4310
Ind. Eng. Chem. Res. 1996, 35, 4310-4318
An Equation of State for Associating Fluids Georgios M. Kontogeorgis, Epaminondas C. Voutsas, Iakovos V. Yakoumis, and Dimitrios P. Tassios* Department of Chemical EngineeringsSection II, National Technical University of Athens, Heroon Polytechniou 9, Zographos 15773, Athens, Greece
An equation of state (EoS) suitable for describing associating fluids is presented. The equation combines the simplicity of a cubic equation of state (the Soave-Redlich-Kwong), which is used for the physical part and the theoretical background of the perturbation theory employed for the chemical (or association) part. The resulting EoS (Cubic Plus Association) is not cubic with respect to volume and contains five pure compound parameters which are determined using vapor pressures and saturated liquid densities. Excellent correlations of both vapor pressures and saturated liquid volumes are obtained for primary-alcohols (from methanol up to 1-tridecanol), phenol, tert-butyl alcohol, triethylene glycol, and water. Moreover, excellent prediction of saturated liquid volumes may be obtained from parameters which have been estimated by regressing only vapor pressures. Finally, we suggest a method for reducing the number of adjustable parameters for alcohols to three while maintaining the good correlation of vapor pressures and saturated liquid volumes. We investigate the possibility of using the homomorph approach for estimating the EoS parameters and explain the problems observed. The estimated pure compound parameters have been tested in the prediction of second virial coefficients with satisfactory results. Introduction Species forming hydrogen bonds often exhibit unusual thermodynamic behavior. The strong attractive interactions between molecules result in the formation of molecular clusters. In mixtures, hydrogen-bonding interactions may occur between molecules of the same species (self-association) or between molecules of different species (solvation or cross-association). These interactions may strongly affect the thermodynamic properties of the fluids. For example, if not for its hydrogen bonds, water would be a gas rather than a liquid at room temperature and atmospheric pressure. Thus, the chemical equilibria between clusters should be taken into account in order to develop a reliable thermodynamic model. A very large number of association EoS have been proposed in the literature [for reviews, see e.g. Donohue and Economou (1995)]. Most recently developed models for associating fluids can be divided into three differentsin principlescategories, depending on the method employed for accounting for the extent of hydrogen bonding: (I) Chemical theory (Heidemann and Prausnitz, 1976; Ikonomou and Donohue, 1988; Anderko, 1989 a,b); (II) Perturbation theory (Chapman et al., 1990; Huang and Radosz, 1990); (III) Lattice/quasi-chemical theory (Panayiotou and Sanchez, 1991). Up-to-date discussions of these models are given by Economou and Tsonopoulos (1995) and Economou and Donohue (1991), and they are not further discussed here. Peschel and Wenzel (1984) showed that whatever expression is used for the physical part (a cubic or a noncubic equation of state) the resulting models yield similar results. Thus, the crucial part in describing associating compounds with an EoS is the association term of the compressibility factor. Several investigators have confirmed this conclusion. The aforementioned EoS have been applied with some success to several associating systems, but the picture S0888-5885(96)00020-6 CCC: $12.00
is far from being clear. For example, Economou and Donohue (1992) presented very good LLE predictions (without any interaction parameters) with APACT for the hydrocarbon solubility in water, but more recently Economou and Tsonopoulos (1995) showed that neither SAFT nor APACT can predict the hydrocarbon solubility using an interaction parameter obtained from correlating the water solubility. The hydrocarbon solubility is underestimated by several orders of magnitude. Furthermore, the results are extremely sensitive to the mixing rules used and very much dependent on the hydrocarbon of the water/hydrocarbon binary. Consequently, a single model is not recommended for all water/hydrocarbon binaries. These results demonstrate that the picture on the performance of the association models is not clear. Another example is provided by the works of Suresh and Beckman (1994) and Yu and Chen (1994) published in the same year in Fluid Phase Equilibria. These two independent investigations present essentially different results and conclusions on the application of the SAFT EoS to water/hydrocarbon binaries. Recently, Economou and Donohue (1991) have shown that chemical, perturbation, and lattice EoS yield essentially the same expressions for the association compressibility factor for compounds which form one or two hydrogen bonds (like acids and alcohols). Elliott et al. (1990) have also shown that the chemical and perturbation models give numerically equivalent results. However, for multisite molecules (like water or hydrogen fluoride), the chemical theory does not result in explicit expressions for the association term of the compressibility factor. Perturbation (SAFT EoS) and lattice (Panayiotou and Sanchez, 1991) theories, on the other hand, yield explicit expressions for multisite molecules and seem, thus, to be more reliable than chemical theory. An additional problem of many association models (for example APACT, SAFT) is the complexity of the EoS used for the physical part. However, on the basis of a careful consideration of existing models/theories for associating fluids, we concluded that it is theoretically sound to combine a simple © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4311
cubic equation of state like the SRK EoS (Soave, 1972) with an association term similar to that used in the SAFT EoS. The purpose of this paper is to develop a reliable equation of state suitable for associating fluids and their mixtures based on the perturbation theory. We report, here, the performance of the resulting EoS (CPA) in correlating the pure compound vapor pressures and liquid densities. This paper is organized as follows: first, the proposed EoS is presented and the number and nature of the EoS parameters are discussed. Then, a short theoretical discussion of the chemical and perturbation theories and their similarities and differences is given. Next, pure compound vapor pressure and saturated liquid volume calculations are presented for primary-alcohols (from methanol up to 1-tridecanol), phenol, tert-butyl alcohol, triethylene glycol, and water. Then, we suggest a method for reducing the number of adjustable pure compound parameters from five to three, and we discuss the possibility of using the homomorph approach in the parameter estimation and its problems. Sample predictions of second virial coefficients are also presented. A discussion on the values of the resulting EoS parameters is also provided. We end with our conclusions. The Cubic Plus Association (CPA) Equation of State The CPA EoS for pure compound is given by the equation
P)
RT
a
RT
+
V-b
F V
V(V + b)
∑ A
[ ]
1 ∂XA
1
-
XA
2
∂F
(1)
where the physical term is that of the SRK EoS and the association term is taken from SAFT (Huang and Radosz, 1990). (It is evident that, for nonassociating substances, the association term is equal to zero.) The summation is over all association sites, and the mole fraction XA of molecules not bonded at site A can be rigorously defined as A
X ) (1 + F
X ∑ B
B
AB -1
∆ )
g(d)seg ≈ g(d)hs )
2-η 2(1 - η)3
(2)
(3) (4)
In the SAFT model the reduced fluid density η is defined through the temperature dependent segment diameter d and the segment number m
η)
πNAV Fmd3 6
(5)
The temperature independent diameter σ is related to another adjustable pure compound parameter, the soft core volume, v*
v* )
πNAV 3 σ 6τ
d -3u° ) 1 - C exp σ kT
(
)
(7)
where the ratio u°/k in eq 7 is the well depth, a temperature-independent energy parameter, characteristic of nonspecific segment-segment interactions. Further details for the SAFT EoS are given by Huang and Radosz (1990) and Chapman et al. (1990). A direct combination of the SRK EoS with the association term of SAFT (eqs 3-7) would lead to a model that contains a mixture of molecular parameters (a and b) and segment parameters (u°/k, m, v*, AB, and κAB). Since, however, our target was to have a molecularbased (and not a segment-based) EoS utilizing the density dependence of the association term of SAFT, we decided to develop the CPA EoS using the following definition of the reduced density which is equivalent to eq 5 since b ) 2πNAVd3/3
η)
b 4V
(8)
Equation 8 is employed in the Carnahan-Starling and related reduced equations of state (for example, PHCT). Finally, the energy parameter of the EoS, a, is defined using a Soave-type temperature dependency
a ) a0(1 + c1 (1 - xTr))2
(9)
and the covolume parameter b is assumed to be temperature independent, in agreement with most published equations of state. The association strength, ∆AB, is defined in CPA, in analogy to SAFT, as
∆AB ) g(d)seg[exp(AB/RT) - 1]βb b 4V ≈ g(d) ) b 3 214V
(10)
2-
The key quantity in the CPA and SAFT EoS is the association strength ∆, which is approximated by the equation
∆AB ) g(d)seg[exp(AB/kT) - 1](σ3κAB)
Finally, the temperature dependence of the segment diameter is given by the equation
(6)
seg
g(d)
hs
(
)
(11)
where the differences from the corresponding equation of the SAFT EoS (eqs 3 and 4) are the following: (1) η is given by the ratio b/4V. (2) The product σ3κAB is substituted by thesequivalent in our termssproduct bβ. The explanation for using the radial distribution function of the hard sphere fluid is presented in Appendix A. Theoretical Discussion of Association Models An important assumption in all aforementioned association theories concerns the association scheme that should be employed or, in terms of perturbation theory, the number of bonding sites. This depends on the associating compound. For example, carboxylic acids are known to form (cyclic) stable dimers; thus the monomer-dimer model (chemical theory) and the onesite model (perturbation theory) should be used. Alcohols (as well as a variety of compounds, e.g., phenol, amines, pyridine bases etc.) are known, through independent spectroscopic measurements, to form higher
4312 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
oligomers besides dimers (e.g. trimers, tetramers, etc.), and the linear infinite equilibrium model (chemical theory) or the two-site model (perturbation theory) needs to be used. Water forms three-dimensional clusters, and no rigorous association solution can be provided for the chemical theory. An approximate equation has been proposed by Anderko (1991). In a more rigorous way, the perturbation theory uses either three-site or four-site models for water. A three-site model is also needed for ammonia. More advanced models are needed when branched chains, closed rings, or cyclic oligomers are formed, as for example is the case of hydrogen fluoride which forms cyclic hexamers (Lencka and Anderko, 1993; Anderko and Prausnitz, 1994; Economou and Peters, 1995). Economou and Donohue (1991, 1992) were the first to show that chemical, perturbation, and lattice EoS yield essentially the same expressions for the association compressibility factor and the mole fraction of monomers (n1/n0) on the basis of the superficial number of moles. Elliott et al. (1990) have also shown that the chemical and perturbation models give numerically equivalent results. The expressions of zassoc for APACT and SAFT/CPA in the case of the linear infinite equilibria (two-site) model of the Kempter-Mecke type are
APACT EoS zassoc )
1 - x1 + 4KRTF 1 + x1 + 4KRTF
(12)
SAFT/CPA EoS zassoc )
(
)(
)
1 - x1 + 4∆F ∆ + F∆′ ∆ 1 + x1 + 4∆F
(13)
where ∆′ is the derivative of ∆ with respect to density. In the chemical equations of state, the association is expressed through the equilibrium constant K, which is independent of density
ln K )
-∆Hassoc ∆Sassoc + RT R
(14)
where ∆Hassoc and ∆Sassoc are, respectively, the enthalpy and entropy of hydrogen-bonding formation. In the perturbation EoS (SAFT, CPA) the association strength is expressed through the ∆-function
∆)
(
)
2-η [exp(AB/RT) - 1]VAB 2(1 - η)3
(15)
where, as explained above, the only difference between CPA and SAFT lies in the definition of the reduced density η and the association volume. Unlike K, ∆ is a function of density. However, the density dependence of the association compressibility factor is much more affected by the external density dependence of the mole fraction of nonbonded molecules (first bracketed term of eq 13) than the density dependence of ∆. Thus, if the density-dependent part of ∆ is ignored, eqs 14 and 15 are identical if we set
KRT ) ∆
(
w exp
assoc
-∆H RT
)
assoc
+
∆S R
It is obvious from eq 17 that we can identify direct relations between the energy of association with the enthalpy of hydrogen bonding as well as of the association volume with the entropy of hydrogen bonding. Parameter Estimation The CPA EoS, as described above, has five pure compound parameters, three from the physical part (a0, c1, b) and two association-based parameters, AB and β. The covolume parameter b appears in both the physical and the association parts of the CPA EoS. These five parameters have been estimated using vapor pressures and saturated liquid volumes following four different approaches: (1) Global estimation; (2) Three-parameter estimation; (3) Through the homomorph approach; (4) Optimization based solely on vapor pressures. The first two approaches are presented next; the other two are given in Appendices B and C. As with other complex noncubic EoS, like the GCFlory (Berg, 1993) and the Group-Contribution lattice fluid (High and Danner, 1990), the CPA has a “nearcubic” behavior. This means that, although it is noncubic with respect to volume, three real positive roots are typically obtained (or one for temperatures above the critical) by solving the volume equation: the smallest is the liquid and the highest is the vapor root. The following criteria have been chosen for establishing the suitability of the model (and for proceeding further to mixture calculations): (1) Performance of the EoS in the correlation of both vapor pressure and saturated liquid density; (2) Prediction of saturated liquid volume using parameters obtained solely from vapor pressure; (3) Prediction of second virial coefficients; (4) Trends of the parameters within the homologous series; (5) Magnitude of the parameters and especially of the association parameters, which as mentioned above are related to measurable quantities (enthalpy and entropy of hydrogen bonding); (6) The possibility of using only three adjustable parameters. Although final choice of the suitability of parameters would be via the mixture calculations, the above six factors are adequate to establish the suitability of the model in the estimation of pure compound parameters. This is necessary in order not to “incorporate” in the mixture calculations “errors” stemming from (possibly) accurate but physically not meaningful pure compound parameters (accurate in this context means good representation of both vapor pressure and liquid density). All experimental data for vapor pressures, liquid densities, and second virial coefficients have been taken from the DIPPR data compilation (Daubert and Danner, 1989). 1. Global Estimation. Using this approach all five pure compound parameters are left adjustable. The optimization is based on both vapor pressures and saturated liquid densities. The objective function which has been used is the following
(16)
(
) exp
AB
)
AB
V -1 (17) RT RT
F)
(
∑
) (
Psexptl - Pscalcd
2
+
Psexptl
∑
)
Vlexptl - Vlcalcd Vlexptl
2
(18)
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4313 Table 1. Correlation of Vapor Pressure and Saturated Liquid Volume with CPA EoSa AB
b
a0
c1
∆P (%)
∆V (%)
0.03218 0.03300 0.03214
4.4130 4.7052 4.2930
0.6518 0.9037 0.7974
Methanol 218.31 0.02320 183.36 0.04490 206.65 0.03371
0.332 0.846 0.496
0.023 0.076 0.354
0.0474 0.0479 0.0471
6.729 7.311 6.620
0.950 0.920 0.899
Ethanol 213.92 0.0180 207.64 0.0160 222.08 0.0160
0.17 0.47 0.30
0.71 0.80 0.74
0.0639 0.0644 0.0645
11.214 11.528 11.638
0.878 1.040 1.019
1-Propanol 225.10 0.00700 194.88 0.01230 197.21 0.01120
0.39 0.37 0.36
0.44 0.46 0.45
0.0808 0.0808 0.0798
16.637 16.369 14.885
0.912 0.997 0.979
1-Butanol 219.11 0.0046 204.08 0.0069 219.25 0.0082
0.31 0.41 0.93
0.50 0.49 0.49
0.0976 0.0975 0.0976
22.868 22.985 23.110
1-Pentanol 1.030 189.05 0.0045 1.026 190.19 0.0041 1.028 190.68 0.0038
0.32 0.22 0.18
0.54 0.56 0.53
0.1135 0.1137 0.1136
27.117 27.230 27.221
0.988 0.925 0.913
1-Hexanol 209.72 0.0042 223.29 0.0033 225.80 0.0032
0.18 0.26 0.30
0.58 0.53 0.53
0.1488 0.1490 0.1491
41.645 41.765 41.522
0.974 0.995 1.037
1-Octanol 262.98 0.0005 265.25 0.0004 251.49 0.0005
0.33 0.26 0.65
0.68 0.58 0.56
0.1643 0.1641 0.1642
47.102 47.225 47.085
1.041 1.040 1.023
1-Nonanol 258.51 0.0006 262.82 0.0005 260.59 0.0006
0.52 0.45 0.54
0.38 0.60 0.58
0.1843 0.1843 0.1838
53.645 54.247 54.119
1.159 1.143 1.223
1-Decanol 243.43 0.0007 262.18 0.0004 275.92 0.0001
0.50 0.17 0.23
0.69 0.61 0.56
0.2013 0.2014 0.2010
62.173 62.188 62.230
1-Undecanol 1.038 283.37 0.0003 1.045 286.49 0.0002 1.039 293.41 0.0002
0.72 0.64 0.58
1.02 0.94 1.01
0.2219 0.2215 0.2243
69.783 69.658 70.678
1-Dodecanol 1.123 283.98 0.0003 1.144 283.27 0.0003 1.115 289.88 0.0003
0.72 0.64 0.30
1.14 1.13 1.05
0.2330 0.2329 0.2328
75.586 75.620 75.750
1-Tridecanol 1.095 321.47 0.0001 1.103 323.20 0.0001 1.090 325.83 0.0001
0.24 0.16 0.26
0.43 0.43 0.45
0.0815 0.0814 Å0.0806
21.403 21.064 18.842
0.939 0.896 0.909
Phenol 138.16 155.60 174.88
0.0379 0.0319 0.0453
0.99 0.92 1.27
0.51 0.53 0.47
0.0820 0.0800 0.0819
16.810 15.880 16.810
tert-Butyl Alcohol 1.089 185.39 0.0016 1.049 184.96 0.0030 1.116 180.73 0.0017
0.18 0.94 0.23
1.31 2.74 1.28
0.1259 0.1249 0.1234
28.300 26.910 24.740
Triethylene Glycol 1.090 297.19 0.0280 1.167 292.30 0.0350 1.272 290.79 0.0450
0.49 0.43 0.38
0.82 0.54 0.22
0.01566 0.01521 0.01570
3.038 2.554 3.110
Water 155.95 174.03 160.92
0.445 0.342 0.423
0.823 0.414 0.941
0.7199 0.7654 0.6687
β
0.05452 0.05950 0.04631
a All five pure compound parameters have been left adjustable. The Tr range is 0.55-0.9.
In Table 1 sample results for normal alcohols (from methanol up to 1-tridecanol), tert-butyl alcohol, triethylene glycol, phenol, and water are presented.
Table 2. Range of the Obtained Values of EAB and Their Comparison with Experimental Data compound
experimental values (K)
range of AB/R (K)
methanol ethanol 1-propanol 1-butanol 1-pentanol 1-hexanol phenol water
1700a,b/2630c 2526-3007d
2086-2726 2497-2715 2343-2707 2320-2645 1976-2369 2401-2905 1933-2309 987-2458
2343e 1051a/1813b
a Value for association enthalpy obtained by theory (Koh et al., 1993). b Experimental value of enthalpy of the hydrogen bond (Koh et al., 1993). c Nath and Bender (1981). d General range of experimental values (obtained from thermodynamic, spectroscopic, dielectric, etc. data) according to Pimental and McClellan (1960). e Ksiazczak and Moorthi (1985).
Excellent correlation of vapor pressures and saturated liquid volumes is obtained with errors well within the experimental accuracy. The two-site model is used for alcohols and phenol and the three-site approximation for water. Moreover, three of the EoS pure compound parameters (b, AB, β) seem to have a physical meaning and are related to experimental measured quantities. Especially, the final value of the b seems to be independent of the initial estimates. Furthermore, b is directly related to the van der Waals volume (Vw) as estimated by the group increcements given by Bondi (1968)
b ) λ*Vw
(19)
An average λ for all alcohols is approximately 1.52, and this information will be used later in developing a method for reducing the number of parameters to three. Different λ values should be used for phenol and water. The obtained association parameters (AB, β) have reasonable values within experimental range as shown in Table 2. 2. Three-Parameter Estimation. On the basis of the investigation presented in the previous section, it is reasonable to assume that for all normal alcohols (up to 1-tridecanol) b ) 1.52Vw. Furthermore, and as explained in a previous section, it is valid to set AB ) ∆Hassoc.. Thus, only three parameters (a0, c1, β) need to be determined from experimental data. The value of AB should be close to the experimental enthalpy of hydrogen bonding. However, although numerous data on thermodynamic functions of association have been published, they often disagree with each other and cannot be unambiguously used to estimate AB. Thus, a common AB value for all alcohols may be used equal to 2526 K or 210 bar cm3/mol. In Table 3, sample results for normal alcohols (from methanol up to 1-tridecanol) with three pure compound adjustable parameters are presented. Very satisfactory correlation (within the accuracy of DIPPR correlations) of vapor pressures and saturated liquid volumes is achieved. Moreover, the optimized values of a0, c1, β are unique, independent of the initial estimates, which is not the case when all five parameters are estimated simultaneously from vapor pressures and saturated liquid volumes. The three-parameter estimation procedure, proposed here, is based on the equation b ) 1.52Vw which is found to provide reasonable vapor pressures and saturated liquid volumes for alcohols. We expect that similar
4314 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 Table 3. Correlation of Vapor Pressure and Saturated Liquid Volume for 1-Alcohols with the CPA EoSa b
∆P (%)
∆V (%)
4.9023
0.84
0.57
7.6125
Ethanol 0.8780 210 0.0137
0.63
0.29
0.0641
11.3713
1-Propanol 0.9601 210 0.0093
0.33
0.47
0.0797
15.1250
1-Butanol 1.0140 210 0.0090
0.76
0.79
0.0952
22.159
1-Pentanol 0.9260 210 0.0037
0.42
2.54
0.1108
27.177
1-Hexanol 0.9805 210 0.0033
0.34
0.90
0.1527
41.389
1-Octanol 1.1461 210 0.0015
2.16
2.51
0.1682
46.698
1-Nonanol 1.1577 210 0.0021
1.82
2.38
0.1838
52.734
1-Decanol 1.2307 210 0.0016
1.46
0.80
0.1993
59.838
1-Undecanol 1.1879 210 0.0014
3.00
1.57
66.463
1-Dodecanol 1.2624 210 0.0014
2.72
3.46
71.882
1-Tridecanol 1.2480 210 0.0019
3.21
1.41
0.0485
0.2149 0.2304
c1
AB
Methanol 0.6556 210 0.0211
0.0330
a0
β
Figure 1. Prediction of second virial coefficients for methanol.
a
Three pure compound parameters have been left adjustable. We set b ) 1.52Vw and AB ) 210. The Tr range is 0.55-0.9.
relationships will hold for other homologous series, e.g., acids, amines, pyridines, etc. Prediction of Second Virial Coefficients Using the CPA EoS and Recommended Set of Pure Compound Parameters The prediction of second virial coefficients through an EoS is a very strict test of its performance. It is wellknown that cubic EoS do not provide satisfactory second virial coefficients, especially at low temperatures, where an extreme underestimation in the absolute value of the second virial coefficients is obtained. Thus, the prediction of second virial coefficients of normal alcohols (from methanol up to 1-butanol), phenol, and water is used in this study to test the model and, in addition, to identify the best and physically meaningful set of parameters that should be used for phase equilibria calculations. In Figures 1-4 the performance of the CPA and of the SRK EoS in the prediction of second virial coefficients is presented for methanol, ethanol, 1-propanol, and water. Very satisfactory predictions are generally obtained with the CPA EoS with parameters obtained either from the global estimation or from the threeparameter estimation, correcting the erroneous behavior given by the cubic EoS. Similar results are obtained for 1-butanol and phenol. The sets of parameters which yield very good second virial coefficients are presented in Table 4. The performance of the CPA EoS in the prediction of second virial coefficients depends significantly on the pure compound parameters. We have found that several sets of pure compound parameters which yield an excellent correlation in vapor pressures and saturated liquid densities may result in poor second virial coefficient prediction, especially at low temperatures. Thus, the satisfactory prediction of second virial coefficients may
Figure 2. Prediction of second virial coefficients for ethanol.
Figure 3. Prediction of second virial coefficients for 1-propanol.
serve as criterion for choosing the best set of the EoS parameters. There are no experimental second virial coefficients for alcohols above 1-butanol. Thus, the reasonable set of parameters, presented in Table 4, is chosen on the basis of their trend within the homologous series established up to n-butanol and shown in Figure 5, where the parameters are plotted as a function of the van der Waals volume. Conclusions A new equation of state (CPA) for associating compounds is presented, which combines the simplicity of
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4315 R ) gas constant T ) temperature Tr ) reduced temperature V ) molar volume Vl ) saturated liquid volume Vw ) van der Waals volume v* ) segment molar volume in a close-packed arrangement XA ) mole fraction of the compound not bonded at site A u°/k ) temperature-independent dispersion energy of interaction between segments z ) compressibility factor Greek Letters
Figure 4. Prediction of second virial coefficients for water.
a cubic equation of state with the theoretical background of the perturbation theory. The CPA EoS provides an excellent correlation of vapor pressures and saturated liquid volumes using three different approaches for the estimation of the EoS parameters. The most accurate methodology is the global estimation of the five pure compound parameters. For the members of a homologous series we suggest, as shown here for alcohols, using the three-parameter evaluation based on constant values for the covolume parameter (b ) λVw) and for the association energy parameter AB (almost equal to the experimental enthalpy of hydrogen bonding of alcohols). The optimized association parameters have reasonable values and are related to the enthalpy and entropy of hydrogen bonding. Finally, using as a criterion the quality of second virial coefficient predictions, the parameters presented in Table 4 are recommended. The extension of the CPA EoS to (binary) mixtures is under development and will be presented in a forthcoming publication. Acknowledgment The authors are very grateful to Shell/KSLA for financing this work as part of a research project on “mixtures of associating fluids”. We want, also, to acknowledge H. Meijer (Shell/KSLA), J. Walsh (Shell/ Houston), E. Hendriks (Shell/KSLA), and A. Moorewood (Infochem) for useful discussions and I. Economou (Democritos NRC) for many valuable suggestions, for animating discussions, and for making some work of his available to us prior to publication. Nomenclature a ) energy parameter a0 ) parameter in the energy term b ) covolume parameter C ) integration constant (in SAFT equation) c1 ) parameter in the energy term c2 ) parameter in the energy term (used in the homomorph approach) d ) temperature-dependent segment diameter g(d) ) radial distribution function K ) equilibrium constant k ) Boltzmann’s constant m ) effective number of segment within the molecule (segment number) NAV ) Avogadro’s number P ) pressure Ps ) vapor pressure
β ) parameter in the association term of CPA (related to ∆Sassoc) ∆AB ) strength of interaction between sites A and B ∆Hassoc ) enthalpy of hydrogen-bonding formation ∆Sassoc ) entropy of hydrogen-bonding formation AB ) association energy of interaction between sites A and B η ) reduced density κAB ) volume of interaction between sites A and B λ ) 1.52, constant F ) molar density σ ) Lennard-Jones segment diameter (temperature independent) ∑A ) summation over all the sites (starting with A) on the molecule τ ) 0.74048, constant Subscripts A, B ) for site A, B on the molecule Av ) Avogadro calcd ) calculated exptl ) experimental r ) reduced Superscripts seg ) segment hs ) hard sphere assoc ) association chem ) chemical Abbreviations AAD ) Average Absolute Deviation APACT ) Associated-Perturbed Anisotropic Chain Theory CPA ) Cubic Plus Association equation of state CS ) Carnahan-Starling EoS ) equation of state hs ) hard sphere LLE ) liquid-liquid equilibria SAFT ) Statistical Associating Fluid Theory SRK ) Soave-Rendlich-Kwong equation of state vdW ) van der Waals VLE ) vapor-liquid equilibria
Appendix A. The Radial Distribution Function of the CPA EoS Figure 6 presents plots of the repulsive term of the SRK EoS, which is of course the van der Waals (vdW) one, and of the hard sphere fluid (hs) versus the fluid density for n-hexane. Two curves are shown for the vdW term: in the one (curve 1) the covolume parameter is obtained from the critical properties (b ) bc ) 0.121 L/mol); in the other (curve 2), the covolume parameter is that obtained by fitting the saturation pressures and liquid volumes as done in this study (b ) 0.108 L/mol). The diameter of n-hexane for the hs fluid (curve 3) is the one used in the SAFT EoS (Huang and Radosz,
4316 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 Table 4. Suggested Pure Compound Parameters for Normal Alcohols (from Methanol up to 1-Tridecanol), tert-Butyl Alcohol, Triethylene Glycol, Phenol, and Water Obtained with the Global Parameter Evaluation compound
b
a0
c1
AB
β
∆P (%)
∆V (%)
methanol ethanol 1-propanol 1-butanol 1-pentanol 1-hexanol 1-octanol 1-nonanol 1-decanol 1-undecanol 1-dodecanol 1-tridecanol tert-butyl alcohol triethylene glycol phenol water
0.0330 0.0479 0.0644 0.0795 0.0976 0.1137 0.1490 0.1641 0.1843 0.2011 0.2243 0.2329 0.0820 0.1234 0.0801 0.0152
4.7052 7.3110 11.7185 14.8841 23.1101 27.230 41.765 47.225 54.247 62.230 70.678 75.620 16.810 24.740 18.842 2.5547
0.9037 0.9200 0.9473 0.9792 1.0283 0.9247 0.9946 1.0402 1.1430 1.0387 1.1152 1.1033 1.0890 1.2720 0.9087 0.7654
183.36 207.64 208.78 219.25 190.68 223.29 265.25 262.82 262.18 293.41 289.88 323.20 185.39 290.79 174.88 174.03
0.0449 0.0160 0.0083 0.0086 0.0038 0.0033 0.0004 0.0005 0.0004 0.0002 0.0003 0.0001 0.0016 0.0450 0.0453 0.0595
0.85 0.48 0.35 0.93 0.18 0.25 0.25 0.43 0.17 0.58 0.31 0.16 0.18 0.38 1.27 0.34
0.08 0.80 0.48 0.48 0.53 0.53 0.58 0.61 0.61 1.01 1.05 0.43 1.31 0.21 0.48 0.41
Figure 5. Trend of pure compound parameters for primaryalcohols (from methanol up to 1-butanol) using the global estimation.
Figure 7. Comparison of the radial distribution functions obtained from SRK, SAFT, and CPA EoS for n-hexane and methanol.
Figure 6. Comparison of the repulsive terms obtained from SRK, SAFT, and CPA EoS for n-hexane and methanol.
1990). Notice that use of the fitted b value gives a curve that is close to that of the hs fluid, while use of b ) bc deviates, as it is well-known, substantially from it. The same similarity between the hs fluid compressibility factor and that obtained with a fitted b value in the vdW repulsive term was also observed by Wong and Prausnitz (1985). In Figure 7 we present the radial distribution for n-hexane as obtained from the radial distribution func-
tion of vdW with the fitted b ) 0.108 L/mol value (curve 1); the radial distribution function of the hs fluid again with the fitted b value (curve 2); the radial distribution function of the hs fluid using the diameter of Huang and Radosz (curve 3). Following the behavior of the repulsive term with the fitted b value, we must chose between the functions that give curves 1 and 2 the one that comes closer to that of the hs fluid (curve 3). This is the function that gives curve 2, i.e., that of the hs fluid. Figures 6 and 7 also present the repulsive term and the radial distribution for methanol as a function of density obtained from CPA with the fitted b value along with those of the hs fluid obtained by using the diameter for methanol given by Huang and Radosz. The closeness of the curves for the two repulsive terms indicates that the fitted b value (0.033 L/mol) in the vdW term (curve 4 in Figure 6) approximates again the behavior of the hs fluid (curve 5 in Figure 6), and it gives a radial distribution (curve 5 in Figure 7) reasonably close to that of the hs fluid (curve 4 in Figure 7). Notice, finally, that the range of densities covered for the two compounds is that involved in applying CPA to them.
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4317
Appendix B. The Homomorph Approach
Table 5. Correlation of Vapor Pressure for Methanol and Water with the CPA EoSa
An alternative way of reducing the number of adjustable pure compound parameters to three is provided by the well-known and extensively discussed homomorph approach. With this approach the values of b, a0, and c1 are estimated by regressing vapor pressures and saturated liquid volumes of the homomorph of the associating compound with the SRK EoS, and then the values of c2 (see below), AB, and β are estimated by regressing vapor pressure and saturated liquid volume data for the associating compound. The only change made in the expression of the CPA EoS is in the attractive parameter a which is now given as
a ) a0(1 + c1(1 - xTr) + c2(1 - xTr) )
2 2
(B1)
According to Prausnitz, the homomorph of a polar molecule is a nonpolar molecule having essentially the same size and shape as those of the polar molecule. It has been found in this work that alkanes are not suitable homomorphs for alcohols. Optimization of a0, c1, and b with SRK from the vapor pressure and volume of ethane (homomorph of methanol) yields excellent vapor pressure and saturated liquid volume for ethane. However, simultaneous optimization of c2, AB, β for methanol yields excellent vapor pressures but not reasonable saturated liquid volumes for methanol. This is due to the fact that the performance of the CPA EoS in the calculation of liquid volume is very much determined by the value of the covolume parameter b. The optimum value of b for methanol is approximately 0.033 L/mol (Table 1), much different from the value of b obtained from ethane ()0.045 L/mol). This explains the poor performance of ethane as a homomorph of methanol. Thus, we have established, on the basis of some physical insight, that aldehydes may be suitable homomorphs for alcohols, formaldehyde for methanol, acetaldehyde for ethanol, propionaldehyde for 1-propanol, and so on. The a0, c1, and b of methanol (for the CPA EoS) are obtained from SRK alone by regressing vapor pressure and liquid volume data for its established homomorph, the formaldeyde. Excellent correlation of vapor pressures and reasonably good liquid volumes are obtained for the formaldehyde. Then, using these a0, c1, and b values, simultaneous optimization of vapor pressure and saturated liquid volume for methanol is performed with the CPA EoS for the c2, AB, β parameters. The correlation of the vapor pressure is excellent, but the error in the liquid volume of methanol is around 12%. Again the explanation is given in terms of the b parameter, which is 0.028 L/mol for formaldehyde (and consequently for methanol). This value is much closer to the optimum value for methanol than the b value obtained from ethane, which implies that formaldehyde is a much better homomorph for methanol than ethane. However, the liquid volumes seem to be quite sensitive to the correct covolume value; thus, even small differences in b (between the optimum value and the one calculated with the homomorph) cause quite some difference in the saturated liquid volume calculations. These results demonstrate the problems of using directly the homomorph approach. Many investigators [e.g., Anderko (1992)] have mentionedswithout explicit proofsthese difficulties, and they have abandoned the homomorph approach. The investigation carried out here attributes the problems of the homomorph ap-
b
a0
c1
AB
∆P (%)
∆V (%)
0.381 0.430 0.640
6.292 10.060 1.141
Water 100.48 91.66 152.54
0.558 0.527 0.897
5.347 5.174 2.578
β
0.03365 0.03481 0.03288
3.9045 4.0610 4.3560
Methanol 1.1145 188.79 0.0679 1.0978 190.50 0.0615 0.9629 186.96 0.0515
0.01676 0.01671 0.01558
3.5060 3.4110 3.6220
0.9109 0.9763 0.5667
0.1190 0.1598 0.0300
a The T range is 0.55-0.9. Predictions for the liquid volumes r are also given.
proach to the importance of having correct b for satisfactory regression of saturated liquid volumes. Considering this point, a modified homomorph approach is suggested. The b value of the homomorph is set at 1.52Vw of the associating compound whichs through rather arbitrarysis necessary for reliable saturated liquid volumes. Then the optimum a0 and c1 parameters of the homomorph with the SRK EoS are obtained from vapor pressure and saturated liquid volume. Finally, the c2, AB, and β parameters of CPA for the associating compound are obtained exclusively from vapor pressures. We report here sample results with this modified homomorph method. For methanol (using formaldehyde as homomorph), the absolute percentage errors are 0.08 in vapor pressure and 2.9 for saturated liquid volume. For ethanol (using acetaldehyde as homomorph), the absolute percentage errors are 0.09 in vapor pressure and 1.9 in saturated liquid volume. Finally, for 1-propanol (using propionaldehyde as homomorph), the absolute percentage errors are 0.08 in vapor pressure and 1.0 in saturated liquid volume. These results are similar to the three-parameter method presented in a previous section. Appendix C. Optimization Based Solely on Vapor Pressure Sample results for methanol and water are shown in Table 5. The correlation of vapor pressure is naturally excellent (within the experimental accuracy), but the prediction of liquid densities depends significantly on the final value of b. When the estimated value of b with this approach differs from the estimated value using the global evaluation, the absolute deviation of the vapor pressure remains below 0.2% but the prediction of saturated liquid volumes is rather poor. However, it is possible to arrive at a parameter set capable of satisfactory representing both vapor pressure (below 1%) and liquid volume (3%). Note that the great majority of EoS (including all cubic EoS) are not capable of representing simultaneously liquid volumes and vapor pressures using parameters obtained solely from vapor pressure data. Among the aforementioned four different approaches for the estimation of pure component parameters for associating compounds with the CPA EoS, the more advantageous is the global estimation. This approach can be used for any associating compound (e.g., normal or branched alcohols, phenols, acids, and water) with no modification. Moreover, it yields very good correlation in both the vapor pressures and saturated liquid densities. However, multiple sets of pure compound parameters are obtained with this approach, and some of them have no physical meaning. This shortcoming
4318 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
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Received for review January 19, 1996 Revised manuscript received July 11, 1996 Accepted July 11, 1996X X Abstract published in Advance ACS Abstracts, October 15, 1996.
