Znd. E n g . C h e m . Res. 1995,34, 3856-3864
3856
Size-Dependent Adsorption Models in Microporous Materials. 2. Comparison with Experimental Data Manuela Giustiniani,*ttMassimiliano Giona? and Douglas K. Ludlod Dipartimento di Ingegneria Chimica, Universita d i Roma "La Sapienza", Via Eudossiana 18, 00184 Roma, Italy, Dipartimento d i Ingegneria Chimica, Universita d i Cagliari, Piazza d'Armi, 09123 Cagliari, Italy, and Department of Chemical Engineering, University of North Dakota, Grand Forks, North Dakota 58202-7101
In this article, we complete the study of size-scaling effects and of the Keller model, presenting a detailed analysis of the application of this model to a broad range of experimental data on microporous materials (in particular, activated carbon and zeolites). Particular attention is focused on the temperature dependence of the Freundlich exponents ai, on the functional form of the adsorption energies, and on the application of Keller adsorption isotherms in a predictive way to generic multicomponent mixtures. 1. Introduction
Our previous article in this issue (Giona and Giustiniani, 1995) provides a detailed analysis of the thermodynamic and statistical mechanical derivation of Keller adsorption isotherms (AIS) (Keller, 1988, 1990). As previously discussed (Giustiniani et al., 1995; Giona and Giustiniani, 19951, Keller AIS are particularly interesting from the theoretical and practical points of view since they enable us t o separate the effects of geometric roughness, expressed by means of some intrinsic parameter characterizing the adsorbent (such as fractal dimension), from the role of energetic heterogeneity. In general, Keller AIS take the form
where ni is the concentration in the adphase of the ith species, Pi the partial pressures, f i the fugacities, and T the temperature. The other quantities appearing in eq 1 are as follows: n,(T), the monolayer capacity of the adsorbent, depending only on the temperature T @, a continuous differential function (in the monolayer case @(XI = (1 x)-l) of its argument f *
+
N
f * = Cckfk"" k=l
the quantities C k , related to the adsorption energies Ek = Ek(T,(Pi})
and the exponents G, related to the surface fractal dimension D of the adsorbent and to the molecular radius ri of the admolecules,
' Universita di Roma "La Sapienza". Fax: +39-6-44585339. E-mail:
[email protected]. Universita di Cagliari. Fax: +39-70-280774. E-mail: mafigiona2.ing.uniromal.it. 8 University of North Dakota. Fax: +1-701-777-4838. E-mail:
[email protected].
(4) where the subscript r refers t o an arbitrary component chosen as a point of reference. Throughout this article, we assume ideal gas behavior; f( = P,. The purpose of this article is to complete the analysis of size effects in microporous materials by supporting the theoretical conclusions with experimental results obtained from the analysis of a wide range of literature data on adsorption equilibria in microporous solids. An important point on which analysis of the experimental data may furnish conclusive results is the temperature behavior of the parameters n,(T), ai(??, and ~(5'7appearing in eq 1 (Giustiniani et al., 1995; Giona and Giustiniani, 1995). The temperature dependence of ai is particularly interesting with regard to the physical origin of non-Henry (Freundlich) behavior at low pressures and the possible appearance of a phase transition in the adsorbed phase. Moreover, in the practical application of Keller AIS, it is relevant to analyze the sensitivity of the model with respect to the fractal dimension D found in the expression of the a,, in order to evaluate the error range associated with the estimate of the fractal dimension directly from Keller AIS. This paper provides a detailed comparison of the results obtained through the application of Keller AIS to experimental data and addresses all the problems related t o such application. The article is organized as follows. After a brief discussion of the sources of the experimental data, we analyze the case of singlecomponent adsorption and the temperature dependence of all the relevant parameters appearing in eq 1. The functional dependence of the adsorption energies on the partial pressures is also discussed. We then perform a sensitivity analysis in order to establish the error range in the estimate of D and a by the straight application of Keller AIS to experimental data. Finally, the case of multicomponent mixtures is addressed. We present a comparison of Keller AIS with experimental data on binary mixtures and address the challenging problem of the application of Keller AIS as a tool for the prediction of equilibria in arbitrary multicomponent mixtures. 2. Sources of Experimental Data We analyze literature data of adsorption equilibria of gases and gaseous mixtures at various temperatures
0888-5885/95/2634-3856$09.00/00 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 11, 1995 3857 on microporous materials of different natures, from activated carbon, characterized by irregular pore structure, to different kinds of zeolitic molecular sieves (NaCa-mordenite, H-mordenite, zeolites 13X and 10x1, which now represent the most interesting synthetic materials in separation processes and whose structures display a fairly regular geometry. The adsorption equilibria of single-component,binary, and ternary mixtures are considered in this article. In particular, we focus on the analysis of three different literature sources: (1)The first is adsorption equilibrium data on the hydrocarbon species ethylene, ethane, and isobutane at 298.15, 323.15, and 373.15 K on a 13X molecular sieve and their binary mixtures at 298.15 K and at a constant total pressure of 137.8 kPa obtained by Hyun and Danner (1982). The experimental data were obtained by using a volumetric method. (2)The second is experimental adsorption data on the pure gases nitrogen, oxygen, carbon monoxide, and methane at temperatures of 153, 163, and 173 K obtained by Malara (1993). The adsorbent is a NaCamordenite large port (LP) supplied by PQ Corp. as crystal powder and pelletized with about 25% by weight of alumina as binder in Vs-in. pellets. The NaCamordenite LP was activated in a He stream of 5 cm3 min-l by heating it at a rate of 2 K min-l up to 723 K, and the temperature was maintained for 48 h. Highly pure gases (He, 0 2 , CO, CH4, N2) were obtained from the SI0 Co. and used without further purification. Pure gas adsorption isotherms were determined by means of the volumetric method, with a Sorptomatic 1900 volumetric apparatus supplied by Carlo Erba Instruments. The adsorbent sample was maintained at a set working temperature f0.5 K by means of a KRY-10 cryostat. A flow desorption method was used for measurements of mixed gas adsorption equilibria. The adsorption packed bed was a cylindrical copper column 4.9 (i.d.1 x 30 cm long filled with 3.9 g of 40-60-mesh fractions of NaCa-mordenite LP. The inlet and outlet pressures were measured by means of two transducers. (3) The third is gas-solid adsorption data on the single-component,binary, and ternary mixtures of H2S, COa, and C3H8 on an H-mordenite molecular sieve at 303.15 K, as reported by Talu and Zwiebel(1986). The measurements were obtained by using a volumetric method. The precision of the amount of adsorbed data was estimated t o be about f5%. An analysis of the criteria for the correct determination of adsorption equilibria by volumetric measurements and of some related problems is given by Staudt et al. (199313).
3. Single-Component Adsorption Isotherms In the practical application of the Keller model to experimental data, some assumptions must be made with regard t o the choice of @ and on the functional form of the adsorption energies Ei. In particular, (1) a Langmuir form for the function Wf*) is considered to characterize a monolayer adsorption
wf*) = l+f" 1 and (2) the effect of nonideality due to adsorbentadsorbate and adsorbate-adsorbate interactions is represented by the functional dependence of the adsorption energies on the pressures
N
N
in which a g , bg, and Eoi (i, j = 1, ..., N> depend exclusively on the temperature. This expression was proposed by Keller (1990) and is purely phenomenological. Other expressions for Ei(T,{Pi})have been proposed by Staudt et al. (1993a) based on Lennard-Jones potentials. In the single-component case, eq 6 obviously reduces to
(7) A detailed analysis of the applicability of the Keller model to adsorption equilibrium data on molecular sieves was performed by Staudt et al. (1993a), Staudt (19941, and Giustiniani et al. (1995). Figures 1 and 2 show two examples of the agreement between the Keller model and experimental adsorption data for singlecomponent mixtures. Figure 1 shows in normalized fashion (n,/n,a, vs cLPLa~) the experimental data for methane, ethane, ethylene, and propylene on activated carbon at T = 293.15 K, as obtained by Costa et al. (19811,compared with the Keller model prediction (the solid line y = x@(x)) in the case where adsorption energies are assumed to be independent of pressure. Close agreement can be observed between model and experimental data. The Keller model with adsorption energies independent of pressure is expected to give good results only in a few cases (depending on the experimental conditions, on the nature of the adsorbent, and on the temperature). The overwhelming majority of adsorbate-adsorbent systems analyzed behave in a nonideal way. It is therefore necessary to consider the dependence of the adsorption energies on pressure. Figure 2 shows the data of Malara (1993) for oxygen, nitrogen, carbon monoxide, and methane adsorbed on NaCa-mordenite zeolite at T = 173 K. The Keller model (solid lines) includes the effects of nonideality, i.e., E, = ESP,?'), expressed by means of eq 7. A quantitative measure of the deviation of Keller AI from the experimental data of Figures 1 and 2 is given in Table 1. The close agreement with experimental data shown by the Keller model in Figure 2 has been confirmed by the analysis of other experimental sources and in particular by adsorption data on 5A Linde molecular sieves (Glessner and Myers, 19691,on 5A and 1OX molecular sieves (Danner and Wenzel, 19691, on 13X molecular sieves (Hyun and Danner, 19821, on H-mordenite (Talu and Zwiebel, 19861, on activated carbon (Reich et al., 19801, and on 5A molecular sieves (Verelst and Baron, 1985). One of the reasons that makes the Keller model particularly interesting is the fact that adsorbateadsorbent systems are described by means of characteristic parameters related t o the structure of the adsorbent (the fractal dimension D, the capacity n,, which is a function of the adsorbent structure and of the temperature, and partially also ad. Table 2 lists the most relevant parameters obtained by applying the Keller model t o literature data. The values obtained for the fractal dimension D need further discussion. The values of D obtained for activated carbon agree with previous estimates obtained by means of molecular tiling methods (monolayer coverage
3868 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 I
0.4
,
I
I
-
0.2 -
0.0
0.8
0.4
1.2
1.6 C,P,U'
Figure 1. Single-component adsorption: n,(T,P)/(n,a,)vs cIPlai is shown in order to compare the Keller model (solid line), which is represented by the curve y = x@(x), with the experimental data of Costa et al. (1981) on activated carbon a t T = 293.15 K.
0.0
I
0
, 200
400
600
P
800 [torr]
Figure 2. Single-component adsorption: n(T,P)vs P. Comparison between the Keller model (solid line) and the experimental data of Malara (1993) on NaCa-mordenite a t T = 173 K. Table 1. Average Percentage Error 100/NexpC~~lni,exp ni,theorl/ni,expCalculated from the Data of the SingleComponent Adsorption Equilibrium of Figures 1 and 2" comDonent
fieure
av % error
1 1 1 1 2 2 2 2
5.71 2.05 4.32 3.68 2.87 8.75 0.74 4.86
The data of Figure 1 refer to the simplified model with E, = constant.
analysis; Avnir et al. (1984) report a fractal dimension D = 2.6-2.7 for carbonaceous adsorbents) or by means of other techniques (Jaroniec (1990) reports a fractal dimension D = 2.4-2.5 for activated carbon). For some specific zeolitic-materials, a value of the fractal dimension D was obtained close to 3 (see, e.g., in Table 2 the case of NaCa-morderiite (Malara, 1993)). This result seems to contradict the generally accepted hypothesis that zeolites are highly regular porous materials (for instance, Avnir et al. (1984)found values of D close to two for some zeolites, by examining monolayer-coverage scaling). A physical explanation of these high values can be found in a recent article by Sulikowski (19931, who shows that zeolites (and specifically synthetic faujasite) treated by alumination and dealumination exhibit a higher value of their fractal dimension (D= 2.43) than the original material (D= 2.02), because of the position of extraframework aluminum cations. It should also be borne in mind that in
gas adsorption, the characteristic size of admolecules ranges between 2 and 6 A. Therefore, any disorder present in the framework structure, and giving rise to anomalous fractal scaling, develops in this limited range of length scales. This observation holds true for every study on adsorption and fractal scaling, no matter what experimental and theoretical technique is adopted. On the other hand, another possible explanation of the high values of the fractal dimension found for some microporous materials could be related to the three-dimensional structure of microporous adsorbents. Moreover, as extensively discussed in section 6, the error range in the evaluation of the fractal dimension by means of a straight application of the Keller model can be about f0.3of the actual value of the fractal dimension. This range of uncertainty can also explain the small discrepancies between the fractal dimensions calculated for the same adsorbent a t different temperatures; cf. the data of Malara (1993) and Hyun and Danner (1982)in Table 2. The parameters n, and a,their temperature dependence, and the implications of this dependence are analyzed and discussed in section 5 . To conclude, the high values of the fractal dimensions found for some zeolitic materials on direct application of Keller AI could be attributed to three different factors: substitutional disorder (as pointed out by Sulikowski (1993)); crossover between surface adsorption scaling (D= 2) and volume filling (D= 3); and high error margins in the evaluation of D. It is impossible t o discriminate between these effects by means of thermodynamic analysis, and further experimental work is therefore necessary. 4. Dependence of Adsorption Energies on
Pressure The introduction of the explicit dependence of the adsorption energies on pressure to take into account the nonideality in the adsorbate-adsorbent interactions and lateral interactions between admolecules is one of the most interesting aspects of the Keller model. Indeed, if the Keller model is compared to the classical attempts to describe nonideality in a liquidlike adsorbed phase by means of activity coefficients (as in the case of RAST (Myers and Prausnitz, 1965; Myers, 198311, a clear physical meaning can be attributed to the Keller parameters Ei(T9). An estimate of the functional form of the adsorption energies Ei(TQ) could in principle be obtained by means of molecular simulations. On the other hand, the pressure (or coverage) dependence of the adsorption energies can also be obtained experimentally, e.g., by calorimetric measurements of the isosteric heat of adsorption. If calorimetric data are not available, it is possible to estimate the energy of adsorption starting from the temperature behavior of the adsorption isotherms by means of the relation
E
dlogP
- d(l/T) / n We consider the data of Hyun and Danner (19821, who report adsorption equilibria at three different temperatures. Equation 8 can be compared with the expression for the adsorption energies obtained by means of the Keller model by considering the phenomenological equation (7). Figure 3 shows the behavior of d(EIR)ldP obtained by means of eq 8 and by means of the Keller model. The agreement is fairly satisfactory, especially
Ind. Eng.Chem. Res., Vol. 34, No. 11, 1995 3869 Table 2. Adsorbing Capacity n,(!l'), Fractal Dimension D, Reference Exponent a,, and Absorbate Species Chosen as the Reference Component for the Adsorption Data Analyzed adsorbent material Linde molecular sieve 5A 5A molecular sieve 1OX molecular sieve 13X molecular sieve 13X molecular sieve 13X molecular sieve H-mordenite zeolite H-mordenite zeolite activated carbon activated carbon 5A molecular sieve NaCa-mordenite NaCa-mordenite NaCa -mordenite 1000 p
T,K
ref Glessner and Mvers. 1969 Danner and We6zel; 1969 Danner and Wenzel, 1969 Hyun and Danner, 1982 Hyun and Danner, 1982 Hyun and Danner, 1982 Talu and Zwiebel, 1986 Talu and Zwiebel, 1986 Costa et al., 1981 Reich et al., 1980 Verelst and Baron, 1985 Malara, 1992 Malara, 1992 Malara, 1992
,
I
I
I
308.15 144.26 144.26 198.15 323.15 373.15 303.15 383.15 393.15 301.40 115.55 173.00 163.00 153.00
I
n,, mmoVg 4.038 7.022 8.754 4.776 3.907 2.802 7.273 6.999 11.962 17.234 1.847 12.76 13.33 14.33-17.63 100 n
D
a,
ref compd
2.826 2.99 2.99 2.174 2.198 2.000 2.001 2.001 2.556 2.719 2.00 2.84 2.84-2.99 2.84-2.99
1.029 0.806 0.777 0.851 0.942 0.975 0.561 0.551 0.764 0.695 1.083 0.394 0.389 0.32
coz
I
I
I
,
20
40
60
80
0 2 0 2
COZ COZ CzH4 H2S H2S CH4 '334
Ar 0 2 0 2 0 2 I
I
20
0 -20
0
20
40
60
100
80
P
120
'
0
I
[kPa]
100 P
120
140
[kPa]
Figure 3. d(EIR)ldP vs P for isobutane at T = 373.15 K (data of Hyun and Danner (1982))derived by eq 8 (dots) and by eq 7 (solid
Figure 4. d(EIR)ldP vs P for the adsorption data of Hyun and Danner (1982) a t T = 323.15 K, derived by the functional
line).
expression eq 7 (solid line) and by integrating the Keller isotherm, eq 9 (dots).
considering that eq 8 was calculated only over the three available temperatures T = 298.15, 323.15, and 373.15 K. On the other hand, Keller isotherms can be used to obtain the functional form for the adsorption energies Ei(T,P). This approach is valid only in the case of singlecomponent adsorption and by making use of some approximations. Indeed, in the single-component case, eq 1can be put in the alternative form
(9) where we have written c(T,E), defined by eq 3, to indicate explicitly the functional dependence on the adsorption energies. If all the other parameters are known, eq 9 can be used to estimate the functional form of the adsorption energies directly from the isotherms. In order to achieve this, we estimate the parameters related to the nature of the adsorbent (D,n,(T)) and a,, Pi, appearing in the Keller model, with E = constant, directly from the experimental data (by optimization). We then integrate eq 9 to separate the interactions effects. Equation 9 is a highly nonlinear first-order differential equation. As a starting point, we chose the value of E deriving from eq 3 at P -, i.e., E = Emlb, by rewriting eq 9 as dEld(1lP). Figure 4 shows the comparison of d(E1R)ldP obtained by solving eq 9 (dots in the figure) together with the corresponding expression derived by differentiating eq 7, with the values of a and b obtained from the optimization of this expression with respect t o the experimental data (solid lines in the figure), applied to the data of Hyun and Danner
-
at T = 323.15 K. We prefer to present the results in terms of the derivative of E with respect to P since, unless we consider temperature effects, the group EIRT appearing in eq 3 is defined with an arbitrary additive constant term (since the definition of c involves the multiplicative constant P,). The close agreement between the results deriving from eq 9 and eq 7 indicates that, at least for the data of Hyun and Danner, eq 7 represents an acceptable model for the functional dependence of the energies on pressure. However, it is important to stress that the main theoretical challenge in the application of Keller isotherms to adsorption equilibria is the development of functional expressions for the adsorption energies deriving from some mechanistic model of interaction. This topic also goes beyond thermodynamic analysis. One way to approach it could be extensive molecular simulation of adsorption. We shall return t o this point in section 7. Finally, it should be noted that the same procedure for the estimation of adsorption energies based on eq 9 cannot easily be generalized to multicomponent cases because of the mathematical nature of the problem. In the multicomponent case, the equivalent of eq 9 is a system of N first-order partial differential equations that could be solved only on the assumption that we possess experimental data for ni(T,(Pi}) over the whole range of variation of the partial pressures P,. 5. Temperature Effects
In this section, we analyze the temperature dependence of the parameters ci, n,, and ai appearing in the Keller model. In the case of ci,we consider the behavior
3860 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995
I
2.5 2.6
2.8
3.0
3.4
3.2
I/T
[1031i-l]
Figure 7. Temperature dependence of the adsorbing capacity n=(!F)vs 1/T for a 13X molecular sieve, data from Hyun and Danner (1982) a t T = 298.15, 323.15, and 373.15 K. The solid line is the T-l. interpolating curve n d ?
-
I 0
C,
5.7
5.9
6.1
d
6.3
,
I
I
I
1
,
I
1
I
I
1
,
I
.
6.5
i / ~[1031i-l]
’
12.6 5.7
5.9
1
6.1
6.3
I/T
6.5 [103~-11
Figure 6. Temperature dependence of the parameters cLobtained for the experimental adsorption data of Malara (1993) a t T = 153, 163, and 173 K. The solid line is the corresponding exponential regression.
Figure 8. Temperature dependence of the adsorbing capacity n,(!F) vs 1/T for NaCa-mordenite, data from Malara (1993) a t T = 153, 163, and 173 K. The solid line is the interpolating curve nAT) T-1.
at low coverages; i.e., E, = Eo,,(T). The analysis of the temperature dependences provides a clearer understanding of the role and the physical meaning of the parameters appearing in the model. In particular, as discussed by Giona and Giustiniani (1995), the temperature behavior of the exponents a, helps us t o interpret the physical origin of the Freundlich exponents. 5.1. Dependence of Ci on T. By definition, eq 3, the quantities c, depend on the temperature (Arrhenius behavior) both explicitly and implicitly through the weak dependence of the group Elazon the temperature. As a test of consistency, Figures 5 and 6 show the dependence of c, on 1IT in a log-normal scale. Figure 5 shows this dependence for the species ethylene, ethane, and isobutane at T = 298.15,323.15, and 373.15 K (data of Hyun and Danner (1982)). The same plot with the experimental data of Malara (1993) for CH4, CO, 0 2 , and N 2 at T = 153,163, and 173 K is shown in Figure 6. In this case, the temperature range is unfortunately extremely narrow. The slope in the log-normal plot of e, vs 1IT allows us to estimate (if the group E,a, depends weakly on temperature) the adsorption energies at low pressures (i.e., at low coverage). For example, the values of E,a, calculated for the equilibrium data of Hyun and Danner = 0.43 at 298.15 (1982)are 4.0 kcaVmol for C4H10 ((Ic~H~~ K), 7.5 kcaVmol for C2H4 (ac2a = 0.71 at 298.15 K), and about 8.0 kcallmol for CzHs ( ( I c ~ H ~= 0.65 a t 298.15 K). It should again be stressed that the slope of log c1 vs 1/T, according to the Keller model, is equal to E,aJR and not t o EJR. This observation is particularly important since it represents an aspect peculiar to the
Keller model deriving from the constraints of thermodynamic consistency (see also Giona and Giustiniani (1995)). Therefore, the effective adsorption energy evaluated from the analysis of the plot ci vs 1IRT is greater than the slope of this curve by a factor equal to 11%. Failure to consider this effect may lead to erroneous underestimation of the effective adsorption energies. 5.2. Dependence of n, on T. The introduction of the parameter n,(T) (adsorption capacity) depending exclusively on the adsorbent and on the temperature is another interesting aspect of the Keller model. The parameter n,(T) does not characterize the entire adsorbenvadsorbate system but is an intrinsic property of the adsorbing material at a given temperature. This aspect can be fully appreciated when the Keller model is compared with the brute-force application of the three-parameter Langmuir-Freundlich adsorption isotherms (see, e.g., Carra and Morbidelli (1987)),which are not thermodynamically consistent and in which the adsorbing capacity is obliged to depend on the adsorbed species in order to fit the experimental data. More precisely, n,(T) represents the adsorbing capacity of the material for an adsorbate with a unitary value of a, since the adsorbing capacity for each species is given by n,(T)ai. This result is again a consequence of thermodynamic consistency (the Maxwell equations). Figures 7 and 8 show the temperature behavior of n,(T) vs T for the literature data available on different temperatures. Figure 7 refers to a 13X molecular sieve at temperatures of 298.15,323.15, and 373.15 K (Hyun and Danner, 1982) and Figure 8 to NaCa-mordenite at T = 153, 163, and 173 K (Malara, 1993). In both
Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 3861
9
0.38 ai
0.34
-
o'61 1 1 0.3 -
t
0.4
290
310
330
350
T
370 (k.1
'
0.26 150
I
155
165
160
170 T [IC]
I 175
Figure 9. Temperature dependence of the exponents calculated from the pure-component adsorption data of Hyun and Danner (1982) a t T = 298.15, 323.15, and 373.15 K on a 13X molecular sieve. The solid lines are the corresponding linear regressions.
Figure 10. Temperature dependence of the exponents a,calculated for the pure-gas adsorption data of Malara (1993) a t T = 153, 163, and 173 K on NaCa-mordenite. The solid lines are only a visual help for the reader.
cases, L(T) behaves like
(1982) and Malara (1993)confirms the complex interaction between energetic structure and geometric roughness, giving rise t o nonunitary ai, monotonically increasing with temperature and monotonically decreasing with the radius of the admolecules. An explanation of this behavior has been proposed by Giona and Giustiniani (1995)in terms of surface diffusion effects. This approach should, however, be regarded as an initial theoretical step requiring further development.
n,(T)
- T1
This result implies that the group n,(T)RT is approximately constant and that the extension area A (see eq 12 of Giustiniani and Giona (1995)) is weakly dependent on the temperature. 5.3. Dependence of ai on T. The temperature dependence of the exponents ai is a useful test to understand the physical origin of Freundlich behavior. As discussed by Keller (1990) (see also Giona and Giustiniani (1995)), the presence of these exponents is necessary to render the mathematical structure of Keller isotherms thermodynamically consistent. According to Keller's analysis, these exponents are purely geometric parameters depending solely on the radius of the molecule and on the fractal dimension characterizing the solid surface. We have already discussed the fact that the presence of nonunitary ai cannot be attributed solely to geometrical factors (e.g., roughness of surface or fractality) but is necessarily related to a heterogeneous energetic structure of the adsorbent. Previous analyses on the exponents ai (Rudzinski and Everett (1992); see also Giona and Giustiniani (1995) for further discussion) point out that they depend on the temperature in approximately linear fashion. Figure 9 refers to the experimental data of Hyun and Danner (1982)for C2H4, C2H6, and C4H10 at T = 298.15, 323.15, and 373.15 K on a 13 X molecular sieve. The functional dependence observed can be approximated by means of a linear expression, ai T. The case of the experimental data of Malara (1993)for the adsorption equilibria of 0 2 , CO, CH4, and N2 at T = 153, 163, and 173 K on NaCamordenite is reported in Figure 10. An increasing dependence of the ai on temperature is observed, but the behavior is no longer linear. These results provide further confirmation of the hypothesis that the exponents ai do not represent a purely geometric effect but are associated with a heterogeneous distribution of adsorption energies. On the other hand, for each adsorbent, the curves ai vs T of each adsorbing species are practically parallel; i.e., ail a, is practically independent of temperature. This observation indicates that the fractal scaling for aJa, is an intrinsic geometric property of the adsorbent material, a t least in the temperature range considered. To conclude, analysis of the temperature behavior of the exponents ai based on the data of Hyun and Danner
-
6. Sensitivity Analysis and Prediction of Fractal Dimension In section 3, we discussed the physical meaning of the values of the fractal dimension calculated by means of the Keller model from literature data. In particular, we found values close to 3 for some zeolitic materials and observed small differences in the fractal dimensions calculated for the same solid adsorbents at different temperatures (see Table 2). In order to understand the error range in the estimate of D and a, it is necessary to perform a kind of sensitivity analysis of the Keller model with respect to these parameters. As a sensitivity function @(D,ar), we chose
W,aJ=
1
cc
NspeclesNexplni,exp- nr,theor/
NspeciesNexp 2 = 1
~1
(10) nr,exp
where all the other parameters appearing in the Keller model are kept fxed and equal to their optimal values. In eq 10,Nspecies is the number of species considered and N e x p is the number of experimental points for each species. The experimental data analyzed in Figures 11 and 12 refer to the species CH4, CO, 0 2 , and N2 at T = 153 K, adsorbed on NaCa-mordenite (Malara, 1993); the reference component is 0 2 . Figure 11 shows the sensitivity of the error function @ with respect to the fractal dimension D , keeping a, fixed to its optimal value, a, = 0.32. As can be observed, @ exhibits a broad local minimum in the interval 2.7 ID I 3. This is indicative of great uncertainty in the estimate of the fractal dimension obtained by regressing the Keller model toward the experimental data. On the contrary, the Keller model is highly sensitive to the choice of a, and, consequently, on the choice of all the exponents a, as can be seen from Figure 12, in which D = 2.845. These results can also explain the differences between the values of the fractal dimensions calculated for the same adsorbent at different temperatures (see, in Table 2, the data of Hyun and Danner (1982) at T = 298.15,
-
3862 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995 0.12
i
HiS
Oo
0.11
0.1
0.09 0.08 I
0.07 2.0
2.4
3.2
2.8
,
nn -
I
0
4.0
3.6
D Figure 11. Error function $J(D,a,)vs D , eq 10, at a, = 0.32 for the adsorption data of Malara (1993) a t T = 153 K. The reference component is 0 2 .
0.4 1
0.3
0.5
I
0.7
0.9 0,
Figure 12. Error function $J(D,a,)vs a,, eq 10, a t D = 2.845 calculated for the adsorption data of Malara (1993) a t T = 153 K. The reference component is 0 2 .
12
16
[@a1
Figure 13. Application of the Keller model (solid lines) to binary mixture equilibria: adsorption of COz and H2S a t T = 303.15 K and P = 15.3 kPa on H-mordenite, data from Talu and Zwiebel (1986). The x axis reports the corresponding partial pressures of the two components.
D = 2.845
I
8
pco,
4J
0.0 I 0.1
4
i
0.0
I/v
d 0
1
10
I
20
I
I
PCIHS
d 40
30
[kpa]
Figure 14. Same as Figure 13 for the binary system C~HE-COZ a t T = 303.15 K and P = 40.06 kPa on H-mordenite, data from Talu and Zwiebel(1986).
323.15, and 373.15 K and the data of Malara (1993) at T = 153, 163, and 173 K). Therefore, the value of the fractal dimension obtained through direct application of the Keller model to experimental data should be regarded as purely indicative and can be susceptible of approximately 10% uncertainty.
7. Multicomponent Adsorption 7.1. Two-ComponentMixtures. In this section, we present the results obtained by making use of the Keller model t o interpret adsorption equilibrium data for binary mixtures. Molecular interactions between different species are taken into account by the functional expression eq 6 for the adsorption energies, in which four binary parameters ( a i , be, i tj ) are introduced for each system. Some of the results obtained are shown in Figures 13-15, where the experimental data for three binary mixtures, obtained by Talu and Zwiebel (1986) a t T = 303.15 K and H-mordenite and a t constant total pressure, are compared with the Keller model. As pointed out by Talu and Zwiebel (19861, these adsorption mixtures display highly nonideal behavior. Figure 13 shows the system C02-H2S adsorbed at a total pressure of about P = 15.3 Wa, Figure 14 the adsorption equilibria of the system C3Hg-CO2 at P = 40.06 kPa, and Figure 15 the adsorption data for the system CsHs-HzS at a constant total pressure of P = 8.13 Wa. In all cases, the Keller model shows fairly close agreement with the experimental data. Similar results have been obtained for other binary systems, see, e.g., Giustiniani et al. (1995), always showing close agreement between theory and experiments.
0.8
0.0 0.0
1
2.0
4.0
6.0 pC3H.
8.0
[kpa]
Figure 16. Same as Figure 13 for the binary system C ~ H E - H ~ S at T = 303.15 K and P = 8.13 kPa on H-mordenite, data from Talu and Zwiebel (1986).
7.2. Prediction of Multicomponent Mixtures. So far, the article has focused on the capacity of the Keller model to interpret single-component and multicomponent data. The other, highly significant step in engineering practice is the application of the Keller model in a predictive way in order to forecast multicomponent data from the knowledge of single-component and binary system equilibria. In the Keller formalism, the prediction of multicomponent equilibria implies the prediction of the expression for the adsorption energies E,= Ei(T,{Pi}) from single-component and binary contributions, since all the nonideality effects are expressed through the dependence of the adsorption energies on the partial pressures. Let us frame this problem in a general formal way.
Ind. Eng. Chem. Res., Vol. 34,No. 11, 1995 3863
1 I
1.6
,
I
,
I
t
%heor
0
t 0.8
I
I
4 1
0
/
1L
/ 0 0
"."
0.0 0.41/, 0.0
I
I
0
0.4
,
0.8
,
,
,:
a
4
Fs' 1 HzS
coz
C3Ha
1.2 1.6 neTp Immollg]
Figure 16. Prediction of ternary mixture equilibria by applying the Keller model: ntheor vs nexp for the system H ~ S - C O Z - C ~ H a t~ T = 303.15 K on H-mordenite zeolite, data from Talu and Zwiebel (1986). The experimental data refer to the concentration in the adsorbed phase of the three components and are obtained at different total pressures. The solid line is the curve ?Itheor = nexp.
Let us consider for the adsorption equilibria the following (quite general) expression:
where the N functional expressions Hi(x1,...J XF)are chosen a priori and the functional form of the functions f , h ( T , { p j ) ) (h = 1, ..., F) is expressed by a h e a r combination of elementary contributions q i h l (chosen a priori), each depending solely on the partial pressure Pr and on a given set of parameters a d T ) , N fih
=xqihl(pl,aihl(n)
13-15. The x axis in Figure 16 is the experimental result nexpfor all three concentrations, and the y axis is the Keller model, ntheor. The comparison between theory and experiment is fairly satisfactory, considering also that the experimental data range over a significant interval of total pressures and that the predictions make use of the simple form of eq 6 for the adsorption energies. Indeed, the results shown in Figure 16 could probably be significantly improved by assuming a more refined model for the functional form of the adsorption energies.
(i = 1,...,N h = 1,..., F)
1=1
(12)
From single-component experiments, we obtain the F x N functions ?&hi or, rather, the values of the corresponding sets of parameters a i h i ( T ) (i = 1, ..., N,h = 1, ..., F). From each binary adsorption experiment (the number of independent binary experiments is N(N 1)/2) involving the mth and the nth species, we obtain the set of 2F functions V m h n and I/Jnhm (the values for the parameter sets a m h n ( T ) and a n h m ( T ) ) . This information completely specifies the system of F x N 2F x N(N - 1)/2 = F P parameter sets a i h l defining the functional form of the adsorption energies. This means that, if the Keller model is t o be fully predictive (without any kind of adjustable parameter) for multicomponent mixtures (starting from the knowledge of single-component and binary equilibrium data), it is necessary that the adsorption energies should satisfy the composition condition eq 11and the additive conditions eq 12. For example, the functional expression eq 6 satisfies eqs 11 and 12, since F = 2 and
+
A comparison of the predictions of the Keller model in the case of ternary mixtures is shown in Figure 16 for the ternary system C&-C02-H2S on H-mordenite molecular sieve a t T = 303.15 K (data from Talu and Zwiebel (1986)). All the parameters appearing in the Keller isotherms are derived from pure gas isotherms and from the three binary systems shown in Figures
8. Concluding Remarks
In this article, we have presented a detailed analysis of the application of the Keller model to experimental data, of its capacity to interpret and predict adsorption equilibria, and of the temperature dependence of all the parameters appearing in the model, discussing what kind of physical information can be obtained from temperature behavior. The work of Keller (1990) represents an outstanding breakthrough in the theoretical analysis of adsorption equilibria by virtue of its physical simplicity and clarity. All the parameters appearing in the model are clearly related t o the physical properties of the adsorbent (D, noo,and partially ar),to the adsorbate size (r,),and to the adsorbate-adsorbent and adsorbate-adsorbate interactions (E,(T,{P,})). In our previous article in this issue (Giona and Giustiniani, 1995), we developed in detail the implications of thermodynamic consistency associated with Freundlich behavior a t low pressures, showing that this behavior is not inconsistent with thermodynamic constraints and is associated with complex phase properties (order-disorder transitions) of the admolecular system. The purpose of the theoretical work developed by Giona and Giustiniani (1995)is to demonstrate that the Keller model is completely thermodynamically consistent and well-formulated. The temperature behavior of the exponents a, shown in this article is in agreement with previous results reported by Rudzinski and Everett (1992) and reinforces the view that the exponents a, are related to complex energetic interactions in the adsorbed phase. The theoretical analysis of this topic is the new challenge in research on size-scaling effects in adsorption. Application of the Keller model over a broad spectrum of experimental conditions in microporous materials clearly shows its practical validity. This is further highlighted by the analysis developed in section 7.2 on the prediction of multicomponent mixtures. The method proposed, which is naturally related to the functional form of the Keller isotherms, represents a new and completely different approach to the thermodynamic prediction of multicomponent equilibria. This method is directly connected with the functional form of the adsorption energies, and it can therefore be expected that detailed study of the functional form of the adsorption energies will bridge the gap between the study of molecular dynamic of adsorption and macroscopic thermodynamics. Of course, refined molecular simulations are needed to complete the analysis from the microscopic point of view.
Acknowledgment We thank J. U. Keller, R. Staudt, A. Viola, and C. Malara for useful discussions.
3864 Ind. Eng. Chem. Res., Vol. 34, No. 11, 1995
Nomenclature n, = concentration in the adphase of the ith species [mol/g of adsorbent] P, = partial pressure in the gas phase of the ith species [atml fi = fugacity in the gas phase of the ith species T = temperature [Kl n, = surface adsorption capacity [mol/g of adsorbent] E, = adsorption energy of the ith species c, = Keller parameters defined in eq 3 D = fractal dimension PI, = Keller parameters defined in eq 3 R = universal gas constant a , = binary energetic parameters; see eq 6 b, = binary energetic parameters; see eq 6 Greek Letters @ = continuous function; see eq 5 a, = Keller parameters defined in eq 4
Subscripts
r = reference component
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Hyun, S. A,; Danner, R. P. Equilibrium Adsorption of Ethane, Ethylene, Isobutane, Carbon Dioxide and Their Binary Mixtures on 13X Molecular Sieves. J . Chem. Eng. Data 1982,27, 196200. Jaroniec, M. Evaluation of the fractal dimension of microporous activated carbons. Fuel 1990, 69, 1573-1574. Keller, J. U. Interrelations Between Thermodynamic Equations of State of Single- and Multi-Component Adsorbates. Ber. Bunsenges. Phys. Chem. 1988,92, 1510-1516. Keller, J . U. Equations of State of Adsorbates with Fractal Dimension. Phys. A 1990,166, 180-192. Malara, C. Multicomponent adsorption equilibria for the separation of gas mixtures by physical adsorption on microporous solids. Ph.D. Dissertation in Chemistry, University of Antwerp, Antwerp, Belgium, 1993. Myers, A. L.; Prausnitz, J. M. Thermodynamics of Mixed Gas Adsorption. M C h E J . 1965, 11, 121-127. Myers, A. L. Activity Coefficients of Mixtures Adsorbed on Heterogeneous Surfaces. M C h E J . 1983,29, 691-693. Reich, R.; Ziegler, W. T.; Rogers, K. A. Adsorption of Methane, Ethane, and Ethylene Gases and Their Binary and Ternary Mixtures and Carbon Dioxide on Activated Carbon a t 212-301 K and Pressures t o 35 Atmospheres. Ind. Eng. Chem. Process Des. Dev. 1980, 19, 336-344. Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic: New York, 1992. Staudt, R. Analytische und experimentelle Untersuchungen von Adsorptiongleichgewichten von reinen Gasen und Gasgemischen an Aktivhkohlen und Zeolithen. Ph.D. Dissertation, University of Siegen, Siegen, Germany, 1994. Staudt, R.; Saller, G.; Dreisbach, F.; Keller, J. U. Correlation of Multicomponent Adsorption Equilibria Data using a New Generalized Adsorption Isotherm (AI) with Fractal Dimension. Presented a t the AICHE 1993 Annual Meeting, St. Louis, Nov 7-12, 1993a, Paper 187b. Staudt, R.; Saller, G . ; Tomalla, M.; Keller, J. U. A Note on Gravimetric Measurements of Gas-Adsorption Equilibria. Ber. Bunsenges. Phys. Chem. 1993b, 97, 98-105. Sulikowski, B. The Fractal Dimension in Molecular Sieves: Synthetic Faujasite and Related Solids. J . Phys. Chern. 1993, 97, 1420-1425. Talu, 0.; Zwiebel, I. Multicomponent Adsorption Equilibria of Nonideal Mixtures. M C h E J . 1986, 32, 1263-1276. Verelst, H.; Baron, G. V. Adsorption of Oxygen, Nitrogen, and Argon on 5A Molecular Sieve. J . Chem. Eng. Data 1985, 30, 66-70.
Received for review December 2, 1994 Revised manuscript received J u n e 13, 1995 Accepted J u n e 27, 1995@
and Fractals in Chemical Engineering, First Italian Conference; Biardi, G., Giona, M., Giona, A. R., Eds.; World Scientific: Singapore, 1995; pp 49-65. Glessner, A. J.; Myers, A. L. The Sorption of Gas Mixtures in Molecular Sieves. Chem. Eng. Prog. Symp. Ser. 1969,65, 7379.
IE9407159
Abstract published in Advance A C S Abstracts, September 15, 1995. @
