I n d . Eng. Chem. Res. 1995,34, 1897-1909
1897
A Simplified SAFT Equation of State for Associating Compounds and Mixtures Yuan-Hao Fu and Stanley I. Sandler* Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
A simplified statistical associating fluid theory (SAFT)equation of state is developed for associating and nonassociating compounds. The new equation of state has the same number of adjustable parameters as the original SAFT equation, but is simpler. For pure compounds, this equation of state can correlate the vapor pressure and liquid density with errors equal to those of the original SAFT equation. Both self-associating and cross-associating binary mixtures were used to test the simplified SAFT equation of state. With a single binary adjustable parameter, the new equation results in better correlations of vapor-liquid equilibria than the original SAFT equation for most mixtures. However, for some mixtures, both the original and simplified SAFT equations of state fail to provide a satisfactory description. The optimized binary parameter of the simplified SAFT equation of state is found to be linearly dependent on temperature for each mixture. Consequently, the binary parameter computed from several data sets can be used to accurately predict the vapor-liquid equilibrium at temperatures other than those for which experimental data are available.
Introduction Strong attractive forces, such as hydrogen bonding, affect the physical properties of associating compounds. For example, the boiling and critical points of such compounds are higher than those of similar size nonassociating compounds. Also associating compounds may form highly nonideal mixtures. In the thermodynamic modeling of associating compounds, the association is typically considered to be a chemical reaction, with the strength of association given by the value of the chemical equilibrium constant. On the other hand, in statistical thermodynamics, hydrogen bonding is regarded as a short-ranged, directional, and strongly attractive interaction between molecules. On the basis of thermodynamic perturbation theory, Wertheim (1984a,b, 1986a,b) developed a theory of associating fluids. Wertheim’s association theory has been tested against Monte Carlo simulation for associating hard spheres (see, for example, Chapman et al. (1986),Joslin et al. (1987), and Jackson et al. (1988)) and shown to be quite good. Chapman et al. (1989) extended Wertheim’s theory to develop the statistical associating fluid theory (SAFT) equation of state for associating fluids. Huang and Radosz (1990) assumed the segmentsegment interaction to be described by the square-well potential to improve the SAFT equation of state for real compounds. This equation of state has been applied to both pure nonassociating and associating components. Huang and Radosz (1991) then extended the SAFT equation to mixtures containing associating compounds. In the SAFT equation of state of Huang and Radosz, the residual Helmholtz energy for a pure component has four contributions: the hard sphere, dispersion, chain, and association terms. The hard sphere, chain, and association terms were derived from statistical thermodynamics. For the dispersion term, Huang and Radosz used a double power series in temperature and density which had been fit to argon physical property data by Chen and Kreglewski (1977). Here we simplify the
SAFT equation of state by using the single attraction term of Lee et al. (1985) for the square-well fluid to replace the multiterm double series dispersion term, while keeping the original hard sphere, chain, and association terms. This results in a much simpler equation of state. The pure component parameters of this simplified SAFT (SSAFT)equation of state are obtained by fitting vapor pressure and liquid density data, and are found to scale with molecular weight for the alkanes, as with the original SAFT equation. We have applied the SSAFT equation of state to calculate the vapor-liquid equilibria of both self-associating and cross-associating binary mixtures using one binary parameter, and the correlations we obtain with the SSAFT equation of state are compared with those of the SAFT equation.
Equation of State A. Pure Components. In SAFT model, molecules are considered t o be composed of equal-size, spherical segments. Different molecules have different segment diameters and numbers of segments. We develop the equation of state by forming molecules from single hard spheres using the procedure shown in Figure 1. Initially, the fluid is assumed to consist of equal-size single hard spheres, and only hard sphere effects are considered. Next, the segment-segment attractive forces are added to each sphere; the dispersion energy can be described by any appropriate potential, such as the square-well or Lennard-Jones potentials. Then chain sites are added to each sphere and chain molecules are formed by the bonding of chain sites. Finally, association sites are added and the effect of association is included. There is a free energy contribution for each step in this procedure. Therefore, in the SAFT equation of state, the residual Helmholtz energy per molecule for a pure component has hard sphere, dispersion, chain, and association contributions and is written as
* Author t o whom correspondence should be addressed. 0888-5885/95/2634-1897$09.00f0
0 1995 American Chemical Society
1898 Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995
segment, c is a parameter which we take to be 0.333, and uo is the temperature independent interaction energy between segments. The value of c was obtained by fitting ethane vapor pressure and liquid density data using the simplified SAFT equation of state. It is more convenient to use molar volume rather than the diameter of a segment as an adjustable parameter. The temperature independent segment molar volume in a closed-packed arrangement, uoo, is
where z = 0.740 48. The dispersion contribution t o the Helmholtz free energy is adisp
= mad"sp
(8)
0
where a? is the dispersive Helmholtz energy of one segment. Here and in the original SAFT equation the attraction between segments is assumed to be the square-well potential. In SAFT the equation of Chen and Kreglewski (1977) was used for which is a double power series containing 24 coefficients
up,
I
L
I
I
(4
(e)
Figure 1. Procedure to form a molecule in the SAFT model. (a) The proposed molecule. (b)Initially the fluid is a hard sphere fluid. (c)Attractive forces are added. (d) Chain sites are added and chain molecules appear. (e) Association sites are added and molecules form association complexes through association sites.
Each compound is assumed to be a chain with m segments. The hard sphere Helmholtz energy is given by
a's = ma:
(2)
where a$ is the Helmholtz free energy for a hard sphere in a hard sphere fluid at the same packing fraction as in the chain fluid (see below). Since there are m segments in a molecule, the total hard sphere contribution of a molecule is mu:. The single hard sphere contribution at;."t o the Helmholtz free energy is calculated from the expression of Carnahan and Starling (1969)
(3) with
17 = hNA46)@,d3
(4)
where ea is the molar density of hard spheres and d is the effective hard sphere diameter of a segment. The molar density of hard spheres is equal to
e, = me
(5)
where e is the molar density of chain molecules. Following Barker and Henderson (1967),the temperature dependence of the effective hard sphere diameter is given by
d = o[l - c exp(-3u0/kr)]
(6)
where CT is the temperature independent diameter of a
RT where D,j. are the universal constants fit to thermodynamic data of argon and u is the temperature dependent square-well depth. The equation we test here results from the replacement of this dispersion term with the simpler equation of Lee et al. (1985). The Helmholtz free energy for the dispersion term derived by Lee et al. (1985) for the square-well fluid is
(10) with
Y = ex&+)
-1
where ZMis the maximum coordination number, which was taken t o be equal to 36 in the simplified perturbed hard chain theory (SPHCT) of Kim et al. (19861, us is the molar volume of a segment, u* = N*,d3/& is the close-packed molar volume of a segment, and u is the depth of the square-well potential. We assume that the square-well-potential depth depends on temperature as given by Chen and Kreglewski (1977)
u = uo[l
+( e / k ~ ) ~
(12)
We set elk = -10 K based on fitting the vapor pressure and liquid density of ethane. The Helmholtz energy of chain formation achainis estimated from Wertheim's association theory by setting the association energy to infinity. In the SAFT equation of state, the Helmholtz energy of chain formation is given by
1 - (1/2)q -= (1- m ) In RT (1 - VI3 achain
(13)
Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995 1899 Finally, the association Helmholtz energy aassoc due to hydrogen bonding is also estimated from Wertheim’s association theory, and is given by
aassoc ,=T[(lnxA-E)+:M]
result of Mansoori et al. (1971)for hard sphere mixtures, the Helmholtz energy is
(14) 2
where XAis the fraction of association sites A that are not bonded, and M is the number of association sites in a compound. The summation is over association sites, and XAis given by
with
(15) B
where Am is the association strength given by
Am = ghs(d)[exp(em/kT)- l l d 3 ~ m
(16)
Here P and K~ are the association energy and volume for the interaction between association sites A and B, and gh”(d)is the hard sphere radial distribution function at contact given by
ghs(d)=
1 - (1/2)q (1 - q)3
(17)
where Q is the total molar density of molecules, xi is the mole fraction of compound i, mi is the number of segments per molecule i, and d i is the temperature dependent segment diameter. Next we consider the attractive force between segments, again assuming that the attractive potential is the square-well potential. Extending the equation derived by Lee et al. (1985) for pure components to mixtures, the dispersion term becomes adisp
The simplified SAFT equation of state for pure fluids obtained from the volume derivative of the Helmholtz free energy is a sum of the compressibility factors from each of the contributions above and is given by
z =P V = zhs+ Zdisp + rhain + zasBoc + 1 (18)
-- -mzM1n(
RT
us
VS
)
+ (V*V
(26)
where us is the total molar volume of a segment which is equal to us = llQm, and ZM = 36. We choose the following as our mixing rules for this equation
RT with and
(20)
(v*y> =
i
j
i
j
(28)
with
In the SSAFT equation of state, there are three adjustable parameters for a nonassociating compound: the temperature independent square-well depth uo, the segment molar volume in a closed-packed arrangement uoo, and the number of segments m in a compound. For associating compounds, there are two additional parameters: the association energy cABand association volume P. B. Mixtures. For mixtures, we use the same procedure t o develop the equation of state as for a pure component. The total residual Helmholtz energy in the SSAFT equation of state is again given by ares
= ahs
+
adisp + achain
+ aassoc
and
(30) where K i is taken as an adjustable parameter in binary mixtures. Next chain molecules in the system are formed by the bonding of chain sites. The chain term for a mixture derived by Chapman et al. (1990) is -chain
(23)
In the hard sphere term the segments are not bonded to form chains, but are single hard spheres; therefore, the mixture is a hard sphere mixture. Based on the
where giihs((dii) is the pair correlation function a t contact in a mixture of hard spheres. As derived by Mansoori et al. (19711,guh”(d,j.)is given by
1900 Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995 Table 1. Parameters for the SSAFT Equation of State compound uolk (K) uoo (mL) m @lk (K) 1 0 0 0 ~ ~
For segments of the same diameter, this equation becomes
(33) Finally, the effect of association is considered. The association term for a mixture given by Chapman et al. (1990) is
where Mi is the total number of association sites in compound i, and X4 is the mole fraction of sites Ai which are not bonded and is given by
ethane propane butane pentane hexane heptane benzene acetone chloroform tetrachloromethane acetic acid propanoic acid butanoic acid methanol ethanol propanol butanol water
90.529 92.571 102.823 101.564 99.724 98.650 129.471 111.463 118.961 132.828 118.088 111.788 109.916 89.135 93.131 104.651 103.80 188.231
16.236 16.507 19.263 18.574 17.657 16.981 17.807 11.962 14.547 20.933 16.236 16.500 15.753 14.000 16.000 16.000 16.000 6.560
2.022 2.735 2.962 3.708 4.548 5.380 3.206 3.794 3.429 2.970 2.282 3.017 4.000 1.989 2.540 3.316 4.006 2.000
5810.5 5240.2 4584.6 2710.4 2802.7 2511.7 2458.6 825.9
28.840 19.998 26.399 119.107 56.634 41.420 49.971 119.930
pressure and liquid density are shown in Table 2. For small n-alkanes, we find that the number of segments m increases linearly with molecular weight. From a least squares analysis, we find that the equation relating the two is M
+
= 0.047Mw 0.511
(42)
Similarly, we find that where
ej
A, and Bj given by AA.B.
= gp(d,)[exp(cACBJ/kT) - 1](d,)3~AiBj(36)
where €AB, and K ~ are ~ Jthe association energy and volume for an interaction between site Ai and Bj. The final form of the SSAFT equation of state for mixtures is
z= Z h s
+ Zdisp + Zchain
+ zassoc +1
(37)
with
53523
]
( 1 - 5313
(38)
(39)
Results A. Pure Components. The parameters uo,uoo, and m of the SSAFT equation of state for pure nonassociating components were obtained by simultaneously fitting vapor pressures and liquid densities, and these are listed in Table 1. The correlation errors in the vapor
+
(43)
+
(44)
muoo= 0.835Mw 8.206
is the molar density of compound j equal to ~j
= qe, and AkBj is the association strength between site
and mu'lk = 4.912Mw 30.318
These correlations are plotted in Figure 2. With these correlations it is possible to estimate the values of the parameters of larger n-alkane compounds for use in the SSAFT equation of state. For associating compounds, we have to specify the number of association sites and the site-site interactions for each compound. Table 3 lists the types of association that may occur. On the basis of spectroscopic data, acids usually associate to form dimers. Therefore, we assume that acids contain only one association site. We also assume that alcohols contain two association sites: one at the hydrogen atom (site A) and the other at the unbonded electron pair on oxygen (site B). However, site A can only form a hydrogen bond with site B. Water has four association sites: two at the unbonded electron pairs and two at the hydrogen atoms. Only the unbonded electron pair will form a hydrogen bond with a hydrogen atom. Some experimental results (Wei et al., 1991)suggest that only three sites of water form hydrogen bonds. We chose the two hydrogen atoms and the one unbonded electron pair as the association sites since steric hindrance of the two association sites on an oxygen atom may not allow the formation of two hydrogen bonds. Table 1 also lists the values of the parameters uo, m, ea, and K~ for the associating compounds obtained by fitting vapor pressures and liquid densities of the compounds studied. Except for small compounds, the segment volume uoo was fixed at 16 for simplicity. The errors in the correlation for the vapor pressures and liquid densities are shown in Table 2, which also lists errors after parameter optimization for the original SAFT equation of state. It should be noted that since the parameters reported by Huang and Radosz were
Ind. Eng. Chem. Res., Vol. 34,No. 5 , 1995 1901 Table 2. Errors in Vapor Pressure and Liquid Densities for Pure Components Resulting from Correlations with the SSAFT and SAFT Equations of State compound
SSAFT SAFT temp range (K) error % pressure error % density error % pressure error % density
ethane 150-305.5 propane 189.5-367.18 butane 213.15-323.15 pentane 233.15-463.15 hexane 243.15-503.15 heptane 273.15-533.15 benzene 300-540 223.15-493.15 acetone chloroform 213.15-343.15 tetrachloromethane 273.15-543.15 acetic acid 273.15-573.15 propanoic acid 293.15-603.15 butanoic acid 303.15-623.15 methanol 253.15-493.15 ethanol 250-510 propanol 260-530 butanol 270-550 water 283-613 av error over all compds av error for nonassociating compds av error for associating compds
0.53 1.13 0.27 1.47 2.06 2.14 0.90 2.41 1.40 1.13 2.87 0.42 0.51 1.65 1.28 0.22 0.21 2.22 1.27 1.34 1.17
2.30 1.56 0.83 0.97 0.89 1.02 1.08 2.77 1.23 1.20 0.96 1.13 0.93 1.05 1.46 1.52 1.23 3.30 1.41 1.385 1.45
2.02 2.00 0.84 1.90 2.05 1.78 1.40 2.78 0.87 1.50 2.35 3.26 2.83 1.10 0.34 0.70 0.30 1.30 1.63 1.71 1.52
data
sources
4.77 2.05 1.51 2.72 3.31 3.35 2.10 1.67 1.50 2.51 1.34 1.38 1.24 0.85 1.17 1.53 2.63 3.20 2.16 2.55 1.67
1 1 1 1 1 1 1 2 2 3 3 3 3 2 3 3 3 1
a (1) Vargaftik, N. B. Table on the Thermophysical Properties of Liquids and Gases; John Wiley & Sons: New York, 1975. (2) Thermodynamic Tables for mn-Hydrocarbons; Thermodynamic Research Center, Texas A&M University: College Station, TX. (3) Daubert, T. E. Data Comuilation Tables of Prouerties of Pure Comuounds; Design Institute for Physical Property Data, American Institute of Chemical Engineers: New York, ‘1985:
obtained using a different optimization procedure than we did, and that the temperature ranges of their correlations are different than used here, to be fair in our comparisons we also redetermined the parameters for the SAFT equation of state. We see that, in general, the SSAFT equation is slightly more accurate than the original equation, though the difference is not large. In Table 1,we see that for the SSAFT equation the segment interaction energies u0/k are close to 100 K for both nonassociating and associating compounds, except for water. This suggests that the dispersion term is correctly accounting for only the weak attraction forces, and explains why the segment interaction energies of associating and nonassociating compounds have similar values. However, the segment interaction energy of water is larger than that of other compounds, which may be due to the high polarity of water. Table 1 also shows that the association energies em/k for acids and alcohols decrease as molecular weight increases. Finally, the association energy for an acid is approximately twice as large as for an alcohol, which is reasonable since an association site of an acid involves two hydrogen bonds, while alcohol association sites involve only a single hydrogen bond. B. Mixtures. Since we are interested in associating mixtures, we have computed the phase diagrams only for such mixtures. The SSAFT equation of state parameters of the compounds were obtained from the pure component properties as discussed in the previous section. For self-associating mixtures, there is only one additional adjustable parameter kv, and this binary parameter was obtained by fitting binary mixture vapor-liquid equilibrium data. The calculated results for the SSAFT equation of state are listed in Table 4 and compared with those of the SAFT equation of state. For alcohol-alkane mixtures, both the simplified and original SAFT equations of state represent the phase diagrams quite well. For acids, the SSAFT equation of state produces small errors in pressure, as does the SAFT equation of state. However, the SSAFT results in smaller errors in the vapor phase mole fractions than
Table 3. Types of Association compound acid
formula
association type
17-- -H-o \
c
cAA# 0 and KAA t 0
- -01:
‘0-H-
site A
ne M
‘H
site A
does the original SAFT equation. The reason may be that the correlation of the data with the original SAFT equation of state results in less association for acids than with the SSAFT equation. Figures 3 and 4 are examples of phase diagrams calculated using both the simplified and original SAFT equations of state. Cross-association occurs in associating compoundassociating compound mixtures. The cross-association parameters cannot be estimated from pure component information. Therefore, we have had to make assumptions about the type of association and values of the cross-association parameters. Table 5 lists the types of association assumed for different cross-associating mixtures. Explanations of the assumptions made and the results obtained are given below. For acid-acid mixtures, acids 1 and 2 each have single association sites, A1 and Az, respectively. In these mixtures, cross-association occurs, and the cross-association energy and volume are assumed to be given by €AiAz
and
-4
=
(454
1902 Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995 300
6
0
experiment
5
m
4,
3,
2, 40
60
100
0.00
1:
0.20
molecular weight
0.40
0.60
0.80
1
mole fraction of Benzene Figure 3. Binary mixture of acetic acid and benzene at 50 "C. The points are the experimental data, the solid line is the result of the original SAFT equation, and the dashed line is results for the SSAFT equation proposed here.
cn
2
100
Q
20
40
60
80
100
120
molecular weight
I-,: :,
.....
100
100 0.00
0.20
0.40
,
,
0.60
0.80
1 1.00
90
mole fraction of propanol Figure 4. Binary mixture of propanol and heptane at 60 "C.
80
Legend as in Figure 3. 70
m
VI
60 50
40
30 20
40
60
80
100
120
molecular weight Figure 2. Correlation of the segment number m, mu0/k,and muoa in the SSAFT equation with molecular weight for n-alkanes. The points are the optimum values found from correlating data, and the lines are the result of a least squares fit.
Therefore acid-acid mixtures also have only a single adjustable parameter per binary in the dispersion term. The results for acid-acid mixtures are given in Table 6, and Figure 5 is an example of a vapor-liquid phase diagram for one such mixture. The SSAFT equation of state can represent this phase diagram very well, while the SAFT equation results in larger errors in the vapor phase mole fractions. For alcohol-alcohol mixtures, we assume that each alcohol has two association sites A1 and B1 on alcohol 1, and sites A2 and B2 on alcohol 2, and that AlB2 and AzBl cross associations can occur. The cross-association parameters are assumed to be given by
Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995 1903 Table 4. Errors in Total Pressure and Vapor Phase Mole Fraction for Self-AssociatingMixtures SSAFT
kij 0.038 0.028 0.042 0.040 0.040 0.030 0.002 0.011 0.023 0.028 0.028 0.016 0.020
mixture
+ +
benzene acetic acid at 45 "C benzene acetic acid at 50 "C acetic acid tetrachloromethane at 35 'C heptane propanoic acid at 25 "C heptane propanoic acid at 45 "C heptane propanoic acid at 50 "C benzene propanoic acid at 40 "C propanoic acid tetrachloromethane a t 30 "C hexane ethanol at 25 "C hexane+ ethanol at 40 "C ethanol heptane at 40 "C hexane propanol a t 50 "C propanol heptane at 60 "C
+ + + + + + + + + +
SAFT
error P (mmHg)
error y
kii
1.54 1.95 2.03 0.40 0.36 1.52 2.05 8.17 2.12 1.92 2.15 3.04 2.71
0.0163 0.0102 0.0073 0.0166 0.0094 0.0122 0.0662 0.0041 0.0092 0.0070 0.0182 0.0106 0.0119
0.040 0.031 0.035 0.015 0.016 0.010 0.005 0.017 0.020 0.024 0.023 0.013 0.018
2.30
0.0153
av error over all mixtures
error P ( " H E )
error Y
data sourcea
3.20 4.46 1.36 0.55 1.77 0.95 2.02 14.75 2.35 1.47 2.25 2.89 2.70
0.0231 0.0166 0.0360 0.0349 0.0311 0.0402 0.1220 0.0022 0.0098 0.0065 0.0148 0.0086 0.0103
1 1 2 3 3 1 1 2 1 1 1 1 1
3.13
0.0273
a (1)DECHEMA Chemistry Data Series: Vapor-Liquid Equilibrium Data Collection; DECHEMA: Frankfurt, 1977. (2) Tamir, A.; Dragoescu, C.; Apelblat, A.; Wisniak, J. Fluid Phase Equilib. 1983,10,9-42. (3) Lark, B. S.; Banipal, T. S.; Singh, S. J. Chem. Eng. Data 1985,30,286-288.
Table 5. Type ~- of Association and Cross-AssociationParameters for Cross-Associating - Mixtures mixture acid-acid
component 1
component 2
]
1r
,."
\ 0-H
alcohol-alcohol
site Al
site A~
H-o,
€AIAI #
association type 0
cross-association parameters
C;,
0
site B1 C-ti
site B2 C-ti
\H
\H
site AI
site A2
}
acid-alcohol CI/" \ 0-H
site B2 site A1
C-ti \H
site A2
water -acid
site C1 H-ti
site B ~ \ ~ site A, water-alcohol
..
site C1 H-0
H-0
site A2
[ )
site B2 C-ti
site B ~ \ ~ site Al
\H
site A2
Table 6. Binary Interaction Parameters and Errors in Total Pressure and Vapor Phase Mole Fractions for Cross-AssociationMixtures ~~
SAJ?T
SSAFT mixture acetic acid propanoic acid at 30 "C acetic acid propanoic acid at 40 "C ethanol propanol a t 70 "C ethanol propanol at 40 "C ethanol butanol a t 70 "C ethanol acetic acid a t 35 "C butanol acetic acid at 35 "C ethanol water at 50 "C water propanoic acid at 60 "C water acetic acid a t 40 "C acetone chloroform a t 25 "C acetone + chloroform at 40 "C acetone + chloroform at 50 "C
+
+ + + + + + + + +
+
av error overall data sets a
kij
0.013 0.013 0.027 0.005 0.011 -0.070 -0.055 -0.180 -0.147 -0.179 0.001 -0.001 -0.006
error P (mmHg) 0.10 0.09 3.72 0.78 4.18 1.08 0.39 3.59 3.34 2.09 3.87 3.27 6.02 2.50
error y 0.0134 0.0084 0.0128 0.0066 0.0054 0.0249 0.0112 0.0066 0.0273 0.0290 0.0120 0.0082 0.0129 0.0137
kij 0.014 0.021 0.025 0.004 0.010 -0.023 -0.046 -0.120 -0.085 -0.134 0.001 0.000 -0.003
error P (mmHg) 0.24 0.19 2.35 0.12 1.21 1.16 0.43 3.56 5.77 2.74 3.86 3.41 5.89 2.38
error y 0.0820 0.0774 0.0096 0.0051 0.0051 0.0110 0.0213 0.0080 0.0278 0.0376 0.0125 0.0084 0.0122 0.0244
data sourcea 1 1 1 1 1 1 1 1 1 1 1 1 1
(1)DECHEMA Chemistry Data Series: Vapor-Liquid Equilibrium Data Collection; DECHEMA: Frankfurt, 1977.
Therefore, again, there is only one adjustable binary parameter for alcohol-alcohol mixtures. Table 6 lists the correlated results for the simplified and original SAFT equations of state. Figure 6 shows the correlated phase diagram for one system, the ethanol-butanol
mixture. Both the SAFT and SSAFT equations of state produce small errors in the pressure and vapor phase mole fractions, and the difference between the two versions of SAFT is small. However, because of its relative simplicity and fewer terms in the dispersion
1904 Ind. Eng. Chem. Res., Vol. 34,No. 5 , 1995 40 0
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Figure 5. Binary mixture of acetic acid and propanoic acid at 40 "C. Legend as in Figure 3. 0
-"
1.b
mole fraction of acetlc acid
I
600
15
400
300
200
Figure 7. Binary mixture of butanol and acetic acid at 35 "C. Legend as in Figure 3.
rameter in the dispersion term in the equation of state for acid-alcohol mixtures. The errors for the simplified and original SAFT equations of state are given in Table 6, and Figure 7 is the phase diagram for butanol-acetic acid mixture. The SSAFT equation can more accurately correlate the vapor-liquid equilibrium data for this mixture than the original SAFT equation, especially in the high butanol concentration region. In water-alcohol mixtures, water has three association sites and an alcohol has two, but only the unbonded electron pair can form a hydrogen bond with a hydrogen atom. In this case, for simplicity, all the cross-association energy and volume parameters are taken to be equal and are estimated as follows
100
0 0.00
and 0.25
0.50
0.75
1.00
mole fraction of ethanol Figure 6. Binary mixture of ethanol and butanol at 70 "C. Legend as in Figure 3.
term, SSAFT equation of state is easier to use for these calculations. For acid-alcohol mixtures, the acid is assumed to have one association site A1 and the alcohol to have two association sites A2 and Bz,and that site A1 can crossassociate with both sites A2 and B2. In the mixture, the chain length of association is shorter than for a pure alcohol, since if an alcohol and acid associate, the chain terminates. The cross-association parameters are assumed to be
We use only the one adjustable parameter kq in the dispersion term for the water-alcohol mixture. Figure 8 is the vapor-liquid equilibrium phase diagram of the water-ethanol mixture correlated using the original and simplified SAFT equations of state. Both equations of state represent the phase diagram well, though the SSAFT equation of state better describes the mixture in the azeotropic region. For water-acid mixtures, we assume that all three sites of water can associate with an acid, and that the values of the cross-association parameters are
and KA~Az= K B ~ A z= KC1A2 = (KAIC1
and
Consequently, we again have only one adjustable pa-
+ KA2-4z)/2
(49b)
Figure 9 shows the comparison of the correlated vaporliquid equilibrium results from the original and simplified SAFT equations of state for the water-propanoic acid mixture. A better correlation results from the
Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995 1905 60 0
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mole fraction of water
mole fraction of ethanol Figure 8. Binary mixture of water and ethanol at 50 "C. Legend as in Figure 3.
Figure 10. Binary mixture of water and acetic acid a t 40 "C. The value of parameter for simplified SAFT with one parameter is -0.179. The values of parameters for simplified with two parameters are -0.179 in dispersion term and 0.09 in association term.
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SSAFT equation of state in the azeotropic region than from the SAFT equation. Figure 10 shows the correlated results for both the simplified and original SAFT equations of state for the water-acetic acid mixture. In this case, neither of the equations of state can describe this phase diagram satisfactorily. Further, in an attempt to improve the correlation of the water-acetic acid mixture, we added an second binary parameter in the cross-association energy. The correlated results with two binary parameters are also shown in Figure 10; the improvement is small. Finally, we consider a mixture containing compounds that cannot form a hydrogen bond by themselves, but can only cross-associate with other compounds. For example, acetone has unbonded electron pairs on its oxygen atom but does not have a proton donor. In contrast, chloroform is a proton donor but not an electron donor. Neither acetone nor chloroform can self
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Figure 11. Binary mixture of acetone and chloroform at 50 "C. Legend as in Figure 3.
hydrogen bond; however, a hydrogen bond can form by cross-association. In this case the association parameters in either of the SAFT equations cannot be obtained from only the physical properties of pure compounds. Consequently, we fit one set of isothermal binary mixture vapor-liquid equilibrium data to obtain the binary parameter kjj and the two cross-association parameters. Since the association parameters are temperature independent by assumption in the model, after
1906 Ind. Eng. Chem. Res., Vol. 34, No. 5 , 1995
Table 7. Binary Interaction Parameters and Errors in Pressure and Vapor Mole Fractions for High Pressure Mixtures SSAFT SAFT mixture temp ("C) k, error % pressure error y kij error 5% pressure error y data sourcea propane
+ methanol
pentane
+ methanol
methanol + benzene
butane
+ ethanol
pentane
+ ethanol
methanol +water
ethanol
+ water
2-propanol
methanol
+ water
+ ethanol
39.95 69.95 99.95 100 125 150 25 100 120 140 160 180 200 220 25 50.6 72.5 100 125 150 25 100 150 200 250 25 150 200 250 275 300 325 25 150 200 250 275 25 100 120 140 160
0.056 0.063 0.067 0.070 0.071 0.078 0.017 0.020 0.012 0.005 -0.004 -0.010 -0.025 -0.017 0.032 0.034 0.037 0.032 0.035 0.035 -0.235 -0.202 -0.184 -0.166 -0.147 -0.202 -0.141 -0.121 -0.100 -0.081 -0.069 -0.056 -0.162 -0.096 -0.075 -0.051 -0.028 0.021 0.034 0.025 0.023 0.019
all data
4.92 2.33 1.45 1.90 1.46 1.99 0.46 2.50 2.76 4.47 4.63 4.76 4.68 2.73 1.75 0.35 1.20 0.34 0.92 0.56 3.96 2.08 0.70 0.38 0.73 1.57 1.25 1.06 0.86 1.35 1.25 2.08 1.11 1.13 1.85 2.69 2.31 0.28 0.57 0.30 0.17 0.32
0.0066 0.0126 0.0159 0.0273 0.0283 0.0258 0.0064 0.0107 0.0130 0.0226 0.0234 0.0202 0.0158 0.0156
0.0039 0.0109 0.0080 0.0131 0.0148 0.0181 0.0177 0.0440 0.0177 0.0091 0.0110 0.0186 0.0402 0.0507 0.0702 0.0097 0.0136 0.0117 0.0190 0.0280 0.0035 0.0054 0.0058 0.0052 0.0060
1.77
0.0180
0.036 0.042 0.043 0.053 0.052 0.050 0.011 0.012 0.007 -0.003 -0.012 -0.012 -0.020 -0.015 0.027 0.031 0.034 0.028 0.034 0.031 -0.137 -0.108 -0.080 -0.070 -0.057 -0.135 -0.087 -0.077 -0.066 -0.066 -0.067 -0.083 -0.126 -0.076 -0.068 -0.059 -0.046 0.005 0.008 0.003 0.002 0.002
5.25 2.64 1.66 2.27 2.46 3.71 0.51 2.09 2.25 3.81 3.87 3.67 3.38 1.83 2.29 1.31 2.84 0.50 0.86 1.30 0.76 1.57 0.83 1.06 2.30 0.38 1.15 1.06 3.15 4.84 5.39 7.76 2.03 1.49 2.94 4.31 4.36 0.19 0.42 0.27 0.14 0.25
0.0080 0.0185 0.0257 0.0173 0.0171 0.0140 0.0060 0.0095 0.0112 0.0205 0.0181 0.0152 0.0107 0.0130
0.0034 0.0112 0.0064 0.0036 0.0087 0.0130 0.0116 0.0296 0.0028 0.0051 0.0040 0.0079 0.0188 0.0178 0.0239 0.0125 0.0153 0.0106 0.0097 0.0113 0.0033 0.0029 0.0024 0.0019 0.0025
2.27
0.0114
1 1
1 2 2 2 3 4 4 4 4 4 4 4 5 5 5 6 6 6 3 7 7 7 7 3 8 8 8 8 8 8 3 8 8 8 8 3 9 9 9 9
(1)Galivet-Solastiouk, F.; Laugier, S.; Richon, D. Fluid Phase Equilib. 1986,28, 73-85. (2) Wilsak, R. A,; Campbell, S. W.; Thodos, G. Fluid Phase Equilib. 1987, 33, 157-171. (3) DECHEMA Chemistry Data Series: Vapor-Liquid Equilibrium Datu Collection; DECHEMA: Frankfwrt, 1977. (4) Butcher, K. I.; Medani, M. S. J. Appl. Chem. 1986,18,100-107. (5) Holderbaum, T.; Utzig, A.; Gmehling, J. Fluid Phase Equilib. 1991, 63, 219-226. (6) Campbell, S. W.; Wilsak, R. A.; Thodos, G. J. Chem. Thermodyn. 1987,19, 449-460. (7) Griswold, J.; Wong, S. Y. Chem. Eng. Prog. Symp. Ser. 1952,48(3), 18-34. (8) Barr-David, F.; Dodge, B. F. J . Chem. Eng. Data 1959,4, 107-121. (9) Butcher, K. L.; Robinson, W. I. J . Appl. Chem. 1966, 16, 289. a
determining the cross-association parameters, we then correlate vapor-liquid equilibrium data for the same mixture at other temperatures with only the one binary parameter ki. Figure 11 shows the correlations obtained by using the original and simplified SAFT equations of state and the experimental data a t three temperatures. We see that both equations of state can accurately describe the acetone chloroform mixture. C. High Pressure Vapor-Liquid Equilibrium. To test the utility of the SSAFT equation of state for high pressure vapor-liquid equilibria, we consider nine self-associating and cross-associating binary mixtures for which data are available over a large temperature and pressure range. We use the same association types and mixing rules as we did for low pressure mixtures. Therefore, there is only one adjustable binary parameter kg in the equation of state. Table 7 lists the results of correlations with both the SAFT and SSAFT equations of state; the average percentage errors in pressure are found to be smaller
+
than 5 % and the average errors of vapor phase mole fractions are small except a t very high temperature. Figures 12a and 13a are examples of the correlations of both equations of state compared with the experimental data. Both the original and simplified SAFT equations of state lead to good correlations of the experimental high pressure vapor-liquid equilibrium data for mixtures containing alcohols, though the errors become larger in the near-critical regions. Inspection of the optimized parameters in Table 7 for the SAFT and SSAFT equations shows that the binary parameter ku is temperature dependent. We find that for water-alcohol mixtures, the binary parameter k i in the SSAFT equation depends linearly on temperature as follows:
k,=A+BT
(50)
as shown in Figure 14a. Using the least squares
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methanol
Figure 12. Binary mixture of methanol and water several temperatures. (a) Correlations using the SAFT and SSAFT equations of state compared to experimental data (points). (b) Predictions using the SAFT and SSAFT equations of state. Table 8. Coefficients for the Equation k" = A + BT ("C) mixture A x lo2 B x 104
+ + + + + + + + +
propane methanol pentane methanol methanol benzene butane ethanol pentane ethanol methanol water ethanol water 2-propanol water methanol ethanol
4.9176 5.3000 2.8662 2.9174 2.6500 -24.292 -21.427 -17.531 2.2052
1.8333 1.6000 -2.1305 1.0451 0.6000 3.8705 4.7928 5.1618 0.0
method, we obtained the parameters A and B for each system given in Table 8. In Figures 12b and 13b, we show the vapor-liquid equilibrium coexistence curves for the methanol-water and the ethanol-water mixtures, respectively, obtained using values of KG from the above equation. For these mixtures, the calculations using the SSAFT equation of state are in good agreement with the experimental data. In contrast the values of the binary parameter K,j. obtained by correlation of these systems with the original SAFT equation are not such simple functions of temperature, as shown in Figure 14b, and consequently the predictions in Figures 12b and 13b with the SAFT equation are not quite as accurate as those with the SSAFT equation.
Conclusions A simpler SAFT equation of state has been developed. For pure compounds, this simplified SAFT equation of
state can correlate the vapor pressure and liquid density for both nonassociating and associating compounds very well. The errors in vapor pressure and liquid density calculated using the simplified SAFT equation of state are generally similar to, or slightly smaller than, those from the original SAFT equation of state. For self-associating mixtures, with only a single adjustable parameter, both the SAFT and the simplified SAFT equations of state produce small errors in pressure and vapor phase mole fractions for mixtures containing alcohols, though the simplified SAFT equation better correlates the vapor-liquid equilibrium of mixtures containing an organic acid. For cross-associating mixtures, the type of cross-association and the cross-association parameters have to be specified. Both models have only one adjustable parameter; the simplified SAFT equation of state usually leads to better correlated results than the original SAFT equation, and is simpler and easier to use because it contains many fewer terms. However, for some mixtures, such as the water-acetic acid mixture, neither equation of state correlates the data very well with only one adjustable binary parameter. High pressure binary vapor-liquid equilibrium data were also used to test the simplified SAFT equation of state. With one adjustable binary parameter, the simplified SAFT equation of state can be used t o accurately correlate such data for binary mixtures. Also
1908 Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995 100000
b looooo
a 200'C
lOOOo+
1000
j
100
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150'C
150'C
15'C
1
100 '23'C
0
experime
0
experiment
0
expcrime
0
cxpcrimcnt
- SAFT
._..SSAFT
10
1
1
10 0.00
I
0.25
0.50
0.75
1.00
fraction of ethanol Figure 13. Binary mixture of ethanol and water at several temperatures. (a)Correlations using the SAFT and SSAFT equations of state compared to experimental data (points). (b) Predictions using the SAFT and SSAFT equations of state. mole
fraction
of
mole
ethanol
a
b
0.000
-
/. I
-
-0.050
--
-0.100-
Y
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-
-0.200
X
-0.250 0.00
I 100.00
200.00
300.00
400.00
-0.250
I
I
I
i
l.00
Figure 14. Binary parameter k" vs temperature for water-alcohol mixtures. The points are correlated results, and the lines are calculated by linear least squares method. (a) SSAFT equation of state. (b)SAFT equation of state.
it is found that a linear relation exists between the
binary interaction parameter in the SSAFT equation
and temperature which allows this model to be used for extrapolation and prediction.
Ind. Eng. Chem. Res., Vol. 34,No.5, 1995 1909
Acknowledgment This research was supported, in part, by Contract No. DE-FG02-85ER13436 from the U.S. Department of Energy and Grant No. CTS-91123434 from the U.S. National Science Foundation, both to the University of Delaware. We also wish to thank Dr. Hasan Orbey for his advice and comments on this work.
Literature Cited Barker, J. A.; Henderson, D. J. Perturbation Theory and Equation of State for Fluids 11: a Successful Theory of Liquids. J.Chem. Phys. 1967,47,4714-4721. Carnahan, N. F.; Starling, K. E. Equation of State for Nonattracting Rigid Spheres. J. Chem. Phys. 1969,51,635-636. Chapman, W. G.; Gubbins, K. E.; Joslin, C. G.; Gray, C. G. Theory and Simulation of Associating Liquid Mixtures. Fluid Phase Equilib. 1986,29,337-346. Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. SAFT: Equation of State Solution Model for Associating Fluids. Fluid Phase Equilib. 1989,52,31-38. Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Znd. Eng. Chem. Res. 1990,29,1709-1721. Chen, S. S.;Kreglewski, A. Applications of the Augmented van der Waals Theory of Fluids. I. Pure Fluids. Ber. Bunsen-Ges. Phys. Chem. 1977,81,1048-1052. Huang, S. H.; Radosz, M., Equation of State for Small, Large, Polydisperse, and Associating Molecules. Znd. Eng. Chem. Res.
Joslin, C. G.; Gary, C. G.; Chapman, W. G.; Gubbins, K. E. Theory and Simulation of Liquid Mixtures 11.Mol. Phys. 1987,62,843-
860. Kim, C. H.; Vimalchand, P.; Donohue, M. D.; Sandler, S. I. Local Composition Model for Chainlike Molecules: a New Simplified Version of the Perturbed Hard Chain Theory. MChE J. 1986,
32,1726-1734. Lee, K.H.; Lombardo, M.; Sandler, S.I. The Generalized van der Waals Partition Function. 11. Application to the Square-Well Fluid. Fluid Phase Equilib. 1986,21,177-186. Mansoori, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W. Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres. J. Chem. Phys. 1971,54,1523-1525. Wei, S.;Shi, Z.; Castleman, A. W., Jr. Mixed Cluster Ions as a Structure Probe: Experimental Evidence for Clathrate Structure of ( H ~ O ) 2 f l +and (H20)21H+. J. Chem. Phys. 1991, 94,
3268-3270. Wertheim, M. S.Fluids with Highly Directional Attractive Forces. I. Statistical Thermodynamics. J. Stat. Phys. 1984a,35, 19-
34. Wertheim, M. S.Fluids with Highly Directional Attractive Forces. 11. Thermodynamics Perturbation Theory and Integral Equations. J.Stat. Phys. 1984b,35,35-47. Wertheim, M. S.Fluids with Highly Directional Attractive Forces. 111. Multiple Attraction Sites. J. Stat. Phys. 1986a,42, 459-
476. Wertheim, M. S.Fluids with Highly Directional Attractive Forces. IV.Equilibrium Polymerization. J.Stat. Phys. 1986b,42,477-
492. Received for review September 23, 1994 Accepted February 2, 1995
1990,29,2284-2294. Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules. Extension to Fluid Mixtures. Znd. Eng. Chem. Res. 1991,30,1994-2005. Jackson, G.; Chapman, W. G.; Gubbins, K. E. Phase Equilibria of Associating Fluids Spherical Molecules with Multiple Bonding Sites. Mol. Phys. 1988,65, 1-31.
@
IE9405605 @
Abstract published in Advance A C S Abstracts, April 1,
1995.
