249
Ind. Eng. Chem. Fundam. 1986, 25, 249-258
Srnnh, J. M. “Chemical Engineering Kinetics”; McGraw-Hill: New York, 1981. Smoluchowski, M. Ann. Phys. (Leipig) 1910, 33, 1559-1586. Sparrow, E. M.; Lln, S . L. Int. J. Heat Mass Transfer 1985, 8 , 769-779. Strider, W.; Prager, S. J. Phys. Fluids 1988, 1 7 , 2544-2548. Torrance, K. E.: Sparrow, E. M. J. Heat Transfer 1985, 8 7 , 283-292. Torrance, K. E.; Sparrow, E. M. J. Heat Transfer IW8, 88, 223-230. Torrance, K. E.: Sparrow, E. M. J. Opt. SOC. Am. 1987, 5 7 , 1105-1114.
Tsai, D. S.;Ho, F. G.; Strieder, W. Chem. Eng. Sci. 1984, 3 9 , 775-779. Tsai, D. S.; Strider, W. Chem. Eng. S d . 1985, 4 0 , 170-173. Young, L. C.; Finlayson, B. A. AIChE J . 1978, 22, 331-353.
Received f o r reoiew July 10, 1984 Accepted May 16, 1985
Chromate Ion Exchange Mechanism for Cooling Water Arup K. Sengupia’+ and Dennis Clifford Environmental Engineering Program, University of Houston -University
Park, Houston, Texas 77004
The chromate ion exchange recovery process for a cooling tower blowdown is unique due to the early, gradual breakthroughof highly preferred Cr(V1) from fixed-bed columns for all types of anion-exchange resins. I t is shown that the early Cr(V1) breakthrough is not due to poor column kinetics but is predictable from an equilibrium model by using the appropriate exchange reaction involving both HCr0,- and Cr,O?-. At acidic pH, HCr0,- is practically the only Cr(V1) species in the aqueous phase, while in the exchanger phase both HCr0,- and Cr20,2- exist. The presence of Cr20?- in the exchanger’s solid phase causes a positively curved (concave upward) isotherm at relatively low Cr(V1) loading of the resin, and this equilibrium property is primarily responsible for the unusual, gradual breakthrough of Cr(V1). The presence of Cr,O,- in the solid phase may be viewed as the dimerization of HCr0,-, according to the Donnan equilibrium principle.
Introduction Related Studies and Gradual Breakthrough. In spite of very high selectivity of Cr(V1) anions by commercial organic anion-exchange resins, recovery of chromate from cooling tower blowdown is not yet commercially popular. Possible oxidation of the resins by Cr(V1) and consequent decrease in the resin-exchange capacity have been traditionally regarded as the prime obstacles to more frequent application of the process. Nevertheless, this recovery process is potentially important in light of the fact that other inorganic and/or organic corrosion inhibitors for cooling water systems are not as efficient as synergistic corrosion control by zinc and hexavalent chromate. In recent years there have been significant improvements in manufacturing processes for anion-exchange resins, especially in their ability to withstand physical attrition and chemical oxidation. Yamamoto et al. (1975) observed only a 5% decrease in the exchange capacity of anion resins after 12 months of operation for chromate recovery from cooling tower water. In open, recirculating cooling water systems, sulfuric acid is normally added to the make-up water to avoid concentration of scale-forming bicarbonate and carbonate ions. Sulfate is, therefore, without exception, the most predominant anion in cooling water (500-4000 mg/L) followed by chloride. Chromate, on the other hand, is present only to the extent of 5-20 mg/L Cr and as such may be regarded as a trace species in the cooling tower blowdown. Despite the severe competition from sulfate and chloride, which are present in concentrations several orders of magnitude higher than Cr(VI), several authors (Kunin, 1976; Yamamoto et al., 1975; Newman and Reed, 1980; Richardson et al., 1968; Miller, 1978) have confirmed the viability of the chromate-exchange process a t acidic pH, primarily due to the
‘Present address: Environmental Engineering, Fritz Engineering Laboratory, Lehigh University, Bethlehem, PA 18015.
trememdously high affinity of chromate for anion-exchange resins. The regeneration process has also been found to be highly efficient (Kunin, 1976; Newman and Reed, 1980) for both weakly basic and strongly basic anion resins. Hexavalent chromium, Cr(VI), may exist in several different anionic forms as will be shown later. For convenience in discussion, we will represent the total of all the chromate species in water as Cr(V1) or “chromate” while each individual species will be represented by its true chemical formula. In all the above-mentioned studies, the phenomenon of gradual &(VI) breakthrough at acidic pH during fiied-bed column experiments has not been addressed. Figure 1 shows Cr(V1) breakthrough in a typical column run at pH 3.9. Note that Cr(V1) breakthrough is very gradual, i.e., non-self-sharpening. This same gradual breakthrough has been observed by all the above-mentioned researchers in fixed-bed column tests. In fact, in order to overcome this problem, a “merry-go-round” system for treating cooling tower blowdown has been proposed by Kunin (1976). During column runs, a preferred species, in general, shows sharp breakthrough characteristics as demonstrated by Clifford (1982). In a binary system, the preferred species is the one for which the ratio of the equivalent fraction distribution between the exchanger phase and the aqueous phase ( y / x ) is higher than unity. Various experiments conducted in our laboratories for the past 30 months provide sufficient evidence that such early Cr(V1) breakthrough is not due to poor column kinetics or co-ion invasion or channeling in the column. From the application viewpoint, the phenomenon of gradual breakthrough bears significant importance because a column run is always terminated a t some chosen value of Cr(V1) exit concentration (normally less than 0.5 mg/L Cr). Therefore, the total available chromate removal capacity of the resin cannot be fully utilized in single fixed-bed runs. Another questionable aspect of the previous studies, including the study of Arden and Giddings (1961), is the
0196-4313/86/1025-0249$01.50/00 1986 American Chemical Society
250
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986 Hours - 21-0
10 0
40
6 0-
-
-
8 0 100 120 140 160 180 .- _ / T-7
2
6 - 10
--
10 0 m g / L
h
>
401
Cr (VI)
~ H = 3 9
6
///i
/
v
SLV = 2 15 meter/hr
0
(Re), = 0 2 9 5 EBCT = 4 2 6 min
I
Resin IRA 94, Sulfate form (STY - DVB, W B A )
201
1
-Ad
ji = i L - - L _ _ _ I - -I._0
340
680
1020
I- L - i - i - ~ 1360 1 7 0 0 2 0 4 0 2 3 6 0 2 7 2 0 3 0 6 0
B e d Volumes
Figure 1. Gradual breakthrough of Cr(V1) during a typical column run a t acidic pH.
assumption that Cr2072-is the only counterion exchanged a t acidic pH, as indicated in the following equations: for strongly basic resins: (R4N+)zS02-+ Cr20T2-F! (R4N+)2Crz072+ S042-(la) for weakly basic resin: (R3NH+)2S0:-
+ Cr20T2-e (R3NH+)2Cr2072+ SO-:
(1b) It will be seen later, however, that HCr04- is the predominant Cr(V1) species present in the cooling water at acidic pH; Cr2OY2-is practically absent. Furthermore, when Cr(V1) is at trace concentrations in both the aqueous and the resin phases, the chromate/sulfate or chromate/chloride isotherms a t acidic pH have been found to be concave upward (the unfavorable type) in the aqueous-phase concentration range of interest (0-20.0 mg/L Cr(V1)). For a trace species that is preferred by the resin, on the other hand, the isotherm is, in general, either linear or convex upward (Langmuir type). Chromate Chemistry. Chromate ions exist in the aqueous phase in different ionic forms with total chromate concentration and pH dictating which particular chromate species will predominate. The following are the important equilibrium reactions (Butler, 1967; Tong and King, 1953): H2Cr0, e H+ + HCr0,- (log K = -0.8) (2) HCr04- 2 H+ 2HCr0,-
F'
+ Cr04'-
(log K = -6.5)
Cr207'- + H 2 0 (log K = 1.52)
HCr207-2 H+ + C!r20T2- (log K = 0.07)
(3) (4)
(5)
It may be noted that reaction 4 does not contain H+; i.e., its equilibrium is independent of pH and depends only on total chromate concentration. This may be regarded as a dimerization reaction for chromate at acidic pH. Higher forms of chromate polymers have also been report,ed (Arden and Giddings, 1961; Kirk-Othmer, 1964) in acid solution a t high concentration. Cr2O7'- + H+ + HCr04- +zCr3Olo2-+ H,O (6) Cr401B2+ H20 (7) Equilibrium constants for these reactions are not available in the literature. However, these chromate polymers have been reported to exist (Ardens and Giddings, 1961) at concentrations higher than 0.1 M and their salts have been prepared synthetically. Since the distribution among chromate species is dependent on both the pH and total Cr(V1) concentration, a predominance diagram (Figure 2) has been drawn with Cr,O,:-
+ H+ + HCr0,-
cn
H2Cr04
HCr04-
Cr0:-
-*--- _---_-_ - - - - - - 1-- --_ _- __-----_: ____-3-
0-4
'
'
-Lo- -
20.0
mgIL
mg/L
I
I
I
'
'
'
'
I
-
'
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986 251
The objective of this study is to identify the exchanger-phase Cr(V1) speciation in order to characterize the chromate ion exchange mechanism. The mechanism devised must accommodate the anomalous chromate ion exchange characteristics a t acidic pH, namely, gradual breakthrough, concave upward isotherm as cited before.
Theory Methods for explaining and interpreting experimental data are generally based on the chromatographic theories, thermodynamic principles in relation to activity coefficients in the aqueous and exchanger phases, consideration of electroneutrality, the law of mass action, and the Donnan equilibrium theory. Type of Isotherm and Breakthrough Profile. A fixed-bed run is terminated when the concentration of the species to be removed reaches a predetermined value. This time and related bed volume throughput can be predicted from a knowledge of the concentration velocity of the species in question. Any concentration velocity, uci, for a species i is a function of the liquid velocity, uo, and the local concentration gradient between the exchanger phase and aqueous phase (Helfferich and Klein, 1970)
(i = 1, ..., n) (11) Equation 11 is derived from a mass balance on species i, neglecting axial dispersion and considering only convective flux. To facilitate the mathematical treatment, normalized concentrations may be used. xi = CJCT (12) Yi
=
Ci/Q
(13)
For a particular ion-exchange resin and specific liquidphase concentration, Q and CT are constant. In a binary system, an “isotherm” is a plot of yi vs. x ior Ci a t a fixed temperature. For a favorable (Langmuir type or convex upward) isotherm
Hence, with instantaneous equilibrium the concentration velocity, uci, in a fixed-bed column for a favorable isotherm always decreases with decrease in concentration which results in a sharp breakthrough. Similarly, for a linear isotherm, a sharp breakthrough will result because all concentration velocities of a given species are equal. For an unfavorable (concave upward) isotherm a2yi/aX? > o (15) Here, the concentration velocity increases with decreasing concentration, resulting in a gradual breakthrough. A Trace Species and Its Characteristics. Assuming that (a) the ion-exchange resin is a single homogeneous phase, (b) the aqueous-phase concentration is constant, and (c) there is negligible swelling or shrinking during ion exchange, the exchanger-phase activity coefficient of species i a t constant temperature and pressure may be expressed mathematically (Chu and Sposito, 1981; Currie and Curtis, 1976) as an infinite, convergent series by including the interaction terms among all other speices G,K,Z) present in the exchanger phase. After taking into consideration the constraints of Raoult’s law and the GibbsDuhem relation, the infinite series becomes In fi = cijojyk + d’jkfljyol + (16) ,
j,k#i
~,k,l#i
Here cijk and di,kl are the adjustable parameters for a multicomponent system a t a given temperature and pressure. If i is the trace species in a binary ion exchange (only i and j ) , it may be shown (Currie and Curtis, 1976) fi = constant (174 and f j = 1.0 By use of eq 17, it may again be shown (Sengupta, 1984) that the separation factor cyij and distribution coefficient Xi in a binary system are also constants when i is a trace species, i.e. Yixj
cyij
= - = constant Yjxi
Yi
Xi - = constant
(18b)
Xi
Experimental Details Resins. Four commercially available anion-exchange resins were used in the study: (a) IRA-458 (acrylic matrix, gel type, strongly basic), (b) IRA-68 (acrylic matrix, gel type, weakly basic), (c) IRA-900 (Sty-DVB matrix, macroporous, strongly basic), and (d) IRA-94 (Sty-DVB matrix, macroporous, weakly basic), where Sty-DVB stands for styrene-divinylbenzene. All these resins were from one manufacturer (Rohm and Haas Co.). The resins were screened prior to use to remove the larger particle sizes. The average particle size used was 500 f50 pm. The resins were conditioned following the standard procedure of cyclic exhaustion with 2 N hydrochloric or sulfuric acid and regenerations with 2 N sodium hydroxide. Finally, the resins were converted into air-dried SO:and C1forms. We were aware that weak- and strong-base acrylic anion resins are inherently much less chromate selective at acidic pH than their counterparts with styrene-divinylbenzene (Sty-DVB) matrices. However, in this study the acrylic resins offered the advantage of verifying certain theoretical predictions developed in relation to trace species, because chromate was present as a trace species in acrylic resins in the presence of competing sulfate and/or chloride. Chromate Isotherms. Chromate/sulfate and chromate/chloride isotherms (23 f 2 “C) were determined for different resins a t pH 4.0. Separation factors and distribution coefficients for Cr(V1) were determined from the isotherm data and used to verify certain theoretical predictions in relation to the chromate ion exchange mechanism. Isotherm data were generated by using a batch equilibration technique where a weighed amount of resin (chloride or sulfate form) was gently agitated for 4-6 h with a fixed volume of solution containing sulfate and Cr(V1) or chloride and Cr(V1) of known initial composition. At the end of the equilibration, solution composition was determined again to calculate Cr(V1) uptake. Column Runs. Column runs were carried out a t room temperature (23 f 2 “C) with Plexiglas columns and constant-flow, positive-displacement pumps. Since the study was aimed at investigating ion-exchange mechanisms at equilibrium, the superficial liquid-phase velocity (SLV) was kept lower than is normally used in commercial practice. For all the column runs, the SLV and particle Reynolds number, (Re) , were 2.15 m/h and 0.295, respectively. Unless stateBotherwise, the effluent pH in all the column runs was within f0.2 unit of the influent pH, and all column runs were conducted at 23 f 2 “C.
252
Ind. Eng. Chem. Fundam., Vol. 25, No. 2 , 1986
Table I. Chromium Atoms per Unit Charge or Exchange Site species chromium atoms species chromium atoms Cr0,20.5 Cr0,Cl1.0 HCrO; 1.0 Cr30102 1.5 Cr2OT21.0 Cr,0,322.0 CrSO-? 0.5
The column runs were lengthy (1-20 days), and no fluctuation in the influent flow rate was observed. Ionexchange resins were in either the sulfate or chloride form initially. Following each run, the columns were completely regenerated with an excess of 4% NaCl or 1% NaOH or both. Mass balance checks on chromium in the spent regenerant and chromium removed in the exhaustion cycle were within f 5 % . Chromium was analyzed, after the necessary dilutions had been made, by using a Perkin-Elmer Model 372 atomic absorption spectrophotometer with a graphite furnace accessary. Sulfate and chloride were analyzed by using a Model 16 Dionex ion chromatograph with 250-mm separator columns and the standard anion eluent. The total chromium analyses by atomic absorption spectroscopy were verified using the colorimetric diphenylcarbazide method for chromium (APHA, 1980); all the chromium in the aqueous phase was found to be in the hexavalent form. Other experimental details, including formation of some trivalent chromium inside the ion-exchange resins and mass balance checks on total chromium, are given by Sengupta (1984). A Cr(V1) effluent history for each of the column runs was drawn by plotting Cr(V1) concentration in the effluent (ordinate) against bed volumes of treated water (abscissa). Since the Cr(V1) breakthrough for all the runs at acidic pH was very gradual, the term "equilibrium breakthrough bed volumes" was introduced in the study to compare Cr(V1) removal among different column runs. In most of the effluent histories, equilibrium breakthrough bed volumes have been shown by a vertical line with arrows at the ends. Equilibrium breakthrough bed volumes for a column run indicate the calculated number of bed volumes treated or the length of time the column would have run under the conditions given, assuming a self-sharpening breakthrough of Cr(V1).
Results and Discussion Higher Chromate Polymers (Cr,O,:- and Cr4012-) in the Exchanger. The information given in Table I citing the number of chromium atoms per exchange site is used here to interpret the experimental results in order to find the possible existence of Cr301t' and/or Cr4OI3*in the exchanger phase. Column runs with 20.0 mg/L Cr(V1) alone in the influent were conducted at pH 4.0 and 9.5 with strongly basic IRA-900 resin in the chloride form. Excepting pH, all other operating conditions were alike. The effluent histories are shown in Figure 3. Since the aqueous-phase concentration was very low (corresponding to 20.0 mg/L Cr(VI)), any electrolyte penetration has been ignored. The total Cr(V1) capacity in this case is only the ion-exchange capacity, provided specific interactions and adsorption of the chromate species in the resin are absent. Note that the total number of equilibrium breakthrough bed volumes (27 800) a t acidic pH (4.0) was about twice the bed volumes (13 800) at alkaline pH (9.5). Since the influent was only Cr(V1) in these two runs, all the ionexchange sites were occupied by chromate species at exhaustion. At alkaline pH (9.5), Cr0:- is the only chromate species present in both the water and the resin phase based on known dissociation equilibria. A t acidic pH (4.0) in the
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Figure 3. Cr(V1) effluent histories for the column runs with IRA900 showing the effect of pH on Cr(V1) removal with Cr(V1) as the only species in the influent. 1 2 0 00
. I j 1 8 0 (1 41600
_1
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Figure 4. Identical chromate (pH 4.0 and 9.5) and sulfate breakthrough curves for IRA-900 on equivalent capacity basis, one-component feed.
exchange phase, only HCr0,- and/or CrzO:- exist because, as shown in Table I, they require half the number of ionexchange sites compared to Cr0:- for the same amount of Cr(V1) exchange. If there were significant amounts of Cr3012- (1.5 Cr atoms per exchange site) and/or Cr40132(2.0 Cr atoms per exchange site) in the exchanger phase, the number of bed volumes at equilibrium breakthrough obtained at acidic pH would have been appreciably greater than twice the number obtained at pH 9.5. Thus, at pH 4.0, with 100% chromate loading, higher polymers of chromate ions are considered to have been absent in the exchanger phase. An identical column run was carried out at pH 3.0, and the equilibrium breakthrough bed volumes remained exactly the same as for pH 4.0, indicating that no polymeric chromate was formed in the exchanger phase in the normal acidic pH range (3.0-5.0) for cooling water ion-exchange processes. Dilute hydrochloric acid was used to reduce the pH of the Cr(V1) solution to 3.0 for the second run; chloride ions were, therefore, obviously present in the influent. No difference in the equilibrium breakthrough bed volumes between pH 4.0 and 3.0 indicates that because of unusually high Cr(V1) selectivity, chloride ions were practically absent in the exchanger phase. The same column (chloride form) was subsequently exhausted with 200 mg/L sulfate solution at neutral pH. Figure 4 depicts the effluent histories of chromate and sulfate fed to the column as single components. It may be seen that, with Cr042-as the sole exchanger-phase chromate species at alkaline pH and HCr04- and/or Cr,072- as the exchanger-phase species a t acidic pH, the exchange capacity of the resin was found to be identical with that for sulfate. Had any Cr(V1) adsorption or specific reaction with the resin taken place, the Cr(V1) sorption capacity would have been different from the sulfate-exchange capacity. This confirms that, under the operating conditions in question, Cr(V1) uptake, both at alkaline and acidic pH, is solely due to ion exchange. Although HCr04- and Crz072-were the only Cr(V1) species
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986
253
Table 11. Experimentally Calculated Separation Factors and Distribution Coefficients for Cr(V1) at Acidic pH (4.0)
IX resin IRA-458 (acrylic, gel SBA)
competing ion concn, Ci 2000 mg/L chloride 2000 mg/L sulfate
IRA-900 (Sty-DVB, macro, WBA)
2000 mg/L sulfate
present in the resin at acidic pH, it is not possible to distinguish one from the other on the basis of stoichiometry; both species provide one Cr(V1) atom per ion-exchange site. Figure 3 is quite conspicuous because of the fairly sharp breakthrough of Cr(V1) a t acidic pH (4.0) in contrast to that shown in Figure 1. The gradual breakthrough of Cr(V1) is observed only in the presence of the competing species, namely, sulfate and chloride. This aspect will be addressed in the latter part of this study. HCr04- as the Sole Exchanger-Phase Species. Ion-exchange reactions with HCr04- as the sole species in both the phases may be written as follows: HCr04- RHCr0, + C1(194
m+
R$04
Ycr 0.007 0.0495 0.122 0.0061 0.0154 0.031 0.081 0.0362 0.085 0.172
XCr
5.38 x 1.65 x 2.60 x 7.60 x 1.54 x 2.44 x 4.33 x 3.83 X 5.90 x 1.04 x
J
20 0 I8OC
10-4 10-3 10-3 10-4 10-3 10-3 10-3 lod 10-4 10-3
-
*
140
"0
SLY
=
=
IRelo=
13.1 31.5 53.0 8.06 10.2 13.1 20.3 98.0 157 199
8
'2 4 9 5 minYfei
i".l
2 15 m/hr 295
42
84
126 BED VOLUMES
-4
210
158
242
-
HRS
*
+ 2HCr0,-
%/i
HRS
4
EBCT
5
ACr
13.0 30.0 47.0 8.01 10.0 12.7 18.7 94.5 144.0 165.0
4
12
8
16
2RHCr04 + S042- (19b)
A series of experiments with different resins were conducted for Cr(VI)/sdfate and Cr(VI)/chloride equilibrium a t constant pH and constant liquid-phase concentration. In all these experiments, chromate Cr(V1) concentrations were kept low enough to treat them as trace species. Since the liquid-phase concentration was kept constant and &(VI) was a trace species, any swelling or shrinking of the resin due to chromate exchange was assumed to be negligible. The thermodynamic equilibrium constants for eq 19a and 19b after algebraic manipulation are given by
Since HCr0,- is assumed to be the only exchanger-phase Cr(V1) species Ycr = Y H C ~
(21)
where subscript HCr represents HCr04-. Chromate being a trace species, the exchanger-phase activity coefficients (fi) remain constants. Under experimental conditions, the liquid-phase electrolyte concentration, CT,and the total exchange capacity of the resin, Q, are also constants. Since ionic strength remains unchanged due to chromate exchange, aqueous-phase activity coefficients are also constants. Hence, eq 20a and 20b take the general form ycr = constant-xcr (22) Therefore, separation factors (aC+ or cycrlcl) or distribution coefficients for the ion-exchange reactions, as shown by eq 19a and 19b, should also be constants. Table I1 shows the experimental data, including the calculated separation factors and distribution coefficients. In all the cases, both separation factors and distribution coefficients increase considerably with an increase in aqueous-phase Cr(V1) concentration, all other parameters remaining unchanged. This contradicts the theoretical predictions of
CO INFLUENT COMPOSITION
02
42
-
84
126
168
210
252
BED VOLUMES
Figure 5. (A) IRA-68 Cr(V1) effluent histories at varying influent chromate concentrations at acidic pH, effluent pH within f 0 . 2 unit of the influent pH. (B) Cr(V1) effluent histories of the column runs in (A) (IRA-68) with normalized concentration scale (C/C,) and vertical arrows indicating the calculated equilibrium breakthrough bed volumes.
eq 22 for trace-component ion exchange. Equation 11 indicates that, for a linear or favorable isotherm (Langmuir type), the breakthrough of the more selective component should be of t3heself-sharpening type assuming instantaneous mass transfer. If eq 19 represents the chromate-exchange equilibrium a t acidic pH, a selfsharpening front should result. During the past 30 months of work in this study, a t least 20 column runs have been carried out a t acidic pH (3.0-4.0) for chromate/chloride, chromate/sulfate, and chromate/nitrate systems at different concentrations with four different ion-exchange resins. For all of them, the breakthrough of Cr(V1) was gradual like the ones shown in Figure 5A,B for the chromate/sulfate system a t three different aqueous-phase Cr(V1) concentrations. In Figure 5B, the concentration axis has been normalized and it may be noted that the number of equilibrium breakthrough bed volumes increases with increasing Cr(V1) concentration in the aqueous phase. Again, that is an unusual characteristic which tends to indicate that, at acidic pH (3.0-5.0), Cr(VI), though favored by commercial anion-exchange resins compared to competing sulfate or chloride, probably gives rise to an unfavorable type isotherm (concave upward). Electrolyte Penetration and Poor Kinetics. Both electrolyte penetration (co-ion invasion) and poor mass transfer may lead to gradual breakthrough in the chromate-exchange process at acidic pH, and intuitively, they
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986
254
HRS
P
3.2
40/48
56
t4
--
72
1
INFLUENT COMPOSITION
1
SULFATE; 5 0 0 m q / L CHLORiDE:SOOrnq/L I CHROMATEsS O m q / L I RESIN. IRA-458 IACRYLIC,SBA)
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c ?=IO5
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..
666
-
DVE. SEA)
pH = 10 5
0
- B E D VOLUMES
Figure 6. Acidic vs. alkaline p H comparison of the nature of Cr(V1) breakthrough during column runs with competing sulfate and chloride species.
--
5.0
I
I
2-
pH
-
4'0
INFLUENT: 2000 m g l L Sulfate 2000 m a l l Chloride 5.0 mg1L Cr(Vi)
520 1040 15602080260031203640416046805200
BED VOLUMES
Figure 8. Acidic vs. alkaline p H comparison between the chromate removal capacities for a IRA-900 resin with sulfate and chloride as competing species.
Hours
240
120
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,
I
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,
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gradual breakthrough during the column run. Intuitively, in the chromate-exchange process Cr(V1) concentration is too low to produce any significant penetration. However, its high selectivity warrants investigation as to the possibility of electrolyte penetration. For chromate exchange, HCr0,- and Cr0:- may be regarded as the prime counterions a t acidic and alkaline pH, respectively. Assuming sodium to be the co-ion in both the cases, Donnan equilibrium gives -(Na+)(HCr04-)= {Na+){HCr04-)(at acidic pH) (23a)
--
= (Na+)2(Cr042-)(at alkaline pH) (23b) INa+)2(Cr042-) 0
L
L
b
-
125
-
-
I
250
-
_
375
-
L
500
~
625
L__1 d
750
675
1000
Bed Volumes
Figure 7. Acidic vs. alkaline pH comparison of the nature of Cr(V1) breakthrough with chloride as the competing anion.
appear to be the only candidates for such nonsharpening ion-exchange fronts. The experimental results, as interpreted here, provide evidence suggesting that the above factors are not responsible for the gradual breakthrough. Such support was obtained by comparing the effluent histories of the column runs at acidic pH (4.0) and alkaline pH (10.5)while other parameters were kept constant. The strongly basic acrylic resin IRA-458, which offers fairly comparable Cr(V1) removal capacity both a t acidic and alkaline pH, was used for the column runs. The breakthrough histories of Cr(V1) are plotted in Figures 6 and 7 . It may be seen that the breakthrough at pH 10.5 is much sharper than that a t pH 4.0. At pH 10.5 the counterion Cr042-is divalent, while a t pH 4.0 the counterion HCr0,- is monovalent. For inorganic ions, in general, the diffusivities in the aqueous and exchanger phases decrease with an increase in the valence or they are a t least comparable for both film and pore-diffusion-controlled mass transfer (Kunin, 1972; Newman, 1973). From the kinetic viewpoint, acidic pH is, therefore, likely to produce sharper Cr(VI) breakthroughs in column runs compared to alkaline pH under otherwise identical conditions. This prediction is, however, in contrast to the experimental resulta shown in Figures 6 and 7 . Referring back to Figure 3, it may also be seen that when competing species are absent, the column breakthrough of Cr(VI), a t acidic pH and almost identical (Re),, is very sharp. This indicates that, a t acidic pH, the predominant Cr(V1) species are not inherently slow. In the presence of competing sulfate and chloride the gradual breakthrough of Cr(V1) at acidic pH may not, therefore, be attributed to poor kinetics even though in real systems kinetic limitations are always present and tend to diffuse the ion-exchange wave fronts. Strong electrolyte penetration tends to produce a concave isotherm (Pepper et al., 1952) which in turn leads to
where ( ) represents the activity of the species in question. Thus (Na+J/{Na+) = (HCrO,-)/(HCrO,-) (at acidic pH) (24a)
-
(Na+)/(Na+)= [(Cr04z-)/(CrO~-)]''z(at alkaline pH) Wb) The left-hand side of eq 24 is a measure of the electrolyte penetration of sodium and is lower than unity because the exchanger-phase chromate concentration is much greater than its aqueous-phase counterpart. Electrolyte penetration is likely to be greater at alkaline pH because eq 24b contains an additional square root operator for a term less than unity. Thus, from eq 24, Cr(V1) penetration should be higher a t alkaline pH than a t acidic pH because penetration increases with an increase in the valence of the counterion. In the event of significant electrolyte penetration, Cr(V1) breakthrough history is, therefore, likely to be more diffuse at alkaline than at acidic pH. This, again, is contrary to the experimental results shown in Figures 6 and 7. Electrolyte penetration cannot, therefore, be the underlying cause for gradual breakthrough. The above comparisons have also been carried out for ion-exchange resins with styrene-divinylbenzene matrices, and the results led to the conclusion stated above. Thus, operation of columns at alkaline pH should be more effective owing to sharper Cr(V1) breakthrough. However, this is not the case. In spite of gradual breakthrough a t acidic pH, the Cr(V1) removal capacity of styrene-divinylbenzene resins at any predetermined Cr(VI) breakthrough is much higher at acidic than at alkaline pH as shown in Figure 8. That is why acidic pH operation is practiced universally for Cr(V1) removal. Cr2O7'- (Dichromate) in t h e Exchanger Phase. Chromate-exchange reactions a t acidic pH, with dichromate as the only exchanger-phase Cr(VI) species, may be written as follows: 2RC1 + 2HCr04- ~t RzCrz07+ 2C1- + H 2 0 (25a)
R2S04+ 2HCr04- 2 RZCr2O7+ SO-:
+ HzO
(25b)
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986 xCr 40.0
2.3
1
xCrx1o4
lo3 4.6
6.9
11.8
9.2
A (3.9)
Chromate-Sulfate Equil. (Acrylic. WBA)
Sulfate
255
- 2000 mg/L
Chromate-Chloride Equil. Resin: IRA-66 (Acrylic .WEA) pH-4.0 0
:
25.01 I
(15.0)
J
Yc
,= 0.00 18 4 C '='- 5 6I
X L
>"
5.0
0
10.0
15.0
20.0
C d V I ) in Solution (mg/L)
Figure 9. IRA-68 chromate/sulfate isotherm at 23 f 2 O C with values in parentheses indicating distribution coefficients ( y a / x ~ ) for each experimental data point.
These reactions are based on the assumption that HCr04in the aqueous phase dimerizes in the exchanger phase to produce Cr2072-there. Assuming constant aqueous-phase concentration and negligible swelling or shrinking due to chromate exchange where Cr(V1) is a trace species, the equilibrium constants for eq 25a and 25b are given by KCr/Cl
=
YCrfCr2
Xa2YC12
1 -
Yc12fc12
XCr2YHCr2
Q
-
(264
Derivations of eq 26 assume that HCr04- and Cr2072- are the only Cr(V1) species in the aqueous and exchanger phase, respectively, i.e. XHCr (27) and YCr2 = Ycr (28) where subscript Cr, represents Cr2072-. As before, Q and CT are constants, and the aqueousphase and exchanger-phase activity coefficients of all the counterions are also constants. Equations 26a and 26b then have the following general form ycr = constant.xc,2
(29)
Equation 29 is the equation of a parabola which has a strong positive curvature and thus gives rise to the unfavorable (concave upward) isotherm, i.e., for eq 29 either a2y/aX2 or a2y/dC2 > o (30) Note that Cr3OlO2-or Cr4O1S2-,if present in the exchanger phase, would have likewise produced a strongly unfavorable isotherm. However, as has been shown, these Cr(VI) polymers are absent in the Cr(V1) concentration range of interest. In a fmed-bed column run, breakthrough of Cr(VI) even under the conditions of instantaneous equilibrium will, therefore, be gradual because of the unfavorable nature of chromate isotherms. In a similar way, as may be seen from eq 29, the chromate distribution coefficient increases
C r W in Solution (mg/L)
Figure 10. IRA-68 chromate/chloride isotherm at 23 f 2 " C with values in parentheses indicating distribution coefficients (ycr/xcr) for each experimental data point. Table 111. Best-Fit Equations for Various Isotherms at pH 4.0 competing anion IX resin concn, Ci Y c r = ax$ IRA-458 (acrylic, SBA) 2000 mg/L sulfate ycr = 127~c,1.~' 2000 mg/L chloride ycr = ~ ~ O O X C , ~ . ~ IRA-900 (Sty-DVB, SBA) 2000 mg/L sulfate ycr = 7 4 3 0 ~ ~ : ~ ~ 2000 mg/L chloride ycr = 2 9 3 0 ~ ~ : ~ ~
with an increase in xcr in qualitative conformity with the experimental results. Isotherms for Cr(VI)/sulfate and Cr(VI)/chloride binary systems were experimentally determined, as shown in Figure 9 and 10. Both the isotherms have positive curvature, and the distribution coefficient at each data point (indicated by the numbers in parentheses) is greater than unity, indicating preferential selectivity for Cr(V1) species. In order to find quantitative agreement with eq 29, the data in the isotherms were fitted to the equation of the form Ycr = axcrb (31) where a and b are constants for a particular set of experiments. The best-fit equations for the two isotherms were as follows: (a) competing sulfate concentration: 2000 mg/L, pH 4.0 ycr = 31.5~C:'~~ (b) competing chloride concentration: 2300 mg/L, pH 4.0 ycr = 6 0 1 . 0 ~ c , ~ . ~ ~
(32)
(33)
It may be seen in eq 32 and 33 that the exponents of x C r are less than 2.0 and, therefore, do not agree well with eq 29. Such isotherms were also determined for strongly basic acrylic and styrene-divinylbenzene resins with quaternary amine functionalities. In all the cases where Cr(V1) tends to be a trace species, the exponent b was found to lie between one and two. Table I11 provides the best-fit equations for different isotherms. I t was observed earlier from eq 22 that a linear relationship exists between ycr and x C r when HCr04- is assumed to be the only Cr(V1) species in the exchanger
256
Ind. Eng. Chem. Fundam., Vol. 25. No. 2, 1986
phase. This suggests that both HCr04- and Cr2072-are present in the exchanger phase, and the presence of the latter causes positive curvature in the chromate isotherms. More general equations for overall chromate-exchange reactions have been developed later in this discussion; they explain why exponent b in eq 32 and 33 always lies between one and two when Cr(V1) tends to be a trace species. High selectivity of Cr,0,2- may be explained as follows: (a) A t the prevailing ionic strength (less than 0.5 M), resin prefers divalent Cr20Y2-to monovalent HCr04- because of the electroselectivity effect (Helfferich, 1962). (b) From related ion-exchange studies (Miller, 1978), it has been observed that Cr,O:has much higher selectivity than other divalent anions like S042-and C032-. This indicates that the resins prefer Cr20Y2-,resulting in low exchanger-phase activity coefficient (fcrC,, . This is due to dichromate's relatively less hydrated, two-tetrahedra structure (Hutchinson, 1959). In terms of activity, the Donnan equilibrium principle gives rise to the relationship
or (35) where K4 is the equilibrium constant for the aqueous-phase dimerization reaction given by eq 4. Assuming the water activity to be unity in both aqueous and exchanger phases, eq 35 stands for the reaction 2HCr04- F? Crz072-+ H 2 0 (36) This is a dimerization reaction in the exchanger phase. The Donnan principle helps extend eq 4 from the aqueous to the exchanger phase, whereby the presence of Cr20Y2along with HCr0,- in the exchanger phase may be viewed as the result of reaction 36. Now it needs to be explained why the breakthrough of Cr(V1) at acidic pH in the absence of competing sulfate and chloride is so sharp as is shown in Figure 3. When Cr(V1) is the only species in the feed, equivalent Cr(V1) fractions both in the aqueous and exchanger phase, are unity; i.e., this corresponds to the highest possible Cr(V1) loading. Since both HCr04- and Cr,072- are highly preferred over chloride anions, the chromate isotherms at ycr values close to unity are always a favorable type leading to sharp breakthrough barring mass-transfer limitations. The above observations indicate that binary chromate isotherms at acidic pH have points of inflection (Sengupta and Clifford, in press). At very low chromate loadings the isotherms have positive curvature (unfavorable type), while a t high chromate loadings they have negative curvature (favorable type). Due to the very high concentrations of competing sulfate and chloride ions in cooling water, chromate loadings in the exchanger phase are normally very low. Due to the Donnan exclusion effect, the pH inside the anion-exchange resin will be higher than that in the aqueous phase. Therefore, the possibility of Cr02- in the exchanger phase may not be ruled out. Chromate removal capacities in the presence of competing sulfate and chloride were determined under identical conditions with pH varying from 3.0 to 5.0 (Sengupta, 1984; Sengupta and Clifford, 1986). No change in chromate removal capacity was observed. If Cr042-were present in the exchanger phase, the Cr(V1) removal capacity would decrease with an increase in pH because Cr02- occupies fewer chromium
atoms per ion-exchange site than HCr04- or Crz072-. Under experimental conditions, i.e., at pH 4.0, no Cr042was present in the exchanger phase. Secondly, the presence of Cr02- in the exchanger phase cannot explain the phenomenon of early, gradual Cr(V1) breakthrough during column runs. Mononuclear Chromate Complexes (CrSOY2-, CrO,Cl-) in the Exchanger Phase. A t acidic pH, high sulfate and chloride concentrations are capable of forming complexes as shown in eq 8 and 9. However, in cooling water systems we seldom encounter ionic strengths greater than 0.5 which favor the formation of such complexes. In the aqueous phase at pH 3-5, they are practically absent. According to the Donnan principle, these monovalent or divalent complexes cannot compete with HCrOL and Cr2072-for ion-exchange sites in the exchanger unless there is a specific preference by the exchanger phase for these complexes. If Cr03C1- and/or CrS0:- are present in the exchanger phase and if Cr(V1) is considered to be a trace species in both phases when sulfate and chloride are present, a prediction of the equilibrium distribution using the previous treatment for HCr0,- and Cr20Y2-gives the relationship (37) yCr03CI or yCrSO,= constantq, This is the equation of a linear isotherm and can never lead to gradual Cr(V1) breakthrough during a fixed-bed column run. Secondly, from eq 8 and 9 we may note that if Cr03C1and/or CrS0:- are the preferred Cr(V1) species in the exchanger phase, hydrogen ion concentration in the aqueous phase will decrease due to the chromate-exchange process, during both batch equilibrium and column runs. In other words, the chromate-exchange process will always be associated with an increase in the aqueous-phase pH, provided no buffer is present. Altogether, about 30 column runs and batch equilibrium tests were carried out with different ion-exchange resins under this study in the absence of any buffer. No increase in pH was even observed. Mononuclear chromate complexes are, therefore, considered to be absent in the exchanger phase. Overall Chromate-Yxchange Equilibrium at Acidic pH. To provide a general treatment of the chromate-exchange equilibrium at acidic pH where Cr(V1) is not necessarily a trace species, Cr(VI)/sulfate exchange will be considered as the example. However, the equilibrium needs to be expressed in terms of Cr(V1) concentrations in both the phases because only Cr(V1) is experimentally measured. At acidic pH (2-5) in the aqueous phase [Cr(VI)] = 2[Cr207'-] + [HCrO,-]
[HCr04-]
(38)
All concentration terms from eq 38 onward are in molar units. From eq 4 and 38 [Crzo72-]= K'[cr(vI)Iz
(39)
YI2
Derivation of eq 39 assumes that the aqueous-phase activity coefficients depend only on ionic charges of the species. According to the Davies equation (Stumm and Morgan, 1981), divalent ion activity coefficients, yz,are related to monovalent ion activity coefficients, yl,by the relationship Yz =
YI4
(40)
The chromate-exchange reactions with the sulfate form of the resins may be considered to be taking place with
Ind. Eng. Chem. Fundam., Vol. 25, No. 2, 1986 257
both HCrOL and Cr2OV2-. Thus, for counterion HCr0,RzS04+ 2HCrOp e 2RHCr04 + SO?(41)
By use of eq 38 and 40, the thermodynamic equilibrium constant for eq 41 is given by
or [RHCrO,] =
(43)
Similarly, for the counterion Cr2072(44) R2S04+ Cr2072-Q R2Cr207+ SO:In a similar way, using eq 39, the thermodynamic equilibrium constant for eq 44 is
KII =
fc,~R2CrzO7l [S042-]y12
f s[R,S041
K , [ ~(VI) r 12
or
using eq 43 and 45 [Cr(VI)] = [RHCrO,]
+ 2[R2Cr207]
=[ Equation 46 is the equation of the chromate-exchange isotherm where the total resin-phase concentration of hexavalent chromium, [Cr(VI)], is given in terms of the total aqueous-ph&e concentration of hexavalent chromium [Cr(VI)] and parameters K', K",K4,fs,yl, fc,,, and fHa. When Cr(V1) is a trace species and the concentration of the competing species (SO,,-) is constant, all the terms on the right-hand side of eq 46 become constant except for [Cr(VI)]. Therefore, eq 46 may be written as ]oC r[ = B,[C~(VI)I+ B ~ [ C ~ ( V I ) ] ~ (47) Since the equivalent weights of HCr04- and Crz072- as Cr are the same, eq 47 may be converted to the following form in terms of equivalent fractions: (48) YcI = B3XC1 + 84xCr 2 The first term on the right-hand side represents the contribution of HCr04-, and the second term represents the contribution of Cr2072-in the exchanger phase and accounts for the concave nature of the binary isotherm. Equation 48 also helps explain why exponent b in eq 31 lies between one and two when obtained from the experimental data. Equation 46, in general, provides knowledge regarding how variables like concentration of competing species (sulfate in this example) and ionic strength (shown by activity coefficient, yl)influence the chromate-exchange equilibria.
Conclusion and Remarks The results show that certain characteristics of the chromate-exchange process a t acidic pH are unusual and possibly unique. Though chromate is highly preferred to chloride and sulfate by ion-exchange resins, the chromate-exchange process during a column run exhibits a very
gradual breakthrough of Cr(VI), independent of masstransfer limitations. The chromate/sulfate and chromate/chloride equilibria at acidic pH show unfavorable isotherms which cause such breakthrough. Dichromate (Cr20:-), though practically absent in the aqueous phase, is present in the exchanger phase. This may well explain the concave type isotherm in the presence of competing species and consequent gradual breakthrough of Cr(V1) during a column run. More theoretically sound attempts can now be made to predict the chromate removal capacity as well as breakthrough of Cr(V1) during column runs. Sengupta (1984) demonstrated the same with acrylicmatrix anion-exchange resins. Lastly, from the application viewpoint, the results of this study are particularly significant because they explain the reasons underlying early Cr(V1) breakthrough previously reported. Conventional fixed-bed column runs are usually terminated when a predetermined level of Cr(V1) has broken through. In the case of chromate exchange, such breakthrough is due to its unique equilibrium ion-exchange mechanism and is not caused solely by mass-transfer limitations. Continuous countercurrent ion-exchange systems (Gold and Sonin, 1975; Higgins and Chopra, 1970) appear to be an attractive alternative to conventional fixed-bed operations for chromate removal. Theoretically, due to the unfavorable nature of the chromate isotherms, continuous countercurrent systems will offer higher chromate removal capacity compared to single fixed-bed units. The "merry-go-roundn system (Kunin, 1976), which is a compromise between the continuous and the conventional fixed-bed systems, also has obvious advantages over the conventional single-unit fixed-bed process.
Nomenclature = concentration of species i in the aqueous phase, mol/L = concentration of species i in the exchanger phase, mol/L = activity of species i in the aqueous phase, mequiv/L = activity of species i in the exchanger phase, mequiv/g or mequiv/mL yi = activity coefficient of species i in the aqueous phase fi = activity coefficient of species i in the exchanger phase C, = concentration of species i in aqueous phase, mequiv/L or mg/L C, = concentration of species i in exchanger phase, mequiv/g or mequiv/mL K = thermodynamic equilibrium constant CT = total liquid-phase concentration, mequiv/L Q = resin exchange capacity, mequiv/mL or mequiv/g x , = equivalent fraction of species i in the liquid phase y i = equivalent fraction of species i in the exchanger phase all = separation factor of i with respect to j , dimensionless A, = distribution coefficient of species i vci = concentration velocity of species i at concentration C, m/h v,, = liquid velocity in the column, m/h t = time, s or h I = ionic strength, mol/L [Cr(VI)] = total hexavalent chromium concentration, mg/L - = denotes exchanger phase Abbreviations SBA = strongly basic anion resin WBA = weakly basic anion resin BV = bed volumes SLV = superficial liquid velocity (Re) = particle Reynolds number EB8T = empty bed contact time Registry NO. IRA-458,9084-78-0;IRA-900,9050-97-9 IRA-68, 9056-59-1; C r , 7440-47-3.
Literature Cited Arden, T. V.; Giddings, M. J . Appl. Chem. 1961, 7 7 , 229. Butler, J. N. Ionic €qui/i6r/urn;Addison-Wesley: New York, 1967
258
Ind. Eng. Chem. Fundam. 1986, 25, 258-265
Chu, S.; Sposito, G. Soil Sci. SOC.Am. J . 1981. 45(6),1984. Clifford, D. Ind. f n g . Chem. Fundam. 1982. 2 1 , 141. Currie, K. L.; Curtis, L. W. J . W l . 1976, 84,179. Gold, H.; Sonin, A. I n Froceedlngs, Symposlum on Adsorption and Ion Exchange; AIChE: New York, 1975;Vol. 71,p 152. Haight, G. P.; Richardson, D.; Coburn, N. Inorg. Chem. 1964, 3 . 1777. Heifferich, F. Ion Exchange ; Xerox University Microfilms: Ann Arbor, MI,
1961. Helfferich, F.; Klein, G. Multicomponent Chromatography: Theory of Interference; Marcel Dekker: Ann Arbor, MI, 1970. Higgins, I.; Chopra, R. I n Proceedings. Symposium on Ion Exchange in the Process Industries; Society of Chemical Industry: London, 1970;p 121. Hutchinson, E. Electrons, Nements and Compounds ; W.B. Sanders: Philadelphia, 1959. Kirk-Othmer Encyclopedia of Chemical Technology; Interscience: New York, 1969;Vol. 5. Kunin, R. Amber /+/-LitesNo. 15 1 ; Rohm and b a s : Philadelphia, May 1976. Miller, W. S. Ion Exchange for Pollution Control; CRC Press: Boca Raton. FL, 1978;Vol. 1, p 191. Newman, J.; Reed, L. I n Proceedings, Water-1979; AIChE: New York. 1980;Voi. 197(76).
Newman. J. S. Nectrochemical Systems; Prentice-Hall: Enalewood Cliffs, NJ, 1973;Table 75-1. Pepper, K.; Reichenberg, D.; Hale, D. J . Chem. SOC.1952, 3129. Richardson, E.; Stobbe, E.; Bernstein, S. Envlron. Sci. Technol. 1968, 2(1 I),
1006. Sengupta, A. K. Ph.D. Dissertation, University of Houston-University Park, TX, 1984. Sengupta, A. K.; Clifford, D. React. Polym., Ion Exch., Sorbents, in press. Sengupta, A. K.; Clifford, D. Environ. Sci. Technol. 1986, 2 0 , 149. Standard Methods for the Examination of Water and Wastewater; APHAAWWA-WPCF: Washington, D.C., 1980. Stumm, W.; Morgan, J. Aquatic Chemistry; Wiley: New York, 1981. Tong, J. Y.; Johnson, R. L. Inorg. Chem. 1986, 5 , 1902. Tong, J. Y.; King, E. L. J . A m . Chem. SOC. 1953, 75, 6180. Yamamoto, D.; Kolchi, Y.; Osamu, A. I n Proceedings, Cooling Tower Institute Annual Meeting, Houston, TX. 1975.
Received for review July 20, 1984 Revised manuscript received April 9, 1985 Accepted July 12, 1985
Multicomponent Reaction and Diffusion in a Tubular Reactor: An Operator-Theoretic Solution Brlan G. Turner and Doralswaml Ramkrlshna' School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
Mathematical models describing isothermal tubular flow reactors in which first-order chemical reactions and uncoupled mutticomponent diffusion occur are solved by a linear operator formalism. A pseudohomogeneous axial dispersion model accounting for both bulk-fluid-phase dispersion and intraparticle diffusion is solved to examine the effect of widely disparate configuration diffusion coefficients on selectivity in the reactor. The operator formalism is shown to obtain also solutions for the mathematically more complicated situation of a wall-catalyzed reactor with a nonuniform, fully developed flow profile. Computations presented for the axial dispersion model demonstrate that intraparticle diffusional hindrances in molecular sieve catalysts can drastically affect the selectivity of the xylene isomerization system. The influence of reactor parameters (Peclet and Damkohler numbers) and reactor feed conditions on product selectivity is investigated in order to elucidate the conditions under which optimum selectivity can be expected.
Introduction The effect of mass diffusion on the behavior of an isothermal tubular flow reactor in which a single irreversible chemical reaction occurs has been studied by numerous researchers. The interplay of diffusion and chemical reaction in multicomponent systems has received less attention because the governing differential equations (or boundary conditions) are, in general, coupled and therefore not amenable to analytical solutions. Simplifications (such as the requirement of equal diffusivities) have been entertained by Wei (1966) and others (Ramkrishna and Amundson, 1974), but progress made thus far has been confined to numerical solutions (Solomon and Hudson, 1971). We will employ linear operator theory to solve analytically a pseudohomongeneous axial dispersion model with a source term. One advantage of the operator formalism is that, having identified the solution t~ this model, a method for solving the mathematically more difficult case of a wall-catalyzed reactor with nonuniform, fully developed flow profile immediately suggests itself. The other desirable feature of the framework is that the effect of intraparticle diffusion on product selectivity during xylene isomerization is easily studied for the pseudohomogeneous axial dispersion model. The computations presented show
that variations in the reactor parameters (Peclet and Damkohler numbers) and the feed conditions influence product distributions in unusual ways.
Analysis The Reaction System. Consider an n-component chemical reaction mixture in which each species can undergo reversible first-order chemical reactions to produce every other species, viz. A i z A j , i , j = l , 2,..., n
where kj,represents the rate constant or the conversion of species i to species j . These rate constants form a matrix K that has been investigated by Wei and Prater (1962). Subsequently, Ramkrishna and Amundson (1973) showed that the principle of microscopic reversibility renders the unsymmetric matrix K self-adjoint with respect to the inner product
where {aj)are the equilibrium mole fractions of the mixture satisfying n
* To whom
correspondence
should b e addressed.
(1)
C k i j a , = 0, k i j a j = kj,iai
j=1
0196-4313/86/1025-0258$01.50/00 1986 American Chemical Society
