-exchange sorption of Ca> on polyacrylic acid (Na>) gel-coated on silica.
2. Experimental procedure 2.1. Sorbent Polyacrylic acid (average mol. wt. 90 000, Aldrich Cat. No. 19,205-8) was used for gel-coating on silica (Aldrich Cat. No. 21,442-6) of 12}24 mesh size. Initially, a thin barrier layer of poly(ethylene-co-vinyl acetate) (EVA) was applied on silica in order to impart alkali resistance. In a typical procedure for gel-coating, 20 g polyacrylic acid dissolved in 400 ml water was added to 200 g of the EVA-coated silica gel taken in an evaporating dish and heated on a steam bath with continuous mixing till water was evaporated (I). An acetone solution of epoxy resin, made by dissolving 5 g of diglycidyl ether of bisphenol-A (Araldite LY 556 from Hindustan Cibatul Limited, Bombay) of epoxy equivalent 180 g in 400 ml acetone containing 1 g of N,N-dimethylbenzylamine (Aldrich Cat. No. 18,558-2) was added to I and evaporated to dryness by heating on water bath with continuous mixing. For further curing of the coated resin the sorbent was heated in an air oven at 703C for 12 h. The coated sorbent was thoroughly washed with large volumes of water in order to leach out any uncrosslinked poly(acrylic acid). The gel-coated sorbent prepared in this way was designated SiO )) [PA ) XE] in accordance with the system of nota tion introduced by Warshawsky and Upson (1989). The product had an elemental composition of C 7.70%, H 0.77%, and O 4.94% (the remaining elements being Si and O of SiO ) and a carboxylic functionality of 1.38 meq/g dry sorbent. The gel resin content of the sorbent as determined by completely removing the gel-coated resin by heating in concentrated chromicsulfuric acid was 0.131 g (dry) resin/g (dry) sorbent. The properties of the sorbent SiO )) [PA ) XE] are given in Table 1.
M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733 Table 1 Properties of sorbent SiO )) [PEI ) XE)] used for Ca> sorption Water content (g/g wet sorbent) Particle diameter (mm) Wet Dry B.E.T. surface area (m/g dry sorbent) Pore volume (cm/g dry sorbent) Carboxylic content (meq/g dry gel-coat resin)
0.82 0.9}1.5 0.8}1.2 315 0.76 10.6
Estimated from water imbibed by dry sorbent.
For the purpose of comparison, a conventional weakly acidic cation resin Amberlite IRC-50 (Sigma Chemical, St. Louis, USA) was used. The resin (wet mesh range 20}50), a macroreticular methacrylic acid-divinyl benzene copolymer, had 69% w/w moisture and a total capacity of 5.6 meq/g dry resin. For ion-exchange sorption experiments, both SiO )) [PA ) XE] and Amberlite IRC-50 were converted to Na> form by shaking with an excess of 0.3N NaOH followed by thorough wash with water to remove free alkali. The resulting products are designated as SiO )) [PA(Na>) ) XE] and IRC-50(Na>), respectively. 2.2. Sorption experiments For measurement of equilibrium sorption of Ca>, small scale dynamic contacts between the sorbent SiO )) [PA(Na>) ) XE] and Ca(NO ) solutions of speci "ed compositions were e!ected in tightly stoppered #asks at 253C for 20 h on a gyrotory shaker. The extent of sorption was calculated from the residual concentration of Ca> in solution. For comparison, sorptions were also measured under similar conditions on the weak acid macroreticular acrylic resin Amberlite IRC-50 used in sodium ion form. The usual batch technique (Hel!erich, 1962) of ionexchange rate measurements in which a known amount of ion exchanger is contacted under agitation with a solution of the sorbate of known concentration and volume in a closed vessel, is not suitable for measurement of fast reaction kinetics which may require very short contact times between the sorbent and the solution. In such cases, the Kressman and Kitchener (1949) reactor which holds the sorbent in a platinum-wire basket forming the center block of a centrifugal stirrer is very convenient as the ion exchanger and solution can be separated almost instantaneously by raising the (still rotating) stirrer out of the solution. The centrifugal force produces a rapid circulating #ow of solution entering the wire basket at the bottom and leaving through the wall and the radial holes in the casing.
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In the present work, batch technique was employed and a mesh basket was used, instead of a closed vessel, for holding the sorbent and providing stirred contact with the solution. A rectangular basket (20 mm;10 mm ;40 mm) made of polypropylene screen (0.5 mm opening) was used to hold the granular sorbent. The basket was rotated by attaching it to the shaft of a rotor while the sorbate solution was brought into contact for a speci"ed period. This arrangement allowed, like the aforesaid Kressman}Kitchener reactor, instantaneous separation of the sorbent from the sorbate solution at any speci"ed time and analysis of the residual concentration of the sorbate to determine the rate of sorption. A fresh amount of sorbent from the same stock was used for each experiment. Dynamic contacts between sorbent and solution were e!ected at di!erent stirring speeds using a low solution concentration (5.0 mmol/l) to determine the minimum speed above which the sorption rate was independent of agitation and hence not in#uenced by "lm di!usion. All kinetic experiments were performed at stirring speeds (200}300 rpm) well above this minimum. The basket reactor used was specially suitable for performing the so-called &interruption test' which is known as the best technique for distinguishing between particle and "lm di!usion control (Hel!erich, 1962). The sorbent basket could be removed from the sorbate solution for a brief period of time and then re-immersed to determine the e!ect of the interruption on the sorption rate. As shown by the results in Fig. 1 for Ca> sorption on SiO )) [PA(Na>) ) XE], the in terruption causes a change in the momentary sorption rate, clearly indicating particle di!usion control of sorption rate under the experimental conditions employed.
Fig. 1. Interruption test for particle-di!usion control of sorption on SiO )) [PA(Na>) ) XE]. Sorbent loading: 40 g (wet)/l; sorbent particle size 0.9}1.5 mm; initial concentration of Ca> in external solution: 10 mmol/l; pH 5.0; temperature 253C; vigorous agitation; (* * *) without interruption; (} } } } } }) with interruption.
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M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
3. Results and discussion 3.1. Sorption isotherm The equilibrium sorption of Ca> by the gel-coated PA(Na>) ) XE resin is compared in Fig. 2 with that of the commercial acrylic resin IRC-50(Na>), measured under similar conditions. The signi"cantly higher sorption by the gel-coated PA(Na>) ) XE resin, as compared to beadform IRC-50(Na>), both of which depend on carboxylic sites for cationic exchange, is a clear indication of much greater accessibility of sorption sites in the former contributing to the higher capacity. The equilibrium sorption data in Fig. 2 "tted well to the Langmuir isotherm, yielding correlation coe$cients in the range 0.998}0.999. The parameters A and K , Q @ representing, respectively, the saturation sorption capacity (mmol Ca>/g dry resin) and sorption binding constant (l/mol), the Langmuir isotherm is written as (10\K ) A C* @ Q x*" (1) 1#(10\ K ) C* @ where x* is the equilibrium sorption (mmol Ca>/g dry resin) and C* is the equilibrium sorbate concentration (mmol/l). The values of A and K determined by linear Q @ regression on the equilibrium sorption data are 5.6 mmol/g dry resin and 1.2;10 l/mol for Ca> sorption on SiO )) [PA(Na>) ) XE]. The corresponding values for Ca> sorption on IRC-50(Na>) are 4.0 mmol/g dry resin and 3.4;10 l/mol, respectively. The theoretical capacity of both the gel-coated polyacrylic and beadform polymethacrylic acid resins being about 10 meq/ g dry, based on carboxylic content, the measured
Fig. 2. Sorption isotherms for Ca>/NO\ on SiO )) [PA(Na>) ) XE] (particle size 0.9}1.5 mm) and Amberlite IRC-50(Na>) (particle size 0.5}1.0 mm). Sorbent loading: SiO )) [PA(Na>) ) XE] 40 g (wet)/l; IRC 50(Na>) 4 g (wet)/l; pH 5.0; temperature 253C.
Fig. 3. Comparison of rates of attainment of equilibrium sorption of Ca>/NO\ on (a) SiO )) [PA(Na>) ) XE] (particle size 0.9}1.5 mm) and (b) Amberlite IRC-50(Na>) (particle size 0.5}1.0 mm) in Ca(NO ) solutions of concentration 10 mmol/l at pH 5.0. Sorbent loading: SiO )) [PA(Na>) ) XE] 40 g (wet)/l; IRC-50(Na>) 4 g (wet)/l; temper ature 253C.
Ca> capacity of the gel-coated PA(Na>) ) XE resin (5.6 mmol/g dry) signi"es that all sorption sites in the resin are accessible, while for the bead-form sorbent the saturation capacity is only about 75% of the theoretical capacity. Both PA(Na>) ) XE (acrylic) and IRC-50(Na>) (methacrylic) resins have high equilibrium binding constants for Ca> sorption, though the value is signi"cantly higher for the latter. 3.2. Sorption kinetics For comparing the sorption rate behavior of SiO )) [PA(Na>) ) XE] and IRC-50(Na>) the fractional attainment of equilibrium sorption was measured as a function of time under identical conditions. The results plotted in Fig. 3 show that the sorption is much faster on the gel-coated resin, the t value for 10 mM Ca> being only 14 s compared to 65 s on the bead-form resin. The e!ect of external concentration on the sorption rate of the two sorbents is shown in Figs. 4 and 5. The sorption experiments were performed using a su$ciently high degree of agitation to eliminate the e!ect of external mass transfer resistance. The results show that the rate of sorption on the bead-form sorbent increases with the sorbate concentration within the range studied, but the latter has no e!ect on the sorption rate for the gel-coated sorbent. The concentration independence of sorption rate is inconsistent with the predictions from the shellcore mechanism for gel-coated sorbents (Chanda and Rempel, 1994). While an ordinary particle di!usion control (pdc) model for gel-coated ion-exchange resins is not available, it may be noted that for bead-form resins such a model predicts (Hel!erich, 1962) concentration independence of sorption rate. To test the applicability of ordinary pdc model to the present kinetic data, a "nite
M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
Fig. 4. Rate of attainment of equilibrium sorption of Ca>/NO\ on SiO )) [PA(Na>) ) XE] (particle size 0.9}1.5 mm) in Ca(NO ) solutions of di!erent initial concentrations: 10 mM (䡩); 15 mM (䉭); 20 mM (;); 25 mM (䊐). pH 5.0; sorbent loading 40 g (wet)/l; temperature 253C; vigorous agitation.
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3. The resin layer undergoes negligible swelling or shrinking during reaction. 4. The sorption takes place by di!usion of the sorbate (ionic) species from the external solution into the resin layer and ion exchange at the ionic sites. The exchanged counterion di!uses through the resin layer into the external solution. It is assumed that the e!ects of electric transference and electric coupling are not important. 5. The sorption (ion-exchange) reaction is fast and the overall rate of the process is controlled by the resin phase di!usion for which Fick's law with a constant di!usivity in the resin phase is applicable. 6. Departure from isothermal behavior is negligible. According to the assumptions under 1}6, the concentration "eld of the counterion in the resin layer (gel coat) can be formulated as follows: DM
*CM 2 *CM # , *r r *r
(r )r)r ) Q M
(2)
where CM is the concentration of the counterion in the resin phase, DM is the di!usion coe$cient, r is the radius of Q the solid inert core, and r is the radius of the gel-coated M sorbent. The initial and boundary conditions of Eq. (2) are:
Fig. 5. Rate of attainment of equilibrium sorption of Ca>/NO\ on Amberlite IRC-50(Na>) (particle size 0.5}1.0 mm) in Ca(NO ) solu tions of di!erent initial concentrations: 10 mM (䡩); 15 mM (䉭); 20 mM (;); 25 mM (䊐). pH 5.0; sorbent loading 4.0 g (wet)/l; temperature 253C; vigorous agitation.
CM "0 (r"r , t"0) M
(2a)
CM "CM (r ) r ( r , t"0) M Q M
(2b)
dCM /dr"0 (r"r , t'0) Q
(2c)
(2d)
*CM 3uDM *CM "! (r"r , t'0) M *t r *r M where
bath ordinary pdc model for the gel-coated ion-exchanger is therefore developed in the following. 3.3. Finite-bath model for gel-coated ion-exchanger In order to describe the sorption kinetics in a gel-layer of ion-exchange resin coated on an inert, granular support, in the present work, the following assumptions and simpli"cations are made: 1. The sorbent consists of an ion-exchange resin layer (gel-coat) of uniform thickness on a spherical particle which is inert and impenetrable to substrate solution. 2. The resin layer is considered as quasi-homogeneous phase for mathematical treatment.
u"a / b
(2e)
a"
(2f )
b"(1!r/r) Q M
(2g)
In Eq. (2f ),
(3)
C CM DM t C*" , CM *" , q" C C r M M M
(4)
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M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
P"[j #+3u(1!r*)!j r* ,cos(j (1!r*) L Q LQ L Q
Eq. (2) is transformed to the non-dimensional form *CM * 2 *CM * *CM * " # *q r* *r* *r*
(5)
!+(j (1#3ur*)#3u/(j ,sin(j (1!r*)] (10) L Q L L Q
q"0, r* * 1, CM *"0
(5a)
Q"(j (3u#j ) L L
q"0, r* ) r* ) 1, CM *"1 Q
(5b)
with j obtained as roots of the equation L
dCM * "0 q ' 0, r*"r*, Q dr*
(5c)
q ' 0, r*"1,
*CM * *CM * #(3u) "0. *q *r*
(5d)
For further transformation, let u"CM *r*.
(6)
Eq. (5) then transforms to *u *u " *q *r*
(7)
q"0, r*"1, u"0
(7a)
q"0, r* ) r* ( 1, u"1 Q
(7b)
du u q ' 0, r*"r*, ! "0 Q dr* r*
(7c)
q'0, r*"1,
*u *u #(3u) !u "0. *q *r*
(7d)
An analytical solution of Eq. (7) has been derived by applying operator-theoretic methods (Ramkrishna, 1984) as shown in Appendix A. The solution may be given as
uJ ,
u(r*, q) u(1, q)
u(1!r*) Q " 1#u(1!r*) Q
(j[3u!(3u#j) r*] Q tan(j(1!r*)" Q 3u#j(1#3ur*) Q
1 1 1/A" (1!r*)! sin 2(j (1!r*) L Q L Q 2 4(j L 3u ! [1!cos 2(j (1!r*)] L Q 2(3u#j ) L 9uj L (1!r*) # Q 2(3u#j ) L
9uj L sin 2(j (1!r*) # L Q 4(3u#j ) L 3uj L . # (3u#j ) L
(13)
The fractional attainment of equilibrium sorption (XM ) has been de"ned as u(r*, 0)!u(r*, q) Q Q XM (q)" u(r*, 0)!u(r*, R) Q Q
(14)
where u(r*, q) and u(r*, R) from Eq. (8) are Q Q r*u(1!r*) Q u(r*, q)" Q Q 1#u(1!r*) Q
r* 1
!
3u(j L cos(j (1!r*) sin(j (1!r*)! L L 3u#j L ; 3u(j L ! 3u#j L ;e\HLO
(12)
and A from L
# A[I(j , r*)] sin(j (1!r*) L L Q L Q L
# A[I(j , r*)] L L Q L
3u(j L cos(j (1!r*) L Q 3u#j L
r*u(1!r*) Q u(r*, R)" Q . Q 1#u(1!r*) Q
(8)
where I(j , r*)"P/Q L Q
(11)
(9)
Resin conversion as a function of time (t) can be calculated from Eqs. (14)}(16) for any assumed value of DM . The best "t of the model equation to the data of Fig. 4 was obtained with DM "7.4;10\ cm/s for resin conversions up to about 70% (Fig. 6). Since the sorbent used for rate measurements was a mixture of particles of di!erent shapes, though belonging to a narrow size range, the radius of an equivalent sphere (Aris, 1957) having volume equal to the average volume of particles in the
M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
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The gel-coated acrylic resin exhibits remarkably faster kinetics than the bead-form acrylic resin in ion-exchange sorption with a t value about th that for the bead form resin. The sorption on the gel-coated poly(acrylic acid) resin appears to be particle di!usion controlled with the sorption rate being nearly independent of substrate concentration. A model has been developed and solved analytically to describe sorption by gel-coated ion-exchange resins in "nite bath. The model shows good "t to the experimental rate data of Ca> sorption on the gelcoated resin at resin conversions less than 70% of the equilibrium value, yielding a value of 7.4;10\ cm/s for the resin-phase di!usivity. Fig. 6. Test of mathematical model Eqs. (14)}(16) for "nite bath sorption on gel-coat sorbent SiO )) [PA(Na>) ) XE] with data of Fig. 4.
size fraction was used for the calculation of DM . As shown in Fig. 6, the experimental values of fractional attainment of equilibrium is seen to be smaller than the model predicted values at resin conversions greater than 70%. This may be indicative of a decrease in resin di!usivity at higher conversions due to the Donnan e!ect. The empirical equation presented by Mackie and Meares (1955) relating the resin di!usivity, DM , to the free solution di!usivity, D, by the fractional void volume of the resin (satisfactorily approximated by weight fraction of imbibed solvent), e , is given below : P e P DM "D . (17) (2!e ) P This equation is widely used in ion exchange (Hel!erich, 1962). The factor e /(2!e ) is simply a geometrical &torP P tuosity factor' accounting for the obstruction of the resin matrix, but not for any e!ect of interaction between the sorbate species and the functional groups in the resin matrix. Using a value of 0.82 for e of the gel-coat resin P and a typical free solution di!usivity value of 2;10\ cm/s, Eq. (17) gives DM K8;10\ cm/s which compares well with the DM of 7.4;10\ value obtained above corresponding to the solution concentration in the range 10}25 mM.
4. Conclusions Poly(acrylic acid) has been coated on high surface area silica from an aqueous solution by evaporation and insolubilized by reacting with a low molecular weight diepoxide of bisphenol A in the presence of benzyldimethylamine catalyst. The resulting gel-coated sorbent, designated as SiO )) [PEI ) XE)] and used in Na> form, shows about 40% higher capacity than the commercial poly(methacrylic acid) bead-form resin IRC-50 in Ca> sorption from its nitrate solution.
Acknowledgements The "nancial support of research from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. We thank Prof. D. Ramkrishna of Purdue University and Dr. Shreekumar of the Indian Institute of Science, Bangalore, for assisting us in arriving at an analytical solution of the "nite-bath model equation for gel-coated ion-exchange resins.
Notation aN A L A Q C C* CM CM * CM D DM JM K @ n r r* r M r Q r* Q t u <
surface area of each sorbent particle, cm parameter de"ned by Eq. (13) saturation sorption capacity of sorbent, mmol/g (dry) sorbate concentration in solution, mmol/l sorbate concentration (dimensionless) in solution sorbate concentration in resin, mmol/l sorbate concentration (dimensionless) in resin initial sorbate concentration in solution, mmol/l solution di!usivity, cm/s resin di!usivity, cm/s #ux of sorbate species across particle (resin) surface, mmol/cm/s sorbate binding constant, l/mol total number of sorbent particles in system radial position in sorbent, cm radial position (dimensionless) in sorbent outer radius of sorbent particle, cm radius of solid inert core of sorbent, cm radius (dimensionless) of solid inert core of sorbent time, s CM *r* [Eq. (6)] volume of external solution, l total volume of resin layer in particles, l
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x* XM
equilibrium sorption, mmol/g dry resin fractional attainment of equilibrium sorption
We can de"ne the operator L as *u 0 *r* L, * 3u !3u *r* * P such that !
Greek letters a b e P j L u
(A.6)
Lu, 3u Appendix A
*u *r*
*u ! *r* P*
*u(r*, q) *q " *u(1, q) !3uu" * ! P *q !
*uJ "! . *t
A.1. Derivation of Eq. (2d)
(A.7)
From the material balance dC nJM aN "< dt
(A.1)
where n is the total number of sorbent particles in the system, JM is the #ux of the sorbate species across the resin surface, aN is the surface area of each particle, < is the total volume of solution, and C is the concentration of the sorbate in solution. Let
(A.2)
where
aN 1 " JM (
dC . dt
dC dt
where u"a/b. Solution of Eq. (7) De"ne
u(q),
u(r*, q) u(1, q)
.
f,
P*Q
f (r*)
f (r*) g(r*) dr*#r f (1) g(1)
, gJ ,
f (1)
g(r*) g(1)
.
We can choose r so as to make the operator L self adjoint. For any vectors u, v e H,
1Lu, v2"
*u *u ! v dr*#3r u !u v(1, q) *r* *r* * * P PQ
"!v(r*)u (r*)"*# u (r*) v (r*) dr* PQ PQ* #3ur [u (1)!u(1)]v(1).
(A.8)
Using the homogeneous boundary condition (7c), we obtain
(A.4) 1Lu, v2!1u, Lv2"[u(r*) v (r*)
where b"(1!r/r). Assuming equilibrium at the interQ M face of the resin and solution (Hel!erich, 1962), C(t)"CM (t) at t'0 and r"r . Eq. (A.4) thus becomes M 3u *CM *CM " ! DM *t r *r M
1 fI , gJ 2"
(A.3)
For spherical particles of radius r Eq. (A.3) simpli"es to M 3 1 JM " rb a M
The vector u is now de"ned on the Hilbert space H"L [r*, 1] the inner product on which is de"ned by Q
(A.5)
!u (r*) v(r*)]*#3ur [u (1) v(1)!u(1) v (1)] PQ "[u(1) v (1)!u (1) v(1)][1!3ur ].
(A.9)
The above expression vanishes if we choose r "1/3u. Thus, L is self-adjoint under the inner prod uct,
f (r*) f (1)
,
P*Q
,
g(r*) g(1)
1 f (r*) g(r*) dr* # f (1) g(1). 3u
(A.10)
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M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
3u(j L cos(j (1!r*) dr* ! L 3u#j L
Also, L has positive eigenvalues which can be calculated using the boundary conditions given by Eq. (7c) and (d). To calculate the eigenvalues,
r*f (r*)"f (r*) Q Q Q 3uf (1)!3uf (1)"jf (1).
1 1 A\" (1!r*)! sin 2(j (1!r*) L Q L Q 2 4(j L
Let f"A sin(j(1!r*)#B cos(j(1!r*) . Then r*[!A(j cos(j(1!r*)#B(j sin(j(1!r*)] Q Q Q !A sin(j(1!r*)!B cos(j(1!r*)"0 (A.11) Q Q 3uA(j#(3u#j) B"0. (A.12)
3u ! [1!cos 2(j (1!r*)] L Q 2(3u#j ) L 9uj L (1!r*) # Q 2(3u#j ) L
For non-trivial solution, the determinant is
A 3u(j L . # L 3u 3u#j L Solving,
df "!jf dx
!r*(j cos(j(1!r*)!sin(j(1!r*) r*(j sin(j(1!r*)!cos(j(1!r*) Q Q Q Q Q Q 3u(j 3u#j "0
.
(A.13)
Thus the characteristic equation is 9u(j L sin 2(j (1!r*) # L Q 4(3u#j ) L
!(3u#j)[r*(j cos(j(1!r*)#sin(j(1!r*)] Q Q Q !3u(j[r*(j sin(j(1!r*)!cos(j(1!r*)]"0 Q Q Q that is, (j[3u!(3u#j) r*] Q . tan(j(1!r*)" Q 3u#j(3ur*#1) Q Let
3uj L . # (3u#j ) L
(A.14)
(A.15)
(A.16)
The solution is given by u(q)"1u(0), f 2f # 1u(0), f 2f e\HLO. M M L L L
f (r*) A sin(j (1!r*)#B cos(j (1!r*) L L L f, L " L L f (1) B L L 3u(j L cos(j (1!r*) sin(j (1!r*)! L L 3u#j "A L 3u(j L L ! 3u#j L
(A.17)
The characteristic equation (A.14) has a root, j "0 for M which A becomes in"nite as can be seen from Eq. (A.15). We therefore determine the product A (j which must M M be "nite as j 0. M From Eq. (A.15), we can write
f (r*) f, M M f (1) M
sin(j (1!r*) 3u M ! cos(j (1!r*) M 3u#j (j (1!r*) M "A (j . M M M 3u ! 3u#j M (1!r*)
where A is obtained from Eq. (A.10) as follows : L
1"A L
PQ*
sin(j (1!r*) L
As j 0, M
! r* f A (j . M M M !1
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M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
# A [I(j , r*)] L L Q L
To calculate A (j , we use Eq. (A.10) as before M M 1"Aj M M
1 u(1!r*)#1 Q r* dr*# "Aj . M M 3u 3u * PQ
3u(j L cos(j (1!r*) sin(j (1!r*)! L L 3u#j L ; 3u(j L ! 3u#j L
Therefore, 3u Aj " M M u(1!r )#1 Q
;e\HLO
and
(A.18)
where
3u f "! M u(1!r*)#1 Q
r* 1
.
I(j , r*)"P/Q L Q P"[j #+3u(1!r*)!j r*,cos(j (1!r*) L Q LQ L Q
Hence,
r*
1u(0), f 2f " M M ;
0
! r*
,!
r* 3u u(1!r*)#1 1 Q
3u u(1!r*)#1 Q
!1
1 r* dr*# (0) (1) 3u P*Q
3u " u(1!r*)#1 Q
r* 1
r* u(1!r*) Q " . u(1!r*)#1 1 Q Similarly,
r* A sin(j (1!r* ) L L PQ*
1u(0), f 2f " L L
3u(j L cos(j (1!r* ) dr* ! L 3u#j L 3u(j L cos(j (1!r*) sin(j (1!r*)! L L 3u#j L ;A . L 3u(j L ! 3u#j L Substituting in Eq. (A.17) and evaluating the integrals one obtains the solution
u(q),
u(r*, q) u(1, q)
"1u(0), f 2f # 1u(0), f 2f e\HLO M M L L L
r* u(1!r*) Q " 1#u(1!r*) 1 Q
!+(j (1#3ur*)#3u/(j ,sin(j (1!r*)] L Q L L Q Q"(j (3u#j ). L L
References Aris, R. (1957). On shape factors for irregular particles } I. The steady state problem. Di!usion and reaction. Chem. Engng Sci., 6, 262. Chanda, M., O'Driscoll, K.F., & Rempel, G.L. (1983). Sorption kinetics of phenolics onto cross-linked poly(4-vinyl pyridine). React. Polymers, 1, 281. Chanda, M., O'Driscoll, K.F., & Rempel, G.L. (1985). Sorption of phenolics and carboxylic acids on polybenzimidazole. React. Polymers, 4, 39. Chanda, M., & Rempel, G.L. (1993). Poly(4-vinyl pyridine) gel-coated on silica. High capacity and fast kinetics in uranyl sulfate recovery. Ind. Engng Chem. Res., 32, 726. Chanda, M., & Rempel, G.L. (1994). Quaternized poly(4-vinyl pyridine) gel-coated on Silica. Fast kinetics of di!usion-controlled sorption of organic sulfonates. Ind. Engng Chem. Res., 33, 623. Chanda, M., & Rempel, G.L. (1995). Polyethyleneimine gel-coat on silica. High uranium capacity and fast kinetics of gel-coated resin. React. Polymers, 25, 25. Chanda, M., & Rempel, G.L. (1997). (a) Chromium (III) removal by epoxy-crosslinked polyethyleneimine used as gel-coat on silica. Sorption characteristics. Ind. Engng Chem. Res., 36, 2184; (b) Chromium (III) removal by epoxy-crosslinked polyethyleneimine used as gel-coat on silica. A new kinetic model. Ind. Engng Chem. Res., 36, 2190. Harris, W.I., Lindy, L.B., & Dixit, R.S. (1986). Kinetic models for uranium recovery from sulfuric acid leach streams by strong-base anion resins. React. Polymers, 4, 99. Hel!erich, F.G. (1962). Ion exchange. (pp. 250}309). New York: McGraw-Hill. Hel!erich, F.G. (1965). Ion exchange kinetics. V. Ion exchange accompanied by reactions. J. Phys. Chem., 69(4), 1178. Hel!erich, F.G., & Hwang, Y.-L. (1991). Ion exchange kinetics. In K. Dorfner, (Ed.), Ion Exchanger, Berlin: Walter de Gruyter. Kressman, T.R.E., & Kitchener, J.A. (1949). Cation exchange with a synthetic phenolsulfonate resin. (V) Kinetics. Discuss. Faraday Soc., 7, 90.
M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733 Mackie, J.S., & Meares, P. (1955). The di!usion of elctrolytes in a cation exchange membrane. Proc. Roy. Soc. (¸ondon), A232, 498. Ramkrishna, D. (1984). Operator theoretic methods in heat and mass transfer problems. In A.S. Mujumdar, & R.A. Mashelkar, (Eds.), Advances in transfer processes, (Vol. 3, pp. 387}439). New Delhi: Wiley Eastern. Streat, M. (1984). Kinetics of slow di!using species in ion exchanger. React. Polymers, 2, 79. Tayot, J.L., Tardy, M., Gattel, P., Plan, R., & Roumiantze!, M. (1978). Industrial ion exchange chromatography of proteins on DEAE
3733
dextran derivatives of porous silica beds. In R. Epton, (Ed.), Chromatography of synthetic and biological polymers, (Vol. 2, pp. 95}110). Chichester: Ellis Horwood. Van der Wiel, J.P., & Wesselingh, J.A. (1989). Continuous adsorption in biotechnology. In A.E. Rodrigues, M.D. Le Van, & D. Tondeur, (Eds.), Adsorption: science and technology NA¹O ASI series, series E: applied sciences, (pp. 427}469). Dordrecht: Kluwer Academic Publishers. Warshawsky, A., & Upson, D.A. (1989). Zero-valent metal polymer composites. I. Metallized beads. J. Polym. Sci. Part A Polym. Chem. Ed., 27, 2963.
Gel-coated ion-exchange resin: A new kinetic model Manas Chanda , Garry L. Rempel * Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 Received 16 June 1998; accepted 17 October 1998
Abstract Regenerable &gel-coated' cationic resins with fast sorption kinetics and high sorption capacity have application potential for removal of trace metal ions even in large-scale operations. Poly(acrylic acid) has been gel-coated on high-surface area silica (pre-coated with ethylene}vinyl acetate copolymer providing a thin barrier layer) and insolubilized by crosslinking with a lowmolecular-weight diepoxide (epoxy equivalent 180 g) in the presence of benzyl dimethylamine catalyst at 703C. In experiments performed for Ca> sorption from dilute aqueous solutions of Ca(NO ) , the gel-coated acrylic resin is found to have nearly 40% higher sorption capacity than the bead-form commercial methacrylic resin Amberlite IRC-50 and also several times higher rate of sorption. The sorption on the gel-coated sorbent under vigorous agitation has the characteristics of particle di!usion control with homogeneous (gel) di!usion in resin phase. A new mathematical model is proposed for such sorption on gel-coated ion-exchange resin in "nite bath and solved by applying operator-theoretic methods. The analytical solution so obtained shows good agreement with experimental sorption kinetics at relatively low levels ((70%) of resin conversion. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Gel-coated resin; Poly(acrylic acid); Weakly acidic resin; Sorption kinetic model
1. Introduction While the sorption selectivity of a resin is basically a property of the functional group it contains, the sorption rate behavior is determined largely by the physical characteristics of the sorbent. Moreover, the sorption capacity of bead-form resins is often found to be much less than the theoretical capacity based on the content of functional groups and this is usually due to the inaccessibility of many sorption sites buried inside the resin matrix. This is especially true for resins synthesized from functional group containing monomers. For example, poly(4-vinyl pyridine) has a measured proton capacity of 5.7 meq/g dry resin (Chanda et al., 1983), as compared to the theoretical proton capacity of ca. 8.5 meq/g dry resin. Similarly, commercial polybenzimidazole has a measured proton capacity of 4.5 meq/g dry resin (Chanda et al., 1985), as compared to the theoretical capacity of 6.5 meq/g dry resin. The discrepancy between measured
*Corresponding author. Tel.: #1 519 888 4567; fax: #1 519 746 4979; e-mail: [email protected].
and theoretical capacities becomes more pronounced for larger size sorbents. Practical applications of ion-exchange separation processes are often limited by slow kinetics of sorption and elution. The rate determining step in most cases has been established to be either external "lm di!usion or intraparticle di!usion. The chemical reaction on the "xed charge is usually too fast to control the sorption rate unless chemical modi"cations occur during the sorption (Hel!erich and Hwang, 1991). While "lm di!usional resistance can be eliminated or minimized by e!ective agitation of the external liquid, intraparticle di!usional resistance is largely in#uenced by the physical characteristics of the resin matrix. As the di!usion path becomes longer and more tortuous with increasing conversion of the resin bead, depending on the nature of the matrix, the rate of sorption may decrease with progressive resin conversion, necessitating prolonged contact with the sorbent to reach equilibrium. As notable examples may be cited the sorption of uranium on Duolite ES467 which requires 10}12 h and the sorption of plutonium nitrate onto a weak-base anion-exchange resin which requires in excess of 500 h to reach equilibrium (Streat, 1984).
0009-2509/99/$ } see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 2 8 7 - 5
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M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
Since both the lower attainable capacity and slower rate of sorption of bead-form sorbents stem from relative inaccessibility of sorption sites in the interior of resin beads, a signi"cant improvement in both these respects would be expected to result from the preparation of the sorbent as a thin gel layer on a high surface area substrate. The sorption rate may also be enhanced by preparing the gel with a relatively high water content. Thus, according to the model of Mackie and Meares (1955), which relates di!usion coe$cient of a species in ion exchanger (DM ) to the di!usion coe$cient in solution (D) by DM "D[e/(2!e)], where e is the fractional intraparticle void volume (satisfactorily approximated by weight fraction of the imbibed solvent), larger amount of water in the resin contributes to greater di!usivity. Whereas an ordinary gel resin with a large volume of imbibed water is not stable and tends to deform and agglomerate, a gel-coated sorbent made by depositing a thin layer of resin on a granular high surface area solid is, however, found to have enhanced stability and abrasion resistance, facilitating its use in the same way as a conventional granular sorbent (Chanda and Rempel, 1993, 1994, 1997). The advantages of surface-coated adsorbents have led to their commercial development. Mention may be made of the products Spherosil and Spherodex of Sepracor which are ion exchangers coated on silica. These adsorbents "nd use not only in high performance liquid chromatography but also on a large scale, an example being the puri"cation of human albumin (Tayot et al., 1978; Van der Wiel and Wesselingh, 1989). A drawback of gel-coated sorbents is the need to employ relatively large quantities in view of the reduced overall capacity which limits their application in largescale commercial separation processes. However, the signi"cantly faster kinetics of these sorbents often become invaluable when sorbates present in trace concentrations in large volumes of substrate are to be separated. Earlier we have developed processes for making gelcoated sorbents from chelating resins such as poly(4vinyl pyridine) (Chanda and Rempel, 1993) and polyethyleneimine (Chanda and Rempel, 1995). In the present paper, we describe a process for making a gel-coated ion-exchange sorbent by partially crosslinking of polyacrylic acid on high-surface area silica. In ion-exchange operations with a high degree of agitation of the external solution, the process is often found to be particle-di!usion controlled (pdc). In the ordinary pdc model, di!usion of the sorbate is assumed to occur throughout the resin bead, whereas in the shell-core model the di!usion is con"ned to the reacted layer as the boundary between the reacted and unreacted layers moves toward the center of the bead. For spherical beadtype sorbents, "nite bath sorption kinetics have been formulated both for the ordinary pdc model with homogeneous di!usion in the resin (Hel!erich, 1962) and for
the pdc model with shell-core mechanism (Hel!erich, 1965; Harris et al., 1986). However, kinetic models for gel-coated sorbents are not readily available. Earlier we reported (Chanda and Rempel, 1994) an analytical solution of the "nite bath model for gel-coated sorbents when pdc with shell-core mechanism is applicable. Recently, we also reported (Chanda and Rempel, 1997) a new mathematical model for gel-coated chelating sorbent in "nite bath when sorption is characterized by ordinary pdc with homogeneous di!usion behavior. Operator theoretic methods (Ramkrishna, 1984) were applied yielding an analytical solution which was tested with the experimental rate data of Cr(III) sorption on gel-coated polyethyleneimine. In the present paper, the method has been further extended to gel-coated ion-exchange sorbent and the resulting analytical solution has been tested with experimental kinetic data of ion-exchange sorption of Ca> on polyacrylic acid (Na>) gel-coated on silica.
2. Experimental procedure 2.1. Sorbent Polyacrylic acid (average mol. wt. 90 000, Aldrich Cat. No. 19,205-8) was used for gel-coating on silica (Aldrich Cat. No. 21,442-6) of 12}24 mesh size. Initially, a thin barrier layer of poly(ethylene-co-vinyl acetate) (EVA) was applied on silica in order to impart alkali resistance. In a typical procedure for gel-coating, 20 g polyacrylic acid dissolved in 400 ml water was added to 200 g of the EVA-coated silica gel taken in an evaporating dish and heated on a steam bath with continuous mixing till water was evaporated (I). An acetone solution of epoxy resin, made by dissolving 5 g of diglycidyl ether of bisphenol-A (Araldite LY 556 from Hindustan Cibatul Limited, Bombay) of epoxy equivalent 180 g in 400 ml acetone containing 1 g of N,N-dimethylbenzylamine (Aldrich Cat. No. 18,558-2) was added to I and evaporated to dryness by heating on water bath with continuous mixing. For further curing of the coated resin the sorbent was heated in an air oven at 703C for 12 h. The coated sorbent was thoroughly washed with large volumes of water in order to leach out any uncrosslinked poly(acrylic acid). The gel-coated sorbent prepared in this way was designated SiO )) [PA ) XE] in accordance with the system of nota tion introduced by Warshawsky and Upson (1989). The product had an elemental composition of C 7.70%, H 0.77%, and O 4.94% (the remaining elements being Si and O of SiO ) and a carboxylic functionality of 1.38 meq/g dry sorbent. The gel resin content of the sorbent as determined by completely removing the gel-coated resin by heating in concentrated chromicsulfuric acid was 0.131 g (dry) resin/g (dry) sorbent. The properties of the sorbent SiO )) [PA ) XE] are given in Table 1.
M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733 Table 1 Properties of sorbent SiO )) [PEI ) XE)] used for Ca> sorption Water content (g/g wet sorbent) Particle diameter (mm) Wet Dry B.E.T. surface area (m/g dry sorbent) Pore volume (cm/g dry sorbent) Carboxylic content (meq/g dry gel-coat resin)
0.82 0.9}1.5 0.8}1.2 315 0.76 10.6
Estimated from water imbibed by dry sorbent.
For the purpose of comparison, a conventional weakly acidic cation resin Amberlite IRC-50 (Sigma Chemical, St. Louis, USA) was used. The resin (wet mesh range 20}50), a macroreticular methacrylic acid-divinyl benzene copolymer, had 69% w/w moisture and a total capacity of 5.6 meq/g dry resin. For ion-exchange sorption experiments, both SiO )) [PA ) XE] and Amberlite IRC-50 were converted to Na> form by shaking with an excess of 0.3N NaOH followed by thorough wash with water to remove free alkali. The resulting products are designated as SiO )) [PA(Na>) ) XE] and IRC-50(Na>), respectively. 2.2. Sorption experiments For measurement of equilibrium sorption of Ca>, small scale dynamic contacts between the sorbent SiO )) [PA(Na>) ) XE] and Ca(NO ) solutions of speci "ed compositions were e!ected in tightly stoppered #asks at 253C for 20 h on a gyrotory shaker. The extent of sorption was calculated from the residual concentration of Ca> in solution. For comparison, sorptions were also measured under similar conditions on the weak acid macroreticular acrylic resin Amberlite IRC-50 used in sodium ion form. The usual batch technique (Hel!erich, 1962) of ionexchange rate measurements in which a known amount of ion exchanger is contacted under agitation with a solution of the sorbate of known concentration and volume in a closed vessel, is not suitable for measurement of fast reaction kinetics which may require very short contact times between the sorbent and the solution. In such cases, the Kressman and Kitchener (1949) reactor which holds the sorbent in a platinum-wire basket forming the center block of a centrifugal stirrer is very convenient as the ion exchanger and solution can be separated almost instantaneously by raising the (still rotating) stirrer out of the solution. The centrifugal force produces a rapid circulating #ow of solution entering the wire basket at the bottom and leaving through the wall and the radial holes in the casing.
3725
In the present work, batch technique was employed and a mesh basket was used, instead of a closed vessel, for holding the sorbent and providing stirred contact with the solution. A rectangular basket (20 mm;10 mm ;40 mm) made of polypropylene screen (0.5 mm opening) was used to hold the granular sorbent. The basket was rotated by attaching it to the shaft of a rotor while the sorbate solution was brought into contact for a speci"ed period. This arrangement allowed, like the aforesaid Kressman}Kitchener reactor, instantaneous separation of the sorbent from the sorbate solution at any speci"ed time and analysis of the residual concentration of the sorbate to determine the rate of sorption. A fresh amount of sorbent from the same stock was used for each experiment. Dynamic contacts between sorbent and solution were e!ected at di!erent stirring speeds using a low solution concentration (5.0 mmol/l) to determine the minimum speed above which the sorption rate was independent of agitation and hence not in#uenced by "lm di!usion. All kinetic experiments were performed at stirring speeds (200}300 rpm) well above this minimum. The basket reactor used was specially suitable for performing the so-called &interruption test' which is known as the best technique for distinguishing between particle and "lm di!usion control (Hel!erich, 1962). The sorbent basket could be removed from the sorbate solution for a brief period of time and then re-immersed to determine the e!ect of the interruption on the sorption rate. As shown by the results in Fig. 1 for Ca> sorption on SiO )) [PA(Na>) ) XE], the in terruption causes a change in the momentary sorption rate, clearly indicating particle di!usion control of sorption rate under the experimental conditions employed.
Fig. 1. Interruption test for particle-di!usion control of sorption on SiO )) [PA(Na>) ) XE]. Sorbent loading: 40 g (wet)/l; sorbent particle size 0.9}1.5 mm; initial concentration of Ca> in external solution: 10 mmol/l; pH 5.0; temperature 253C; vigorous agitation; (* * *) without interruption; (} } } } } }) with interruption.
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M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
3. Results and discussion 3.1. Sorption isotherm The equilibrium sorption of Ca> by the gel-coated PA(Na>) ) XE resin is compared in Fig. 2 with that of the commercial acrylic resin IRC-50(Na>), measured under similar conditions. The signi"cantly higher sorption by the gel-coated PA(Na>) ) XE resin, as compared to beadform IRC-50(Na>), both of which depend on carboxylic sites for cationic exchange, is a clear indication of much greater accessibility of sorption sites in the former contributing to the higher capacity. The equilibrium sorption data in Fig. 2 "tted well to the Langmuir isotherm, yielding correlation coe$cients in the range 0.998}0.999. The parameters A and K , Q @ representing, respectively, the saturation sorption capacity (mmol Ca>/g dry resin) and sorption binding constant (l/mol), the Langmuir isotherm is written as (10\K ) A C* @ Q x*" (1) 1#(10\ K ) C* @ where x* is the equilibrium sorption (mmol Ca>/g dry resin) and C* is the equilibrium sorbate concentration (mmol/l). The values of A and K determined by linear Q @ regression on the equilibrium sorption data are 5.6 mmol/g dry resin and 1.2;10 l/mol for Ca> sorption on SiO )) [PA(Na>) ) XE]. The corresponding values for Ca> sorption on IRC-50(Na>) are 4.0 mmol/g dry resin and 3.4;10 l/mol, respectively. The theoretical capacity of both the gel-coated polyacrylic and beadform polymethacrylic acid resins being about 10 meq/ g dry, based on carboxylic content, the measured
Fig. 2. Sorption isotherms for Ca>/NO\ on SiO )) [PA(Na>) ) XE] (particle size 0.9}1.5 mm) and Amberlite IRC-50(Na>) (particle size 0.5}1.0 mm). Sorbent loading: SiO )) [PA(Na>) ) XE] 40 g (wet)/l; IRC 50(Na>) 4 g (wet)/l; pH 5.0; temperature 253C.
Fig. 3. Comparison of rates of attainment of equilibrium sorption of Ca>/NO\ on (a) SiO )) [PA(Na>) ) XE] (particle size 0.9}1.5 mm) and (b) Amberlite IRC-50(Na>) (particle size 0.5}1.0 mm) in Ca(NO ) solutions of concentration 10 mmol/l at pH 5.0. Sorbent loading: SiO )) [PA(Na>) ) XE] 40 g (wet)/l; IRC-50(Na>) 4 g (wet)/l; temper ature 253C.
Ca> capacity of the gel-coated PA(Na>) ) XE resin (5.6 mmol/g dry) signi"es that all sorption sites in the resin are accessible, while for the bead-form sorbent the saturation capacity is only about 75% of the theoretical capacity. Both PA(Na>) ) XE (acrylic) and IRC-50(Na>) (methacrylic) resins have high equilibrium binding constants for Ca> sorption, though the value is signi"cantly higher for the latter. 3.2. Sorption kinetics For comparing the sorption rate behavior of SiO )) [PA(Na>) ) XE] and IRC-50(Na>) the fractional attainment of equilibrium sorption was measured as a function of time under identical conditions. The results plotted in Fig. 3 show that the sorption is much faster on the gel-coated resin, the t value for 10 mM Ca> being only 14 s compared to 65 s on the bead-form resin. The e!ect of external concentration on the sorption rate of the two sorbents is shown in Figs. 4 and 5. The sorption experiments were performed using a su$ciently high degree of agitation to eliminate the e!ect of external mass transfer resistance. The results show that the rate of sorption on the bead-form sorbent increases with the sorbate concentration within the range studied, but the latter has no e!ect on the sorption rate for the gel-coated sorbent. The concentration independence of sorption rate is inconsistent with the predictions from the shellcore mechanism for gel-coated sorbents (Chanda and Rempel, 1994). While an ordinary particle di!usion control (pdc) model for gel-coated ion-exchange resins is not available, it may be noted that for bead-form resins such a model predicts (Hel!erich, 1962) concentration independence of sorption rate. To test the applicability of ordinary pdc model to the present kinetic data, a "nite
M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
Fig. 4. Rate of attainment of equilibrium sorption of Ca>/NO\ on SiO )) [PA(Na>) ) XE] (particle size 0.9}1.5 mm) in Ca(NO ) solutions of di!erent initial concentrations: 10 mM (䡩); 15 mM (䉭); 20 mM (;); 25 mM (䊐). pH 5.0; sorbent loading 40 g (wet)/l; temperature 253C; vigorous agitation.
3727
3. The resin layer undergoes negligible swelling or shrinking during reaction. 4. The sorption takes place by di!usion of the sorbate (ionic) species from the external solution into the resin layer and ion exchange at the ionic sites. The exchanged counterion di!uses through the resin layer into the external solution. It is assumed that the e!ects of electric transference and electric coupling are not important. 5. The sorption (ion-exchange) reaction is fast and the overall rate of the process is controlled by the resin phase di!usion for which Fick's law with a constant di!usivity in the resin phase is applicable. 6. Departure from isothermal behavior is negligible. According to the assumptions under 1}6, the concentration "eld of the counterion in the resin layer (gel coat) can be formulated as follows: DM
*CM 2 *CM # , *r r *r
(r )r)r ) Q M
(2)
where CM is the concentration of the counterion in the resin phase, DM is the di!usion coe$cient, r is the radius of Q the solid inert core, and r is the radius of the gel-coated M sorbent. The initial and boundary conditions of Eq. (2) are:
Fig. 5. Rate of attainment of equilibrium sorption of Ca>/NO\ on Amberlite IRC-50(Na>) (particle size 0.5}1.0 mm) in Ca(NO ) solu tions of di!erent initial concentrations: 10 mM (䡩); 15 mM (䉭); 20 mM (;); 25 mM (䊐). pH 5.0; sorbent loading 4.0 g (wet)/l; temperature 253C; vigorous agitation.
CM "0 (r"r , t"0) M
(2a)
CM "CM (r ) r ( r , t"0) M Q M
(2b)
dCM /dr"0 (r"r , t'0) Q
(2c)
(2d)
*CM 3uDM *CM "! (r"r , t'0) M *t r *r M where
bath ordinary pdc model for the gel-coated ion-exchanger is therefore developed in the following. 3.3. Finite-bath model for gel-coated ion-exchanger In order to describe the sorption kinetics in a gel-layer of ion-exchange resin coated on an inert, granular support, in the present work, the following assumptions and simpli"cations are made: 1. The sorbent consists of an ion-exchange resin layer (gel-coat) of uniform thickness on a spherical particle which is inert and impenetrable to substrate solution. 2. The resin layer is considered as quasi-homogeneous phase for mathematical treatment.
u"a / b
(2e)
a"
(2f )
b"(1!r/r) Q M
(2g)
In Eq. (2f ),
(3)
C CM DM t C*" , CM *" , q" C C r M M M
(4)
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M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
P"[j #+3u(1!r*)!j r* ,cos(j (1!r*) L Q LQ L Q
Eq. (2) is transformed to the non-dimensional form *CM * 2 *CM * *CM * " # *q r* *r* *r*
(5)
!+(j (1#3ur*)#3u/(j ,sin(j (1!r*)] (10) L Q L L Q
q"0, r* * 1, CM *"0
(5a)
Q"(j (3u#j ) L L
q"0, r* ) r* ) 1, CM *"1 Q
(5b)
with j obtained as roots of the equation L
dCM * "0 q ' 0, r*"r*, Q dr*
(5c)
q ' 0, r*"1,
*CM * *CM * #(3u) "0. *q *r*
(5d)
For further transformation, let u"CM *r*.
(6)
Eq. (5) then transforms to *u *u " *q *r*
(7)
q"0, r*"1, u"0
(7a)
q"0, r* ) r* ( 1, u"1 Q
(7b)
du u q ' 0, r*"r*, ! "0 Q dr* r*
(7c)
q'0, r*"1,
*u *u #(3u) !u "0. *q *r*
(7d)
An analytical solution of Eq. (7) has been derived by applying operator-theoretic methods (Ramkrishna, 1984) as shown in Appendix A. The solution may be given as
uJ ,
u(r*, q) u(1, q)
u(1!r*) Q " 1#u(1!r*) Q
(j[3u!(3u#j) r*] Q tan(j(1!r*)" Q 3u#j(1#3ur*) Q
1 1 1/A" (1!r*)! sin 2(j (1!r*) L Q L Q 2 4(j L 3u ! [1!cos 2(j (1!r*)] L Q 2(3u#j ) L 9uj L (1!r*) # Q 2(3u#j ) L
9uj L sin 2(j (1!r*) # L Q 4(3u#j ) L 3uj L . # (3u#j ) L
(13)
The fractional attainment of equilibrium sorption (XM ) has been de"ned as u(r*, 0)!u(r*, q) Q Q XM (q)" u(r*, 0)!u(r*, R) Q Q
(14)
where u(r*, q) and u(r*, R) from Eq. (8) are Q Q r*u(1!r*) Q u(r*, q)" Q Q 1#u(1!r*) Q
r* 1
!
3u(j L cos(j (1!r*) sin(j (1!r*)! L L 3u#j L ; 3u(j L ! 3u#j L ;e\HLO
(12)
and A from L
# A[I(j , r*)] sin(j (1!r*) L L Q L Q L
# A[I(j , r*)] L L Q L
3u(j L cos(j (1!r*) L Q 3u#j L
r*u(1!r*) Q u(r*, R)" Q . Q 1#u(1!r*) Q
(8)
where I(j , r*)"P/Q L Q
(11)
(9)
Resin conversion as a function of time (t) can be calculated from Eqs. (14)}(16) for any assumed value of DM . The best "t of the model equation to the data of Fig. 4 was obtained with DM "7.4;10\ cm/s for resin conversions up to about 70% (Fig. 6). Since the sorbent used for rate measurements was a mixture of particles of di!erent shapes, though belonging to a narrow size range, the radius of an equivalent sphere (Aris, 1957) having volume equal to the average volume of particles in the
M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
3729
The gel-coated acrylic resin exhibits remarkably faster kinetics than the bead-form acrylic resin in ion-exchange sorption with a t value about th that for the bead form resin. The sorption on the gel-coated poly(acrylic acid) resin appears to be particle di!usion controlled with the sorption rate being nearly independent of substrate concentration. A model has been developed and solved analytically to describe sorption by gel-coated ion-exchange resins in "nite bath. The model shows good "t to the experimental rate data of Ca> sorption on the gelcoated resin at resin conversions less than 70% of the equilibrium value, yielding a value of 7.4;10\ cm/s for the resin-phase di!usivity. Fig. 6. Test of mathematical model Eqs. (14)}(16) for "nite bath sorption on gel-coat sorbent SiO )) [PA(Na>) ) XE] with data of Fig. 4.
size fraction was used for the calculation of DM . As shown in Fig. 6, the experimental values of fractional attainment of equilibrium is seen to be smaller than the model predicted values at resin conversions greater than 70%. This may be indicative of a decrease in resin di!usivity at higher conversions due to the Donnan e!ect. The empirical equation presented by Mackie and Meares (1955) relating the resin di!usivity, DM , to the free solution di!usivity, D, by the fractional void volume of the resin (satisfactorily approximated by weight fraction of imbibed solvent), e , is given below : P e P DM "D . (17) (2!e ) P This equation is widely used in ion exchange (Hel!erich, 1962). The factor e /(2!e ) is simply a geometrical &torP P tuosity factor' accounting for the obstruction of the resin matrix, but not for any e!ect of interaction between the sorbate species and the functional groups in the resin matrix. Using a value of 0.82 for e of the gel-coat resin P and a typical free solution di!usivity value of 2;10\ cm/s, Eq. (17) gives DM K8;10\ cm/s which compares well with the DM of 7.4;10\ value obtained above corresponding to the solution concentration in the range 10}25 mM.
4. Conclusions Poly(acrylic acid) has been coated on high surface area silica from an aqueous solution by evaporation and insolubilized by reacting with a low molecular weight diepoxide of bisphenol A in the presence of benzyldimethylamine catalyst. The resulting gel-coated sorbent, designated as SiO )) [PEI ) XE)] and used in Na> form, shows about 40% higher capacity than the commercial poly(methacrylic acid) bead-form resin IRC-50 in Ca> sorption from its nitrate solution.
Acknowledgements The "nancial support of research from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. We thank Prof. D. Ramkrishna of Purdue University and Dr. Shreekumar of the Indian Institute of Science, Bangalore, for assisting us in arriving at an analytical solution of the "nite-bath model equation for gel-coated ion-exchange resins.
Notation aN A L A Q C C* CM CM * CM D DM JM K @ n r r* r M r Q r* Q t u <
surface area of each sorbent particle, cm parameter de"ned by Eq. (13) saturation sorption capacity of sorbent, mmol/g (dry) sorbate concentration in solution, mmol/l sorbate concentration (dimensionless) in solution sorbate concentration in resin, mmol/l sorbate concentration (dimensionless) in resin initial sorbate concentration in solution, mmol/l solution di!usivity, cm/s resin di!usivity, cm/s #ux of sorbate species across particle (resin) surface, mmol/cm/s sorbate binding constant, l/mol total number of sorbent particles in system radial position in sorbent, cm radial position (dimensionless) in sorbent outer radius of sorbent particle, cm radius of solid inert core of sorbent, cm radius (dimensionless) of solid inert core of sorbent time, s CM *r* [Eq. (6)] volume of external solution, l total volume of resin layer in particles, l
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M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
x* XM
equilibrium sorption, mmol/g dry resin fractional attainment of equilibrium sorption
We can de"ne the operator L as *u 0 *r* L, * 3u !3u *r* * P such that !
Greek letters a b e P j L u
(A.6)
Lu, 3u Appendix A
*u *r*
*u ! *r* P*
*u(r*, q) *q " *u(1, q) !3uu" * ! P *q !
*uJ "! . *t
A.1. Derivation of Eq. (2d)
(A.7)
From the material balance dC nJM aN "< dt
(A.1)
where n is the total number of sorbent particles in the system, JM is the #ux of the sorbate species across the resin surface, aN is the surface area of each particle, < is the total volume of solution, and C is the concentration of the sorbate in solution. Let
(A.2)
where
aN 1 " JM (
dC . dt
dC dt
where u"a/b. Solution of Eq. (7) De"ne
u(q),
u(r*, q) u(1, q)
.
f,
P*Q
f (r*)
f (r*) g(r*) dr*#r f (1) g(1)
, gJ ,
f (1)
g(r*) g(1)
.
We can choose r so as to make the operator L self adjoint. For any vectors u, v e H,
1Lu, v2"
*u *u ! v dr*#3r u !u v(1, q) *r* *r* * * P PQ
"!v(r*)u (r*)"*# u (r*) v (r*) dr* PQ PQ* #3ur [u (1)!u(1)]v(1).
(A.8)
Using the homogeneous boundary condition (7c), we obtain
(A.4) 1Lu, v2!1u, Lv2"[u(r*) v (r*)
where b"(1!r/r). Assuming equilibrium at the interQ M face of the resin and solution (Hel!erich, 1962), C(t)"CM (t) at t'0 and r"r . Eq. (A.4) thus becomes M 3u *CM *CM " ! DM *t r *r M
1 fI , gJ 2"
(A.3)
For spherical particles of radius r Eq. (A.3) simpli"es to M 3 1 JM " rb a M
The vector u is now de"ned on the Hilbert space H"L [r*, 1] the inner product on which is de"ned by Q
(A.5)
!u (r*) v(r*)]*#3ur [u (1) v(1)!u(1) v (1)] PQ "[u(1) v (1)!u (1) v(1)][1!3ur ].
(A.9)
The above expression vanishes if we choose r "1/3u. Thus, L is self-adjoint under the inner prod uct,
f (r*) f (1)
,
P*Q
,
g(r*) g(1)
1 f (r*) g(r*) dr* # f (1) g(1). 3u
(A.10)
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M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
3u(j L cos(j (1!r*) dr* ! L 3u#j L
Also, L has positive eigenvalues which can be calculated using the boundary conditions given by Eq. (7c) and (d). To calculate the eigenvalues,
r*f (r*)"f (r*) Q Q Q 3uf (1)!3uf (1)"jf (1).
1 1 A\" (1!r*)! sin 2(j (1!r*) L Q L Q 2 4(j L
Let f"A sin(j(1!r*)#B cos(j(1!r*) . Then r*[!A(j cos(j(1!r*)#B(j sin(j(1!r*)] Q Q Q !A sin(j(1!r*)!B cos(j(1!r*)"0 (A.11) Q Q 3uA(j#(3u#j) B"0. (A.12)
3u ! [1!cos 2(j (1!r*)] L Q 2(3u#j ) L 9uj L (1!r*) # Q 2(3u#j ) L
For non-trivial solution, the determinant is
A 3u(j L . # L 3u 3u#j L Solving,
df "!jf dx
!r*(j cos(j(1!r*)!sin(j(1!r*) r*(j sin(j(1!r*)!cos(j(1!r*) Q Q Q Q Q Q 3u(j 3u#j "0
.
(A.13)
Thus the characteristic equation is 9u(j L sin 2(j (1!r*) # L Q 4(3u#j ) L
!(3u#j)[r*(j cos(j(1!r*)#sin(j(1!r*)] Q Q Q !3u(j[r*(j sin(j(1!r*)!cos(j(1!r*)]"0 Q Q Q that is, (j[3u!(3u#j) r*] Q . tan(j(1!r*)" Q 3u#j(3ur*#1) Q Let
3uj L . # (3u#j ) L
(A.14)
(A.15)
(A.16)
The solution is given by u(q)"1u(0), f 2f # 1u(0), f 2f e\HLO. M M L L L
f (r*) A sin(j (1!r*)#B cos(j (1!r*) L L L f, L " L L f (1) B L L 3u(j L cos(j (1!r*) sin(j (1!r*)! L L 3u#j "A L 3u(j L L ! 3u#j L
(A.17)
The characteristic equation (A.14) has a root, j "0 for M which A becomes in"nite as can be seen from Eq. (A.15). We therefore determine the product A (j which must M M be "nite as j 0. M From Eq. (A.15), we can write
f (r*) f, M M f (1) M
sin(j (1!r*) 3u M ! cos(j (1!r*) M 3u#j (j (1!r*) M "A (j . M M M 3u ! 3u#j M (1!r*)
where A is obtained from Eq. (A.10) as follows : L
1"A L
PQ*
sin(j (1!r*) L
As j 0, M
! r* f A (j . M M M !1
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M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733
# A [I(j , r*)] L L Q L
To calculate A (j , we use Eq. (A.10) as before M M 1"Aj M M
1 u(1!r*)#1 Q r* dr*# "Aj . M M 3u 3u * PQ
3u(j L cos(j (1!r*) sin(j (1!r*)! L L 3u#j L ; 3u(j L ! 3u#j L
Therefore, 3u Aj " M M u(1!r )#1 Q
;e\HLO
and
(A.18)
where
3u f "! M u(1!r*)#1 Q
r* 1
.
I(j , r*)"P/Q L Q P"[j #+3u(1!r*)!j r*,cos(j (1!r*) L Q LQ L Q
Hence,
r*
1u(0), f 2f " M M ;
0
! r*
,!
r* 3u u(1!r*)#1 1 Q
3u u(1!r*)#1 Q
!1
1 r* dr*# (0) (1) 3u P*Q
3u " u(1!r*)#1 Q
r* 1
r* u(1!r*) Q " . u(1!r*)#1 1 Q Similarly,
r* A sin(j (1!r* ) L L PQ*
1u(0), f 2f " L L
3u(j L cos(j (1!r* ) dr* ! L 3u#j L 3u(j L cos(j (1!r*) sin(j (1!r*)! L L 3u#j L ;A . L 3u(j L ! 3u#j L Substituting in Eq. (A.17) and evaluating the integrals one obtains the solution
u(q),
u(r*, q) u(1, q)
"1u(0), f 2f # 1u(0), f 2f e\HLO M M L L L
r* u(1!r*) Q " 1#u(1!r*) 1 Q
!+(j (1#3ur*)#3u/(j ,sin(j (1!r*)] L Q L L Q Q"(j (3u#j ). L L
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M. Chanda, G.L. Rempel/Chemical Engineering Science 54 (1999) 3723}3733 Mackie, J.S., & Meares, P. (1955). The di!usion of elctrolytes in a cation exchange membrane. Proc. Roy. Soc. (¸ondon), A232, 498. Ramkrishna, D. (1984). Operator theoretic methods in heat and mass transfer problems. In A.S. Mujumdar, & R.A. Mashelkar, (Eds.), Advances in transfer processes, (Vol. 3, pp. 387}439). New Delhi: Wiley Eastern. Streat, M. (1984). Kinetics of slow di!using species in ion exchanger. React. Polymers, 2, 79. Tayot, J.L., Tardy, M., Gattel, P., Plan, R., & Roumiantze!, M. (1978). Industrial ion exchange chromatography of proteins on DEAE
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