2388
Ind. Eng. Chem. Res. 1992,31,238&2394
Altiokka, M. R. Flow Pattern in the Annulus of the Scraped Surface Heat Exchange Type Reactors. Submitted for publication in htrkieh J. Eng. Environ. Sci. 1991. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. The Equations of Change for Isothermal Systems. In Transport Phenomena; Wiley: New York, 1960; Chapter 3, pp 82-101. Bremter, D. B.; Niasan, A. H. The Hydrodynamicsof Flow between Horizontal Concentric Cylinders-1. Chem. Eng. Sci. 1958, 7, 216211. Dykes, D. J. Studies of the Sulphation of a-Olefins. Ph.D. Thesis, Aston University, 1980. Froishteter, G. B.; et al. Relationships of Laminar Flow and Power Consumption in Scrape Surface Equipment. J. Appl. Chem. USSR 1978,51 (l), 100-105.
Leung, L. S. Power Consumption in a Scraped Surface Heat Exchange. Trans. Znst. Chem. Eng. 1967,45, 179-181. Skelland, A. H. P.; Leung, L. S. Power Consumption in a Scraped Heat Exchange. Br. Chem. Eng. 1962, 7,264-267. Trommelen, A. M. Physical Aspects of Scraped Surface Heat Exchange. Thesis, Technische Hogeschool Delft, 1970. Trommelen, A. M.; Boerema, S. Power Consumption in a Scraped Surface Heat Exchanger. Trans. Znst. Chem. Eng. 1966, 44, 329-334. van de Westelaken, H. C. Votator Performance-12. Unilever Research. Cited in Dykes, D. J. (1980).
Received for review February 5, 1992 Accepted July 2, 1992
Equation of State with Multiple Associating Sites for Water and Water-Hydrocarbon Mixtures Ioannie G. Economou and Marc D. Donohue* Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218
The aesociated-perturbed-anisotropic-chaintheory (APACT) of Ikonomou and Donohue is extended to compounds with three associating sites per molecule. The number of associating species is calculated using an expression based on the fvstcorder thermodynamic-perturbationtheory (TPT-1) of Wertheim as simplified by Gubbins, Radosz, and co-workers. The new equation predicts accurately the thermodynamic properties of pure water from the triple point to the critical point. It is capable of describing quantitatively vapor-liquid equilibria (VLE),liquid-liquid equilibria (LLE) and vapor-liquid-liquid equilibria (VLLE)for mixtures of water with aliphatic and aromatic hydrocarbons over a wide range of temperature and pressure with one adjustable binary interaction parameter that is independent of temperature and density.
Introduction Modeling the thermodynamic properties and phase equilibria of mixtures of water with aliphatic and aromatic hydrocarbons is a difficult and challenging problem since such systems show extremely nonideal behavior that resulta in limited miscibility over a broad range of conditions. Hydrogen bonding between water molecules results in clusters that can extend over many coordination shells. Each water molecule has four hydrogen bonding sites, and so each molecule can form one to four bonds. The strong hydrogen bonding tendency of water results in unusual thermodynamicproperties of pure water compared with other compounds of similar size. In mixtures, the clustering of water molecules results in limited mutual solubility of water with components that do not hydrogen bond (referred to here as diluents). Many attempts have been made to describe the thermodynamic properties and phase equilibria of water mixtures. Cubic equations of state, such as the RedlichKwong-Soave and the Peng-Robinson equations of state, and Camahan-Starling-van der Waals type equations of state that are commonly used in industry have been applied widely to these systems (Peng and Robinson, 1976; Whiting and Prausnitz, 1982;Heidman et al., 1986;Luedecke and Prausnitz, 1985;Michel et al., 1989). In addition, activity coefficient models such as UNIFAC have been used for vapor-liquid equilibrium (VLE) and liquid-liquid equilibrium (LLE) calculations (Hooper et al., 1988). These simple models can describe quantitatively the phase behavior of water systems only with the use of many temperature- and density-dependent, adjustable parameters since the strong local composition effects
* Author to whom correspondence should be addressed.
caused by hydrogen bonding interactions are not accounted for explicitly in these equations. As a result, these models can be used only over limited ranges of conditions and considerable experimental data are needed to regress the parameters. Theoretically-based equations of state are expected to improve the accuracy of phase equilibrium calculations for systems with strong specific interactions such as hydrogen bonding. Heidemann and Prausnitz (1976)solved the equations for chemical equilibria analytically and incorporated the reault into a Camahan-Starling-van der Waals model to obtain a c l d - f o r m equation of state. Ikonomou and Donohue (1986)applied this approach using the perturbed-anisotropic-chain theory (PACT) and obtained the associated PACT (APACT). In the APACT, two different chemical equilibrium models were derived. The first model is for monomer-dimer equilibria for components with one bonding site per molecule, and the second model is an infinite equilibrium model (capable of predicting the formation of chains of molecules) for components with two bonding sites per molecule. The monomerdimer model is applicable to organic acids such as acetic acid whereas the infinite equilibrium model better deacribea the behavior of alcohols. The infinite equilibrium model also was used for pure water and water mixtures (Ikonomou, 1987);however phase equilibrium calculations for these mixtures were not as accurate as the calculations for mixtures containing alcohols. It is believed that this difference in accuracy is due to the fact that the two-site inifite equilibrium model is inadequate for water. Wertheim (1986a,b),using perturbation theory with a potential function that mimics hydrogen bonding, developed a statistical mechanical model for systems with a repulsive core and multiple hydrogen bonding sites. Gubbins, Radosz and co-workers (Chapman et al., 1990;
oaaa-5aa5/92/ 2631-23aa$o3.00/0 0 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992 2389 Huang and Radosz, 1990, 1991) simplified Wertheim’s theory and developed an equation of state, the statistical-associating-fluidtheory (SAFT),that is useful for phase equilibria calculations for associating systems. Recently, Economou and Donohue (1991) showed that the hydrogen bonding term in APACT is essentially identical to the hydrogen bonding term of SAFT for both the monomerdimer and infiite equilibrium models. In this paper, we extend the APACT to systems with three hydrogen bonding sites per molecule. We choose a three-sitemodel for water, rather than the more intuitively appealing four-site model, because recent experimental results (Wei et al., 1991) show that in clusters of water molecules only three out of the four hydrogen bonding sites per water molecule are bonded. While one could argue that one should not expect complete bonding at liquidlike densities, the experimental evidence suggests that there are geometric constraints against having more than three sites bonded per molecule. The association model presented here is based on the first-order thermodynamic perturbation theory of Wertheim. Wertheim’s fmt-order theory does not impose any constraints on the activity of a specific site due to the bonding of the other sites on the molecule. Therefore, we believe that it is more appropriate to use a three-site model rather than a four-site model. The expressions for the three-site hydrogen bonding term are derived phenomenologically based on the expressions from SAFT given by Huang and Radoaz (1990). The new equation is accurate in predicting thermodynamic properties of pure water such as vapor pressures and liquid densities from the triple point to the critical point. More importantly, it is capable of describing accurately phase equilibria for mixtures of water-aliphatic hydrocarbons and watel-aromatic hydrocarbons with only one temperature- and density-independent adjustable parameter. We believe the model can be used to make estimates where no experimental data are available. Comparisons are made with the two-site APACT using the infinite equilibrium model.
Equation of State The APACT is a closed form equation of state that accounts explicitly for hydrogen bonding, dispersion, polar, and induced-polar forces. Hydrogen bonding is taken into account through chemical theory that assumes that hydrogen bonds result in the formation of new species. Repulsive interactions are calculated from the CarnahanStarling equation of state for spherical molecules, and attractive forces are calculated as a perturbation over the reference fluid. Generalization of the equation of state to chain molecules was made by the introduction of a third parameter, c, which takes into account the nonspherical shape of the molecules. Originally, this generalization to chain molecules was made on the basis of phenomenological arguments and, following Prigogine, the parameter c was referred to as one-third the total number of external degrees of freedom per molecule. However, recently, the generalized Flory equation of state (Dickman and Hall, 19861, which is identical in form to the repulsive term in perturbed-hard-chain theory (PHCT) and PACT, showed that the parameter c can be related rigorously to the molecular geometry and the ratio of the excluded volume of a chain molecule to the excluded volume of a sphere. The equation of state for a pure component can be written in terms of the compressibility factor 2 as a sum of the contributions from these particular interactions: 2 = 1 + 2- + 2 r e p + zat* (1) where 2- is given by Ikonomou and Donohue (1986) and
Economou and Donohue (1991) and z”p and z”th are given by Vimalchand (1986). Attractive interactions are calculated as a sum of the isotropic interactions (Lennard-Jones interactions) and anisotropic interactions (polar interactions). The Lennard-Jones interactions are calculated as a perturbation expansion over the hard-sphere potential whereas the anisotropic interactions are calculated as a perturbation over the Lennard-Jones potential. The repulsive and the attractive terms in APACT are aesociation independent because of the assumptions made about the variation of the parameters of the associating species with the extent of association (Ikonomou and Donohue, 1986; Economou and Donohue, 1992). The 2- term for one and two bonding sites per molecule is evaluated from the material balances and expressions for the chemical equilibria (Ikonomou and Donohue, 1986). This formalism cannot be applied for components with three or more sitea per molecule because it is incapable of describing the three-dimensional structure of the resulting clusters. Consequently, we are not able to present a single chemical reaction scheme that describes the formation of these clusters. Wertheim (1986a,b) developed a perturbation theory for systems of hard spheres with one or more hydrogen bonding sites based on a simple potential function that is expected to mimic hydrogen bonding behavior. This theory was simplified for chainlike and treelike clusters into the SAFT equation of state (Chapman et al., 1990; Huang and Radosz, 1990,1991). SAFT accounts explicitly for hydrogen bonding, repulsive and attractive interactions and is given by eq 1, with
and EeP and Zattrare given elsewhere (Chapman et al., 1990; Huang and Radosz, 1990). 1 is the reduced density calculated from the expression 7 = (?rN,/6)pd3m where d is the segment diameter and m is the number of segmenta per molecule. XAis the fraction of molecules not bonded a t site A and the summation is over all the sites of the molecule. The quantity XAis calculated from the mass-action equation according to the expression XA = (1
+ CpXBAm)-’ B
(3)
where p is the total density and Am is a function characterizing the association strength similar to the equilibrium constant K in chemical theory (Economou and Donohue, 1991). For a molecule with three bonding sites, namely A, B, and C, where only bonding of the types A-C and B-C is allowed and also AAC= ABC = A, one obtains (Huang and Radosz, 1990) XA = XB = xc =
1 - pA
+
2 [ l + 6pA
+
PA)^]'/^
2
1 + pA
+ [l + 6pA +
PA)^]'/^
(4)
(5)
From eqs 4 and 5, the fraction of the molecules that are not bonded in any of their sites (monomers) is obtained:
1
+ 4pA -
2 + (1 + pA)[l
+ 6pA +
PA)^]'/^
(6)
Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992
2390
Table I. Molecular Parameters for Water Fitted to the Two-Site APACT and to the Three-Site APACT and % AAD in Vapor P ~ S Eand U ~Liquid Molar Volume"
%AAD P ,K
2-siteAPACT 3-siteAPACT
u*,
cm3/mol
219.3 183.8
11.73 11.64
C
AH",kJ/mol
AP/R
P t
"h
1.00 1.00
-20.10 -20.10
-10.28 -10.97
0.570 (8) 0.425 (8)
0.966 (60) 1.040 (60)
" Numbers in parentheses show the number of the data points used to fit the parameters. In addition, the following expression for 2by substituting eqs 4 and 5 into eq 2: 2888"
4PA
=-
1
is obtained
+ 3pA + [ l + 6pA + P PA)^]'/^ f(d
(7)
where
f(d =
2 + 2q - 32 2 - 31 + q 2
Economou and Donohue (1991) showed that APACT and SAlW have equivalent expressions for nl/no and 2for components with one and two bonding sites per molecule and that A, a function in perturbation theory characterizing the strength of association, is equivalent to KRT@where K is the equilibrium constant for hydrogen bonding and g is a function of density that depends on the equation of state and on the expressions used to calculate the molecular parameters of the associated species. In general,g is evaluated from the expression (Economou and Donohue, 1992)
Here, we apply this equivalence of chemical and perturbation theories for one and two sites and assume that the three-site results from SAFT can be used in APACT as well. As a result, one can express nl/no and 2- for APACT with the expressions derived from SAFT (eqs 6 and 7 by substituting A with the equivalent term KRTeS: nl = 2/[1+ no
(1 2-w
-
4KRTeSp - (KRT~SP)~ +
+ KRTegp)[l + 6KRTegp + (KRT~SP)~]'/~] (9)
=
4KR Tegp (10) 1 + 3KRTegp + [ l + 6KRTeSp + (KRT~SP)~]'/~
In the case of a mixture of component M with three bonding sites per molecule with components that do not hydrogen bond, the mole fraction of the monomers of M and the term are calculated from the expressions
where x i is the mole fraction of component i in the absence of any essociation and j is the total number of componenta. For xMequal to unity, eqs 11 and 12 reduce to the expressions for the pure associating component. Also for xM equal to zero, eq 12 gives the result that 2- is zero. The mixing rules for the repulsive and the attractive terms for a mixture of componenta are given by Vimalchand (1986). Similarly, one can develop expressions for nl/no and 2"" for the four-site model.' Again, these expressions are based on the expressions proposed from Huang and Radosz (1990) for SAFT by substituting A with the equivalent term KRTeS. For a pure associating component with four sites, it is n1 = 2/[1+ 8KRTeSp(l+ KRT@p) + n0 (1 + 4KRTeSp)[l + 8KRTeSp]'/2] (13) p s o c
=-
1
+
8KRTeSp (14) 4KRTeSp + [ 1 + ~KRT@P]'/~
Equations 9-14 were derived based on the APACT equation of state. However, these equations are general and can be used in other chemical-theory-baaedequations of state for associating fluids such as the COMPACT equation of state (Ikonomou and Donohue, 1987) and the ESD equation of state (Elliott et al., 1990). The difference among these theories would be the expression used to evaluate g. Properties of Pure Components The three-site APACT has been used to calculate the thermodynamic properties of pure water. In fact, water molecules have four hydrogen bonding sites per molecule. However, there is experimental evidence (Wei et al., 1991) that only three out of the four sites are bonded in water clusters. The equation of state was fitted to experimental vapor-pressure and liquid-density data from the triple point to the critical point to regress the molecular parameters for the pure component. The three-site APACT is a five-parameter equation of state. In addition to c that was defined above, a characteristic energy parameter, P , and a characteristic size parameter, u*, are used, together with two parameters characterizing the association forces, the standard enthalpy of hydrogen bond formation, AH", and the standard entropy of hydrogen bond formation, AS". It is AHo AS" In K = + (15)
In Table I, the parameters for the three-site APACT for water are shown together with the percentage average absolute deviation (% AAD) between the equation of state predictions and the experimental values for the vapor pressure and the liquid molar volume. The temperature range is from the triple point to the critical point, and the pressure range is from low pressure up to lo00 bar. In addition,the parameters for the twesite APACT for water and the percentage average absolute deviation for the vapor pressure and the liquid molar volume data are given
Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992 2391 0.0
Table 11. Molecular Parameters for Nonpolar and Polar Hydrocarbons Fitted to the PACT and % AAD in Vapor Pressure and Liauid Molar Volume' comDonent nonpolar methane propane butane n-hexane n-octane n-decane cyclohexane polar benzene toluene m-xylene ethylbenzene 1-methylnaphthalene l-ethylnaphthalene
V*,
T+. K cm3/mol
% c
P t
AAD
-0.4
UIiq
146.7 260.7 287.3 326.5 347.1 362.1 372.3
22.29 42.95 53.32 73.84 95.52 118.06 63.77
1.OOO 1.325 1.504 1.817 2.186 2.542 1.543
0.359 (6) 0.331 (12) 0.375 (8) 0.226 (53) 0.331 (10) 0.864 (12) 0.155 (25) 0.276 (23) 0.195 (7) 0.372 (13) 0.216 (12) 0.939 (12) 0.169 (16) 0.273 (22)
381.3 387.0 396.8 399.0 496.3
56.41 64.83 74.57 74.56 91.75
1.327 1.621 1.807 1.756 1.760
0.285 (8) 0.359 (17) 0.500 (21) 0.364 (17) 0.259 (12) 0.230 (29) 0.i40 (17) 0.396 (16) 0.036 (8) 0.183 (16)
477.9
99.61
2.057 0.063 (6) 0.316 (9)
'Numbers in parentheses show the number of the data points used to fit the parameters.
for comparison. The threesite APACT percentage average absolute deviation is smaller for the vapor-pressure data whereas the percentage average absolute deviation for the liquid molar volume is similar for the two models. The difference between the two models will be obvious when these models are used for mixture phase equilibrium calculations. We attempted to use the four-site APACT to correlate data for pure water. However, the results obtained from the four-site APACT were worse than the results from the two-site APACT and the three-site APACT and the regressed values of the association parameters AH" and AS" were physically unrealistic. In the two-site APACT and the three-site APACT the assumption is made that cifor the associated cluster of size i is given by the expression ci = ic, (16) The physical picture of the three-site model is a threedimensional cluster, and so one might assume that hydrogen bonding results in spherical clusters. Spherical molecules have a c parameter equal to unity, and so it might be better to assume that ci = c1 (17) We tried this assumption to the threesite APACT in order to calculate thermodynamic properties of pure water. The fit of the experimental data was worse than the fit obtained using eq 16, and in addition physically incorrect optimized values were obtained for the molecular parameters of water. Other expressions for cicould be tried as well [see, for example, Economou and Donohue (1991)], however, using eq 16 we obtain the additional simplification that function g, defined by eq 8, has a value equal to zero for the entire density range. Molecular parameters for the equation of state also were calculated for a number of nonpolar and polar hydrocarbons. For components that do not hydrogen bond, the APACT reduces to the PACT equation of state that has three molecular parameters for each component ( P ,u*, and c). In Table I1 the molecular parameters for the hydrocarbons examined in this work and the ?& AAD for the vapor pressure and liquid molar volume are shown. The values of these parameters are very-similarto the values originally proposed by Vimalchand et al. (1988). Generalized correlations for the molecular parameters of various types of fluids were developed by Vimalchand et al. (1988)
-0.1) 0
z
0
-1.2
-2.0
0.0
I 0.2
I
I 0.4
t
I 0.1)
I
I 0.8
,
1.0
XM Figure 1. 2"" vs xM for a binary mixture of an associating component and a diluent at 298 K from the two-site APACT, the three-site APACT, and the four-site APACT.
and can be used in this work. For the aromatic hydrocarbons, only the quadrupole moment was taken into account using a quadrupolar interaction energy as described in detail by Vimalchand and Donohue (1985). Inclusion of the dipole moments for these compounds (which are low) did not improve the phase equilibrium calculations.
Aqueous Mixtures The three-site APACT has been applied to several binary mixtures of water with nonpolar and polar hydrocarbons of different sizes and shapes and over a wide range of temperature and pressure. Different types of phase equilibria were examined including VLE,LLE,and W E . Calculations also were performed with the two-site APACT for comparison. In order to test the difference between the expressions for hydrogen bonding from the two models, sample calculations were performed for a binary system at 298 K where one of the components, M, hydrogen bonds and the other does not. RTp was set equal to 1300, which corresponds to a liquid-phase value for the molar volume of water at 298 K. In Figure 1,the quantity eis plotted as a function of the superficial mole fraction of component M. Similar results were obtained for the mole fraction of the monomer of component M, nM,/nM.The difference between the two models does not appear large; however the chemical potential depends strongly on the mole fraction of the monomers and therefore the difference between the two models affects phase equilibrium calculations considerably. In Figure 1,calculations for 2- are shown also for the four-site model. The effect of association predicted from the four-site model is much stronger than that predicted from the other two models. We believe thisis why the four-site model does not correlate accurately the phase equilibria for watel-hydrocarbon mixtures, but it is not clear whether Wertheim's theory overpredide the effect of hydrogen bonding as the number of associating sites increases. However, Wertheim's firsborder thermodynamic perturbation theory does ignore the effect of steric self-hindrances and therefore neglects the effect of the activity of a particular site to the activity of the surrounding sites (Economou and Donohue, 1991). In Table 111,the binary mixtures examined in this work are presented. For most of the nonpolar systems, the three-site APACT can predict quantitatively the phase equilibria without any adjustable parameter. For the other
2392 Ind. Eng. Chem. Res., Vol. 31, No. 10,1992 *
I
'
I
'
I
'
I
I
P a l e r in Vapor P a l e r in Prapane-Rlch Liquid Propane in Paler-Rich Liquid 2-site APACT ( k p - 0 . 0 5 )
10-5
250
280
310
340
370
400
I
'
'
I 340
300
I
'
I
'
I
W a t e r in a-Hexaae 0 n-Hexane i n Water P-siLt A P K T
I 380
T (K)
I 420
i 460
'
I
1500
T (K)
Figure 2. VLLE for the water (1)-propane (2) mixture. Experimental data (points) and calculated results (lines) from the two-site APACT and from the three-site APACT.
Figure 4. Mutual solubilities of water (1) and n-hexane (2). Experimental values (points) and predicted ( k =~ 0) rmults (lines) from the two-site APACT and from the three-site APACT.
XI 0.9990
0,9992
0.9886
0.9890
'
1.0000
1 ' Haler in
1
'
1
'
1
I 340
I
I
I
380
420
460
Cyclohexane 0 Cyclohexane in Water
K APACT (kl,=-0.0255) - 3-silt APACT (k,)=-0.023) 0
---
8
0.9994
i77.6 2-siLe
300
1500
T (K)
Figure 5. Mutual solubilities of water (1) and cyclohexane (2). Experimental values (points) and predicted (kij = 0)results (lines) from the two-site APACT and from the three-site APACT.
XI
Figure 3. Experimental (points)and calculated (lines) bubble-point values for the VLE of water (1)-butane (2) mixture at 344 and 477.6 K.
mixtures, a small adjustable parameter that is temperature- and density-independent is needed in order to get good agreement between the experimental data and the model calculations. For the polar hydrocarbons only the quadrupole moment was taken into account through an expression developed by Vimalchand and Donohue (1985). Inclusion of the dipole moment of the aromatic hydrocarbons did not improve the agreement between calculations and experimental data and higher values for kij were needed for both the two-site and the three-site APACT. In Figure 2, VLLE are shown for the water (1)-propane (2) system at the range 278-370 K. A small binary adjustable parameter ie needed to obtain reasonable agreement for all the three phases. Three-site APACT is in good agreement with the experimental data for the vapor phase and the propane-rich liquid phase, whereas for the water-rich liquid phase it is in good agreement with the experimental data only at high temperatures. Two-site APACT calculatm accurately the vapor-phase composition, but it fails in calculating correctly the liquid-phase compositions. At low temperatures, both the two-site APACT and the three-site APACT predict that the propane solubility continues to decrease as temperature decreases. The experimental data show an increase in the propane solubility at low temperatures. We were unable to fit the data
but note that this behavior was unique for all the systems examined. Bubble-point calculations for the water (1)-butane (2) system at 344 and 477.6 K and with pressures up to 700 bar are shown in Figure 3. The three-site APACT is in excellent quantitative agreement with the experimental data over the entire range of pressure, whereae the two-site APACT is in good agreement with the data at high pressure but it fails at low pressure. To obtain a good agreement at low pressure as well, a temperature-dependent binary parameter is needed for the two-site APACT. LLE predictions for the water (1)-hexane (2) and the water (1)-cyclohexane (2) systems are shown in Figures 4 and 5, respectively. For both systems, three-site APACT predicts accurately the solubility of water in the organic phase and the solubility of the hydrocarbon in the aqueous phase. The two-site APACT overpredicts the solubility of water in the organic phase and underpredicts the solubility of the hydrocarbon in the aqueous phase. Phase equilibrium calculations for the water-polar hydrocarbon systems are shown in Figures 6 and 7. In Figure 6,phase equilibria are shown for the water (1)-benzene (2) mixture at 498 K. At low pressure, VLE between a vapor phase and a water-rich liquid phase and between a vapor phase and an organic-rich liquid phase occur. At 48 bar a three-phase VLLE point exists, and at higher pressure the vapor phase disappears and LLE occurs. Three-site APACT predicts the VLE and the VLLE quite accurately. The caldationa based on the two-site APACT for the VLE are similar to those obtained from the three-site APACT except for the high-pressure VLE where
Ind. Eng. Chem. Res., Vol. 31, No. 10,1992 2393 Table 111. Binary Water-Hydrocarbon Mixture6 Examined in This Work kij
hydrocarbon nonpolar methane propane butane n-hexane n-octane n-decane cyclohexane polar benzene toluene m-xylene ethylenebenzene 1-methylnaphthalene 1-ethylnaphthalene
typeofequilib
T,K
P,bar
2-site APACT
3-site APACT
VLE VLLE VLE LLE LLE LLE VLE LLE LLE
344-510 278-370 344-477 290-410 298-473 310-550 573-613 573-613 313-473
1-700 5-45 1-700 1-35 0.4-35 0.1-89 13-92 92-230 0.3-30
0.0 -0.05 -0.0255 0.0 0.0 0.0 0.0 0.0 0.0
-0.03 -0.023 0.0 0.0 0.0 0.0 0.0 0.0
VLE LLE VLLE LLE LLE LLE LLE LLE
498-579 283-498 498 373-473 373-473 310-568 273-550 311-550
20-160 0.3-60
0.0 0.02 0.0 0.025 0.018 0.02 0.06 0.06
0.0 0.055 0.0 0.045 0.037 0.035 0.10 0.095
48
1.5-24 1-20 1-106 2-64 2-62
data source! 1 2 3 4 5-7 8 9 9 5
0.0
10 4, 5 10 11 11 8 12,13 12,13
(1) Olds, R.H.; Sage, B. H.; Lacey, W. N. Znd. Eng. Chem. 1942,34,1223-1227. (2) Kobayashi, R.; Katz, D. L. Znd. Eng. Chem. 1963,46, 440-451. (3) Sage, B. H.; Lacey, W. N. Some Properties of the Lighter Hydrocarbons, Hydrogen Sulfide, and Carbon Dioxide, API: New York, 1955. (4) Baumgaertner, M.; Moorwood, R. A. S.; W e n d , H. ACS Symp. Ser. 1980,133,415-434. (5) Tsonopoulos, C.; Wilson, G. M. AIChE J. 1983,29,990-999. (6) Black, C.; Jorie, G.G.; Taylor, H. S. J. Chem. Phys. 1948,16,537-543. (7) McAuliie, C. J. Phys. Chern. 1966, 70,1267-1275. (8) Heidman, J. L.; Tsonopoulm, C.; Brady, C. J.; Wilson, G. M. MChE J. 1986,31,376-384. (9) Wang, Q.; Chao, K. C. Fluid Phase Equilib. 1990,59,207-215. (10) Rebert, C. J.; Kay, W.B. AIChE J. 1969,5,285-289. (11)Anderson, F. E.; Prausnitz, J. M. Fluid Phase Equilib. 1986,32,63-76. (12) Brady, C. J.; Cunningham, J. R.;Wilson, G.M. Gas Processors Association Reeearch Report-62, Tulsa, OK, 1982; pp 18-19. (13) Soerensen, J. M.; Ark, W. LLE Data Collection; Dechema: Frankfurt, Germany, 1979; Vol. V, Part 1. 70
100
I 0
60
- 3-site
10-1
-
10-2
-
I
I
3
-
.-:
APACT
c) U
2
c.
-
-- -
I
Water i n m-Xjlcne m-Xylene in Water E l i t e APACT (k,,=0.018) 3-rile APACT (klr0.037)
3
50
a 0)
2 40 .I
a
10-4
350
380
410
440
470
500
T (K)
Figure 7. Mutual solubilities of water (1) and m-xylene (2). Experimental values (pointa) and calculated resulta (lines) from the two-site APACT and from the three-site APACT.
30
20
0.0
0.2
0.4
0.6
0.6
1.0
mole fraction Figure 6. Phase equilibria for water (l)-benzene (2) at 498 K. Experimental values (points)and predicted (ki, = 0) results (lines) from the two-site APACT and from the three-site APACT.
the compositions predicted for the organic-rich liquid phase are lower than experimental values. At higher pressures, the solubility of water in benzene predicted from the two-site APACT is higher than that predicted from the three-site APACT. Three-site APACT predictions are in good agreement with LLE experimental data over a temperature range from 283 K up to 498 K. In Figure 7,LLE calculations are presented for water (1)-m-xylene (2)mixture. Three-site APACT calculates accurately the solubility of water in m-xylene at low temperatures, whereas for temperature greater than 420 K it underpredicts it. The slope of the solubilities of the mxylene in water as calculated from the three-site APACT is lower than the slope from the experimental data. Calculations based on the two-site APACT are in good
agreement with the experimental data except at low temperatures where the solubility of water in m-xylene is overpredicted. Many models in the literature have been applied to water-hydrocarbon systems. For example, a UNIFAC activity coefficient model with parameters regressed specifically for UE was pro@ by Magnuasen et d. (1981) and used widely for calculating mutual solubilities. However, Hooper et al. (1988)showed that this model fails to predict LLE for water-hydrocarbon systems using temperature-independent parameters. Agreement between model correlations and experimental data could be obtained only by using multiple temperature-dependent parameters (Hooper et al., 1988). We feel that it is inap propriate to compare our equation of state predictions with correlations obtained from a model using several temperature-dependent adjustable parameters. Hence, no comparisons are presented with other models.
Summary The APACT equation of state was extended to components with three associating sites per molecule. Phase
2394 Ind. Eng. Chem. Res., Vol. 31, No. 10,1992
equilibrium calculations for watel-hydrocarbon mixtures showed that the three-site APACT is very accurate in predicting phase equilibria of different types for aqueous mixtures of nonpolar hydrocarbons with no adjustable parameters. For mixtures with polar hydrocarbons, the three-site APACT needs a binary parameter for accurate eatimation of the phase equilibria. For most of the systems examined, the three-site APACT was in better agreement with the experimentaldata than the two-site APACT. The advantage of the proposed model compared to other models widely used is that it provides excellent estimation of the LLE for water-hydrocarbon systems with one temperature-independent parameter and therefore it can be extrapolated to give accurate predictions at conditions where no experimental data are available. The formalism used in this work for aqueous mixtures can be used for systems containing other components that have multiple hydrogen bonding sites per molecule, such as ammonia. Acknowledgment Support of this research by the Division of Chemical Sciences of the Office of Basic Energy Sciences, US. Department of Energy, under Contract No. DE-FGOP87ER13777 is gratefully acknowledged. Nomenclature 3c = total number of external degreea of freedom per molecule d = segment diameter AHa = standard enthalpy of association K = equilibrium constant m = number of segments per molecule N,, = Avogadro number no = superficialnumber of moles nl = number of monomers R = gas constant ASo = standard entropy of association T = temperature P = characteristic energy parameter u* = characteristic volume parameter X = fraction of molecules not bonded I: = mole fraction 2 = compressibility factor Greek Letters A = association strength in perturbation theory r ) = reduced density p = molar density Subscripts A, B, C = hydrogen bonding site A, B, C Superscripts
assoc = association attr = attractive rep = repulsive Registry No. HzO, 7732-18-5.
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Received for review February 19,1992 Revised manuscript received June 10,1992 Accepted June 26, 1992
