3916
I n d . Eng. C h e m . Res. 1995,34, 3916-3924
SEPARATIONS A New Model for the Simulation of Ion Exchange Equilibria Stefan0 Melis and Giacomo Cao* Dipartimento d i Ingegneria Chimica e Materiali, Universita d i Cagliari, Piazza d’Armi, 09123 Cagliari, Italy
Massimo Morbidelli* Dipartimento di Chimica Fisica Applicata, Politecnico di Milano, Piazza Leonard0 d a Vinci 32,
20133 Milano, Italy
A model based on the mass action law for the prediction of multicomponent ion exchange equilibria is developed by assuming ideal behavior for both the solution and the solid phase, and the existence of a distribution of functional groups with different adsorption energies (and then equilibrium constants). The reliability of the model, which is characterized by two parameters for each binary system, is tested by comparison with a remarkably large set of literature experimental data. It is shown that ion exchange equilibria in systems involving a total number of N , counterions can be predicted from the knowledge of the experimental behavior of only (N, - 1)binary systems, thus in agreement with the so-called triangle rule. This greatly reduces the experimental and modeling effort needed to develop a multicomponent model with respect to other approaches which require the investigation of all possible pairs of counterions present in the system, i.e., N&N, - 1)/2 binary systems.
Introduction Ion exchange resins are extensively used in industry for various separation processes (cf. Helfferich, 1962). In particular, the optimal design of industrial fixed bed ion exchange units requires the accurate simulation of multicomponent ion exchange equilibria. The typical approach is to develop thermodynamic models which, based on binary equilibrium data, can predict multicomponent equilibria. The models proposed in the literature can be divided in two main groups: those describing the ion exchange process in terms of the law of mass action and those regarding the ion exchange as a phase equilibrium. In early works, which belong t o the first group (cf. Dranoff and Lapidus, 1957; Pieroni and Dranoff, 19631, it was assumed that the ion exchange equilibrium is ideal, i.e., the activity coefficients of all components equal unity both in solution and in the solid phase. This model corresponds to assuming that the presence of other counterions does not affect the equilibrium exchange between two particular ions, which implies constant selectivity coefficients. Of course the behavior of systems where the selectivity coefficientschange with the resin composition cannot be described on the basis of these models (cf. Helfferich, 1962). For this we need to introduce nonidealities in the mixture behavior resulting from the interaction among the counterions in the liquid and/or in the solid phase. Several approaches of this type have been proposed in the literature, which differ in the selection of the model used to describe the nonideal behavior of the liquid and the adsorbed phases. Shallcross et al. (1988) reviewed several such models and, on the basis of an earlier model by Smith and Woodburn (1978), proposed to use the Pitzer (1979) equation to describe single ion activity coefficients in the liquid phase and the Wilson model for the activity coefficients in the adsorbed phase.
The latter involves binary interaction parameters (two for each binary system) which need experimental equilibrium data to be estimated. Thus, for a multicomponent system involving N , components, this model involves N d N , - 1) parameters which describe the interaction between the N J N , - 1)/2 different couples of ions present in the system. For evaluating these parameters, in addition to the equilibrium constants, we need to investigate experimentally the corresponding N,(N, - 1)/2 binary systems. In particular, Shallcross et al. (1988) investigated the equilibrium of the ternary system Na+/Ca2+/Mg2+ on Amberlite 252. For this they estimated the six binary interaction parameters of the Wilson model, using the experimental data relative to the corresponding three binary systems. A similar model has been used by de Lucas Martinez et al. (1992, 1993) for various binary and multicomponent systems on Amberlite IR-120, involving the ions H+, Na+, K+, Ca2+, and Mg2+. A method to reduce the number of model parameters has been developed by Allen et al. (1989) and Allen and Addison (1990) by forcing the Wilson parameters to satisfy some constraints based on empirical relationships. Although the behavior of binary systems was satisfactorily described, the prediction of the third binary according to the triangle rule was not always in agreement with the experimental data. A modification in the procedure for evaluating the model parameters has been employed by Mehablia et al. (1994). This involves the estimation of the equilibrium constants through the Gaines and Thomas method, i.e., by an appropriate integral of the corresponding binary data, followed by the usual nonlinear regression of the equilibrium data for estimating the activity coefficient parameters. Also the approach of Soldatov et al. (19871, which implies the derivation of explicit expressions for the equilibrium constants as a function of the adsorbed phase composition (see also Marton and Inczedy (1988)
0888-5885/95/2634-3916$09.00l0 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 11,1995 3917 and Kol'nekov (1987)), requires the same experimental effort in characterizing binary systems. In the models of the second group, ion exchange is treated as an adsorption process (cf. Novosad and Myers, 1982; Myers and Byington, 1986). In these models, deviations from the ideal adsorption behavior are explained in terms of the energetic heterogeneity of the functional groups of the ion exchanger. In particular, Myers and Byington (1986) considered a binomial distribution of the adsorption energies, while assuming that each adsorption site behaves ideally. The model involves three adjustable parameters per each binary system, which correspond to the average adsorption energy, the standard deviation, and the skewness of the energy distribution. Thus the number of parameters is the same as that in the models based on the mass action law discussed above (cf. Shallcross et al., 1988). However, the main difference in this approach is that it accounts for solute-adsorbent rather than solute-solute interactions, and takes them as responsible for the deviation from the ideal behavior. The consequence is that we do not need to investigate all the binary systems formed by all possible pairs of N , counterions present in the multicomponent system, i.e., NdN, - 1112,but rather only one binary system per each counterion. This is usually done by considering the (N, - 1) binary systems formed by each one of the N , counterions with the same one taken as reference ion. This sigdicantly reduces the experimental effort needed to develop a multicomponent equilibrium model as well as the overall number of adjustable parameters involved. Myers and Byington (1986) reported several applications of this model to ternary literature data by following the triangle rule; i.e., the behavior of the binary system A/C as well as of the ternary one AB/C is predicted on the basis of experimental measurements of the exchange equilibria of the binary systems A 5 and BK. More recently, this model has been used, in combination with the description of the dissociation equilibria in solution, to correlate binary uptake data of amino acids and predict the corresponding multicomponent adsorption equilibria (cf. Saunders et al., 1989; Dye et al., 1990). The same model was also applied by Jones and Carta (1993)to describe the uptake equilibria of dipeptides on cation resins with varying degree of cross-linking. In all cases rather satisfactory results were obtained. In all previous studies, some aspects, such as the equilibrium behavior of systems involving both monoand bivalent ions a t various values of the solution normality, have not been considered. These are specifically addressed in this work where, in addition to the Myers and Byington model, we consider a new model based on the law of mass action. This model differs from all models of the first group discussed above in that it assumes that both the fluid and the solid phases behave ideally, while it acknowledges the heterogeneity of the adsorption sites. In other words, in this approach we assume that the effect of mixture nonidealities is smaller than that due t o the resin heterogeneity and can be neglected. This corresponds to introducing the idea of Myers and Byington (1986) accounting for the heterogeneous nature of the solid in the framework of the equilibrium models based on the law of mass action. This is indeed similar to the approach of Barrer and Meyer (1959) who, after recognizing the presence in a specific zeolite of two different sites, described the
adsorption equilibria in the case of binary systems of monovalent cations through the superimposition of two ideal mass action laws. It should be noted that the model developed in this work leads to the same equations as those of Myers and Byington (1986) in the case of ions with equal valence, while they are not the same in the case of ions with different valence. We will discuss these differences in detail by comparing the model predictions with appropriate experimental data.
The Ion Exchange Equilibrium Model Let us consider the global exchange process for a binary system occurring on a cation exchange resin:
PASU+ + a, p+ p q + + &,p+
(1)
where A"+ and W+ are the two exchanging ions while r and s indicate the resin and the solution phases, respectively. It is worth mentioning that a cationic resin is considered here without any loss of generality. The thermodynamic equilibrium constant for such a reaction is given by (a++Y(aB/+)a
K=
(2) (aAsa+Y(aB,)"
in terms of the activities of the involved ionic species. This quantity is defined by the thermodynamic relationship:
AG" = -RT In K
(3)
where AG" is the standard free-energy change associated with the ion exchange process (1). By assuming that the exchange equilibrium is ideal, i.e., the activity coefficients of all components equal unity both in solution and in the solid phase, and neglecting the effects due to the resin swelling and hydration, the equilibrium constant may be written as (cf. Pieroni and Dranoff, 1963)
(q++Y(cB/+)'
K=
(4) (CA,aiY(qB$+)a
where qi and Ci are the concentrations in the resin and in the solution phase, respectively. Let us now introduce the ionic fraction of the generic ith ionic species, siv1+:
v;c,
v;c;
where Ci represents the concentration of Siv'+in solution and vi its charge, N , is the total number of counterion species, N is the total ionic concentration in the solution phase or the solution normality, qi is the solute concentration in the solid phase, and q o is the total capacity
3918 Ind. Eng. Chem. Res., Vol. 34,No. 11, 1995 of the resin. Substituting in eq 4, the equilibrium constant becomes
10 the equilibrium constants Kj can be related directly to the parameters which characterize the energy distribution:
(7) It should be noted from the equation above that the equilibrium distribution is affected by changes in the total resin capacity or in the solution normality, only in the case of counterions with different valences (i.e., a f PI. Moreover, the equilibrium constant K,which is a function only of temperature, in general cannot be computed from eq 3, since the corresponding values of the standard free-energy changes are not available. Thus, K is usually treated as a fitting parameter (cf. Pieroni and Dranoff, 1963), and its value is determined by direct comparison with equilibrium experimental data. We now introduce the heterogeneity of ion exchange functional groups, by assuming a given distribution for the standard free-energy change of the ion exchange process, AGO. In general, this is characterized by an average value, to which corresponds the average equilibrium constant E = e x p ( - s / R T ) and a standard deviation 0, Here we are interested not in considering the real energy distribution of the functional groups, but rather in having an approximation effective from the computational point of view. Thus, we take the simplest situation, where we assume that the resin is characterized by only two types of functional groups. Accordingly, the total resin capacity, 40 is given by
s,
40 = 40,l + 40,2
(8)
where qo,1 and 40,2 represent the capacity of the resin with respect to the functional groups of type 1 and 2, respectively. As mentioned above, this distribution can be represented in terms of average value and standard deviation, as follows: __
Ni--2
AGO =
pjAGj" = p l A G l " ip2AG2"
(9)
The overall resin composition is obtained by considering for each component the quantities present on each type of functional group: Ni2
(16) The developed model allows us to compute, for a given composition of the fluid phase Xi,the corresponding equilibrium composition of the resin phase Yi. This is obtained by first evaluating the composition of the phase adsorbed on each type of functional group. Thus, for a binary system and for given values of the model parameters E and y , or alternatively K1 and K2, we solve eq 13 together with the stoichiometric condition:
(17) for the unknowns and E;.,Bd+, for each value o f j independently. Then we compute the overall resin phase composition through eq 16. For multicomponent systems containing N, counterions we proceed along similar lines. In this case, we need to first select a reference counterion, say the Ncth ion SN:N,+, and then consider the (N, - 1)equilibrium relations between this and each one of the other counterions, Siv'+with i = 1, 2, ..., (N, - 1). Similarly t o eq 13 we introduce for each of these the equilibrium constant on sites of type j :
j=l
Nf2
C pj(AGjo-
2=
= p p 2 ( A G , " - AG2"I2 (10)
j=1
for j = 1,2 and i = 1 , 2 , ..., ( N , - 1) (18)
where
p J. = q o Jqo .l
j=1,2
(11)
and AGj" represents the standard free-energy change of the ion exchange process on the j t h functional group. Similarly to eqs 6 and 7, we define the ionic fraction of the generic ionic species Sivi+ on the j t h functional group:
"iqj,i y,.=J4 40 j
j=1,2
(12)
and the equilibrium constant of the exchange process occurring on the same functional groups:
where Kj = exp(-AGj"/RT). Note that using eqs 9 and
These provide (N, - 1)equations which in addition to the stoichiometric condition: (19) lead to the composition of the phase adsorbed on sites of each type, i.e., the values of the concentrations YJ,s,"i+ = YJ,,. The overall resin phase concentrations are then computed by summing up all concentrations relative to the individual sites, as indicated by eq 16 for the component Aa+. Note that in this case the model requires the 2(N, - 1) binary parameters Kj,2 (or alternatively Ei = (K1,2P1(K2,,P~ and y 2 = (K1,J K2,2)(p1P2)1'2), which arise from the characterization of the binary exchange equilibria of each of the counterions SZv1+ with the reference one. Moreover, the model requires the value of the parameter p1 (pz = 1 - PI), which should be the same for all binary systems.
Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 3919
As an example let us consider the case of N , counterions with the same charge, i.e., vi = v for i = 1, 2, ..., N,. Using eqs 18 and 19, we obtain the overall composition of the resin phase:
Nc
-
where since i = N , indicates the reference counterion, we have from eq 18 that KjaC= 1. From the relation above we can readily derive an explicit expression for the separation factor Si,&
where Zj = c~~l(q,k)l’v&., for j = 1, 2. Note that in the case of a binary system, i.e., N , = 2, the equation above reduces to the two-site model developed earlier by Barrer and Meier (1959) with reference to a specific zeolite where the actual existence of two different types of sites was identified through a structural analysis.
Comparison with Other Equilibrium Models Myers and Byington (1986) have developed a theory of ion exchange based on the thermodynamics of phase equilibria by introducing the energetic heterogeneity of the resin functional groups. The energy distribution was simulated through the discrete binomial distribution, which is characterized by the total number of site types and a skewness parameter, p . In the applications of this model, also subsequent to the original work, usually only two types of sites have been considered, in similarity with the model developed above. In this case, the skewness parameter becomes the fraction of sites of one type defined by eq 11, i.e., p = p l (and p2 = 1 PI. In the case where only two types of sites are present, the original derivation of the model of Myers and Byington (1986) can be briefly summarized as follows. Let us assume that the adsorption process on each type of sites can be described by the ideal Langmuir isotherm:
-
-_
where Si,k = (Ki/Kk)l’”,Wi,k = ( y i / Y k ) l i v , V = ( 1 - P ) / [ p ( l - p)]0.5, u = -p / [ p ( l - P ) ] O . ~ , and p = P I . This is the form of the multicomponent model of Myers and Byington (1986) usually adopted in applications. This has been derived by Saunders et al. (1989) by noting that since the functional groups of the ionic resin of interest in applications are generally completely dissociated, it is more appropriate to describe the single site exchange equilibrium through the stoichiometric isotherm:
(25) 1=1
rather than through the Langmuir isotherm given by eq 22. This leads directly t o eq 24 (or 21) for the separation factor, without need t o invoke large values for the solution normality. The Myers and Byington model given by eq 24 has been applied successfully to several binary and ternary systems, as discussed in the Introduction, involving both mono- and bivalent ions. However, we see that it does not account for the effect of the solution normality, which is instead included in the model developed in this work. In particular, the new model reduces to eq 24 (or 21) only in the case where ions of equal valence are involved, which is the situation where the solution normality does not affect the equilibrium distribution. On the other hand, when ions with different valences are involved, in general the model cannot be written in closed form, but indeed the equilibrium distribution is affected by changes in the solution normality as can be readily seen by inspection of the constitutive equations. For example, in the case of a binary exchange between a bivalent ion A and a monovalent ion B, the model can be again solved analytically leading to an explicit expression for the separation factor:
where Bj,i is the Langmuir constant of adsorption of the ionic species i on the sites of type j . Recalling that qi = CET2q,,i and using eq 11, the following expression for the separation factor is obtained:
where zj = &Bj,& It can be seen that for large values of the solution normality, N , eq 23 becomes identical to eq 21, where the equilibrium dissociation constants ( q , i ) l / ”are replaced by the Langmuir adsorption constants, Bji. The obtained expression, i.e., eq 21, can be rewritten in the form:
where CS = 2qdN. This expression is different from the corresponding one given by the model of Myers and Byington (1986),i.e., eq 24, and explicitly accounts for the solution normality. Finally, note that the effect of the solution normality is not properly accounted for by eq 23, which implies that this effect is present only in the case where the functional groups are not all dissociated. This effect should then vanish in the case of strong resins, in contrast with experimental evidence.
3920 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995
0.6 -
0.4
0.0
0.2
0.6
0.4
0.8
1.0
X
Figure 1. Comparison between calculated (HMAM) and experimental values (de Lucas Martinez et al., 1993) of the ionic fraction in the resin phase as a function of the ionic fraction in the solution phase for the ion exchange equilibrium Na+/H+.
Comparison with Experimental Data The reliability of the model developed in this work, in the following referred to as the heterogeneous mass action model (HMAM), is tested in this section by comparison with equilibrium experimental data available in the literature. The comparison is extended also to previous models, such as the ideal mass action model (IMAM)and the heterogeneous adsoption model (HAM) proposed by Myers and Byington (1986). Following previous studies in the literature, in the heterogeneous models (HMAM and HAM) we will assume that the two types of sites are equally abundant, i.e., p1 = p z = 0.5. This decreases the number of adjustable parameters. Let us first consider the fitting of binary systems, where the HMAM and the HAM models require the estimation of two parameters, i.e., K and y o r SI,^ and W ~ , Zrespectively. , Since as shown above for systems involving counterions of equal valence the HMAM and the HAM coincide, the analysis of monovalent ions is reported below jointly for the two models. In Table 4 of the original work of Myers and Byington (1986) it is shown that the two models provide an accurate interpretation of a significative number of systems including only monovalent cations. In addition to those, we consider here the binary systems Na+/H+and K+/H+ on Amberlite IR-120, investigated experimentally by de Lucas Martinez et al. (1993). The values of the HMAM parameters, i.e., and y , can be estimated by fitting the experimental data through a nonlinear least squares procedure. The comparison between HMAM results and experimental data is shown in Figures 1 and 2, while the estimated values of E and y are summarized in Table 1, together with the corresponding values of the average percentage error, E , arising from the fitting procedure. For both systems the agreement with the experimental data is satisfactory. It is worth noting that when the value of qo was not available in the original work used as a source of experimental data, the corresponding value reported in Table 1 has been assumed. This does not cause any loss of accuracy of the obtained results since, if the corrected value of qo would have been used, a change in the fitted parameters would have occurred, but the capability of the HMAM in describing the experimental data would have not changed.
-
0.0
0.6 0.8 1 3 X Figure 2. Comparison between calculated results of both IMAM and HMAM and experimental values (de Lucas Martinez et al., 1993) of the ionic fraction in the resin phase as a function of the ionic fraction in the solution phase for the ion exchange equilibrium K+/H'. 0.2
0.4
The behavior of the systems Na+/H+ and K+/H+ reported in Figures 1and 2 is not ideal, as indicated by the value larger than 1of the heterogeneity parameter y obtained through the fitting procedure. The HMAM reduces in fact to the IMAM when the two functional groups present in the resin exhibit the same exchange characteristics. In this case from eqs 9-11 and 14 and 15, it follows that = AGO1 = A P z , o2 = 0, y = 1, and = K I = Kz.Clearly, the IMAM cannot describe the experimental behavior of these systems. This may be seen from Figure 2 where the accuracy of the IMAM in reproducing the experimental data of the system K+/ H+ is compared with that of the HMAM. In this case the improvement given by the use of the HMAM as compared to the IMAM, whose single parameter value and average percentage error arising from the fitting procedure are given by K = 2.37 and E = 6.4%, respectively, is modest due t o the relatively small value of the heterogeneity parameter y . It is worth noting that in the original work of de Lucas Martinez et al. (1993) the experimental data shown in Figures 1 and 2 were interpreted using an equilibrium adsorption model involving three parameters, namely, the equilibrium constant and the two binary interaction parameters of the Wilson model. The obtained average percentage errors were equal to 7.3% and 4.2%)respectively. This accuracy is comparable to that obtained with the HMAM, which uses only two adjustable parameters, Le., E = 3.2%and 2.42%, respectively. This indicates that the developed HMAM exhibits at least the same flexibility in reproducing the experimental data even though it includes a lower number of adjustable parameters. A good agreement between HMAM results and experimental data is also obtained for the case of the binary systems Ag+/H+ (data of Dranoff and Lapidus (1957))) K+/H+ and Na+/H+ (data of Manning and Melshmeier (1983)))and Na+/H+ (data of Pieroni and Dranoff (1963)) as may be seen from the values of the average percentage error, E , reported in Table 1. The comparison between model results and experimental data for these systems is not shown in detail. Although the HMAM is developed by considering an exchange process occurring on a cation exchange resin, it can also be applied to describe processes occurring on anionic resins. As an example, for the case of the
Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 3921 Table 1. Estimated Values of HMAM Parameters and Average Percentage Errors for Model Predictions
a -
reference
system
de Lucas Martinez e t al. (1993) de Lucas Martinez et al. (1993) Dranoff and Lapidus (1957) Dranoff and Lapidus (1957) Dranoff and Lapidus (1957) Manning and Melshmeier (1983) Manning and Melshmeier (1983) Manning and Melshmeier (1983) Manning and Melshmeier (1983) Manning and Melshmeier (1983) Manning and Melshmeier (1983) Khoroshko et al. (1974) Khoroshko e t al. (1974) Pieroni and Dranoff (1963) Pieroni and Dranoff (1963) Pieroni and Dranoff (1963) Sengupta and Paul (1985) Sengupta and Paul (1985) Sengupta and Paul (1985) Sengupta and Paul (1985) Subba Rao and Davis (1957) Shallcross e t al. (1988) Shallcross et al. (1988) Shallcross et al. (1988) Shallcross et al. (1988) Shallcross et al. (1988) Wheaton and Bauman (1951)
Na+/H+
K+/H+ Ag+/H+
Cu2+/Ag+ CUZ+/H+
K+/H+ Na+/H+ Ca2+/H+ Ca2+/Na+ K+/Na+ CaZ+/K+ Mg2+/K+ Ca2+/K+ Na+/H+ Cu2+/Na+ CUZ+/H+ Cd2+/H+ ZnZ+/H+ Cd2+/Zn2+ Cd2+/Zn2+/H+ Cu2+/Na+ Ca2+/Na+ Mg2+/Na+(N = 0.05) Mg2+/Na+ Caz+/MgZ+ Ca2+/Me2+/Na+ Cl-IOH'
1.82 2.11 7.58 0.48
Y 1.88 2.42 1.0 1.97
3.24 1.18 4.52
1.0 1.38 1.0
4.76 10.4 2.13 1.02
2.73 1.62 1.41 1.0
7.48 6.61
1.0 1.0
1.72 4.12 1.78 1.70
1.0 1.0 1.0 1.0
0.59
5.17
%
comments
3.2 2.2 5.7 2.3 3.1 5.9 5.7 8.0 1.3 6.7 8.1 1.4 1.9 4.8 3.8 8.1 1.4 1.1 4.0 5.2 2.6 6.0 1.8 2.9 10.6 10.6 2.4
fitting (Figure 1) fitting (Figure 2) fitting fitting predictionn fitting fitting fitting prediction prediction prediction fitting fitting (Figure 8) fitting fitting predictiona fitting fitting predictionn predictionu fitting (Figure 7) fitting (Figure 4) fitting (Figure 5) fitting (Figure 5) predictionn (Figure 9) prediction5 (Figure 10) k t t i n g (Figure3)
E,
Since the value of qo was not available, we assumed qo = 2.94 mol/L. 1.0 -, 0
,'
Exp. data M
-H
, I'
0.6
Id
> ,...,'
0.4
./'
Exp. data; N = 0.05 Exp. data; N = 0.10 0 Exp. data; N = 0.20 Exp. data; N = 1 .OO
0
,..',..."" . .,"
0.2
,'
0 . 0 v . 0.0
9
0.2
'
3
0.4
.
x
I
0.6
'
1
0.8
'
I
1.0
A
./"
- HMAM '
0.0
(
0.2
'
l
0.4
'
(
0.6
.
l
0.8
'
1.0
Figure 3. Comparison between calculated results of both IMAM and HMAM and experimental values (Wheaton and Bauman, 1951) of the ionic fraction in the resin phase as a function of the ionic fraction in the solution phase for the ion exchange equilibrium Cl-IOH-.
X Figure 4. Comparison between calculated (HMAM) and experimental values (Shall'cross et al., 1988) of the ionic fraction in the resin phase as a function of the ionic fraction in the solution phase for the ion exchange equilibrium Ca2+/Na+a t various values of solution normality.
binary system Cl-/OH- (data of Wheaton and Bauman, 1951),it can be seen from Figure 3 that the agreement between model results, with values of the fitting parameters reported in Table l, and experimental data is satisfactory. This set of data allows also demonstration that for highly nonideal systems, characterized by selectivity reversal, the IMAM cannot be used. This is clearly seen in Figure 3 where the capability of the two models in reproducing the experimental data is compared. The value of the equilibrium constant of the IMAM and the average percentage error arising from the fitting procedure are K = 0.98 and E = 20.3%) respectively. We now turn to verify the HMAM capability in describing the experimental data where counterions with different valences are involved. This allows also the demonstration that the HMAM properly accounts for the influence of the solution normality on the equilibrium uptake. Since this effect is not accounted
for by the HAM, this model is not considered in the following. We analyze first the experimental data reported by Shallcross et al. (1988) for the binary systems Ca2+/Na+and Mg2+/Na+. It can be seen from Figures 4 and 5 that a satisfactory agreement between model results and experimental data is obtained, in terms of the ionic fraction of Ca2+and Mg2+in the resin phase as a function of the corresponding value in the solution phase for various levels of the solution normality. For illustrative purposes, the fitting parameters reported in Table 1are evaluated through two different nonlinear least squares fitting procedures for the two binary systems considered. In particular, for the system Ca2+/Na+all the data available at each normality level have been considered simultaneously in the fitting procedure, while for the system Mg2+/Na+only the data available a t a specific value of the solution normality, namely, N = 0.05, have been taken into account. In the latter case the data available at the remaining levels of
3922 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 1.o
1 .o
0.8
0.8
0.6
0.6
Exp. data: N = 0.01
>
>
0 Exp. data; N = 0.1
0.4
A
0.4
0
Exp. data; N = 0.5 Exp. data: N = 1 Exp. data; N = 2
0.2
A X
0.2
,,./
O . O Y ' , 0.0 0.2
.
0.01 0.0 v
.
0.4
X
I
0.6
4
'
0.8
1.0
A
Exp. data; N = 0.05 Exp. data; N = 0.1
0
Exp. data; N = 0.2
x
Exp. data; N = 1
- HMAM
0.0
0.2
0.4
0.6
1
0.2
'
I
.
0.4
I
0.6
HMAM '
I
0.8
. 1.0
'
Figure 5. Comparison between calculated (HMAM) and experimental values (Shallcross et al., 1988) of the ionic fraction in the resin phase as a function of the ionic fraction in the solution phase for the ion exchange equilibrium Mg*+/Na+a t various values of solution normality.
0
.
Exp data, N = 3 Exp. data: N = 4
0.8
X
Figure 6. Comparison between calculated (HMAM) and experimental values (Shallcross e t al., 1988)of the separation factor for the system Mg2+/Na+ as a function of the ionic fraction in the solution phase for various values of solution normality.
solution normality have then been predicted through the HMAM, using the parameter values estimated a t N = 0.05, thus obtaining an average relative error E = 3.0%. It clearly appears that the effect of the solution normality, which increases the equilibrium uptake as the solution becomes more diluted, is properly accounted for by the model. A further test of the model capability in describing the influence of solution normality is obtained by fitting simultaneously all the experimental data available at different solution normalities also for the system Mg2+/Na+(data from Shallcross et al. (1988)). The detailed comparison with the experimental data is not reported since the differences with respect to those shown in Figure 5 would not be distinguishable. This is in agreement with the insignificant variation of the average relative error which in this case becomes 2.9% while in the previous one it was 3.0%. This is confirmed by noting from Table 1 that when fitting the data simultaneously for all normality levels the model parameters remain practically unchanged. A comparison between the experimental data above and the model results is also shown in Figure 6 in terms of separation factor as a function of the ionic fraction in solution for various values of the solution normality.
X
Figure 7. Comparison between calculated (HMAM) and experimental values (Subba Rao and Davis, 1957) of the ionic fraction in the resin phase as a function of the ionic fraction in the solution phase for the ion exchange equilibrium Cu2+/Na+at various values of solution normality.
A satisfactory comparison between model results and experimental data has been also obtained when considering the system Cu2+/Na+reported by Subba Rao and David (19571,where a wider range of solution normality values has been investigated. This may be seen from Figure 7 where model results, with parameter values reported in Table 1, are compared with experimental data. In order to increase the generality of the reliability test of the HMAM, other binary systems available in the literature involving counterions of different valences are considered: Cu2+/Ag+(data from Dranoff and Lapidus (1957)1, Ca2+/H+(data from Manning and Malshmeier (198311, Mg2+/K+and Ca2+/K+(data from Khoroshko et al. (197411, Cu2+/Na+(data from Pieroni and Dranoff (196311, and Cd2+/H+and Zn2+/H+(data from Sengupta and Paul (1985)). For all these systems the HMAM is able to reproduce the experimental data with satisfactory accuracy, as seen from the values of the average relative error arising from the fitting procedure shown in Table 1. In order to investigate the flexibility of the HMAM in reproducing binary equilibrium data, we have compared its performance with that of the HAM, for a case where normality is kept constant. As a typical example, let us consider the binary system Ca2+/K+(Khoroshko et al., 19741mentioned above, where the HMAM exhibits an average percentage error equal to 1.9%(cf. Table 1). The corresponding results of the HAM, where similarly to the HMAM above, we have used p = 0.5 and two adjustable parameters = 70.5, W = 4.35) are compared in Figure 8 and lead to an average percentage error equal to 3.2%. This reduces to 2.3% whenp is also used as an adjustable parameter (p = 0.683, = 62.9, W = 5.30). The conclusion is that, in the case of constant normality, only small differences arise between the two models in reproducing equilibrium data of binary systems involving counterions with different valences. Finally, let us consider the capability of the HMAM in predicting binary as well as ternary experimental equilibrium data according to the triangle rule, Le., knowing the equilibrium behavior of only two of the three possible binary systems. For this we analyze the experimental equilibrium data of the binary system
(s
s
Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 3923 1.0
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-
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........
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.
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Figure 9. Comparison between predicted (HMAM) and experimental values (Shallcross e t al., 1988) of the ionic fraction in the resin phase as a function of the ionic fraction in the solution phase for the ion exchange equilibrium Ca2+/Mg2+.
Ca2+/Mg2+as well as of the ternary one Ca2+/Mg2+/Na+ reported by Shallcross et al. (1988). In both cases the same parameter values estimated above for the systems Ca2+/Na+and Mg2+/Na+and reported in Table 1 have been used. For the system Ca2+/Mg2+the comparison between experimental data and model predictions is shown in Figure 9. It is seen that the agreement is satisfactory (average relative error E = 10.6%). The same conclusion can be reached from the data shown in Figure 10, where the experimental data for the ternary system Ca2+/Mg2+/Na+ are compared with model predictions (average relative error E = 10.6%). It is worth noting that the agreement of these experimental data with the adsorption equilibrium model used by Shallcross et al. (1988) has been quantified in the original work by defining for each component the following objective function:
I
0.4
Y
Figure 8. Comparison between calculated and experimental values (Khoroshko et al., 1974) of the ionic fraction in the resin phase as a function of the ionic fraction in the solution phase for the ion exchange equilibrium Ca2+/II+.
0.8
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.
I
8
0.6
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.
8
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Figure 10. Comparison between predicted (HMAM)and experimental values (Shallcross et al., 1988) of the ionic fraction in the resin phase for the ion exchange equilibrium Ca2+/Mg2+/Na+a t two normality values.
method for describing the nonideal behavior of the fluid phase. These can be compared with the corresponding values obtained through the HMAM, i.e., E N a = 21.69, Eca = 0.37, and EM^ = 0.95. It is seen that the accuracy of the two models is approximately the same. However, it should be noted that, while the use of the HMAM involves only two adjustable parameters for two binary systems, i.e., a total of four parameters, the model used by Shallcross et al. (1988) requires the estimation of three parameters for each of the three binary systems which constitute the ternary one, i.e., a total of nine parameters. Even more important is that, as noted in the Introduction, the use of the latter model requires a greater experimental effort as compared to the HMAM, since it is based on the knowledge of the equilibrium behavior of three rather than only two binary systems. The difference between the two models becomes even more significant when systems involving more than three ionic species are considered. In order to further test the reliability of the proposed HMAM in predicting binary as well as ternary equilibrium data according to the triangle rule, we consider other systems reported in Table 1, namely Cu2+/H+ (data from Dranoff and Lapidus (195711, Ca2+/Na+,K+/ Na+, and Ca2+/K+(data from Manning and Malshmeier (198311, Cu2+/H+(data from Pieroni and Dranoff (196311, and Cd2+/Zn2+and Cd2+/Zn2+/H+ (data from Sengupta and Paul (1985)). For each of these systems we have used the model parameter values estimated above by fitting the relevant binary equilibrium data reported by the same authors and indicated in Table 1. In all cases the model predicted values are in good agreement with the experimental data, with average relative errors, whose values are reported in Table 1, which remain always below about 10%. These results clearly demonstrate the potential of the HMAM in predicting equilibrium ion exchange data for binary as well as multicomponent systems following the triangle rule.
Concluding Remarks
The obtained results are E N a = 22.99, Eca = 1.53, and Pitzer method and EN^ = 23.07, Eca = 1.71, and Ehlg= 7.99 by using the Debye-Huckel Ehlg= 8.09 by using the
A model for the simulation of ion exchange equilibria has been developed. Based on the assumption of ideal behavior for both the solution and the solid phase, the model accounts for the thermodynamic equilibrium of the exchange reactions occurring on two types of functional groups which are assumed to be present in the
3924 Ind. Eng. Chem. Res., Vol. 34,No. 11, 1995
resin. The model takes properly into account the dependence of the exchange process upon the solution normality as well as the selectivity changes with the resin composition. The reliability of the proposed model in describing binary uptake data as well as in predicting the behavior of multicomponent systems has been tested by comparison with a remarkably large set of experimental data available in the literature.
Acknowledgment The financial support of the Italian Consiglio Nazionale delle Ricerche, Progetto Finalizzato Chimica Fine, is gratefully acknowledged.
Nomenclature a, = activity of ith ion BJ,,= Langmuir adsorption constant of the ith component on sites of type j C,= concentration of ith ion in the solution phase, mol/L AGO = standard Gibbs free energy change, kJ/mol AGO = average value of the standard Gibbs free energy distribution, defined by eq 9, kJ/mol K = thermodynamic equilibrium constant KJ = thermodynamic equilibrium constant for functional groups of type j K],,= thermodynamic equilibrium constant between ion i and reference counterion for functional groups of type j E = average value of the equilibrium constant distribution N = total concentration or normality of the solution phase, mom N,= number of counterions in the system Nf= number of types of functional groups pJ = fraction of functional groups of type j q, = concentration of ith ion in the resin phase, moVL qJ,, = concentration of ith ion in the resin phase on functional groups of type j , m o w qo = total ion exchange capacity, moVL qoJ = ion exchange capacity of functional groups of type j , mol& R = ideal gas constant S l , k = separation factor between ions i and k, defined by eq 21 T = temperature, K X , = ionic fraction of ith ion in the solution phase, defined by eq 5 W,,k = heterogeneity parameter in eq 24 Y , = ionic fraction of ith ion in the resin phase, defined by eq 6 yj,, = ionic fraction of ith ion in the resin phase on functional groups of type j , defined by eq 12 Greek Letters a,p,v = stoichiometric coefficients or ion charge y = heterogeneity parameter defined by eq 15 6 = average percentage error uz = variance of the equilibrium constant distribution, defined by eq 10
Subscripts r = resin phase s = solution phase
Allen, R. M.; Addison, P. A.; Dechapunya, A. H. The Characterization of Binary and Ternary Ion Exchange Equilibria. Chem. Eng. J . 1989,40,151-158. Barrer, R. M.; Meier, W. M. Exchange Equilibria in a Synthetic Crystalline Exchanger. Trans. Faraday SOC.1969,55,130141. de Lucas Martinez, A.; Zarca, J.; Caiiizares, P. Ion-exchange Equilibrium of Ca2+,Mg2+,K+, Na+, and H+Ions on Amberlite IR120: Experimental Determination and Theoretical Predictions of the Ternary and Quaternary Equilibrium Data. Sep. Sci. Technol. 1992,27,823-841. de Lucas Martinez, A.; Caiiizares, P.; Zarca, J . Binary Ion Exchange Equilibrium for Ca'+, Mg++, K+, Na+ and H+ Ions on Amberlite IR120. Chem. Eng. Technol. 1993,16, 35-39. Dranoff, J. S.; Lapidus, L. Equilibrium in Ternary Ion Exchange Systems. Ind. Eng. Chem. 1957,49,1297-1302. Dye, S. R.; De Carli, J. P. 11; Carta, G. Equilibrium Sorption of Amino Acids by a Cation Exchange Resin. Ind. Eng. Chem. Res. 1990,29,849-857. Helfferich, F. Ion Exchange; McGraw-Hill: New York, 1962. Jones, L.; Carta, G. Ion Exchange of Amino Acids and Dipeptides on Cation Resins with Varying Degree of Cross-linking. 1. Equilibrium. Ind. Eng. Chem. Res. 1993,32,107-117. Khoroshko, R.; Kol'nenkov, V. P.; Soldatov, V.; Sudarikova, N.; Peryshkina, N. G. Synthetic Culture Media for the Growth of Plants on the Basis of Ion Exchange Materials. Agrokhimiya 1974,10,122. Kol'nenkov, V. P. Interpolation Method of Calculating Three-ion Exchange Equilibria from Data for Binary Exchanges. Russ. J . Phys. Chem. 1987,61, 997-1000. Manning, M. J.; Melshmeier, S. S. Binary and Ternary Ionexchange Equilibria with a Perfluorosulfonic Acid Membrane. Ind. Eng. Chem. Fundam. 1983,22,311-317. Marton, A,; Inczedy, J. Application of the Concentrated Electrolyte Solution Model in the Evaluation of Ion Exchange Equilibria. React. Polym. 1988,7, 101-109. Mehablia, M. A,; Shallcross, D. C.; Stevens, G. W. Prediction of Multicomponent Ion Exchange Equilibria. Chem. Eng. Sci. 1994,49,2277-2286. Myers, A. L.; Byington, S. Thermodynamics of Ion Exchange: Prediction of Multicomponent Equilibria from Binary data. In Ion Exchange: Science and Technology; Rodrigues, A. E., Ed.; NATO AS1 Series E, 107; Martinus Nijhoff: Dordrecht, 1986; pp 119-145. Novosad, J.;Myers, A. L. Thermodynamics of Ion Exchange as a n Adsorption Process. Can. J. Chem. Eng. 1982,60, 500-503. Pieroni, L.; Dranoff, J. S. Ion Exchange Equilibria in a Ternary System. AIChE J . 1963,9,42-45. Pitzer, K. S. Theory: Ion Interaction Approach. In Activity Coefficients i n Electrolyte Solutions; CRC Press: Boca Raton, FL, 1979; Vol. 1, pp 157-208. Saunders, M. S.;Vierow, J . B.; Carta, G. Uptake of Phenylalanine and Tyrosine by a Strong-acid Cation Exchanger. M C h E J . 1989,35,53-68. Sengupta, M.; Paul, T. B. Multicomponent Ion Exchange Equilibria. I. Zn2+-Cd2+-H+ and Cu2*-Ag2+-H+ on Amberlite IR 120. React. Polym. 1986,3,217-229. Shallcross, D. C.; Herrmann, C. C.; McCoy, B. J. An Improved Model for the Prediction of Multicomponent Ion Exchange Equilibria. Chem. Eng. Sci. 1988,43,279-288. Smith, R. P.; Woodburn, E. T. Prediction of Multicomponent Ion Exchange Equilibria for the Ternary System S042--N03--C1from Data of Binary Systems. AIChE J . 1978,24,577-587. Soldatov, V. S.; Bychkova, V. A.; Kol'nenkov, V. P.; Alefirova, T. Ya. Calculation of Ion-exchange Equilibria in System with Three Exchanging Ions of Different Valency. Russ. J . Phys. Chem. 1987,61, 966-968. Subba Rao, H. C.; David, M. M. Equilibrium in the System Cu++Na--Dowex-50. AIChE J . 1967,3, 187-190. Wheaton, R.M.; Bauman, W. C. Properties of Strongly Basic Anion Exchange Resins. Ind. Eng. Chem. 1961,43,1088-1093.
Received for review May 18, 1995 Accepted May 25, 1995@ IE9407480
Literature Cited Allen, R. M.; Addison, P. A. Ion Exchange Equilibria for Ternary Systems from Binary Exchange Data. Chem. Eng. J . 1990,44, 113-118.
Abstract published in Advance A C S Abstracts, August 1, 1995. @
