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Ind. Eng. Chem. Res. 1996, 35, 1624-1629
Efficient Implementation of Wertheim’s Theory for Multicomponent Mixtures of Polysegmented Species J. Richard Elliott, Jr.* Chemical Engineering Department, The University of Akron, Akron, Ohio 44325-3906
A simplification of Wertheim’s theory is described by which the extents of association of all species may be efficiently determined by solution of a single nonlinear equation rather than a nonlinear system of equations. This simplification should facilitate the implementation of Wertheim’s theory into chemical engineering process simulators for equations of state like the statistical associating fluid theory and the simple equation of state for associating mixtures of Elliott et al. (Ind. Eng. Chem. Res. 1990, 29, 1476). The computation speed for an associative equation of state is about 2-3 times slower than that for the Soave equation when this simplification is applied, regardless of the number of components. Introduction Wertheim’s (1984a,b; 1986a,b) first-order thermodynamic perturbation theory (TPT1) forms the basis for many of the most significant advances recently achieved in modeling the thermodynamics of hydrogen-bonding solutions. The theory describes the large nonidealities of mixing for such systems in terms of a very simple intermolecular potential function. The accuracy of TPT1 in reproducing the thermodynamic properties of fluids characterized by such potentials was demonstrated at an early stage through molecular simulations (Joslin et al., 1987; Kolafa and Nezbeda, 1987). Demonstrating the accuracy of such intermolecular potentials in representing real engineering systems awaited development of engineering equations of state (EOS’s) which implemented TPT1. This step began with the development of the statistical associating fluid theory (SAFT) (Chapman et al., 1990) and the EOS of Elliott, Suresh, and Donohue (1990) (ESD). The accuracy of the SAFT EOS has been demonstrated for systems including alcohols, organic acids, and hydrocarbons by Chapman et al. (1990) and for polymer systems by Suresh et al. (1994). The accuracy of the ESD EOS was demonstrated in an extensive evaluation by Puhala and Elliott (1993). The study of Puhala and Elliott included a comparison with the Soave (1972) EOS which demonstrated that incorporating the association term leads to a significant increase in accuracy relative to more conventional engineering equations of state. The association model also provided accuracy equivalent to the UNIFAC method at low temperatures and pressures while providing a consistent framework for extensions to supercritical regions. These developments suggest that the time has come to seriously consider some form of Wertheim’s theory for possible implementation into chemical engineering process simulators. In the present analysis we make comparisons to the Soave equation because it is representative of the class of equations of state which are applicable to a broad range of compounds over a broad range of conditions. Application of the UNIFAC method to supercritical methanol + benzene, for example, is not entirely straightforward. The purpose of the present communication is merely to analyze computational efficiency in relation to a familiar benchmark. Readers interested in evaluations of the * Fax: (216) 972-5856. email:
[email protected].
accuracy of the ESD EOS should refer to the work of Puhala and Elliott (1993). Despites its successes, a problem with Wertheim’s theory becomes apparent when considering it for the kinds of applications encountered in a process simulator. The problem is that the general formulation requires numerical solution of a nonlinear system of equations to derive the extents of bonding for all of the association sites. This nonlinear system is encountered during every iteration on density. Several iterations on density are typically required every time the EOS routine is called. The EOS lies at the heart of many of the process calculations and must be applied repeatedly to obtain the final results. It is generally estimated that EOS calculations are responsible for the majority of the computing time required in a process simulation. This means that EOS calculations must be performed efficiently and reliably. The purpose of this article is to show how the nonlinear system of equations described above can be reduced to a single nonlinear equation regardless of the number of components or the number of hydrogenbonding segments per molecule. The root of this equation will be shown to exist between the values of zero and unity, with these providing accurate initial guesses and rapid convergence. This simplification is generic in the sense that it applies to any implementation of Wertheim’s theory, including both SAFT and ESD. The only limitation is that, at this time, the molecular models satisfy the following criteria: 1. The molecule must be represented by segments with two hydrogen-bonding sites per segment. Thus alcohols, primary amines, secondary amines, diols, and some ketones and nitriles could be treated with a reasonable degree of accuracy. Treatment of ethers and acids might require supplementation of the approach described here, however. 2. The cross-association parameter must be approximated by the square-root rule. In SAFT notation, this means that ∆AiBj ) (∆AiBi∆AjBj)1/2. In ESD notation, this means that Rij ) (RiiRjj)1/2. It should be noted that this approximation removes any dependence of Rij on the mixture pair correlation function gij(σij). This represents a slight difference from the previous calculations with the SAFT EOS. 3. The extent of association is assumed to be the same for all hydrogen bonding segments on the same molecule.
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Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1625
The result of the simplification is that typical applications of Wertheim’s theory can be applied with only a modest increase in computing time relative to that of a conventional cubic EOS. The presentation below first presents a generic treatment which can be applied to either the SAFT EOS or the ESD EOS. An additional section is included to highlight further simplifications which accrue to the ESD EOS and may provide the basis for improved initial guesses in the SAFT EOS. The final section presents several concluding remarks. Simplified Solution for the Extents of Bonding in Polysegmented Mixtures We begin by writing the expression for the Helmholtz energy in a mixture resulting from hydrogen-bonding interactions.
Aassoc ) NkT
∑i xi{Mi/2 + BΓi ∑ [ln XiB - XiB/2]}
(1)
where Mi is the number of bonding sites per molecule and XiB is the mole fraction of bonding sites of type B that are not bonded. The first summation is over all types of molecules, and the second sum is over all sites on molecules of type i. This equation is equivalent to eq 21 of Chapman et al. (1990). For hydrogen-bonding molecules which satisfy the criteria listed in the Introduction, the bonding sites can be classified as proton acceptors (A) and proton donors (D). If we assume that there exist two bonding sites per segment, one acceptor and one donor, then
Aassoc
nc
) NkT
where F is the molar density and
∆AiBj ) d3ijgij(σij)segκAiBj [exp(AiBj/kT) - 1]
σij ) (σii + σjj)/2, σii is the effective diameter of the hydrogen-bonding segment, and gij is the radial distribution function of the hydrogen-bonding segment evaluated at the contact value. κAiBj and AiBj are parameters characterizing the cross association between species, k is Boltzmann’s constant, and T is the absolute temperature in degrees Kelvin. Equation 4 effectively indicates that the probability for transition from the unbonded state (Xi) to the bonded state (1 - Xi) is the sum of transition probabilities over all prospective bonding partners. In principle, eq 4 results in the nonlinear system of equations alluded to in the Introduction. The summation in eq 4 ranges over all components, yielding a system of equations with dimensionality equal to the number of components. The key simplification of this research note is to rearrange eq 4 in accordance with criterion 2 of the Introduction such that
( )
1
1
xRii 1
( ) 1
xRii
Xi
-1 )
xR11
∑j xjNdjRijXj
(3)
(4)
where the summation is over all components. This equation is equivalent to eq 22 of Chapman et al. (1990), subject to the definition that
Rij ) F∆AiBj
X1
-1 )
∑j xjNdjXjxRjj
[
xRjj xRii
for all i (7)
)]
(
-1
1 -1 Xi
(8)
Defining a quantity F and collecting a common denominator,
F≡
NC
F
where Xi ) XiAj ) XiDj is the mole fraction unbonded on the first segment of the ith species. Ndi is the “degree of polymerization” of the ith species (the number of bonding segments). xi is the mole fraction of the ith species based on the molecular weight of a monomer. The material balance relations combine with the law of mass action to become
1 - Xi ) xi
1
Xj ) 1 +
where, for example, the superscript Aj signifies the jth segment on the ith molecule. Noting the symmetry between acceptors and donors and further applying the assumption that the extent of bonding on the first bonding segment is the same as that at any other position on the same molecule gives
∑ xi{Ndi + Ndi(2 ln Xi - Xi) ) ∑ xiNdi(2 ln Xi + 1 - Xi)
(6)
( )
1
Rearranging
xi{Ndi + ∑ ln XiAj - XiAj/2 + ∑ i)1 jg1 (2)
∑j xjNdjXjxRjj
Note that the summation on the right-hand side holds for all species, so
Ndi
Aassoc ) NkT
-1 )
Xi
Ndi
ln XiDj - XiDj/2} ∑ jg1
(5)
∏ j)1
1
xRii
( )
(1 + FxRjj) )
1
Xi
∑j
-1 )
∑j
xjNdjxRjj
1 + FxRjj
xjNdjxRjj
(9)
NC
(1 + FxRkk) ∏ k*j
(10)
This results in a polynomial for F which may be solved by iteration by taking F ) 0 and F ) 1 as initial guesses in a secant iteration. The upper bound for F can be determined by considering the analytical result for a single strongly associating species, Rii f ∞ (i.e., X ) [-1 + (1 + 4R)1/2]/2R, cf. Elliott et al., 1990). Thus,
lim F ) Rf∞
[ ] x 1 1
X
R
-1 f
1
[xR - 1] f 1
xR
(11)
In the limit of R f 0, F f 0 because there is no chance of association. The polynomial form for this function was previously recognized by Suresh and Elliott (1992) for binary mixtures of associating mixtures, in which case a cubic polynomial applies. The present note simply extends that work to encompass any number of associating species and any number of bonding segments per
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1626 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996
molecule. Given a value for F, it is straightforward to solve for Xi of all ith species.
multiplying by -XiA1.
()
η ∂Xi Xi ∂η
Determination of the Contribution of Association to the Compressibility Factor
(1 +
∑j xjNdjRjj(Xj)2) )
( ) F ∂Rij
∑j xjNdjRijXj R
-Xi The next problem is to evaluate the derivatives to obtain the compressibility factor and, finally, in the following section the fugacity coefficients. assoc
Z
(
)η
)
∂Aassoc/NkT ∂η
( )( ) ( ) 2
)
∑i xiNdi X
∑i
-1
∂η η ∂Xi
i
xiNdi(2 - Xi)
( ) 1
Xi
η ∂Xi
Xi ∂η
( )
∑j
( ) η ∂Xj
Rij +
∂η
∑j
()
)
( )
xjNdj
η ∂Rij
(Xi)2 ∂η
∑j
∂η
Xj (14)
[( ) ( )] η ∂Xj
xjNdjRijXj
( ) ( ) [ (
Xj ∂η
)]
()
(Xi)2 ∂η
[( ) ( )]
∑j xjNdjRijXj
(
η ∂Xj
F ∂Rij
+
Xj ∂η
) x (
(20)
The Contribution of Association to the Fugacity Coefficients
Rij ∂F
ln(φassoc )) k
(15)
)
)
∂Aassoc/kT ∂Nk
( ) 1
(16)
( )
Xi
(17)
)
(Xi)2 ∂Nk
∑j
-1 )
Xi
N ∂Xj ∂Nk
( ) x(
) ( ) x(
Rjj -η ∂Xi w Rii (X )2 ∂η i
Rij +
∑j
)
Rjj Xjη ∂Xi (18) Rii (X )2 ∂η i
η ∂Xj ) (Xj) ∂η
Substitution reduces the number of terms involving the jth species
()
-η ∂Xi (Xi)2 ∂η
)
[x
∑j xjNdjRijXj
( ) ( )]
Rjj Xjη ∂Xi
F ∂Rij
+
Rii (X )2 ∂η i
Rij ∂F
Collecting terms identical to the left-hand side and
( )
-N ∂Xi
(Xi)2 ∂Nk
)
[( )
∑j xjNdjRijXj
N ∂Xj
(22)
Nj*k
( ) ( ) N ∂xj ∂Nk
xjNdj
δjk xk
Rij +
N ∂Rij ∂Nk
Xj
-1+
( )] N ∂Rij
Rij ∂Nk
N ∂Xj
Xj ∂Nk
(13)
+
Xj ∂Nk
[( )
∑j xjNdjRijXj
∂Nk
XjNdj
∑j )
∂Xi
-1
∑j xjNdjXjRij
( )
xjNdj
2
Differentiating implicitly gives the derivative for the ith species in terms of the jth species
-η ∂Xj ) (Xj)2 ∂η
(21)
( )( )
∑i
NiNdi
-N ∂Xi
Rjj 1 -1 Rii Xi
(
ln(φkassoc) ) Ndk[2 ln(Xk) + (1 - Xk)] +
Rearranging eq 8,
1 -1 ) Xj
∑j xjNdjRjj(Xj)2)
Rij ∂η
seg seg F ∂ ln gii (σii) ∂ ln gjj (σjj) + 1+ 2 ∂F ∂F
)
∂F
Similar simplifications apply to the derivation of the fugacity coefficients:
η ∂Rij
+
η ∂Rij F ∂Rij ) ) Rij ∂η Rij ∂F
-η ∂Xi
+ ∂F
The right-hand side of eq 20 may now be evaluated completely in terms of the Xi’s previously determined and the result substituted into eq 12 for each term. A further simplification occurs for the ESD EOS that generates an interesting insight, however, as described in the penultimate section.
Collecting terms inside the summation,
-η ∂Xi
)]
∂ ln gjjseg(σjj)
(13)
)
xjNdj
∑j xjNdjRijXj 1 + 2 (1 +
-η ∂Xi (Xi)2 ∂η
[ (
seg F ∂ ln gii (σii)
-Xi
Taking the derivatives implicitly,
w
)
x,T
∑j xjNdjXjRij
-1 )
( )
η ∂Xi
)
(12)
(19)
Rearranging and substituting from eq 15,
Xi ∂η
x,T
∂F
ij
(23)
δjk +
+ xk
( )] N ∂Rij
Rij ∂Nk
(24)
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+
Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1627
(
) x (
1 -1 ) Xj
( ) x(
-N ∂Xj ) (Xj)2 ∂Nk
()
)
Rjj 1 -1 Rii Xi
)
)
( ) x(
2
(Xi) ∂Nk
)
Rjj XjN ∂Xi (25) Rii (X )2∂N i k
[x ( ) ( )]
∑j xjNdjRijXj
approximation that is applied for gijseg(σij). The ESD EOS applies the very simple approximation -1 gseg for all i,j ij (σij) ) (1 - 1.9η)
Rjj -N∂Xi w Rii (X )2∂N i k
N ∂Xj ) (Xj) ∂Nk
-N ∂Xi
(17)
Rjj XjN ∂Xi
Rii (X ) ∂Nk i
N ∂Rij
Rij ∂Nk
(
1+F
)
∂ ln gseg 1 1.9η ij (σij) ) )1+ ∂F 1 - 1.9η 1 - 1.9η
∑j xjNdjRijXj]
)
Xi ∂η
) (1 - 1.9η)(1 +
(26)
( )
Xi ∂Nk
(1 +
∑j xjNdjRjj(Xj)2) ) -Xi
(1 - 1.9η)(1 +
[ ( )]
∑j xjNdjRijXj
N ∂Rij
δjk
+
xk
Rij ∂Nk
Zassoc ) (27)
∑i xiNdi
( )( ) 2 - Xi η ∂Xi Xi
( ) (
)
)
∑j xjNdjRijXj 2
(1 +
(
seg N ∂ ln gii
∑i xiNdi(2 - Xi)XixRii
-F
1+
∑j xjNdjRjj(Xj)
(32)
2
Splitting the summation as follows,
N ∂Xi
Xi ∂Nk
)
∂η
1 - 1.9η
seg N ∂Rij ∂ ln gjjseg N ∂ ln gii ) + Rij ∂Nk 2 ∂Nk ∂Nk
-NdkRikXkXi - Xi
∑j xjNdjRjj(Xj) )
(31)
2
Substituting into eq 12,
The derivative of Rij with respect to Nk is very similar to the derivative with respect to F,
( )
∑j xjNdjRjj(Xj) ) 2
-XiFxRii
Collecting terms identical to the left-hand side and multiplying by -XiA1
N ∂Xi
(30)
[-Xi
η ∂Xi
+
xk
2
The result is that the derivative term in eq 20 can be factored out of the summation because it no longer depends on the index of the summation. Specifically,
( )
δjk
+
(29)
)
∂ ln gjjseg +
∂Nk
∂Nk
∑j xjNdjRjj(Xj)2) (28)
As in the case of the compressibility factor, the derivative quantity is at this point completely determined by the available quantities. Substitution of eq 28 into eq 22 yields the desired fugacity coefficient. These results have been checked against the results previously reported for binary systems by Elliott et al. (1990), and they give identical results. Further Simplifications for the ESD EOS The preceding sections make it clear that numerical solution of a nonlinear system can be avoided for all the relevant terms in any implementation of Wertheim’s theory. Therefore, the computation time can be significantly reduced relative to that of a complete solution of a nonlinear system in an entirely general manner. The purpose of this section is to illustrate further simplifications which accrue when the ESD EOS is applied as the reference equation for the repulsive and disperse attractive terms.
∑i xiNdixRiiXi (2 - Xi) )
∑i xiNdixRiiXi (1 + (1 - Xi))
)
∑i xiNdixRiiXi + ∑i xiNdixRii(Xi)2
(1 - Xi)
)F+
∑i xiNdiRii(Xi)2
∑i xiNdiRii(Xi)
)F+F
xRiiXi
2
) F[1 +
∑i xiNdiRii(Xi)2] (33)
∑i xiNdixRiXi (2 - Xi) w 1+
∑j
)F
(34)
xjNdjRjj(Xj)2
gives the following remarkably simple result,
(
Zassoc ) η
)
-F2 ∂Aassoc/NkT ) ∂η 1 - 1.9η
(35)
Fugacity Coefficient Applying the relation for gijseg implied by the ESD EOS to the fugacity coefficient,
Compressibility Factor The key difference between the ESD EOS and the SAFT EOS relevant to the association term is the
Xi
(1 - Xi)
N
∂ ln gseg 1.9bkF ij ) ∂Nk 1 - 1.9η
(36)
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1628 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 Table 1. Comparison of Computation Speeds for the Soave (1972) EOS versus the ESD EOS CPU time per iterationb (ms) feed composition
typea
system ethanol+ H2O CO2 +H2O + acetonitrile heptane + H2O + glutaral toluene + H2O + MTBEc ethylbenzene + H2O + H2S ethanol + acetone + n-butanol + H2O
BP LL LL TP TP BT
0.5 0.49 0.49 0.45 0.50 0.1
0.5 0.5 0.49 0.45 0.42 0.02
T (K)
P (MPa)
520 315 333 298 368
0.01 0.02 0.10 0.08 0.02
Soave
ESD EOS
1.4 2.0 2.0 3.2 3.7 2.4
4.0 6.1 5.1 5.9 7.2 6.0
15 15
0.86
0.1
a Types of calculations are classified as follows: BP, bubble pressure; LL, liquid-liquid flash; TP, three-phase flash at the bubble pressure; BT, bubble temperature. b CPU times are reported in milliseconds based on an Intel 486DX2-66 system running Microsoft Fortran. c MTBE (methyl tertbutylether) is treated as a nonassociating component. Raymond and Elliott (1994) showed that the ESD EOS provides accuracy equivalent to the UNIFAC-LLE method with this approximation.
Substituting into eq 28,
( )
-NdkRikXkXi - Xi
N ∂Xi
(
1.9bkF
)
∑j xjNdjRijXi 1 - 1.9η
(
∑j xjNdj
-
)
Xi ∂Nk
(1 +
∑j xjNdjRjj(Xj)2)
(2 - Xi)XixRii
1+
) (1 +
∑j
(37)
∑i
( )( ) ( ( ( ( -
Nj*k
2 - Xi
1+
∑j
(2 - Xi)XixRii
-
) ) )
-Ndk(1 - Xk) (40)
2 - Xi N∂Xi Xi
∂Nk
)
Nj*k
) ) Ndk 2 ln(Xk) - F2 ln(φassoc k
1+
∑j xjNdjRjj(Xj)
)
2
+ NdkxRkkXk -F 1 - 1.9η
)
(43)
Benchmark of Computational Efficiency for an Association Model
×
∑i xiNdixRiXi(2 - Xi) 1+
1.9bkF 1 - 1.9η
These are remarkably simple final results.
1.9bkF + NdkxRkkXk ) F 1 - 1.9η
(
)
1 - Xk ) Xk
×
+ NdkRikXkXi ) XiFxRii 1 - 1.9η
1.9bkF
( )( )
xRkk
(
ln(φassoc ) ) Ndk 2 ln(Xk) + Ndk(1 - Xk) k 1.9bkF Ndk(1 - Xk) - F2 (42) 1 - 1.9η
xjNdjRjj(Xj)2
1.9bkF
∑j xjNdj
∑i
xiNdi
1
)
∂Nk
∑i xiNdi
)
1.9bkF -F2 - Ndk(1 - Xk) (41) 1 - 1.9η
2 - Xi N∂Xi Xi
-NdkxRkkXkF ) -NdkxRkkXk
xjNdjRjj(Xj)2)
When this result is substituted into eq 22, the summation on the right-hand side becomes
xiNdji
)(
)(
1.9bkF + NdkxRkkXkF (39) NdkxRkkXk ) - F2 1 - 1.9η
1.9bkF
- NdkRikXkXi -XiFxRii 1 - 1.9η
∑j xjNdjRjj(Xj)2
1.9bkF + F 1 - 1.9η
∑j xjNdjRjj(Xj)2 (38)
Finally, we can get an idea of how much penalty in computational efficiency is involved in implementing Wertheim’s theory by comparing speeds for vaporliquid equilibrium calculations of the ESD EOS with Wertheim’s theory to the simple Soave (1972) EOS. Table 1 shows that including the association term results in roughly 2-3 times slower computation speeds. This decrease in computation speed is directly related to the additional iterations on F that are required during each density iteration in the fugacity subroutine. The routines described here can be obtained at a nominal cost as described in the Chemical Engineering Progress Software Directory. Concluding Remarks Implementation of the above methodology into the SAFT EOS or the ESD EOS is straightforward and
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Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 1629
would clearly result in more efficient implementation of Wertheim’s theory relative to numerical solution of nonlinear systems of equations. For the SAFT EOS, it should be expected that slight changes in the binary interaction parameters of the disperse attraction energy would be necessary to compensate for differences between the square-root rule and what was previously applied. For applications of the ESD EOS like those of Suresh and Elliott (1992) and Puhala and Elliott (1993), the square-root rule was already applied so no reconsideration is necessary. For mixtures which do not satisfy criterion 1, Coleman et al. (1991) have performed a related analysis for binary mixtures of polysegmented species with carboxylic acids and ethers. It is possible that their methodology could be adapted to provide a general solution for carboxylic acids and ethers that is consistent with the present analysis. This extension is the subject of ongoing research. In this manner, the implementation of Wertheim’s theory would be highly efficient for an extremely broad range of mixture types. It should be noted that the inclusion of multiple bonding segments per molecule has an important implication with respect to models of water. A molecular model of water with two acceptors and two donors could be applied which satisfies the three criteria outlined above. The model would simply incorporate two “bonding segments” on the same small repulsive site. Mathematically, this would be identical to two bonding segments on any diol, and all the equations described above would be applicable. In their evaluation of various models of water, Suresh and Elliott (1992) concluded that a model of water with two acceptors and two donors would sacrifice more in computational efficiency than could be justified by the small improvements in accuracy for such a model. This conclusion will need to be reevaluated in light of the simplified computational procedure described above. Finally, the mass balance expressions and law of mass action developed in Wertheim’s theory have a direct correspondence with chemical equilibrium expressions appearing in the electrolyte and reaction chemistry literature (cf. Suresh and Elliott, 1992). To the extent that the chemical equilibria satisfy the criteria listed in the Introduction, they could be solved straightforwardly by the method presented here. This may be especially valuable as a crude model of aqueous polyelectrolytes. Nomenclature A ) Helmholtz free energy bi ) the molecular volume of the ith species F ≡ 1/xRii(1/Xi - 1), the factor describing the association of all components gij ) contact value of the radial distribution function k ) Boltzmann constant M ) total number of bonding sites in a molecule N ) number of molecules N0 ) number of molecules in system neglecting hydrogen bonds but not covalent bonds nc ) number of components Nd ) degree of polymerization, assumed equal to the number of bonding segments per molecule T ) temperature
xi ) mole fraction of the ith component XA ) fraction of unbonded proton acceptor sites XD ) fraction of unbonded proton donor sites X ) fraction of unbonded proton bonding sites, noting that the fraction of unbonded donor and acceptor sites are equal U ) internal energy Z ) compressibility factor Greeks Rij ) F∆AiBj, characteristic association parameter for the i-j interaction in the ESD EOS β ) 1/kT, a reciprocal temperature δjk ) Kronecker delta ∆AiBj ) characteristic association parameter for the i-j interaction in the SAFT EOS (eq 5) η ) dimensionless packing fraction σ ) diameter of chain repulsive site AD ) hydrogen-bonding energy κAD ) hydrogen-bonding volume parameter φassoc ) contribution to the fugacity coefficient of the kth k species resulting from association F ) molar density of hydrogen-bonding segments
Literature Cited Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990, 29, 1709. Coleman, M. M.; Graf, J. F.; Painter, P. C. Specific Interactions and the Miscibility of Polymer Blends; Technomic: Lancaster, PA, 1991. Elliott, J. R., Jr.; Suresh, S. J.; Donohue, M. D. A Simple Equation of State for Nonspherical and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 1476. Joslin, C. G.; Gray, C. G.; Chapman, W. G.; Gubbins, K. E. Theory and Simulation of Associating Liquid Mixtures. Mol. Phys. 1987, 62, 843-860. Kolafa, J.; Nezbeda, I. Monte Carlo Simulations on Primitive Models of Water and Methanol. Mol. Phys. 1987, 61, 161. Puhala, A. S.; Elliott, J. R., Jr. Correlation and Prediction of Binary Vapor-Liquid Equilibrium in Systems Containing Gases, Hydrocarbons, Alcohols, and Water. Ind. Eng. Chem. Res. 1993, 32, 3174. Soave, G. Equilibrium Constants from a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197. Suresh, S. J.; Elliott, Jr., J. R. Multiphase Equilibrium Analysis via a Generalized Equation of State for Associating Mixtures. Ind. Eng. Chem. Res. 1992, 31, 2783-2794. Raymond, M. B.; Elliott, J. R., Jr. Phase Equilibria for Prospective Oxygenated Fuels with Water. Presented at the AIChE Spring National Meeting, Atlanta, GA, March, 1994. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. I. Statistical Thermodynamics. J. Stat. Phys. 1984a, 35, 19. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. II. Thermodynamic Perturbation Theory and Integral Equations. J. Stat. Phys. 1984b, 35, 35. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. III. Multiple Attraction Sites. J. Stat. Phys. 1986a, 42, 459. Wertheim, M. S. Fluids with Highly Directional Attractive Forces. IV. Equilibrium Polymerization. J. Stat. Phys. 1986b, 42, 477.
Received for review September 11, 1995 Accepted February 20, 1996X IE950566+
X Abstract published in Advance ACS Abstracts, April 15, 1996.
