848
Ind. Eng. Chem. Res. 1988, 27, 848-851
Generalized Statistical Model for Multicomponent Adsorption Equilibria on Zeolites Renato Rota, Giuseppe Gamba, Renato Paludetto, Sergio Carrk, and Massimo Morbidelli* Dipartimento di Chimica Fisica Applicata, Politecnico di Milano, Piazza Leonard0 da Vinci 32, 20133 Milano, Italy
T h e statistical thermodynamic approach t o multicomponent adsorption equilibria on zeolites has been extended t o nonideal systems, through the correction of cross coefficients characterizing the interaction between unlike molecules. Estimation of the model parameters requires experimental binary equilibrium data. Comparisons with the classical model based on adsorbed solution theory are reported for three nonideal ternary systems. The two approaches provide comparable results in the simulation of binary and ternary adsorption equilibrium data a t constant temperature and pressure. 1. Introduction The typical structure of zeolitic materials, constituted by ordered ensembles of cages with regular size and more or less negligible interchange of molecules between cages, can be regarded as a real model of a grand canonical statistical system. In this framework, it is natural, though not necessary, to take as subsystems the individual zeolitic cages. Following this approach, several contributions appeared in the literature dealing with the derivation of single-component and multicomponent equilibrium isotherm models from the relevant grand partition function (c.f. Ruthven (1984)). The most serious limitation to the application of this procedure lies in the evaluation of the configuration integrals, which can be evaluated analytically only for very simple cases. Ruthven and Wong (1985) have proposed to take such integrals as adjustable parameters of the model, to be estimated by comparison with experimental data. This modification, on one side, makes this model more empirical in nature but, on the other side, removes the need for evaluating the actual integrals as well as for modeling the adsorbed-phase behavior, making the model more flexible in simulating experimental data. The comparison with experimental data reported by Ruthven and Wong (1985) showed that this approach provides good fitting of single-component data and good predictions of ideal (or at most moderately nonideal) binary mixtures. The aim of this work is to extend this model to highly nonideal systems and to compare its performance with that of other models previously reported in the literature.
2. Adsorption Equilibrium Model In the most direct application of the statistical model, the subsystem is identified with the zeolitic cage, which at most contains m adsorbate molecules. Thus, the relative grand partition function involves m configuration integrals, each accounting for the presence of j molecules in the cage, where j = 2, ..., m (c.f. Ruthven (1984)). When dealing with zeolites characterized by relatively large cage dimensions, such as X or Y, the maximum number of molecules per cage may be as high as m = 4-6 or more. This leads to a large number of configurational integrals, which in the approach mentioned above are taken as adjustable parameters. Since sufficient independent experimental information for the evaluation of all these parameters is usually not available, the relative multivariable optimization problem is ill-conditioned and leads to nonreliable estimations of the parameters. This problem has been discussed in detail by Ruthven and Wong (1985), who
suggested keeping the value of m small by considering as subsystem a half cage rather than an entire cage. A comparison with equilibrium data for the system ethaneethylene on zeolite 13X showed that the two assumptions provide similar results. In the sequel, we are going to take as subsystem any suitable fraction of the cage volume such that at most only two molecules are contained. This choice reduces the number of configuration integrals to be considered, which implies that only binary molecular interactions are included in the model. Accordingly, the adsorption equilibrium isotherms for one- and two-component systems reduce as follows: single component:
binary mixture:
Ni
= [ K i p , + (KiPJ2R1i + WiPJW2P2)R121/
+ KiPi + KzP2 + (K1P1)2R11/2+ (KzPJ2R22/2+ tKiPi)W2P2)Rnl ( 2 )
and similarly for N2. In eq 1 and 2, R , represents the binary interaction parameter between one molecule of type i and one of type J. In general, the interaction parameters between like molecules, such as R,,, are estimated from single-component equilibrium data (i.e., by using eq l), while the cross coefficients characterizing interactions between unlike molecules, Le., R,],are taken equal to the geometric mean of those relative to like molecules, i.e., R,, and Rll. This approximation is expected to apply in the case of ideal mixtures, where the interactions between like and unlike molecules are very similar. Indeed, using this assumption it is possible to predict multicomponent equilibria based only on single-component equilibrium data. In this case the performance of this approach is comparable to that of the ideal adsorbed solution theory model (I.A.S.T.) (Myers and Prausnitz, 1965), which is also based only on single-component experimental data. Following classical developments in equations of state for vapor-liquid equilibria, the ideal geometric averaging rule for R,, can be empirically modified as R,] = ~ ~ 1 1 ~ 1 1 -~ 6,)1 ’ 2 ~ ~ (3) where 6,, is an empirical coefficient to be estimated from binary equilibrium parameters, such that for “ideal” mixtures 6,, = 0. Thus, for computing the adsorption equilibrium for a multicomponent mixture, one first needs to estimate for
0888-S885/SS/2627-0S48$01.5~~Q 0 1988 American
Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 849 each component the parameters Ki and Rii from singlecomponent data using eq 1 and then to estimate for each pair of components the nonideal interaction parameters, 6,, using eq 2 and 3 and suitable binary equilibrium data. In the case of mixtures containing NC components, eq 2 can be readily generalized, leading to the following expression for the number of adsorbate molecules of the generic ith component:
Table I. Average Percentage Differences between Experimental Data (Costa et al., 1984) and Calculated Values modelo systemb a b C d 1-2 1-3 2-3 1-2-3
NC
KiPi(1 + CKjPjRij) j=1
Ni =
NC
NC
k=l
j=l
(4)
1 + CKkpk(1 + ‘/z ZKjPjRkj)
where the Rij parameters are computed by using eq 3. The statistical model used in this form is similar to the real adsorbed solution theory model (R.A.S.T.): they both need binary data for tuning the parameters accounting for nonideal interactions, i.e., 6ij for the first model and activity coefficienh for the second one. With respect to the version of the statistical model proposed by Ruthven and Wong (1985),the present one neglects the presence of interactions between more than two molecules in the subsystems but improves the description of cross interactions. Accounting for both would probably lead to an excessive number of adjustable parameters, thus returning to the abovementioned problems regarding their estimation. Let us briefly analyze in detail the special case where, at the temperature and pressure values of interest, the adsorbent operates under saturation conditions. This situation corresponds to most bulk adsorption separation processes based on the principle of displacement chromatography. Typical examples are aromatics on zeolites X or Y (c.f. Paludetto et al. (1987a,b)). The Henry’s constants of these components are very large, so their experimental evaluation requires extremely low bulk compositions. An alternative way is to resort to binary data, ignoring single-component behavior, and from these predict the equilibrium behavior of ternary and quaternary systems. This approach has been applied, in the context of real adsorbed solution theory, to the equilibrium of various xylene and chlorotoluene isomer mixtures (Paludetto et al., 1987a,b). The same approach can be applied to the statistical model, noticing that when all Henry’s constants are very large the unity terms in eq 4 become negligible, thus leading to NC
2aiP,C ajPjQij j-1
Ni=
NC
NC
k=l
j=l
(5)
C akpk C ajPjQkj
where ai = Ki/K1and Qij = Rij/Rl,,where 1 is any component taken as the reference condition. Recalling from eq 3 that Qij
= (QiiQjj)1’2(1 - aij)
(6)
and a1= Qll = 1, it follows that the adjustable parameters in the model are ai, Qii, and 6, = Sji with i = 2, ...,NC and j = 2, ..., NC with i # j , which are all estimated by comparison with binary equilibrium data, at constant pressure and temperature. 3. Comparison with Experimental Data The reliability of the statistical model has been tested by comparison with three sets of highly nonideal experimental data. For a comparison, the results obtained with
10.05 29.98 26.45 25.79
4.83 26.60 25.90 21.30
5.00 10.19 7.67 13.45
4.83 11.00 5.25 10.80
Models: (a) I.A.S.T., (b) ideal statistical model, (c) R.A.S.T., (d) real statistical model. bSystems: ethane (l), propane (21, ethylene (3).
Table 11. Values of the Adjustable Parameters Used in the Calculations Shown in Table I. Legend as in Table I
i
K,
1
13.779 444.820 478.660
2 3
R,,
i-j
0.2052 5.1328 X 4.5426 X
1-2 1-3 2-3
a,,
A,
-2.3060 -2.4640
-6.49 x 10-4 -3.91 X -2.73 X
o.oooo
the I.A.S.T. or R.A.S.T. model are also reported. This approach requires knowledge of single-component equilibrium behavior, which can be provided using any suitable equilibrium isotherm. In this work eq 1 has been used. Moreover, in the R.A.S.T. model, the Hildebrand model (Hildebrand et al., 1970) has been used for the activity coefficients (thus including one adjustable parameter per each pair of components). Using models with more parameters, such as the Wilson model, does not improve significantly the final result for any of the examined systems. A detailed treatment of these models has been reported by Paludetto et al. (1987b),whose basic equations have been summarized in the Appendix. The first examined system is constituted by ethaneethylene-propane on zeolite 5A at 293 K and 93 kPa, whose experimental data were reported by Costa et al. (1984). In Table I are reported the average percentage errors, in terms of amount adsorbed for each component, using four models: (a) I.A.S.T., (b) ideal statistical model (i.e., eq 3 and 4 with 6,. = 0), (c) R.A.S.T., and (d) real statistical model. All parameters of the first two models are estimated only from single-componentdata, so binary and ternary data are predicted. On the other hand, in the two nonideal models, the interaction parameters are estimated from binary data, so only ternary data are in this case predicted. The estimated values of the adjustable parameters used in the models are summarized in Table 11. Going back to the error values shown in Table I, it appears that the introduction of nonideal interactions (i.e., models c and d) improves significantly the agreement with experimental data (except for the binary system ethanepropane which is ideal). Moreover, for models c and d, it appears that the errors in the prediction of ternary data are comparable to those in the fitting of the relative binary data. It is to be stressed that the classical thermodynamic approach (models a and c) and the statistical one (models b and d) exhibit very similar performances, both in their ideal and nonideal versions. It is also worth mentioning that the performance of ideal model b should be inferior to that of the model proposed by Ruthven and Wong (1985),who accounted for interaction between more than one pair of molecules in the same zeolitic subsystem. The second and third experimental systems to be examined both refer to the case where the adsorbent operates under saturation conditions, so the statistical model is used as in eq 5, and no data relative to single-component behavior are used. The second system is constituted by p-xylene, m-xylene, and toluene on zeolite KY at 423 K
850 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 Table 111. Average Percentage Differences between Experimental Data (Paludetto et al., 1987a,b) and Calculated Values
model" svstemb
a
1-2 1-3 2-3 4-5 4-6 5-6 1-2-3 4-5-6
1.24 3.02 3.80
b 2.16 6.90 5.53
0.50 1.00 1.30 3.32 5.09
0.90 3.43 2.08 5.04 5.16
Models: (a) R.A.S.T., (b) real statistical model. *Systems: (1) p-xylene, (2) m-xylene, (3) toluene, (4) toluene, (5) o-chlorotoluene, (6) p-chlorotoluene.
Table IV. Values of the Statistical Model (Equations 5 and 6) Parameters Used in the Computations Shown in Table 111. Legend as in Table 111
i
mi
Qii
1 2 3
1.0000 0.5691 0.2955 1.0000 0.5081 0.4113
4 5 6
1.0000 0.5085 1.0017
i-j 1-2 1-3 2-3
-0.2334 -0.3695 -0.1175
1.0000 0.9630 0.5963
4-5 4-6 5-6
-0.4701 -1.0220 0.0000
6ij
Table V. Values of the R.A.S.T. Model (Equations A3, A5, and A7) Parameters (uiJ= A A T * ~ ~ / R ) Used T in the Computations Shown in Table 111. Legend as in Table I11 i-i 103~,, 103~,, -0.4435 1-2 -1.7402 -0.7436 1-3 -2.3483 -0.3024 2-3 -0.6081 -0.6269 4-5 -1.0109 -1.0926 4-6 -1.5579 0.0000 5-6 -0.6577
and 101 kPa (Paludetto et al., 1987b) and the third one by o-chlorotoluene,p-chlorotoluene, and toluene on zeolite CaX at 493 K and 101 kPa (Paludetto et al., 1987a). A comparison between the average percentage errors for the R.A.S.T. and the real statistical models (eq 5 and 6) is summarized in Table I11 for both systems. The relative parameters are summarized in Tables IV and V. Again, it appears that the correlative capabilities of the two models with respect to the binary data, as well as the predictive ones with respect to the ternary data, are fully comparable. It is worth noticing that in the formulation given by eq 5 the total loading of the adsorbent is independent of = 2. This is a consequence of ascomposition, i.e., EiNi suming all molecules of the same size, which in principle could be removed. However, in the two cases examined above, this approximation was well satisfied by the experimental data, so it was introduced also in the R.A.S.T. model in order to simplify the computations (see Paludetto et al. (198713)). Finally, the ability of the model (eq 5 and 6) in correlating highly nonideal systems is shown in Figure 1, where the calculated curve has been obtained by direct fitting of the binary equilibrium data for the system benzene-pchlorotoluene on zeolite CaX at 493 K and 101 kPa.
Conclusion The statistical thermodynamic approach to the description of multicomponent adsorption equilibria has been extended to the case of nonideal systems. Following the
VAPOR PHASE MOLE FRACTION OF BENZENE
Figure 1. Equilibrium diagram for the system benzene-p-chlorotoluene on zeolite CaX. (0) Experimental values, (-) eq 5.
approach proposed by Ruthven and Wong (1985), the configuration integrals have been taken as adjustable parameters. The number of these is maintained minimum by considering only interactions between no more than one pair of molecules, while allowance is made to the cross interaction parameters to deviate from the geometric mean rule. This requires the introduction of one empirical correction parameter, whose value can only be estimated from binary equilibrium data. The model performance has been compared with the classical thermodynamic approach for three different ternary systems, at constant temperature and pressure. It is found that the models are fully comparable both in correlating binary equilibrium data as well as in predicting ternary data. It is to be noted that the statistical model leads to a multicomponent isotherm in explicit form, while the R.A.S.T. model requires the solution of a system of nonlinear algebraic equations. This may make the first one preferable when dealing with the simulation of the dynamic behavior of multicomponent fixed-bed adsorbers.
Nomenclature A = surface area of adsorbent, cm2/g A , = Hildebrand activity coefficient parameters K = Henry's constant, molecule/cage atm m = maximum number of molecules per cage, molecule/cage N = adsorbed amount, molecule/cage NC = number of components P = pressure, atm p o L ( x , T )= adsorption equilibrium pressure for the pure component at T and P, identical with that of the multicomponent mixture, atm Q = parameter defined by eq 5 R = ideal gas constant, (atm cm3)/(molK) R,, = interaction parameter y -- vapor-phase mole fraction x = adsorbed-phase mole fraction Greek Symbols = parameter defined by eq 5
LY
y =
activity coefficient in adsorbed phase
r = adsorbed amount, mol/g r" = loading capacity of the adsorbent, mol/g
6 = empirical coefficient = spreading pressure, atm cm x * , = spreading pressure of the pure component at pressure equal to p o ,
P
Superscript O = pure component
Ind. Eng. Chem. R e s . 1988,27, 851-857
Subscripts l,i,j,k = components
Appendix The R.A.S.T. model is constituted by the following equations (Myers and Prausnitz, 1965): equality between the fugacities of each ith component in the gas and adsorbed phases Pyi = POi(a,T)yixi
(AI)
A d a = Yoi(P)d In P RT
(A21
Gibbs isotherm
where Yoi(P)indicates the single-component equilibrium isotherm stoichiometric relationship NC
cxi = 1
i=l
(A31
For given temperature, pressure, and gas-phase composition, from eq Al-A3 the spreading pressure and adsorbed-phase composition at equilibrium conditions can be calculated. To estimate the activity coefficients in eq Al, the Hildebrand model, based on regular solution theory (Hildebrand et al., 1970), can be used: 1 In Yi = i;;;CC(Aji - ( 1 / 2 ) A j k ) @ j @ k (A4) kI k
where 'Pi= ( x i / I ' o i ) / ~ ~ j / l ? o j )The . adjustable parameters are Aij = Aji and Aii = 0. In the case of a binary mixture, this model reduces to 1 In yi = -Al2 @'22 (A5)
roi
85 1
If the adsorbent operates at saturation conditions, it is possible to assume that all pure components, at the operating pressure, have reached their asymptotic region, i.e., Yoi(P)= rmi. In this case, integration of eq A2 from Poi to P leads to (Paludetto et al., 1987a,b) A(T*~- a ) / R T = Ymi In ( P / P i )
(A6)
where a*iis the spreading pressure of the pure component at pressure equal to P. Thus, if eq A6 is substituted in eq A l , the following equation is obtained (i = 1, ...,NC): x i = (yi/yi) exp[-A(Aa - Aa*J/(rmiRT)] (A7)
contaning the NC - 1 adjustable parameters AT*^ = a*i AT*^ = 0, where 1 is any reference component), which can be estimated by comparison with binary experimental data.
- a*l (so that
Literature Cited Costa, E.; Calleja, G.; Cabra, R. "Adsorption Equilibrium of Hidrocarbon Gas Mixtures on 5-A Zeolite". In Fundamental of Adsorption; Myers, A. L., Belfort, G., Eds.; Engineering Foundation: New York, 1984; p 175. Hildebrand, J. H.; Prausnitz, J. M.; Scott, R. L. Regular and Related Solution; Van Nostrand Rheinhold: New York, 1970; pp 107-109. Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 1 1 , 121. Paludetto, R.; Gamba, G.; Storti, G.; C a d , S.; Morbidelli, M. Chem Eng. Sci. 1987, 42, 2713. Paludetto, R.; Storti, G.; Gamba, G.; Carrl, S.; Morbidelli, M. Znd. Eng. Chem. Res. 1987,26, 2250. Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley: New York, 1984; Chapeters 3 and 4. Ruthven, D. M.; Wong, F. Ind. Eng. Chem. Fundam. 1985,24, 27.
Received for review February 10, 1987 Revised manuscript received October 23, 1987 Accepted November 11, 1987
Physicochemical Properties of New Solid Urea-Nitric Phosphate Fertilizers. 1. Products from Nitric Acid, Phosphate Rock, and Urea Jack M. Sullivan,* John H. Grinstead, Jr., Yong K. Kim, and Kjell R. Waerstad Division of Research, National Fertilizer Development Center, Tennessee Valley Authority, Muscle Shoals, Alabama 35660
The physicochemical properties of solid nitrogen-phosphorus (NP) fertilizer products prepared by the processing of phosphate rock with nitric acid and urea are discussed. In general, the physicochemical properties of these products are strongly dependent upon the molar ratios of HN03:Ca0 and urea:CaO employed in their production. HN03:Ca0 acidulation ratios ranging from 1.2 to 2.1 and urea:CaO ratios ranging from 1.6 to 4.0 were investigated. The product grades ranged from 21.5-13.0-0 to 29.3-9.1-0. The water-soluble P205 contents ranged from 69% to loo%, while the neutral ammonium citrate solubilities ranged from 81% to 100%. The critical relative humidities of the products ranged from 34.2% to 60.9%. All of the products were acidic, with the average pH of 1% solutions of the materials being about 2.3. The properties of some individual components and the discovery of a new fertilizer salt, Ca(H2P04)(N03).(NH2)2C0, also are described. The use of nitric acid in conjunction with ammonia for the processing of phosphate rock into nitric phosphate fertilizers is highly developed and extensively used, particularly in the European countries (Slack et al., 1967). The acidulation step employed in nitric phosphate processing schemes generally may be depicted as 20HN03 + CaI,,F2(PO4), 6H3P04 + lOCa(N0A + 2HF (1)
-
The resulting phosphoric acid-calcium nitrate-hydrofluoric acid acidulate then may be ammoniated:
-
6H3P04+ 10Ca(N03)2+ 2HF + 14NH3 6CaHP04 + 14NH4N03+ 3Ca(N03)2+ CaF2 (2) However, a major problem arises because of the presence of highly hygroscopic calcium nitrate, which renders the slurry unsuitable for granulation. Attempts to remove the
This article not subject to U.S.Copyright. Published 1988 by the American Chemical Society
