Ind. Eng. C h e m . Res. 1991, 30, 1994-2005
1994
Registry No. H2S, 7783-06-4; dolomite, 16389-88-1.
Literature Cited Abbasian, J.; Rehmat, A.; Leppin, D.; Banerjee, D. D. Desulfurization of Fuels With Calcium-Based Sorbents. Fuel Process. Techno[. 1990a, 25, (1). Abbasian, J.; Rehmat, A.; Leppin, D.; Banerjee, D. D. An Advanced Coal GasificationJDesulfurization Process. Proceedings of the 25th IECEC, Reno, NV; AIChE: New York, 1990b. Borgwardt, R. H.; Roache, N. F. Reaction of H2S and Sulfur With Limestone Particles. Ind. Eng. Chem. Process. Des. Deu. 1984, 23. Jones, F. L.; Patel, J. G. Performance of Utah Bituminous Coal in the UGAS Gasifier. Presented at the Fifth EPRI Contractor's Conference on Coal Gasification, EPRI, Palo Alto, CA, 1985. Kamath, V. S.; Petrie, T. W. Rate of Reaction of Hydrogen Sulfide-Carbonyl Sulfide Mixture With Fully Calcined Dolomite. Enuiron. Sci. Technol. 1981, 15.
Keairns, D.; Newby, R.A,; O'Neill, E. P.; Archer, D. H. High Temperature Sulfur Removal System Development for Westinghouse Fluidized Bed Coal Gasification Process. Prepr. Pap.-Am. Chem. Soc., Diu. Fuel Chem. 1976, 21, (9). Ruth, L. A,; Squires, A. M.; Graff, R. A. Desulfurization of Fuels With Half-Calcined Dolomite: First Kinetic Data. Enuiron. Sci. Technol. 1972, 6, (12). Squires, A. M.; Graff, R. A,; Pell, M. Desulfurization of Fuels With Calcined Dolomite. 1. Introduction and First Kinetic Results. Chem. Eng. Prog. Symp. Ser. 1971,67. Weldon, J.; Haldipur, G. B.; Lewandowski, D. A.; Smith, K. J. Advanced Coal Gasification and Desulfurization with Calcium Based Sorbents. Prepr. Pap.-Am. Chem. Soc., Diu. Fuel Chem. 1986, 33, ( 3 ) .
Received for review February 1, 1991 Revised manuscript received April 24, 1991 Accepted May 9, 1991
Equation of State for Small, Large, Polydisperse, and Associating Molecules: Extension to Fluid Mixtures Stanley H. Huang and Maciej Radosz* Exxon Research and Engineering Company, Annandale, New Jersey 08801
Statistical associating fluid theory (SAFT)has been extended t o mixtures using rigorous statistical mechanical expressions for the hard-sphere, chain, and association terms. In contrast to previous equations of state, only the dispersion term requires mixing rules (averaging equations for SAFT parameters). We use two approaches t o deriving the mixing rules, a van der Waals one-fluid approximation (vdW1) rooted in conformal solution theory and a volume-fraction (vf) approximation similar to group-contribution equations of state. We test these approaches on 60 phase equilibrium data sets for asymmetric (small large) and associating binaries. While both sets of mixing rules, vdWl and vf, are found t o be adequate away from the critical region, the vf predictions are found t o be more accurate near the critical region.
+
Introduction Computer simulation plays a central role at different stages of process development. Such simulation sensitively depends on the quality of physical concepts underlying various process models. One of these concepts that is frequently invoked is to use an equation of state rooted in molecular thermodynamics to derive chemical potentials or fugacities needed to predict phase equilibria. For example, numerous single-stage and multistage, distillative and extractive separations are commonly predicted on the basis of vapor-liquid and liquid-liquid equilibria. To derive the chemical potentials for such predictions, one has to use an equation of state that is not only valid over the whole density range, from zero to the density of interest, but also applicable to the mixture of interest. We recently reported an equation of state for p u r e components (Huang and Radosz, 1990), based on the statistical associating fluid theory (SAFT). The reference part of SAFT includes the hard-sphere, chain, and association terms. The perturbation part accounts for relatively weaker, mean-field (e.g., dispersion and induction) effects. The reference part can be traced to the ideas and results of Wertheim (1984,1986) and Chapman et al. (1989, 1990), as described by Huang and Radosz (1990). The perturbation part is similar to that proposed initially by Alder et al. (1972) and also used by Chen and Kreglewski (1977) for pure compounds and by Simnick et al. (1979) for mixtures. The key result of our work on pure and 0888-5885191J 2630-1994$02.50f 0 0
polydisperse components was a practical, prototype equation of state with well-behaved, easy to estimate parameters. Our goal is to extend this equation of state to mixtures of small, large, chain, and associating molecules over the whole density range using only one binary adjustable parameter that is temperature independent. For reference, while specialized methods have been proposed for narrow groups of systems (for example, empirical activity coefficient models for associating molecules at low pressures, or cubic equations of state for small nonassociating molecules), there is no single approach available for mixtures of small and large, associating and nonassociating molecules, at low and high pressures. SAFT is unique in this respect. We will define SAFT in terms of the Helmholtz energy and present correlation results for low-pressure associating binaries away from the critical region and for high-pressure binaries near the critical region. In this work, we do not include systems containing cross associations, such as acids + water. These systems will be addressed later in the continuation of this work.
Equation of State The theoretical results underlying the equation of state are given in this section in terms of the residual Helmholtz energy am per mole, defined as am(T,Vm = aba(T,VJV) - aidralgu (T,V,N),a t the same temperature and density. AI1 other thermodynamic quantities can be derived fol1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991 1995 depends on the effective sphere diameter d and on rk. Mixtures of Chains. We make use of the pair correlation function given by eq 9 to determine the Helmholtz energy increment due to bonding:
lowing standard procedure as described, for example, by Topliss (1985). Symbols used here are consistent with those used in the earlier papers by Chapman et al. (1990) and by Huang and Radosz (1990). The residual Helmholtz energy is a sum of reference (major contribution) and dispersion (minor perturbation) parts:
=p
f
--
RT
+ adup
(1) The der, in turn, is a sum of hard-sphere, chain, and association terms: pf=
+ a h i n + aaeaoc
(2) These three reference terms can be extended to mixtures based on rigorous statistical mechanics. Mixtures of Hard Spheres. Our hard-sphere term for mixtures is based on a theoretical result of Mansoori et al. (1971), which for the Helmholtz energy equation is given by
,
- CXi(l - mi) In (gii(dii)b) i
(10)
where gii is evaluated for the interaction of two spheres i in a mixture of spheres, evaluated at the hard-sphere contact, given by eq 9. Equation 10 has been derived on the basis of the associating fluid theory, where the association bonds are replaced by covalent, chain-forming bonds, as explained in Chapman et al. (1990, and earlier papers). Mixtures of Associating Spheres. The Helmholtz energy due to association, used for pure components by Huang and Radosz (1990), is an average that is linear with respect to mole fractions, as described by Chapman et al. (1990):
(l2l3 + 3cic2& - 3tir2(l3)' RT
~p
r30
- r3I2
where XAi,the mole fraction of molecules i not bonded at site A, in mixture with other components, is given by where 5h+3 is a function of the molar density p. Since our reference fluid contains hard spheres, which can be bonded to form chains, we use the f functions proposed by Chapman et al. (1990), which are applicable to bonded spheres:
3k
=
( T N A V / ~P)CXimi(dii)
i
(4)
where X is the mole fraction, m is the number of segments per molecule, and d is the effective, temperature-dependent segment diameter. We note that for pure components, eq 3 reduces to (5)
where CBjmeans summation over all sites on molecule j : Aj, Bj, Cj, ..., and Z j means summation over all components. As we can see, XAidepends on the molar density pj: P j = Xjpmixture (13) and on the association strength A*@,: AAiBj
= gij(dij)"[exp(€AiBj/k~- l ] ( ~ i , ) ~ f i i (14) ~j
+
where aV = (aii u , , ) / 2 . We approximate the segment radial distribution !unction in eq 14 with eqs 8 and 9. Dispersion Term. A general expression for the dispersion term is
where the hard-sphere term for pure segments (ao,per mole of segments) is that proposed by Carnahan and Starling (1969):
- - - m-aOdbP RT RT adisp
(15)
Another useful expression derived by Mansoori et al. (1971), which we will invoke later, is that for the pair correlation function for a mixture of hard spheres (which approximate our hard segments):
where m is the segment number and ao&P is the dispersion Helmholtz energy, per mole of segments. The aOdhPterm is a function of the segment energy u / k T (Huang and Radosz, 1990). Hence, there are two parameters in the dispersion term that have to be generalized, u/kT and m. Our first approach is based on a conformal solution, van der Waals one-fluid theory (vdWl), that defines the molecular energy and size (volume) of a hypothetical pure fluid having the same residual properties as the mixture of interest. The vdWl averaging equations are often referred to as the vdWl mxing rules. Our mixing rule for u / k T is
Equation 8 for like segments becomes gii(dii)e" = g i i ( d i i p =
where
aob -=RT
4q - 3q2 (pure components) (I - 7)'
(6)
and where q is a segment packing fraction (reduced density): q = (?rNAV/6)pd3m (pure components) (7)
The hard-segment distribution function in eqs 8 and 9
uij = (1 - kij)(UiiUjj)l/'
(18)
1996 Ind. Eng. Chem. Res., Vol. 30,No. 8, 1991 Table I. vdWl Mixing Rules Tested at Low Pressures, Away from the Critical Region T,K P , bar kij data no. av dev in X1 n-Paraffins n-hexane (2) + 293.2 0.311-1.909 -0.010 10 n-butane 0.0134 298.2 -0.007 8 n-pentane 0.252-0.633 0.0062 303.2 -0.005 8 0.104-0.228 0.0131 n-heptane 0.311-1.909 0 293.2 10 n-butane 0.0205 0 298.2 8 0.252-0.633 n-pentane 0.0256 0 303.2 8 n-heptane 0.104-0.228 0.0353
av dev in Y1
data source'
0.0030 0.0050 0.0121 0.0097 0.0236 0.0096
1 2 3 1 2 3
Alkanols
methanol (1) + n-pentane n-hexane benzene tetrachloromethane ethanol (1)+ n-hexane n-heptane cyclohexane benzene ethylbenzene tetrachloromethane 1-propanol (1) + n-hexane n-heptane n-decane cyclohexane methylcy clohexane benzene toluene ethylbenzene tetrachloromethane methanoic (1) + benzene toluene ethanoic (1)+ n-octane cyclohexane benzene tetrachloromethane n-propanoic (1) + n-heptane n-octane cyc1ohexane benzene tetrachloromethane dimethylamine (1) + tetrachloromethane diethylamine (1) + tetrachloromethane methanamine (1) + tetrachloromethane
304.1-335.4 293.7-336.7 332.1-349.7 328.2 308.2 329.4-349.3
1.000 1.013 1.013 0.621-0.887 0.2714.378 1.013
0.048 0.044 0.010 0.012 0.012 0.024
14 14 12 6 5 24
0.0067 0.0096 0.0178 0.0226 0.0183 0.0065
0.0485 0.0073 0.0126 0.0122 0.0062 0.0069
4 5 6 7 7 8
331.9-349.2 345.0-365.2 338.7-347.1 341.6-350.1 352.3-393.2 318.15
1.013 1.013 1.013 1.013 1.013 0.257-0.453
0.024 0.030 0.032 0.006 0.018 0.014
11
10 18 10 10 8
0.0047 0.0050 0.0137 0.0346 0.0293 0.0056
0.0147 0.0030 0.0130 0.0214 0.0058 0.0045
9 10 11 12 13 14
339.4-362.8 358.0-362.7 368.2 348.8-369.0 360.9-366.9 350.8-369.3 366.1-377.3 363.2-377.6 370.3-392.1 347.1-364.0
1.013 1.013 0.169-0.913 1.013 1.013 1.013 1.013 0.933 1.013 1.013
O.Oo0
12 7 12 22 7 15 9 11 12 7
0.0569 0.0261 0.0128 0.0252 0.0242 0.0221 0.0226 0.0295 0.0168 0.0180
0.0111 0.0283 0.0058 0.0175 0.0152 0.0182 0.0263 0.0344 0.0127 0.0092
15 16 17 18 19 20 21 22 23 24
0.018 0.015 0.018 0.025 0.004 0.010 0.014 0.010 0.014 Acids
333.1 298.2
0.570-0.672 0.043-0.087
0.132 0.118
5 17
0.0183 0.0121
0.0527 0.1228
25 26
379.4-392.5 352.6-388.2 353.6-391.4 350.2-385.0
1.013 1.013 1.013 1.013
0.070 0.058 0.038 0.050
8 9 12 17
0.0205 0.0125 0.0184 0.0137
0.0653 0.0657 0.0386 0.0448
27 28 29 30
323.2 394.8-413.0 354.0-411.7 353.3-401.9 350.4-410.1
0.031-0.187 1.013 1.013 1.013 1.013
0.010 0.015 0.030 0.012 0.010
15 12 24 13 20
0.0118 0.0109 0.0182 0.0050 0.0072
0.0443 0.0066 0.0332 0.0307 0.0196
31 32 33 34 35
Secondary Amines 293.2
0.134-1.695
-0.045
25
0.0196
0.0147*
36
303.2
0.195-0.389
4.026
11
0.0219
0.0122
37
20 20
0.0269 0.0213 0.0163
0.0049* 0.0046* 0.0057*
38 38 38
Primary Amines 253.2 273.2 293.2
0.025-0.476 0.069-1.205 0.169-2.637
-0.028 -0.031 4.033
20
Data Sources (1)Hoepfner, A,; Kreibich, U. T.; Schaefer, K. Ber. Bunsen-Ges. Phys. Chem. 1970, 74, 1016. (2) Chen, S. S.; Zwolinski, B. J. J . Chem. SOC.,Faraday Trans. 1974, 70, 1133. (3) Smyth, C. P.; Engel, E. W. J. Am. Chem. SOC.1929, 51, 2646. (4) Tenn, F. G.; Missen, R. W. Can. J. Chem. Eng. 1963, 41, 12. (5) Rad, J. D.; Code, R. K.; Best, D. A. J . Chem. Eng. Data 1972, 17, 211. (6) Nagata, I. J. Chem. Eng. Data 1969, 14, 418. (7) Scatchard, G.; Wood, S. E.; Mochel, J. M. J . A m . Chem. SOC.1946, 68, 1957. (8) Hipkin, J.; Myers, H. S. Ind. Eng. Chem. 1954, 46, 2524. (9) Sinor, J. E.; Weber, J. H. J. Chem. Eng. Data 1960, 5, 243. (10) Van Ness,H. C.; Soczek, C. A.; Kochar, N. K. J . Chem. Eng. Data 1967, 22, 346. (11) Yuan, K. S.;Lu, B. C-Y.; Ho, J. C. K.; Keshpande, A. K. J . Chem. Eng. Data 1963,8, 549. (12) Ellis, S. R. M.; Clark, M. B. Chem. Age India 1961, 12, 377. (13) Ellis, S. R. M.; Spurr, J. M. Br. Chem. Eng. 1961, 6, 92.
Ind. Eng. Chem. Res., Vol. 30,No. 8, 1991 1997 Table I (Continued) (14)Barker, J. A.; Brown, J.; Smith, F. Discuss. Faraday SOC.1953,15, 142. (15)Prabhu, P. S.;Van Winkle, M. J. Chem. Eng. Data 1963,8,210. (16)Gurukul, S. M. K. A.; Raju, B. N. J. Chem. Eng. Data 1966,11,501. (17)Ellis, S. R. M.; McDermott, C.; Williams, J. C. L. h o c . Int. Symp. Dist., Inst. Chem. Eng. London 1960. (18)Ocon, J.;Tojo, G.; Bao, M.; Arce, A. An. Real. SOC.Espan. De Fis. Y Quim. 1973,69,1169. (19)Raju, B. N.;Rao, D. P. J. Chem. Eng. Data 1969,14,283. (20)Ocon, J.; Tojo, G.; Bao, M.; Arce, A. An. Real. SOC.Espan. De Fir. Y Quim. 1973,69,1177. (21)Lu, B. C. Y. Can. J. Technol. 1957,34,468. (22)McDermott, L. Thesis, Birmingham, 1964. (23)Ellis, S.R. M.; Froome, B. A. Chem. Ind. 1954,237. (24)Carley, J. F.; Bertelaen, L. W. Ind. Eng. Chem. 1949,41,2806. (25)Vreveky, M. S.;Geld, N. A.; Shchukarev, S. A. Z . Phys. Chem. 1928,133,377. (26)Lakhanpal, M. L.; Mandal, H. G.; Lal, G. Indian J. Chem. 1975,13,1309. (27)Zieborak, K.;Brzostowski, W. Rocz. Chem. 1958,32,1145. (28)Baradarajan, A.; Satyanarayana, M. Indian J. Technol. 1967,5,264. (29)Haughton, C. 0.Br. Chem. Eng. 1967,12,1102. (30)Wisniak, J.; Tamir, A. J. Chem. Eng. Data 1975,20, 168. (31)Schuberth, H.2.Phys. Chem. (Leipzig) 1981,262,945. (32)Johnson, A. I.; Furter, W. F.; Barry, T. W. Can. J. Technol. 1954,32,179. (33)Svitilova, M.; Malijevska, I,; Malijevsky, A.; Pick, J. Coll. Czech. Chem. Commun. 1980,45,1748. (34)Malijevska, I.; Pick, J. Coll. Czech. Chem. Commun. 1978,43,2096. (35)Wisniak, J.; Tamir, A. J. Chem. Eng. Data 1967,21,88. (36)Wolff, H.; Hoeppel, H. E. Ber. Bunsen-Ges. Phys. Chem. 1966,70,874. (37)Kilian, H.,Bittrich, H. J. 2.Phys. Chem. (Leipzig) 1965,230,383. (38)Wolff, H.; Wuertz, R.Ber. Bunsen-Ges. Phys. Chem. 1968,72,101. Data compiled from: Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Collection. Chemistry Data Series; DECHEMA: Frankfurt/Main, Federal Republic of Germany, 1977. bValues of experimental Y1 were calculated by the Wilson model.
where kij is an empirical binary parameter, fitted to experimental data. Since our segment volume parameters Vooand uo are defined on a per-segment basis and hence do not vary much from molecule to molecule, especially for large molecules, we will use the segment number m as a measure of the molecular size. Therefore, our mixing rule for m (the average segment number for mixtures) is
m = CCXiXjmij
(19)
mij = Y2(mi+ mi)
(20)
i J
where
In addition to the vdWl set of mixing rules given above and in the spirit of the group-contribution equations of state of Skjold-Jorgensen (1984) and Pults et al. (1989), we define and explore a modified set with a more general expression for u:
where f is a volume fraction, defined as Ximiuoi fi = C X j m j u o j j
To s u m up, we have introduced two seta of mixing rules: vdWl (eqs 1620) and volume fraction (vf, eqs 18-22), both of which utilize only one adjustable binary parameter kij given in eq 18. We will discuss results obtained for the vdWl and vf mixing rules in the following two sections. We note that in contrast to conventional approaches where the mixing rules have usually been applied to all the terms, we apply the mixing rules to only one term, i.e., the dispersion term. In a sensitivity study, we also tested generalized forms of eqs 20 and 22 for m and u / k T ,respectively. Equation 20 is generalized by introducing a second adjustable parameter 1, for m:
and eq 22 is generalized to Xi(miuoi)y fi = CXj(mjuoj)’
(24)
i
where y is a specified exponent for the molecular volume. In the case that y is specified as 2/3, the fraction f represents surface fraction. These two approaches, utilizing y and lij, are tested on one system for sensitivity analysis. In all other examples in this work we use only the vdWl and vf mixing rules with only one binary parameter, kip The equation of state presented above has been used to correlate vapor-liquid equilibria of many real fluid mixtures. The purpose for the correlation was to test the mixing rules, vdWl and vf, discussed in this section. The method was to minimize the deviations in phase compositions by adjusting the binary parameter kij in eq 18. The computer software developed for this work utilizes a parametrization procedure proposed by Toplias (1985) and described in detail for SAFT in the Appendix. All the pure-component parameters used were those given by Huang and Radosz (1990). The computational requirements are similar to other noncubic equations of state, such as the perturbed hard chain theory, which means that these requirements are acceptable for a vast majority of applications. Although in some simulation applications the computational requirements can be a limiting factor, for example, in reservoir simulation, we are much more concerned with the predictive power and the ease of estimating the equation of state parameters, which are important strengths of
SAFT. vdWl Mixing Rules Tested at Low Pressures, Away from the Critical Region Representative real binary fluids have been selected for low-pressure tests,for example, hydrocarbon-hydrocarbon, alkanol-hydrocarbon, carboxylic acid-hydrocarbon, and amine-hydrocarbon. For each of the selected binaries, Table I lists temperature and pressure ranges, optimized value of kip number of data points used in fitting, average deviation in liquid phase (Xl) and vapor phase (Yl) mole
1998 Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991
0
0.2
0.4
0.6
0.8
1.0
Mole Fraction of (1) in Liquid
1901 0.8
1.o
0.9 Mole Fiaction of Ethene
Figure 1. SAFT correlation for three n-hexane binaries. Curves shown are predictions by optimal k, values. Setting k, equal to zero deteriorates only slightly the correlation quality.
I
1.0,
J 0 0 0 Experimental Data
- SAFT
Prediction
+
0
10
0
0.2
0.4
0.6
0.8
1.0
Mole Fraction of (1) in Liquid
Figure 2. SAFT correlation for three binaries of benzene + alkanols. Solid curves represent SAFT predictions. Dashed portions represent the azeotropic regions, which are not predicted from SAFT, only roughly estimated.
5 0
0.2
Ethene n-Eicorane Data by de Loos et al. (1984) SAFT Prediction
0.4
0.8
0.6
'
3
Mole Fraction of Ethene
Figure 4. Almost no differences can be distinguished at low pressures between the vdW1 (eqs 16-20) and vf (eqs 18-22) mixing rules (part b). However, the vf mixing rules are more accurate near the critical region shown in insert (part a). In both cases we use only one adjustable binary parameter, k,,.
1 .O[
Methane t n-Hexadecane o o Data by Lin et al. (1980) SAFT Prediction 462 K
/
Mole Fraction of (1) in Liquid
Figure 3. SAFT correlation for three binaries of benzene + alkanoic acids. Solid curves represent SAFT predictions. Dashed portions represent the azeotropic regions, which are not predicted from SAFT, only roughly estimated.
fractions, and data source. We note that in all cases only one binary parameter has been used to regress experi-
mental vapor-liquid equilibrium data. As expected, hydrocarbon-hydrocarbon binaries are well represented, as shown in Figure 1for three hexane binaries, with only small value of kip The quality of prediction does
0
0.2
0.4
0.6
0.8
1.0
Mole Fraction of Methane
Figure 5. SAFT with the vf mixing rules correlates methane n-hexadecane system, except in the critical region.
+
Ind. Eng. Chem. Res., Vol. 30, No. 8,1991 1999
+
0
Carbon Dioxide Methanol Data by Ohgaki & Katayama (1976) o n SAFT Prediction
0
0
I
o,-.-o.-
B
0*
298 K
li
/ -
+
Carbon Dioxide 1-Hexanol Data by Nickel & Schneider (1989) SAFT Prediction
o
10
1
100
800
1-Hexanol Concentration. mgfml
I
0
0.2
0.6
0.4
1.0
0.8
Mole Fraction of Carbon Dioxide
200
Figure 6. SAFT with the vf mixing rules correlates COz + methanol system. 6o
1
I
+
o
,/
Carbon Dioxide 2-Propanol Data by Radorz (1986) SAFT Prediction 394
Figure 8. SAFT with the vf mixing rules correlates COO+ 1-hexanol system, except in the critical region. Volumes calculated by SAFT are used to convert the calculated phase compositions to volumetric units (mg/mL). Deviations observed in the vapor phase may result from using calculated densities for the vapor phase.
; ?'
0
0
o o
O
0
.p
0
0 '
\
"gp
.B
7
0
II ?
0
I
Carbon Dioxide
+ 1-Hexanol
o Data by Nickel & Schneider (1989)
- SAFT Prediction
01 0
"
0.2
I
'
0.4
'
'
0.6
'
I
0.8
'
1
1.o
Monomer Fraction
Figure 9. SAFT with the vf mixing rules predicts the monomer fraction in the vapor phase but underpredicta that in the liquid phase. It is possible that the aeeociation parameters obtained from fitting pure-component bulk properties may have to be readjusted to the mixture properties.
0
0.2
0.4
0.6
0.8
1.0
Mole Fraction of Carbon Dioxide
Figure 7. SAFT with the vf mixing rules correlates COP + 2propanol system, except in the critical region.
not deteriorate significantly when kij is set equal to zero. Alkanol-hydrocarbon binaries that exhibit azeotropic behavior are also well represented. This is shown in Figure 2 for three benzene binaries in vapor-liquid mole fraction coordinates. Only the solid parta of the curves have been calculated from SAFT because our preliminary flash program used in this work (to be refined) cannot handle regions near the azeotrope. A similar method of presenting results has been used for benzene binaries with carboxylic acids shown in Figure 3. In general, all the low-pressure vapor-liquid equilibria in the nonassociating and associating binary fluids we have
tested can be accurately predicted using one adjustable parameter, kij. We found that also high-pressure vaporliquid equilibria can be predicted this way, except near the critical region. This is shown in Figure 4 for ethene and eicosane at 423 K. As we approach the critical region the vdWl mixing rules tend to overpredict the equilibrium pressure. This is typical for many other binaries we have tested. While it is recognized that analytical equations of state are not very accurate at near-critical conditions, we found that improvement can be accomplished using the vf mixing rules.
vf Mixing Rules More Accurate Near the Critical Region The degree of improvement at near-critical conditions is illustrated for the ethene-eicosane system in Figure 4. Both vdWl and vf sets of mixing rules are comparable away from the critical region. While not perfect, the vf
2000 Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991 Table 11. vf Mixing Rules Tested on High-pressure Systems T,K P, bar kij data no. av dev in X1 av dev in Y1 I. Binary Systems of Non-Self-Associating Fluids n-decane (2) + 27 0.0098 0.0128 0.085 344-51 1 13.8-327.5 methane 277-510 3.4-110.3 0.030 31 0.0099 0.0141 ethane 0.0174 0.012 21 0.0110 344-511 3.4-68.9 propane 0.150 24 0.0110 0.0128 13.8-172.4 carbon dioxide 344-51 1 n-hexadecane (2) + 0.118 21 0.0158 0.0077 462-623 20.3-252.6 methane n-eicosane (2) + 0.030 16 0.0069 32C-360 11.0-132.6 ethane 0.076 26 0.0184 323-423 9.5-233.6 ethene n-tetracontane (2) + 0.101 25 0.0176 373-448 9.0-377.4 ethene 1-methylnaphthalene (2) + 0.073 25 0.0129 0.0282 464-704 20.5-251.3 methane carbon dioxide 373-523 39.7-159.9 0.145 11 0.0220 0.0019 carbon dioxide (1) + methanol ethanol 1-propanol 2-propanol methane (1)+ ethanol 1-propanol water ethane (1) + ethanol 1-propanol water propane (1) + ethanol
data source
1
2 3 4 5 6 7 7 8 9
11. Binary Systems of Non-Self-Associating (1) + Self-Associating (2) Fluids 298-313 313-333 313-333 317-394
5.8-80.6 5.1-99.5 5.2-104.1 13.8-120.7
0.032 0.062 0.062 0.052
16 12 16
0.0199 0.0161 0.0185 0.0315
0.0034 0.0076 0.0050 0.0194
10 11 11 12
21
313-333 313-333 344-511
18.1-104.6 14.1-102.0 13.8-689.5
0.050 0.025 0.250
10 9 27
0.0012 0.0037
0.0004 0.0005 0.0047
11
313-333 313-333 344-511
13.1-74.8 13.5-67.4 13.8-689.5
0.014 0.005 0.297
11 11 27
0.0168 0.0125
0.0106 0.0023 0.0041
11 11 14
325-400
5.7-46.9
-0.070
16
0.0478
0.0304
15
11 13
Data Sources (1) Reamer, H. H.; Olds, R. H.; Sage, B. H.; Lacey, W. N. Ind. Eng. Chem. 1942,34, 1526. (2) Reamer, H. H.; Sage, B. H. J . Chem. Eng. Data 1962, 7, 161. (3) Reamer, H. H.; Sage, B. H. J . Chem. Eng. Data 1966, 11, 17. (4) Reamer, H. H.; Sage, B. H. J . Chem. Eng. Data 1963,8, 508. (5) Lin, H. M.; Sebastian, H. M.; Chao, K. C. J . Chem. Eng. Data 1980,25, 252. ( 6 ) Peters, C. J.; De Roo, J. L. Fluid Phase Equilib. 1987, 34, 287. (7) de Loos, T. W.; Poot, W.; Lichtenthaler, R. N. Ber. Bunsen-Ges. Phys. Chem. 1984,88, 855. (8) Sebastian, H. M.; Simnick, J. J.; Lin, H. M.; Chao, K. C. J . Chem. Eng. Data 1979, 24, 149. (9) Gregg, C.; Huang, S. H.; Radosz, M., unpublished results. (10) Ohgaki, K.; Katayama, T. J . Chem. Eng. Data 1976,21, 53. (11) Suzuki, K.; Sue, H.; Itou, M.; Smith, R. L.; Inomata, H.; Arai, K.; Saito, S. J . Chem. Eng. Data 1990, 35, 63. (12) Radosz, M. Chem. Eng. Data 1986,31,43. (13) Olds, R. H.; Sage, B. H.; Lacey, W. N. Ind. Eng. Chem. 1942,34, 1223. (14) Reamer, H. H.; Olds, R. H.; Sage, B. H.; Lacey, W. N. Znd. Eng. Chem. 1943, 35, 790. (15) Gomez-Nieto, M. A. Ph.D. Dissertation, Northwestern University, Evanston, IL, 1977.
set predicts a more realistic phase boundary pressure around the critical point. The extent of deviations in the critical region varies from system to system. Representative examples are shown for a methane-hexadecane system (a short-long binary) in Figure 5, for a COzmethanol system (a quadrupolar-dipolar, associating binary) in Figure 6, and for a C02-propanol-2 system (also a quadrupolar-dipolar, associating binary) in Figure 7. In all these cases, pressure is plotted versus mole fraction because the experimental phase compositions have been measured in mole fractions. For a C02-hexanol system shown in Figure 8, Nickel and Schneider (1989) measured phase compositions spectroscopically (near-infrared, an important novel approach) in volumetric units (mg/mL). Therefore, we had to use the equation of state estimated densities in order to convert the predicted mole fractions into mg/mL, to be able to compare them with the experimental data. Since we have no way to verify these densities, especially the vapor-phase density, we consider the quality of the calculated curves to be reasonably good. In addition to the overall hexanol concentration, Nickel and Schneider also measured the hexanol monomer con-
centration. As shown in Figure 9, the equation of state underpredicts the liquid monomer concentration (overpredicts the degree of association). There are several possible sources of this discrepancy that can be considered. For example, uncertainty of the calculated mixture densities, as mentioned above, and the use of the association parameters ( t and K ) derived from the pure-component properties. By contrast, the predicted monomer fraction in the vapor phase is in good agreement with the experiment. Since the monomer fraction in the vapor phase is less sensitive to t and K (because of low density), perhaps the c and K derived for hexanol from its pure-component properties should be readjusted on the basis of the association data obtained for mixtures. In future work, we will address this problem of how to reconcile the predicted monomer fraction with the bulk concentrations. One of the more difficult systems to predict from equations of state is water-hydrocarbon. Although a detailed study of such systems is beyond the scope of this paper, we report testing SAFT on solubility of water in methane and ethane a t different temperatures (we have not tested solubilities of hydrocarbon in water though). As
Ind. Eng. Chem. Res., Vol. 30,No. 8, 1991 2001 344 K
Ii
3ia K
444 K
511 K
l o o o r5
$
I
at 423 K
100
2 50
2
P
10 I
d
I
I
5
0.001
5
0.01
I
0.1
5
1
Mole Fraction of Water in Ethane-rich Phase
+
Figure 10. SAFT with the vf mixing rules correlates solubility of water in ethane-rich phase. Only one adjustable parameter, kij, is used in correlating this binary system of ethane
300r
Ethene n-Eicosane Data by de Loor et al. SAFT Prediction
+ water.
I
1501 0.75
2-
0.000 - !I..I == 0.010
/
d
0.80
0.90
I
1.00
Mole Fraction of Ethene
at 423 K
Figure 12. Sensitivity analysis for 1, (eq 23) for the ethene + ei-
cosane system near the critical region.
Detailed correlation results for these and many other binaries, nonassociating and associating, at high pressures are given in Table 11, which lists temperature and pressure ranges, optimized values of kij, number of data points used in fitting, average deviation in liquid phase (Xl) and vapor phase (Yl) mole fractions, and data source.
+
Ethene n-Eicorane m 0 Data by de Loor et al. (1984)
s
0
0.2
0.4
0.6
0.8
Sensitivity Analysis for Mixing Rules 1.0
Mole Fraction of Ethene
111
0.8
+
,all values, k.. = 0.0 ,= 1.05, k!! = 0.081 , = 1.00, k!! = 0.074 ,= 0.50, k l = 0.039
0.9 Mole Fraction of Ethene
In all the calculations discussed above, we use only one adjustable binary parameter, kij, given by eq 18. In this section, we briefly explore, by way of sensitivity analysis, the other two parameters, y (eq 24) and lij (eq 23). Our sensitivity analysis is based on the results of correlating the ethene n-eicosane binary system at 423 K. The adjustable parameters are obtained by fitting the liquidphase composition at 23 bar; compositions at other pressures are predicted. Results for kii and y (the segment energy, eqs 18 and 24) are presented in Figure 11. €?,esults for Zij (the segment number, eq 23) are presented in Figure 12. The critical region in Figure 11 is enlarged in an insert for details. When kij is equal to zero, the calculated results are practically the same for all y values. The solid curve marked ref shows the common results for reference. However, different y values strongly affect the correlation quality especially in the critical region. Roughly, the larger the y value, the more the two-phase region is expanded. In this work, y = 1 is selected. Figure 12 shows the l!j effects. Because 1, affects the segment number m,which is an independent multiplier for the segment energy uodMp, 1, effects are independent of kij.and y values. In this figure, only the curve of y = 1.05 is shown. Roughly, positive values of 1, suppress the two-phase region near the critical point, whereas they enlarge the two-phase region in the low-pressure region. In other words, lij can be used to adjust the slope of the coexistence (pressure-composition) curves.
1 .o
Figun, 11. Sensitivity analysis for k~ and y (eqs 18 and 24) for the
ethene + eicosane system away from the critical region (top) and near the critical region (bottom).
shown in Figure 10, the solubilities of water in ethane calculated from SAFT are in excellent agreement with the experimental data. A similar agreement has been obtained for methane (Table 11). It is worth noting that only one adjustable, temperature-independent parameter (kij)has been used to predict the ethane-water and methanewater data.
Conclusions The SAFT equation of state has been extended in this work to mixtures of small, large, and associating molecules
2002 Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991
at low and high pressures. This extension has been accomplished using rigorous statistical mechanical expressions for the hard-sphere, chain, and association terms. Only the dispersion term required mixing rules, and only one binary, temperature-independent parameter is required to represent the experimental data that usually are difficult to predict from equations of state. Both sets of mixing rules, the van der Waals one-fluid approximation vdWl and the volume-fraction approximation vf, similar to those used in the group-contribution equations of state, have been found to be adequate away from the critical region. However, near the critical region, the vf mixing rules have been found to be more accurate.
Acknowledgment Helpful comments and stimulating discussions with Professor Keith Gubbins, John Walsh, and Karl Johnson, Cornel1 University, are gratefully acknowledged. We are also grateful to the group of Professor John Prausnitz, the University of California, Berkeley, for letting us use their software package for flash calculations. A preliminary account of this work was presented at the Annual Meeting of the American Institute of Chemical Engineers, Chicago, 1990.
Nomenclature
Lennard-Jones segment diameter (temperature independent), A LA= summation over all the sites (starting with A) u =
Superscripts A, B, C , D, ... = association sites res = residual seg = segment assoc = associating, or due to association hs = hard sphere ideal = ideal gas
Appendix Example of SAFT Parametrization. A general parameterization concept used in this work is that proposed by Topliss (1985). The specific SAFT parametrization procedure is given in this section for the vdWl mixing rules as an example. Since this section is intended to be complete and self-contained, a few of the equations given and referenced earlier may be repeated here. The SAFT equation can be expressed as follows: ares
=
+ achain + aassoc
aseg
(AI) The segment term is expanded into hard-sphere and dispersion parts: = ab + adisp
(A21 For a mixture system, the hard-sphere term ahais given by
a = molar Helmholtz energy (total,res, seg, bond, assoc, etc.),
per mole of molecules a. = segment molar Helmholtz energy (seg), per mole of
segments
d = temperature-dependent segment diameter, A k = Boltzman's constant = 1.381 X J/K m = effective number of segments within the molecule (seg-
ment number) mu" = volume occupied by 1 mol of molecules in a closedpacked arrangement, mL/mol M = number of association sites on molecule molar = molar with respect to molecules N = total number of molecules NAv = Avogadro's number == 6.02 X molecules/mol P = pressure R = gas constant segment molar = molar with respect to segments T = temperature, K u / k = temperature-dependent dispersion energy of interaction between segments, K u o / k = temperature-independent dispersion energy of interaction between segments, K V = total volume u = molar volume, Pq = liquid molar volume, mL/mol of bulk fluid uo = temperature-dependent segment volume, mL/mol of segments uoo = temperature-independent segment volume, mL/mol of segments X = mole fraction XA= monomer mole fraction (mole fraction of molecules not bonded at site A) 2 = Pu/RT, compressibility factor = volume of interaction between sites A and B AAB = "strength of interaction" between sites A and B, A3 cAB = association energy of interaction between sites A and B, per molecule, J 7 = (p/6)pnmd3= (*NA,/6)pmd3 pure-component reduced density, the same for segments and molecules p = pn/NAv,molar density, mol/A3 p n = number density (number of molecules in unit volume), A-3
where
[k
is defined as fk
=
(XNav/
6)P CXimi(dii)
644)
1
The dispersion term adispfor a mixture system is given by adiap/R T =
mC Dij[u/ k TIi[q / T I ] i l
(A51
where Dii represents the universal constants proposed by Chen and Kreglewski (1977). The vdWl mixing rules for m and u l k T are
where =
( U O ) ~ ~
1'/2[(~0)i1/3 + ( ~ ~ ) ~ ~ / ~ ] (A81 1 ~
uij = (1 - kij)(UIiUjj)1/2
Terms for chain formation and association are achin/RT = C X i ( l- mi) In gii(dii)
(A9) (A101
where gii is a special case of (A15) by setting i and j equal;
where X k , the mole fraction of molecules i not bonded at
Ind. Eng. Chem. Res., Vol. 30, No. 8,1991 2003 site A, in mixture with other components, is given by Xi%= [1 + pCCXjXBjAAiBj]-l (A121
partial derivatives are fully discussed by Topliss (1985):
r .
C3
3BC
3BC.
C3
i Bj
The association strength AhBj is A A P j = ( aij)3fiBjgij( dij) [exp (€AiB]/ k 79 - 11 (A13)
L
where aij
=
+ Ujj)/2
(Uii
(A141
The radial distribution function gij(dij) is given by gij(dij) = 3diidjj 52 1
1-5;
+--dii +
djj
(1 - {3)2
(A151 The parametrization of the SAFT equation will introduce eight parameters, A-H, whose definitions are
A = CXimi(di)O
to= (a/G)pA
(A16)
i
3B. 3C2 D3 0 2 1-5
(%),=
A
C3
D
D
3C2
+-(1-02 O2 + In (1 - 5) D2
-5 + 3 2 - 5")
3BC
+
T
--b 2c3
--D3 (1-
+
2c3In ( I - 5)
5)3 D3
(A28)
C3
P - D3 -(2 + n (1 -
n2
+
:)'
(
$).r?jDijGi(
(A29)
F = aCb/RT = C X i ( l - mi) In gii(dii) (A21) i
(c) =1 RT
Although parameters A and E are equivalent, for the convenience of programming, we retain the two different notations. After parametrization, the residual Helmholtz energy can be expressed simply by
F
($,=1
A + - -6BC D
C3
6BC
+
(1 -
-arw --
(y + RT
Ab 1
A)
+ P(-3C3 - 3BC - 2A) + ?(A D2
D
-
b
$)
6C3 I T
(1- 7 1 3
r
where is the abbreviation for {> From this point on, what is needed is to obtain various partial derivatives and code them. Because we are interested in phase equilibrium calculations, we require only derivatives with respect to density and compositions. The relations between thermodynamic quantities and the
i
2C3
T T 5 5 -- O2
D2
o3
+
2004 Ind. Eng. Chem. Res., Vol. 30,No. 8, 1991
(
$,c
3c2 -(2 D3
=
-
3B 3C2 -P - -(2 + !3 D2 D3 +
+
6C2 -i-
1
- -D3 -' (1 - {)3
J
6c2In (1 - {) (A51) D3
(5)cE (g)cF(e)cc(e) =
= 0 (A52)
=
RT
RT
CH
1
r + $1
c3
-
6BC t) -p 0 2
(1 - .02
+
2C3 D3
- -(1+
(1 -
n3
{)
+
+
6BC D3
8C3
-P--i--yP D4 (1 - {I3
2C3 D
6C3
-.?
+- O4 + (1 - !34
6C3
(g)DE( s)' (g)DF (g)Dc( I)' = $)fFjDijGi(
=0
= z)E??ijDijGi-l(
(A541
(A551
(A56)
(A571
(g)AA ( =
RT
=
RT
RT
(e) =0 RT
(A43)
AC
RT
AH
(A45)
(gjBB =0
(A471
cc
-6C/D2 1- {
=-+-
6C/D2 (1 -
{)2
6C
+In (1 - {) D2
Partial derivatives of parameters: (A50)
( A ) , = ( A ) p p= 0
(A65)
Ind. Eng. Chem.Res.,Vol. 30, NO. 8, 1991 2005
+
(Xh)x, = -(XAi)2[pCXBhAAiBk + p z C X j ( X BJ)xhAhBj ' Bh
1 Bj
(A911 Equations 83, 86, and 89 are solved numerically.
Literature Cited Alder, B. J.; Young, D. A.; Mark, M. A. J. Chem. Phys. 1972,56, 3013. Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969,51,635. Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Fluid Phase Eyuilib. 1989,52, 31-38. Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. I d . Eng. Chem. Res. 1990,29, 1709. Chen, S. S.; Kreglewski, A. Ber. Bunsen-Ces. Phys. Chem. 1977,81, 1048. Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29,2284-2294, Mansoori, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W. J. Chem. Phys. 1971,54, 1523. Nickel, D.; Schneider, G. M. J . Chem. Thermodyn. 1989,21,293. Pults, J. D.; Greenkorn, R. A.; Chao, K. C. Chem. Eng. Sci. 1989,44, 2553. Simnick, J. J.; Lin, H.M.; Chao, K. C. Ado. Chem. Ser. 1979, 182, 209. Skjold-Jorgensen, S. Fluid Phase Equilib. 1984, 16, 317. Topliss, R. J. Techniques to Facilitate the Use of Equations of State for Complex Fluid-Phase Equilibria. Ph.D. Dissertation, University of California, Berkeley, 1985. Wertheim, M. S. J. Stat. Phys. 1984, 35, 19, 35. Wertheim, M. S. J. Stat. Phys. 1986,42,459,477. Wertheim, M. S. J. Chem. Phys. 1986,85, 2929. Wertheim, M. S. J. Chem. Phys. 1987,87, 7323.
Received for review December 11, 1990 Revised manuscript received February 15, 1991 Accepted March 13, 1991
