Ind. Eng. Chem. Fundam. 1986, 25, 381-385
Emmett, J. R. Water Chem. Nucl. React. Syst. 1977, No. 4 . Frisch, N. W.; Kunin, R. A I C M J . 1960, 6 , 640. Helfferich. F. 0.;Plesset, M. S. J . Chem. phvs. 19S8, 28, 418. Helfferich, F. G. Angew. Chem., Int. Ed. Engl. 1962, 7 , 440. Helfferich. F. G. Ion Exchange; M&aw-Hill: New York, 1962. Helfferich, F. G. J . Phys. Chem. 1965, 69, 1178. Kataoka, T.; Sato, N.; Ueyama, K. J . Chem. Eng. Jpn. 1968. 7 . 38. Kataoka, T.; Yoshda, H.; Ueyama, K. J . Chem. Eng. Jpn. 1972, 5 , 132. Kataoka, T.; Yoshida, H.; Yamada, T. J . Chem. Eng. Jpn. 1978, 6. 172. Kataoka, T.; Yoshida, H.; Shibahara, Y. J . Chem. Eng. Jpn. 1976. 9 , 130. McGarvey, F. X.; Kunin, R. Ind. Eng. Chem. 1951, 4 3 , 734. Monet, G. P. Chem. Eng. Prog. 1056, 52, 299. Pan, S. H.; David, M. M. AIChESymp. Ser. 1978, 7 4 , 74. Plesset, M. S.; Helfferich, F. G.; Franklin, J. N. Chem. Phys. 1958, 29, 1084.
381
Smith, T. G.; Dranoff, J. S. Ind. Eng. Chem. Fundam. 1964. 3 , 195. Streat, M.; Brhgal, W. J. Trans. I n s t . Chem. Eng. 1970, 48, T151. TMe, K. Proc. Am. Power Conl. 1981, 4 3 , 1128. Van Brocklln, L. P.; David, M. M. Ind. Eng. Chem. Fundam. 1972, 1 7 , 91.
Received for review August 27, 1984 Accepted October 15, 1985
Supglementary Material Available: A more detailed derivation of the model for mixed-bed ion exchange (20 pages). Ordering information is given on any current masthead page.
Mixed-Bed Ion Exchange at Concentrations Approaching the Dissociation of Water. 2. Column Model Applications C. Eugene Haubt and Gary
L. Foutch’
School of Chemical Engineering, oklehotna State University, Stillwater, Oklahoma 74078
Application of the models developed In the accompanying paper to design of mixed-bed ion-exchange columns is examined. The column model accounts for differing cation and anion resin properties and predicts effluent concentrations in the parts per billion range. The model accounts for the effects of a finite exchange rate, dissociation of water molecules, and reversible exchange. Test simulations were made of a laboratory-scale mixed-bed column using typical m i x d b e d parameters with four different cation to anbn ratios. The lowest impurity levels were predicted for a cation to anlon volume ratio of 1.5:l. This is consistent with industrial practice for mixed-bed ion-exchange units used in electrical power generation.
Introduction The desired result from a column calculation is the effluent volume and concentration as a function of time. As discussed by Helfferich (1962a,b), a universal theory of column performance does not exist. Instead, there have been a multitude of equations derived to compute c o l m performance. Depending upon the case at hand, it is of vital importance that the proper assumptions and simplifications be made. An ion-exchange column begins an exchange cycle in a regenerated state. As the column is contacted with solution, the bed is exhausted or converted to another form as the exchange process continues. The exchange wave or boundary between unconverted and exhausted resins is not ideally sharp. Thus, when breakthrough occurs, the layers of resin a t the exit end of the bed are not fully utilized. The degree of column utilization is the ratio of the capacity at breakthrough to the total resin capacity in the column. The selectivity of the resin toward the exchanging ions helps to determine the sharpness of the exchange wave (Helfferich, 1962b). For a column exchanging ions A in the resin phase for ions B in the solution phase, the separation factor cB/cA
aB/A=-=-=-=-
CBCA
%mA
YBXA
(1)
CB/CA CACB * A ~ B YAXB is greater than one when ion B is preferred by the resin phase. In this case, the equilibrium is favorable. Ions A lagging behind the exchange wave are preferentially disProcess Engineering Department, Conoco, Ponca City, OK 74603.
0196-4313/86/1025-0381$01.50/0
placed into the solution by ions B, and the ions A catch up with the exchange wave. Ions B, which are ahead of the exchange wave, are preferentially held until the wave catches them. In this case, the wave maintains a steadystate “constant pattern” form where sharpening effects, due to favorable equilibrium, and spreading effects, due to finite exchange rates, longitudinal diffusion, and hydrodynamic effects, are in equilibrium. When the separation factor is less than one, the exchanger prefers ion A, which is already in the resin, and the equilibrium is said to be unfavorable. In this case, the wave continues to spread as it travels through the column. Mixed-Bed Modeling Mixed-bed deionization technology and industrial practice have been well ahead of the corresponding theory of ion exchange accompanied by chemical reaction since the introduction of mixed-bed units. The application of mixed-bed ion exchange was introduced by Kunin and McGarvey (1951a). The purpose of mixing cation and anion resins was to obtain a neutralization reaction which would make the exchange process irreversible. This was a giant step in water treatment capabilities, as it enabled the obtainment of extremely low impurity levels with a neutralized effluent. The majority of articles on mixed-bed ion exchange have been concerned with proper mechanical design, operation, and maintenance of these units instead of the relevant ion-exchange theory. Of the articles dealing with mixedbed theory, most have been oriented toward the development of correlations relating breakthrough time to fluid flow rates, inlet concentrations, resin capacities, etc. (Kunin and McGarvey, 1951b; Reenta and Kahler, 1951; Thomp0 1986 American Chemical Society
382
Ind. Eng.
Chem. Fundam., Vol. 25, No. 3, 1986
son et al., 1953). The published models representing mixed-bed exchange require additional theoretical development to adequately simulate current operating conditions of mixed-bed exchangers. There have been two major systematic studies published on the ion-exchange theory and modeling of mixed-bed units (Cadell and Moison, 1954; Frisch and Kunin, 1960). Based on earlier ion-exchange investigations, it was known that the exchange kinetics of mixed-bed units were controlled by film diffusion resistance and that the cation and anion resins in mixed beds normally exchanged ions at equivalent rates. Thus, the simple linear driving force concept acB/at = kl'as(CBo - cB*) (2) was used as the constitutive flux equation, and the mixture of cation and anion resins was treated as a single salt removing resin. For the concentrations studied, the exchange isotherm was found to be strongly concentration dependent as well as strongly irreversible. Consequently, the equilibrium was described by
CB* = 0, q~ < Q =
cBo, QB
a ( X , + €c,)
av
ac,
+ -az =o
1- - t a q n (10) at at Equation 10 is now written for the cation exchanger in a mixed bed by defining the variables xn' and f,:
az
where f, is the ratio of cation to total resin volumes and x,' =
c,/c,'
q n = YnQn The variable x,' is commonly based on the total bulk-phase solution concentration. However, this concentration is not constant through a mixed bed due to the neutralization reaction. Dimensionless distance and time coordinates used to simplify eq 11 are as defined by Thomas (1944):
(3)
=Q (4) The above flux equation with the corresponding equilibrium conditions was integrated by using the column material balance given by CB*
-1- aCn + - + - - aCn =o
(5)
to yield the expression for the effluent ion concentration (Vermeulen, 1958; Frisch and Kunin, 1960) as given by ~ g =' exp[N(T, - 1) - 11 (6) This equation predicted mixed-bed breakthrough curves for effluent ion concentrations as low as several parts per million (-0.001 M). The most appropriate effective system diffusivity was found to be that given by Helfferich (1965)for ion exchange followed by a strong neutralization reaction occurring at the particle-film interface De = 2DnDo/(Dn+ Do) (7)
and
The variables k l , d,, and Q have different values for the cation and anion resin. Thus, the cation or anion resin must be selected as a basis for the dimensionless variables so that common increments of 2 and t will be used in the integration of eq 11 for the two resins. The cation resin and sodium inlet concentration were selected as the basis in this development. Values corresponding to the cation and anion resins will be shown with superscripts of 1 and 2, respectively. The relationship between t and r for the two resins is given by the equations
or
De = 2D&h/(Dc
f
Dh)
(8)
Based on the liquid-film neutralization model (developed in paper l ) , the reaction plane for current mixed-bed conditions is nearer to the bulk phase than the particlefilm interface and the effective system diffusivity is not constant.
Column Material Balance The column material balance developed here assumes plug flow and neglects the effects of longitudinal diffusion and finite particle size. This material balance differs from that discussed by Dranoff and Lapidus (1958) and Omatete et al. (1980) in that dimensionless time and distance coordinates are used instead of the throughput parameter as defined by
A separate material balance is required for the cation and the anion resins. Because of this, a new variable accounting for the volume fractions of the respective resins is introduced. The general material balance for a sodium cation exchange column is written as
To simplify eq 11, each of the partials are written in terms of E and 7 for the cation resin:
Substituting eq 16-18 into eq 11yields the final material balance for the cation resin
The rate of exchange is also expressed in terms of 7 by the combination of eq 20 (developed in our companion paper)
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986 383
and eq 18, giving
Table I . System Parameters for Mixed-Bed Simulations initial equivalent fraction of chloride in the anion 0.1
A similar development for anion resin, again based on and rl,leads to the equations
kI2d,lQ'
aYc =-
87'
kl'd,2Q2
6Ri2(CC0/C,f- Cc*/Cnq
['
(23)
With the use of eq 14 and 15, eq 22 and 23 directly reduce to the same form as eq 19 and 21 with f', T', and fc replaced by f 2 , r2,and (1 - fc), respectively. The operation of a mixed-bed column is now simulated by a calculation mesh having f and 7 as the abscissa and ordinate, respectively. Equations 19 and 21-23 are simultaneously integrated along characteristic lines of constant [ and constant 7. Material balances and rate expressions of this form may be integrated simultaneously for as many different types of resins or different resin sizes as desired. (For different resin sizes, fc will represent the volume fraction of any one particular size of resin.) This method of separate resin treatment enables mathematical studies more closely simulating actual column operating conditions and variables. One example includes imperfect resin separation before regeneration. A column can readily be modeled in which a small fraction of the cation resin is regenerated with the anion resin. This resin would be modeled as a third resin and would release sodium ions during the initial phases of exchange. Other possible studies include modeling the effects of feed concentration surges, various cation to anion resin ratios, multiple particle sizes, improper resin mixing or incomplete resin regenerations.
Discussion The column material balance and exchange rate expressions, as previously developed, were used to simulate sodium-chloride mixed-bed systems. The material balances for the cation and anion resin (eq 19 and 21-23) are simultaneously integrated by using the method of characteristics along lines of constant 7 and [ (Acrivos, 1956; Dranoff and Lapidus, 1958). The concentration profiles down the column at a constant 7 are first determined by integrating the material balances with respect to f by using the improved Euler technique. This results in a horizontal sweep across the calculational matrix. The equations are then integrated with respect to 7 by using the backward finite difference method, and another horizontal sweep is made. With this approach, the calculations are continued until the ion concentrations in the column effluent reach a predetermined fraction of the concentrations in the feed solution. The integration increments for 7 and [ were based on the physical properties of the cation resin. Respective dimensionless increments of 0.04 and 0.01 were used in the column simulations. The [ increment corresponds to approximately 0.25 cm, depending on the resin properties. The time increment represented by 7 is inversely proportional to the feed solution concentration (eq 13). The 7 increment of 0.04 corresponds to approximately 12 min for feed concentrations of 0.001 M. For feed concentrations below 0.0001 M, 7 was decreased to 0.004. For a given feed concentration, column integrations were relatively insensitive to the magnitude of 7. The error in
resin initial equivalent fraction of sodium in the cation resin cation resin particle diameter, cm anion resin particle diameter, cm bed void fraction feed solution concentration, mequiv/cm3 volumetric flow rate, cm3/s column diameter, cm height of packed resin, cm cation resin capacity, mequiv/cm3 anion resin capacity, mequiv/cm3 selectivity coefficient for chloride-hydroxide exchange selectivity coefficient for sodium-hydrogen exchange hydrogen diffusivity, cm2/s sodium diffusivity, cm2/s hydroxide diffusivity, cm2/s chloride diffusivity, cm2/s solution viscosity, CP solution density, g/cm cation resin volume fraction a
0.1 0.076 0.063 0.445 0.002 10.32 2.54 40.0 3.0728 2.4334 1.45 1.55 9.35 x 1.35 x 5.23 x 2.03 x 1.0 1.0
10-5 10-5 10-5 10-5
a
Values given in the figures.
predicted concentration profiles gradually increased as 7 was made larger. Column results were very sensitive to variations in the size of f increments. Predicted concentrations were ie error by several percent for f increments of 0.02, and this error rapidly increased for larger increment sizes. These sensitivities are explained by the concentration profiles within the column. Solution concentrations are quickly reduced with progression through the exchange bed. However, the solution concentrations at a given distance from the column inlet are relatively stable with time as the active-ion-exchange zone moves slowly down the column. At each calculational mesh point, the rate of exchange for the two resins is calculated by either the bulk-phase neutralization or film reaction models. The film reaction model is used for the cation and anion resins when the bulk-phase hydrogen concentration is less than 0.5 X M and greater than 1.5 X M, respectively. Otherwise, the bulk-phase neutralization model is used. From the above conditions, the bulk-phase neutralization model is used for both resins when the hydrogen concentration is M. in the range of (0.5-1.5) X Column simulations were performed for laboratory-scale mixed-bed units. System parameters for the mixed-bed simulations are presented in Table I. For parameter studies, a column length of 40 cm was used. The processing time was sensitive to the column length due to the small magnitude of f increments required for computational accuracy. Approximately 70 s of processing time was required on an IBM 3081D computer system to obtain the full breakthrough curve. This time will vary somewhat, depending on the inlet concentration and other system parameters. Plant scale units are normally 1.5-2.5 m in length with fluid linear velocities similar to those used in this study. Thus, a plant-scale simulation may require as much as 10 min of comparable computing time. Because of this excessive processing time, a more elaborate method of integration with respect to distance down the column ( E ) should be implemented so that larger increments may be used without sacrificing accuracy of results. A fourpoint Milne predictor-corrector method may be one such possibility. Program storage requirements were kept to a minimum level so that the program could be readily run on a microcomputer. The concentration profiles through the
384
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986
,
1.0 e
,
1.o
t
v)
2
o_
CATION
0.5
0.10
7
n
w 0.001
W
U
0.0001
D
t
0
0.666
-
i
.
--SODIUM --CHLORIDE
I
20
40 60 SO 100 TIME (MINUTES)
120
140
Figure 1. Sodium breakthrough curve for mixed-bed simulations with varying cation bed fractions.
0.1 :
CATION CURVE FRACTION
-
n W
W L
V 0.000 1 0
-
20
C
0.6
D
0.666
40 60 80 100 TIME (MINUTES)
120
1 140
Figure 2. Chloride breakthrough curve for mixed-bed simulations with varying cation bed fractions.
column are stored for only four consecutive time ( 7 ) increments. With the current method of integration, only the concentration profile from the preceding time increment is required at a given calculation point. More elaborate integration methods, as previously suggested, may require all four of these consecutive profiles. Because of this storage approach, the concentration profiles or breakthrough curve must be printed during the calculation iterations. The simulation program requires less than 400K of storage space. This can be readily reduced by only storing two consecutive concentration profiles if more elaborate integration techniques are not used. The utility of this model development and a typical program application are demonstrated by calculating breakthrough curves for a sodium-chloride mixed-bed unit with varying cation to anion resin ratios. Typical cation to anion resin ratios used in industrial practice range from 1:l to 2:l. The sodium and chloride breakthrough curves are shown in Figures 1 and 2, respectively. A feed con-
Figure 3. Variation of sodium and chloride concentration profiles with time and distance (cation/anion ratios = 1.5).
centration of 0.002 M and a column length of 40 cm were used in these simulations. The other system parameters are typical of mixed-bed units. The breakthrough curves are shown as a ratio of the effluent to feed solution concentrations. The dimensionless time has been converted to minutes based on the elapsed time beginning with the discharge of feed solution from the column. As shown in Figures 1 and 2, the ratio of exchange resins in the unit affects the overall capacity of the exchange bed as well as the lower concentration limits in the effluent solution. The lowest total effluent concentration was obtained for a cation to anion ratio of 1.5:l (case C). This corresponds well with industry practice (Strauss, 1978). Figure 3 shows the sodium and chloride concentration profiles within the mixed bed for case C of Figures 1 and 2. These profiles show the cause of the large deviations between the sodium and chloride breakthrough curves as the resins approach equilibrium with the feed solution. As the resins become saturated, the anion resin capacity is exhausted first. This results in an acidic solution wave as the cation resin continues to exchange hydrogen for sodium ions. The cation resin in this wave is simulated with the bulk-phase neutralization model. When the acidic wave reaches the active anion-exchangezone, the neutralization reaction occurs in the film surrounding the anion resin. The effects of the film neutralization reaction are to increase the anion-exchange rate, equalize the sodium and chloride concentrations, and neutralize the acidic wave. The sharp approach of the chloride concentration profiles to the equilibrium values is due to the finite rate of exchange at equilibrium as predicted by the film reaction model. Detailed mathematical studies of mixed-bed parameters, such as the resin ratio, exchange capacities, and particle sizes, have not been possible with previous models for mixed-bed exchange. Besides the study of the above parameters, this model predicts the lower limit for mixed-bed exchange. In the example application, initial effluent concentrations less than 1.0 X lo4 M were predicted. Mathematical studies of the lower exchange limit have not been made before due to the lack of rate models which are applicable at these concentration levels. Suitable experimental data for confirming the reactive-ion-exchange rate expressions and column simula-
Ind. Eng. Chem. Fundam., Vol. 25, No. 3, 1986 385 tions at concentration levels for which the model was developed were not discovered in the published literature. As previously mentioned, only a very few experimental results of laboratory-quality mixed-bed studies have been published. The experimental breakthrough curves of these studies were modeled essentially by curve-fitting the data points through adjustment of the film thickness, masstransfer coefficients, or dimensionless quantities in the rate expressions. Complete descriptions of the experimental parameters in these studies were not given, and the results were not modeled with the current simulation program. For solution concentrations above 1.0 X lo4 M, the rate equations give results identical with those of Kataoka (1976a) and Helfferich (1965), which have been experimentally confirmed. Conclusions The film reaction model permits the separate treatment of cation and anion resins in mixed-bed units..This in turn presents a method for systematic mathematical studies of the effects of differing cation and anion resin properties on the operation of mixed-bed units. Previous to this development, the cation and anion resins in a mixed bed were most effectively modeled as a single salt removing resin. This model also predicts effluent concentrations on the order of parts per billion while accounting for the effects of a finite exchange rate, dissociation of water molecules, and reversible exchange. For feed solution concentrations of 0.002 M, column integrations using 7 and [ increments of 0.02 and 0.01 predicted results which were within 0.5% of those using respective integration incrementa of 0.001and 0.005. The integration error is very sensitive to the size of the [ increment but relatively insensitive to variations in the 7 increments. These sensitivities result from the rapid decrease of impurity concentrations with solution progression through the column but the relatively slow movement of the active exchange zone down the exchange bed. Test simulations were made of a laboratory-scale mixed-bed column using typical mixed-bed parameters with four different cation to anion ratios. The lowest impurity levels were predicted for a cation to anion volume ratio of 1.5:l. This is consistent with industrial practice for mixed-bed exchange units used by electrical power companies. Nomenclature specific surface area, cm2area/cm3 resin a, concentration of species i, mequiv/cm3 Ci total counterion concentration, mequiv/cm3 ct particle diameter, cm effective liquid-phase diffusivity, cm2/s diffusion coefficient of species i, cm2/s Di fc volume cation resin/total resin volume ionic flux of species i, mequiv/(s cm2) second-order rate constant kl nonionic liquid-phase mass-transfer coefficient, cm/s ionic liquid-phase mass-transfer coefficient, cm/s k i selectivity coefficient for ion B in the solution reKA placing ion A in the resin phase mi molality of species i N number of mass-transfer units = k X Z / w 4 column cross-sectional area
h 2
Qi
8
Ri t
2 V
vb
v,
Xi
Xi
x xi Yi
z
mean resin-phase concentration of species i, mequiv/cm3 total resin exchange capacity, mequiv/cm3 ratio of electrolyte to nonelectrolyte mass-transfer coefficients time, s throughput parameter throughput ratio = (V - eVb)/ V, volume of solution having passed a given column layer, cm3 bed volume, cm3 volume of bulk solutio_n equivalent to exchange capacity of bed (vbx/c,) equivalent fraction of species i in solution equivalent fraction ofspecies i based on the column feed concentration = Ci/C: total ion concentration in resin phase per unit volume of bed, mequiv/cm3 concentration of species i in resin phase per unit volume of bed, mequiv/cm3 equivalent fraction of species i in resin phase distance from column inlet, cm
Greek Letters separation factor e bed void fraction P superficial liquid velocity, cm/s dimensionless distance t bulk density of resin, g/cm3 Pb 7 dimensionless time aB/A
Superscripts refers to resin phase * interfacial equilibrium condition 1 cation-exchange parameter 2 anion-exchange parameter f column feed condition 0 bulk-phase condition
-
Subscripts A ion exiting the resin phase B ion entering the resin phase C chloride ion h hydrogen ion i species i n sodium ion 0 hydroxide ion Registry No. HzO, 7732-18-5. Literature Cited Acrlvos, A. Ind. Eng. Chem. 1968, 4 8 , 703. Cadell, J. R.; Molson, R. L. Chem. €ng. PTog., Symp. Ser. 1954, 50. 1. Dranoff, J. S.; LapMus, L. Ind. Eng. Chem. 1958. 50, 1648. Frisch, N. W.; Kunin, R. AIChEJ. 1980, 6 , 640. Helfferich. F. G. Angew. Chem., Int. Ed. €ng/. 1982a. 7 , 440. Helfferich, F. 0. Ion Exchange; McGraw-Hill: New York, 1982b. Helfferlch, F. G. J . Wys. Chem. 1985, 6 9 , 1178. Kataoka, T.; Yoshlda, H.; Shlbahara, Y. J . Chem. Eng. Jpn. 1976, 9 , 130. Kunin, R.; McGarvey, F. X. U.S. Patent 2578937, 1951a. Kunln, R.; MCaNey, F. X. Ind. Eng. Chem. 1951b, 4 3 , 734. Omatete. 0. 0.; Chzle. R. N.; Vermeulen, T. Chem. Eng. J . (Lausanne) 1980, 79, 229. Reents. A. C.; Kahler, F. H. Ind. Eng. Chem. 1951, 43, 730. Strauss, S. D. Power 1978, 722. 5-14. Thomas, H. C. J . Am. Chem. Soc. 1944, 66, 1664. Thompson, J.; McGarvey, F. X.; Wantz, J. F.; Ailing, S. F.; ollwood, M. E.; Babb, D. R. Chem. Eng. Rog. 1953, 4 9 , 341, 437. Vermeulen, T. A&. Chem. Eng. 1958, 2 , 147.
Received for review August 27, 1984 Accepted October 15, 1985
