4774
Ind. Eng. Chem. Res. 1998, 37, 4774-4782
Nonideal Adsorption Equilibria Described by Pure Component Isotherms and Virial Mixture Coefficients W. Scot Appel and M. Douglas LeVan* Department of Chemical Engineering, Vanderbilt University, Nashville, Tennessee 37235
John E. Finn Regenerative Life Support Branch, NASA Ames Research Center, Moffett Field, California 94035
A novel method is used to describe adsorption equilibria of nonideal mixtures. The twodimensional virial equation of state is reorganized so that contributions of single components are separated from those of mixed components. Existing single component adsorption isotherm equations, which describe pure component data better and with fewer parameters than a truncated virial series, are substituted for the pure component terms. The remaining equation of state then contains terms corresponding to the pure component isotherms and terms with virial cross-coefficients for the mixture. The method is applied first to a mixture of Langmuirian adsorbates to demonstrate a sound thermodynamic and mathematical basis. We then consider nonideal binary and ternary mixtures of carbon dioxide, hydrogen sulfide, and propane on H-mordenite and a highly nonideal mixture of n-hexane and water on activated carbon. Introduction The design of adsorption columns for fixed-bed applications, such as air revitalization and solvent recovery, requires an accurate description of multicomponent adsorption equilibrium. Ideally, one would like to be able to predict multicomponent equilibria using only pure component data and single component isotherm models; however, describing the adsorption equilibria of highly nonideal mixtures can be very difficult even when multicomponent equilibrium data are available. Many theories have been presented for the prediction and correlation of multicomponent equilibria and reviews are available.1-3 Several methods have been used to describe the multicomponent adsorption equilibrium of nonideal mixtures. Models have been developed by extension of the ideal adsorbed solution theory (IAST)4 using both activity coefficients to develop real solution theories5-7 and heterogeneous surface models to describe the adsorbent.8,9 An alternative approach involves use of an equation of state (EOS) to describe the adsorbed phase.10-16 Two recent studies have compared the effectiveness of various models in predicting multicomponent adsorption equilibria. Hu and Do17 used four IAST-based models to analyze binary systems. They concluded that heterogeneous IAST models are more accurate than global IAST. Another study considered five equation of state models which were extended through mixing rules to describe binary systems.15 Most of the models considered were unable to describe accurately the nonideal systems analyzed. Many of the currently available models are not generally useful for describing multicomponent equilibrium data. Modifications of the IAST are able to describe specific binary mixtures on certain adsorbents * To whom correspondence should be addressed: Vanderbilt University, Box 1604, Station B, Nashville, TN 37235. Tel: (615)-322-2441. Fax: (615)-343-7951. E-mail: mdl@ vuse.vanderbilt.edu.
but cannot be generally applied to all systems. Most of the equation of state-based models either have difficulty describing adsorbate-adsorbate interactions or require a large number of parameters in order to obtain reasonable results. Contributions from adsorbate-adsorbate and adsorbate-adsorbent interactions must both be considered when developing a model of nonideal multicomponent adsorption equilibrium. Interactions of adsorbate molecules with surface sites are system specific, and a wide variety of single component equilibrium models are available to describe the observed behavior of various adsorbate-adsorbent pairs. Therefore, incorporating pure component models when describing multicomponent equilibrium will often lead to an accurate description of adsorbate-adsorbent interactions. Nonetheless, during multicomponent adsorption, competition by adsorbates for surface sites can give rise to large deviations from expected behavior because of adsorbate-adsorbate interactions. Furthermore, available areas for adsorption may be different for small and large molecules because of the pore size distribution of the adsorbent. Thus, the combined effects of all possible surface and adsorbate interactions can result in large deviations from ideal solution behavior. In this work we present an approach for describing nonideal multicomponent adsorption equilibria which is robust, thermodynamically consistent, and easy to use. We combine the convenience and effectiveness of pure component isotherm models with the two-dimensional virial equation and its ability to describe adsorbate-adsorbate interactions. Single component equation of state models are substituted for the pure component terms in the two-dimensional virial EOS. The combined EOS is then used to develop multicomponent adsorption isotherms. The method is used to describe the equilibrium data of various binary mixtures and one ternary mixture. First, we consider a theoretical system consisting of a binary mixture of adsorbates that follow the Langmuir
10.1021/ie980257u CCC: $15.00 © 1998 American Chemical Society Published on Web 11/04/1998
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4775
isotherm. The example demonstrates the rigorous thermodynamic and mathematical basis for this work. We then analyze the data of Talu18 and Talu and Zwiebel6 for three nonideal binary mixtures and a ternary mixture of carbon dioxide, hydrogen sulfide, and propane on H-mordenite. Finally, we examine the data of Rudisill et al.19 for the highly nonideal binary mixture of n-hexane and water on BPL activated carbon. Development of the Virial Mixture Coefficient Method Equation of State. One common approach to the modeling of multicomponent adsorption equilibrium data is to use a two-dimensional EOS. Using an EOS to describe the adsorbed phase has some advantages over other modeling techniques. The most important is the assurance of thermodynamic consistency between all isotherms derived from a single EOS. Furthermore, if an EOS has the correct limiting behavior in which spreading pressure approaches zero as loading or partial pressure approaches zero, then the resulting isotherms are assured to have a proper Henry’s law limit. The virial mixture coefficient (VMC) method presented here is based on the EOS approach to develop multicomponent adsorption isotherms. We begin with the two-dimensional virial EOS which can be written16
πA RT
)n+
1
1
∑∑ninjBij + 2∑i ∑j ∑k ninjnkCijk + ... A i j A
mixture. Alternatively, eq 2 can be represented in terms of these contributions as
|
1 1 πA ) n1 + B11n21 + 2C111n31 + ... + RT A A 1 1 n2 + B22n22 + 2C222n32 + ... + A A 2 3 3 B n n + C n n2 + C n n2 ... (2) A 12 1 2 A2 122 1 2 A2 122 1 2 The first, second, and third lines of eq 2 represent, respectively, contributions to the spreading pressure from pure component 1, pure component 2, and the
|
(3)
By writing eq 1 in the form of eq 3, we have illustrated that the mixture virial terms give an adjustment to the spreading pressure contributions from the pure components. The choice of the pure component equations of state is arbitrary. The technique does not require that both pure species be described by the same EOS, only that the EOS used provide acceptable descriptions of the single component equilibrium. The remaining virial coefficients for the mixture constitute a complete polynomial expansion and as such are able to describe any system provided that a sufficient number of terms are included. Additionally, it is important to recognize that if only one component is present, the VMC model reduces to the pure component EOS. Adsorption Isotherm. Description of the adsorbed phase using a two-dimensional EOS means that the fundamental property equations from classical thermodynamics may be applied to determine the relationship between spreading pressure and component fugacities or partial pressures. This approach has been discussed in detail by Van Ness21 and others,15,16 and the appropriate relation is given by
fi ) fi*
ln
(1)
where π is the spreading pressure, A is the specific surface area of the adsorbent, ni is the loading of component i, and Bij, Cijk, ... are the virial coefficients. In this form, the virial coefficients account only for adsorbate-adsorbate interactions in the adsorbed phase. Following Steele,20 all adsorbate-adsorbent interactions are described by Henry’s law coefficients. Approaches using the virial equation are available for both homogeneous and heterogeneous surfaces,16 with the latter requiring only different Henry’s law coefficients for different surface patches. This complication is not necessary here because Henry’s law behavior will be completely captured by closed-form pure component isotherms. In eq 1, interactions between like molecules are represented by virial coefficients with identical subscripts such as B11 and C333, and interactions between dissimilar molecules are given by mixture coefficients of the form B12 and C123. Typical of virialtype coefficients, the order of the subscripts is irrelevant (i.e., B21 ) B12). For our method, we separate single component terms from mixture terms and rewrite eq 1 for a binary mixture as
|
πA πA πA πA ) + + RT RT pure 1 RT pure 2 RT mixture
[(
)
∂(πA/RT) ∂ni
∫A∞
]
dA A
-1
T,A,nj
(4)
where fi is the fugacity and fi* ) ni/(KiA) is the reference state fugacity corresponding to an ideal surface gas at infinite A with Henry’s law constant Ki. Given an equation for πA/RT, eq 4 may be used to obtain the corresponding isotherms. This relation is valid for both single and multicomponent equations of state, and the terms in the resulting isotherm equations derived from virial terms contain loadings raised to powers one less than in the EOS. An alternative, yet equivalent, form of this relation may be developed by using the Gibbs fundamental property relation and is given by21,22
ln
fi ) fi*
[(
∫0π
)
∂(πA/RT) ∂ni
]
-1
T,π,nj
dπ π
(5)
with fi* ) π/(KiRT) ) ni/(KiA). For our purposes a very simple and useful result can be developed by substituting eq 3 into eq 4. This gives
ln
fi ) f i*
[(
∫A∞
)
]
∂(πA/RT|pure i) dA -1 + ∂ni A T,A ∞ ∂(πA/RT|mixture) dA (6) A ∂ni T,A,nj A
(
∫
)
Recognizing that the first integral in eq 6 is just ln(fi/fi*) for pure i and canceling the reference state gives
ln fi ) ln fpure i +
(
∫A∞
)
∂(πA/RT|mixture) ∂ni
dA (7) T,A,nj A
Equation 7 shows that the mixture isotherm for component i will be a combination of the isotherm for pure
4776 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998
i, evaluated at the ni in the mixture, and a term containing the virial mixture coefficients, which is easily evaluated. Description of Multicomponent Adsorption Equilibrium Data. The VMC method is used to describe multicomponent adsorption equilibria by providing an adjustment to existing isotherms. Thus, when using the VMC method to analyze binary data, parameters from the pure component isotherms can be retained and the virial coefficients determined by using the mixture data. Likewise, when analyzing a ternary system, parameters can be retained from both the pure component and the binary systems and only ternary data used to determine the three-component virial coefficients. In this manner, the number of parameters which need to be determined at each step is kept to a minimum. We follow this procedure below. Values for the virial coefficients are found in one of two ways. In our first example, the coefficients are obtained mathematically from a series expansion. In the second and third examples, where we analyze experimental data, the virial coefficients are determined by minimizing the objective function
≡
∑l ∑i [ln pil - ln pilexp]2
(8)
Examples Binary Langmuir Isotherm. To establish a sound basis for the VMC method, we consider a multicomponent system with an exact analytical solution, i.e., a binary mixture of Langmuirian adsorbates. We first consider the exact solution given by the binary Langmuir isotherm
ΓAkipi 1 + k1p1 + k2p2
(i ) 1, 2)
(9)
where k1 and k2 are constants of the Langmuir isotherm, p1 and p2 are component partial pressures, and the familiar monolayer capacity N [mol/kg] has been written as Γ [mol/m2] × A[m2/kg] where both Γ and A are constant. To maintain thermodynamic consistency for the binary Langmuir isotherm, Γ has the same value for both components.23 The EOS corresponding to eq 9 can be found from the Gibbs adsorption isotherm and is given by
(
)
ΓA πA ) ΓA ln RT ΓA - n1 - n2
πA RT
(
)
1 n1 + n2
∞
) ΓA
∑m
m)1
) n1 + n2 +
ΓA
m
n1n2 n22 n31 n21 + + + + 2ΓA ΓA 2ΓA 3(ΓA)2 n21n2
n1n22 n32 + + + ... (ΓA)2 (ΓA)2 3(ΓA)2
(10)
(11)
Equation 11 is in the form of a virial expansion and is exact when an infinite number of terms is included. Making use of eq 4, we can derive multicomponent isotherms from eq 11. Assuming ideal gas conditions so that fi can be replaced by pi, the isotherm of component i in the mixture is given by
( )
ln pi ) ln
n2 n21 n1 n1n2 ni + + + + + 2 ΓAki ΓA ΓA 2(ΓA) (ΓA)2 n22 2(ΓA)2
where pilexp is the measured partial pressure for species i, pil is the calculated partial pressure at the experimental loading using the isotherm obtained from the VMC method, and the outer summation is over the mixture data. As an alternative, an objective function could be written in terms of the difference between measured and predicted loading;16 however, this requires solving coupled nonlinear equations for species loading during each iteration since the isotherm is pressure explicit. This option was explored and resulted in virial mixture coefficients very similar to those obtained with eq 8.
ni )
Equation 10 can be expanded in an infinite series as
+ ... (12)
This approach can be compared to the results obtained by using the VMC method. Application of the VMC method to a mixture of single component Langmuirian adsorbates requires an EOS in the form of eq 3. We begin with the pure component Langmuir isotherm given by
n)
ΓAkp 1 + kp
(13)
The corresponding EOS found from the Gibbs isotherm is
ΓA πA ) ΓA ln RT ΓA - n
(
)
(14)
For the VMC method, the pure terms are written in the form of eq 14 and the mixture terms are from the twodimensional virial EOS. The resulting EOS is given by
(
)
(
)
ΓA ΓA πA + ΓA ln + ) ΓA ln RT ΓA - n1 ΓA - n2 2 3 3 B n n + C n2n + C n n2 + ... (15) A 12 1 2 A2 112 1 2 A2 122 1 2 Expanding the pure component terms using
ΓA ln
(
ΓA
ΓA - ni
)
∞
) ΓA
( )
1 ni
∑ m ΓA
m)1
m
(16)
gives
πA 1 1 ) n1 + B11n21 + 2C111n31 + ... + RT A A 1 1 n2 + B22n22 + 2C222n32 + ... + A A 2 3 3 B n n + C n2n + C n n2 + ... (17) A 12 1 2 A2 112 1 2 A2 122 1 2
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4777
Comparing eqs 11 and 17 indicates that the virial coefficients in eq 15 are given exactly by
B12 )
1 2Γ 1 3Γ2
C112 ) C122 )
(18)
and so on. The isotherm equations from the VMC method are found either by treating eq 15 with eq 4 or, more directly, from eqs 7 and 13. The result is
[
ln p1 ) ln
k1(ΓA - n1)
[
ln p2 ) ln
n1
n2
k2(ΓA - n2)
] ]
2 3 + B12n2 + 2C112n1n2 + A A 3 C112n22 + ... (19) 2A2 2 3 + B12n1 + C112n21 + A 2A2 3 C122n1n2 + ... (20) A2
which after expansion and using eq 18 corresponds exactly to eq 12. To summarize, this example has shown by eq 11 and eqs 17 and 18 that the VMC method gives the exact result. The advantage of the VMC method compared to the original virial equation is that the complete contribution from the pure component isotherms may be included without using an infinite number of terms. The result is improved accuracy using fewer parameters. Adsorption on H-Mordenite. Talu18 and Talu and Zwiebel6 studied the adsorption of carbon dioxide, hydrogen sulfide, and propane on H-mordenite at 30 °C and reported equilibrium data for the single components and their binary and ternary mixtures. The multicomponent equilibria were recorded at both constant gas composition and constant total pressure and have been studied by many researchers. The data have been analyzed by using spreading pressure dependent activity coefficient models,6,24 heterogeneous ideal adsorbed solution theory models,8,9 equation of state-based models,15,16 and others.17,25 The VMC method was used to analyze adsorption equilibrium data for the binary systems and the ternary system. The pure component systems have been described by Valenzuela and Myers26 using Toth isotherms. Their parameters were utilized for this work, and the pure component isotherms are shown in Figure 1. The equation for the Toth isotherm is given by
p)
[
]
b θ -1 -t
1/t
(21)
where θ ) n/ns and ns is the saturation loading. The equation of state corresponding to the Toth isotherm is given by25
πA RTns
)θ-
θ t
∞
ln(1 - θt) -
θjt+1
∑ j)1 jt(jt + 1)
(22)
Equation 22 can be used to develop an EOS in the form of eq 3.
Figure 1. Single component isotherms of CO2, H2S, and C3H8 on H-mordenite at 30 °C (data of Talu and Zwiebel6). Solid curves are Toth isotherms with parameters of Valenzuela and Myers.25
In a manner analogous to the previous example, the isotherms for the components of a binary mixture according to the VMC method may be written as
ln p1 )
ln p2 )
(
)
(
)
b1 1 2 3 ln -t + B12n2 + 2C112n1n2 + t1 A θ1 1 - 1 A 3 C122n22 + ... (23) 2A2 b2 1 2 3 ln -t + B12n1 + C112n21 + t2 A θ2 2 - 1 2A2 3 C122n1n2 + ... (24) A2
where θi ) ni/nis. The leading term on the right-hand side of eqs 23 and 24 represents ln p for the pure components as given by eq 21. Description of the binary data requires one parameter (B12) if the B terms are included, three parameters (B12, C112, and C122) when the C terms are included, and so on. In a similar manner, isotherms for the ternary mixture may be developed. The three-component system is more complex as additional interactions are possible. The partial pressure for component 1 in a ternary mixture is given by
(
)
b1 1 2 2 ln -t + B12n2 + B13n3 + t1 A A θ1 1 - 1 3 3 3 C112n1n2 + C122n22 + 2C113n1n3 + A2 2A2 A 3 3 C n2 + 2C123n2n3 + ... (25) 2 133 3 2A A
ln p1 )
The analogous equations for components 2 and 3 are similar in form and are not shown here. As mentioned earlier, values of the binary mixture coefficients are retained when correlating the ternary data, so only coefficients representing interactions among all components must be determined (C123, D1123, D1223, etc.). Descriptions of the binary mixtures using eqs 23 and 24 are shown in Figures 2-4. The H2S-C3H8 system is shown at constant composition (yH2S ) 0.0366) in Figure 2a and at constant total pressure (P ) 8.13 kPa) in Figure 2b. Solutions are shown for virial mixture coefficients including series through the C terms and
4778 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998
a
b
a
b
Figure 2. Binary adsorption of H2S-C3H8 on H-mordenite (data of Talu and Zwiebel6). Descriptions shown for virial series through C terms and through E terms. (a) yH2S ) 0.0366; (b) total pressure ) 8.13 kPa. Coefficients for the VMC isotherm through C terms are B12/A ) -0.616, C112/A2 ) 0.492, and C122/A2 ) 0.793 with 1 ) C3H8.
Figure 3. Binary adsorption of CO2-H2S on H-mordenite (data of Talu and Zwiebel6). Descriptions shown for virial series through C terms and through E terms. (a) yCO2 ) 0.7845; (b) total pressure ) 15.55 kPa. Coefficients for the VMC isotherm through C terms are B12/A ) 0.840, C112/A2 ) 0.394, and C122/A2 ) 0.103 with 1 ) CO2.
through the E terms with 3 and 10 parameters, respectively. Due to the amount of available data, an exact description could have been obtained with 16 parameters. Figure 2a shows an azeotropic crossover which is modeled well by using as few as three parameters. The CO2-H2S system is given in Figure 3 with the constant composition plane (yCO2 ) 0.7845) displayed in Figure 3a and the constant total pressure plane (P ) 15.55 kPa) shown in Figure 3b. This system is easier to describe than the previous system, and the quality of the descriptions is similar using series through either C terms or E terms. Constant composition (yCO2 ) 0.1680) and total pressure (P ) 40.9 kPa) curves for the CO2-C3H8 system are shown in Figure 4. The data are described well by using a series through either C terms or E terms. The description of the ternary data for the CO2-H2SC3H8 system is shown in Figure 5. The data are presented along three paths representing constant total pressure (P ) 13.35 kPa) and constant composition ratios. Figure 5a illustrates the CO2 path for yC3H8/yH2S ) 1.0621, Figure 5b shows the H2S path for yC3H8/yCO2 ) 0.9635, and Figure 5c gives the C3H8 path for yCO2/ yH2S ) 1.0630. While more accurate descriptions are possible, those shown utilized the binary B and C terms. The ternary data were modeled using either a single parameter (C123) or four parameters (C123, D1123, D1223, and D1233) in addition to the binary coefficients. Figure
5 shows an acceptable description of the ternary system even when only a single coefficient was used. Our description of the binary and ternary systems can be compared to those of previous methods. In doing so, it should be noted that we accepted prior pure component isotherm correlations available in the literature (see Figure 1). In general, the spreading pressure dependent activity coefficient model of Talu and Zwiebel6 used in a correlative mode describes all binary systems and the ternary system quite accurately, roughly equivalent to our highest order results. Our method uses fewer terms to describe the pure component isotherms (see Talu18) and more terms for the mixture at high order. Talu and Zwiebel6 considered some other methods for comparison and showed that they did not perform as well. Several other correlations of the binary systems are available for comparison. For the H2SC3H8 system, errors in xH2S of less than 5% are shown in Figure 2b for the VMC method. Valenzuela et al.9 and Hu and Do17 both show errors up to 20% in xH2S for the methods that they considered. For this binary pair, an accurate correlation of excess Gibbs energy, described by a two constant equation,27 has been obtained by Talu et al.,24 from which activity coefficients can be easily obtained. For the CO2-C3H8 system, Figure 4b shows maximum errors in xC3H8 of only about 2% for the VMC method, with the data described well by either series through C terms or E terms. This can be compared to
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4779
a
a
b
b
Figure 4. Binary adsorption of CO2-C3H8 on H-mordenite (data of Talu and Zwiebel6). Descriptions shown for virial series through C terms and through E terms. (a) yCO2 ) 0.1680; (b) total pressure ) 40.9 kPa. Coefficients for the VMC isotherm through C terms are B12/A ) -0.401, C112/A2 ) 0.583, and C122/A2 ) 0.531 with 1 ) C3H8.
results of Moon and Tien8 and Zhou et al.,15 which both show maximum errors of 10% in xC3H8. Coadsorption of Hexane and Water. Adsorption equilibrium data for hexane, water, and their binary mixture on BPL activated carbon at 25 °C were reported by Rudisill et al.19 Binary data were recorded for water isotherms at three hexane loadings. This system shows very nonideal behavior due to severe interactions between the components. Even at low hexane loadings, the saturation capacity of water is reduced significantly. Additionally, adsorption of water on carbon is characterized by an S-shaped isotherm (type 5). This is in sharp contrast to hexane which shows favorable (type 1) adsorption. Pure component isotherms for hexane and water are shown in Figures 6 and 7, respectively. The adsorption equilibrium data for pure hexane were described with a Toth isotherm. The isotherm is identical to eq 21, and the equation of state is given by eq 22. The pure water equilibrium data were described by Talu and Meunier22 using an isotherm they developed based on association of molecules in the adsorbed phase. The theory gives good agreement with experimental data for water on carbon, and we utilized the parameters obtained by Talu and Meunier. The association theory gives the partial pressure of water as22
pH2O )
()
Ψ KΨ exp s 1 + keΨ n
(26)
c
Figure 5. Ternary adsorption of CO2-H2S-C3H8 on H-mordenite with total pressure ) 13.35 kPa (data of Talu and Zwiebel6). Binary descriptions through C terms, and ternary descriptions shown with single C term and through D terms. (a) yC3H8/yH2S ) 1.0621; (b) yC3H8/yCO2 ) 0.9635; (c) yCO2/yH2S ) 1.0630. The VMC isotherm C term coefficient is C123/A2 ) 0.413 with 1 ) CO2 and 2 ) H2S.
where K is the Henry’s law constant, ke is an equilibrium constant for association, ns is the saturation capacity of water, and Ψ ) πA/RT is given by
-1 + (1 + 4keζ)1/2 Ψ) 2ke
(27)
with ζ ) nsn/(ns - n). Equations 22 and 27 can be combined to give the equation of state for a hexane-water mixture according
4780 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998
Figure 6. Adsorption of hexane on BPL activated carbon at 25 °C (data of Rudisill19). The solid line represents a Toth isotherm with ns ) 3.686 mol/kg, b ) 0.026, and t ) 0.237.
to the VMC method in the form
πA RT
n1
) n1 -
t
[ ( )] n1
ln 1 -
ns1
-1 + (1 + 4keζ)1/2 2ke
∞
t
-
n1 (n1/ns1)jt
∑ j)1 jt(jt + 1)
+
2 3 + B12n1n2 + C112n21n2 + A A2 3 C122n1n22 + ... (28) 2 A
where hexane is component 1, water is component 2, and ζ ) ns2n2/(ns2 - n2). Equation 28 can be used to develop isotherm equations for hexane and water in the binary mixture using eqs 3-6. Alternatively, it is much easier to apply eq 7 directly, thereby never involving spreading pressures of the pure components. By either approach, the resulting isotherms for hexane and water are, respectively,
ln p1 )
(
)
b1 1 2 3 ln -t + B12n2 + 2C112n1n2 + t1 A θ1 1 - 1 A 3 C122n22 + ... (29) 2A2
(
ln p2 ) ln
)
Ψ 2 3 KΨ + s + B12n1 + C n2 + 2 112 1 1 + keΨ A n2 2A 3 C122n1n2 + ... (30) A2
where Ψ is given by eq 27 written for component 2. We have used eqs 29 and 30 to describe the hexanewater system. In Figure 7 we show the VMC description of the hexane-water data. The binary data were correlated by using series through C or E terms which described the data well at hexane loadings of 1.032 and 1.993 mol/kg. At the lowest hexane loading of 0.497 mol/kg, the series through E terms was necessary to model the curvature of the water isotherm adequately. Discussion The VMC method can be used to describe multicomponent adsorption equilibrium data in a thermodynamically consistent manner. Beginning with a pure com-
Figure 7. Binary adsorption of water-hexane on BPL activated carbon at 25 °C (data of Rudisill et al.19). Pure water curve given by Talu and Meunier.22 Coefficients for the VMC isotherm through E terms are B12/A ) 0.651, C112/A2 ) -2.932, C122/A2 ) 0.0844, D1112/A3 ) 2.526, D1122/A3 ) 0.0664, D1222/A3 ) -0.0139, E11112/A4 ) -0.562, E11122/A4 ) -0.0290, E11222/A4 ) 0.000 958, and E12222/ A4 ) 0.000 427 with 1 ) hexane.
ponent equation of state for each species in the mixture, adsorption isotherms are easily developed by using thermodynamic relations. The virial coefficients can be determined either by using physical arguments as in the first example or by minimizing an objective function as in the second and third examples. For the latter case, the resulting parameters are mathematical adjustments and no attempt has been made to assign physical meaning to the values. The technique is robust as it can be applied to a wide variety of systems. At the theoretical limit in which interactions between all adsorbate species are included, the VMC method gives an exact analytical description for the adsorption equilibria of mixtures containing Langmuirian adsorbates. In essence, by making use of the pure component isotherms, the VMC model does not require pure component virial terms. Thus, we are able to describe multicomponent equilibrium with fewer terms than an equivalent series expansion would require. Also, it should be noted that the quality of the VMC description depends to an important extent on the accuracy of the pure component isotherms used. We have adopted previously published correlations where possible, i.e., for four of the five experimental components considered. Utilizing the Toth isotherm to model pure component equilibrium, the VMC method gave accurate descriptions of binary and ternary mixtures of carbon dioxide, hydrogen sulfide, and propane on H-mordenite. The binary systems were described with as few as nine coefficients: three for each pure isotherm and three for the virial mixture B and C terms. Additionally, it is easy to incorporate more coefficients if greater accuracy is required. The ternary system was described through C terms using three parameters for each pure Toth isotherm, three virial coefficients for each binary mixture, and one virial coefficient for the ternary mixture. This is a fairly small number of parameters considering the accuracy of the model, and it is many fewer than would be required for comparable accuracy using a virial expansion to describe the pure component isotherms. The hexane-water equilibrium data of Rudisill et al.19 are more difficult to describe due mainly to the shape of the pure water isotherm. Although Talu and Meunier22 provide a good fit, it is difficult to completely capture the curvature observed in the experimental
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4781
data. Nevertheless, the VMC method was able to provide an accurate description at all hexane loadings when the virial E terms were included. Use of the D terms (not shown) provided little improvement over the C terms, which gave a good description at the higher hexane loadings. The polynomial nature of the virial equation of state introduces both benefits and concerns for the VMC method. The virial equation represents a complete polynomial function. Thus, it can be a powerful tool to describe highly nonideal systems. However, some care must be taken when using the VMC method. As the order of the virial terms increases, the possibility arises for nonphysical oscillations to develop in the isotherm equation, especially outside the region described by the data. So, when using the VMC method or any other method for column design or modeling of adsorption dynamics, it is important to ensure the correct nature of the isotherm equation. Conclusions The multicomponent virial equation will often give an excellent description of equilibrium data provided that the single component isotherms are well described by the pure component virial isotherm. However, the single component virial isotherm is often not the best choice for describing single component adsorption equilibria. In such cases, pure component equations of state can be substituted for the pure component terms in a two-dimensional virial equation of state to develop an improved method for analyzing multicomponent adsorption equilibria. This leads to eq 7, our principal result. With spreading pressure described by a virial equation of state, we have called the approach the virial mixture coefficient (VMC) method since those coefficients are utilized to provide an adjustment to the pure component terms. Description of multicomponent adsorption equilibria using the VMC method leads to thermodynamically consistent adsorption isotherms. The main advantage of this technique over other correlation methods is the potential for accurate descriptions of highly nonideal systems using a fairly small number of parameters. Multicomponent adsorption equilibria were considered for a mixture of Langmuirian adsorbates, binary and ternary mixtures of carbon dioxide, hydrogen sulfide, and propane on H-mordenite, and a mixture of water and hexane on BPL activated carbon. Pure component isotherms from the literature were used for all experimental components except hexane, and the VMC method gave reasonable descriptions when virial mixture C terms were included and more accurate descriptions when virial E terms were used. Acknowledgment We gratefully acknowledge financial support from the National Aeronautics and Space Administration through a Joint Research Interchange and a Graduate Student Research Program fellowship for W.S.A. and from the U.S. Army Edgewood Research, Development and Engineering Center. Nomenclature A ) specific surface area of adsorbent, m2/kg B, C, ... ) virial coefficients
b ) constant in Toth isotherm equation f ) fugacity, kPa K ) Henry’s law constant, mol/(m2 kPa) k ) constant in Langmuir isotherm equation, kPa-1 ke ) equilibrium constant for association, kg/mol n ) loading, mol/kg ns ) saturation loading, mol/kg p ) pressure, kPa R ) gas constant, J/(mol K) T ) temperature, K t ) exponent in Toth isotherm equation x ) adsorbed-phase mole fraction y ) gas-phase mole fraction Greek Letters ) least squares error Γ ) constant monolayer capacity, mol/m2 π ) spreading pressure, N/m Ψ ) spreading pressure group for pure water, mol/kg θ ) fractional loading ζ ) association theory parameter, mol/kg Subscripts i,j,k ) component indices l ) data point index Superscript exp ) experimental
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Received for review April 27, 1998 Revised manuscript received September 9, 1998 Accepted September 16, 1998 IE980257U
