Prediction of Phase Equilibria in Binary Aqueous Systems Containing

The cubic-plus-association (CPA) equation of state (EoS) is applied in this study to ... cubic equation of state for the physical part and perturbatio...
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Ind. Eng. Chem. Res. 1998, 37, 4175-4182

4175

Prediction of Phase Equilibria in Binary Aqueous Systems Containing Alkanes, Cycloalkanes, and Alkenes with the Cubic-plus-Association Equation of State Iakovos V. Yakoumis,†,‡ Georgios M. Kontogeorgis,†,‡ Epaminondas C. Voutsas,*,† Eric M. Hendriks,§ and Dimitrios P. Tassios† Thermodynamics and Transport Phenomena Laboratory, Department of Chemical Engineering, Section II, National Technical University of Athens, Heroon Polytechniou 9, Zographos 15780, Athens, Greece, and Shell Research and Technology Centre, Amsterdam, P.O. Box 38000, 1030 BN Amsterdam, The Netherlands

The cubic-plus-association (CPA) equation of state (EoS) is applied in this study to binary aqueous mixtures containing hydrocarbons. The CPA EoS combines the Soave-Redlich-Kwong (SRK) cubic equation of state for the physical part and perturbation theory for the chemical (association) part. Rigorous expressions for the contribution of the association term to the pressure and to the chemical potential, which do not include any derivatives of the mole fraction of molecules i not bonded at site A (XAi), are presented. Three different association models for water have been considered depending on the number of hydrogen bonding sites per water molecule: the two-, three-, and four-site models. Successful correlation of both vapor pressures and saturated liquid volumes is obtained with all three models. However, satisfactory correlation results of the mutual solubilities of water/aliphatic hydrocarbon systems are obtained only with the foursite model using a single interaction parameter (kij) in the attractive term of the EoS. A generalized expression of kij as a function of the molecular weight of the members of the homologous series is presented, something that allows CPA to be used as a predictive tool. Very satisfactory prediction results are obtained, comparable to the correlation ones of the SAFT EoS for the water solubility in the hydrocarbon-rich phase and orders of magnitude better for the hydrocarbon solubility in the water-rich phase. Satisfactory predictions are also obtained for the vapor-phase compositions and the three-phase equilibrium pressures. Introduction The knowledge of phase equilibria for aqueous mixtures is essential for a variety of chemical engineering separation operations, especially those concerned with water pollution abatement. The most common of these are single and multistage liquid-liquid extraction processes and vapor-liquid stripping. Extraction processes are based on equilibration of hydrocarbon-rich and water-rich liquid streams; these streams occur in petroleum processing, petroleum reservoir production, and coal gasification. Phase equilibria of water/hydrocarbon systems is also very important in reservoir production and drilling operations (Hemptinne, 1997). Modeling of phase equilibria of water/hydrocarbon mixtures, which is the objective of this paper, is a difficult and challenging problem since such systems show extremely nonideal behavior that results in limited miscibility over a broad range of conditions. The phase diagrams of such systems may include two-phase vaporliquid or liquid-liquid and three-phase vapor-liquidliquid regions. The solubilities in the coexisting phases are strongly asymmetric, with the solubility of the hydrocarbon in the water-rich phase being several orders of magnitude lower than that of water in the * To whom all correspondence should be addressed. Email: [email protected]. † National Technical University of Athens. ‡ Present address: IGVP & Associates Engineering Consultants Ltd., 35 Kifisias Ave., 11523 Ampelokipi, Athens, Greece. § Shell Research and Technology Centre, Amsterdam.

hydrocarbon-rich phase. Moreover, the hydrocarbon solubility presents a minimum value at relatively low temperatures, whereas the water solubility is a monotonic function of temperature (Anderko, 1991; Economou and Tsonopoulos, 1997). Cubic equations of state, such as the Soave-RedlichKwong (SRK) and the Peng-Robinson (PR) ones, can describe quantitatively the phase behavior of aqueous systems using the one-fluid van der Waals mixing rules but with different interaction parameters for each phase, which for the water-rich phase they must be temperature-dependent, since the strong local composition effects caused by hydrogen-bonding interactions are not accounted for explicitly in these equations using the conventional van der Waals one-fluid mixing rules (Tsonopoulos and Heidman, 1986). Alternatively, Kabadi and Danner (1985) and Michel et al. (1989) used complex unconventional mixing rules for the attractive term parameter of the EoS in order to describe the phase equilibria of water/hydrocarbon mixtures. Their effort had, however, either limited accuracy or could not be applied to multicomponent mixtures (Michel et al., 1989). From the practical point of view it is desirable to employ a model with a small number of adjustable parameters determined from pure compound and mixture properties, which at the same time account for the association effects. A convenient approach to satisfy this requirement is to employ the concept of association in conjunction with an equation of state. For these reasons a number of extensive and thorough investigations have been carried out during the last 20 years

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4176 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998

toward the development of models suitable for associating fluids [for a review see, e.g., Economou and Donohue, 1996]. The models for associating fluids can be divided into three differentsin principlescategories, depending on the theory employed for accounting for the extent of hydrogen-bonding: (I) chemical theory (Heidemann and Prausnitz, 1976; Ikonomou and Donohue, 1988; Anderko, 1989a,b; Goral, 1996), (II) perturbation theory (Chapman et al., 1990; Huang and Radosz, 1990), and (III) lattice/quasi-chemical theory (Panayiotou and Sanchez, 1991). In all these models there is a physical term accounting for the deviations due to physical forces and an association term accounting for the effect of hydrogen bonding. In some of the models (e.g., Anderko, 1989a; Suresh and Elliott, 1992) the physical term is a cubic function with respect to volume, but in most of them, i.e., associated-perturbed-anisotropic chain theory (APACT) (Ikonomou and Donohue, 1988), statisticalassociating fluid theory (SAFT) (Huang and Radosz, 1990), and simplified statistical-associating fluid theory (SSAFT) (Fu and Sandler, 1996), the physical term is a complex noncubic function. In both chemical and perturbation theories the association term aims to capture the physics of hydrogen bonding. Recently, Hendriks et al. (1997), using cluster partition functions showed that chemical, first-order perturbation, and lattice theories are essentially equivalent. In a series of papers from this laboratory we have developed and successfully applied the cubic-plus-association (CPA) equation of state (EoS) to associating fluids by combining the SRK EoS with the Wertheim perturbation theory, as the latter is used in SAFT [Kontogeorgis et al. (1996) for pure fluids, Yakoumis et al. (1997) for vapor-liquid equilibrium calculations and Voutsas et al. (1997) for liquid-liquid equilibrium calculations]. As shown there, the use of a cubic function for the physical part of the EoS, instead of the complex ones used with other EoS, does not affect the physical meaning of the association term and the resulting equation of state describes successfully pure associating fluids and their mixtures with inert compounds. In this study we apply the CPA EoS to the highly important class of water/hydrocarbon systems. Water molecules are known to form three-dimensional structures through hydrogen bonding. Each water molecule has two proton donors (the two hydrogens) and two pairs of electrons that are able to form hydrogen bonds. As a result, each water molecule is capable of forming up to four hydrogen bonds. Huang and Radosz (1990) proposed a three-site model for water in SAFT, while Economou and Donohue (1992) applied both the twoand three-site models using APACT. These two approaches are in agreement with the suggestion that, because of the geometry of the water molecule and the structure of the three-dimensional networks that are formed, most water molecules are considered to have three or fewer hydrogen bonds (Economou and Donohue, 1996). On the other hand, Galindo et al. (1996) and Economou and Tsonopoulos (1997) modeled water with SAFT as a four-site fluid for pure compound and mixture calculations. In this study water has been treated as a two-, three-, and four-site molecule. The aim of using these three different approaches is to compare them and to conclude which one yields the best results in representing simultaneously pure water properties (vapor pressures

and saturated liquid volumes) and phase equilibria of aqueous binary mixtures containing one aliphatic hydrocarbon. The remaining of the paper is organized as follows: First the CPA EoS is presented briefly, including a redevelopment of the association term, which is much simpler than that introduced by Chapman et al. (1990) since it does not involve derivatives of the mole fraction of the nonassociated molecules. Next, pure compound parameters for water obtained by fitting vapor pressures and saturated liquid volumes with the three associating models are presented and compared in the correlation of liquid-liquid equilibria (LLE) for the water/n-hexane system. Best results are obtained with the four-site model, which is then applied to the prediction of the phase equilibria of the water/alkane, water/cycloalkane, and water/alkene binaries. We end with our conclusions. Materials and Methods The CPA Equation of State: (1) Compressibility Factor. The compressibility factor of the CPA EoS is given by

zCPA ) zSRK + zassoc

(1)

where

zSRK )

V R V - b RT(V + b)

(2)

The contribution of the association term to the compressibility factor is given by

zassoc ) -

(

1 2

1+F

)∑∑

∂ ln g ∂F

i

xid(Ai)(1 - XAi) (3)

Ai

where F is the molar density of the mixture, xi is the (analytical) mole fraction of the component i in the mixture, d(Ai) is the number of association sites of type A on analytic molecule i, and XAi is the mole fraction of the molecules i not bonded at site A, which is the solution of the following equation:

xjXB ∆A B )-1 ∑j ∑ B

XAi ) (1 + F

j

i j

(4)

j

The association strength parameter (∆AiBj) is given by

[ ( ) ]

∆AiBj ) g exp

AiBj - 1 bijβAiBj RT

(5)

where AiBj and βAiBj are the association energy and volume of interaction between site A of the molecule i and site B of the molecule j, respectively. Finally, the radial distribution function (g) for CPA EoS is given by

y 2 g) (1 - y)3 1-

(6)

where

y)

bF 4

(7)

Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4177

and b is the co-volume parameter of the mixture. The derivation of eq 3 has been based on the cluster partition function approach (Hendriks et al., 1997). Although no derivative of XAi is involved, this expression is rigorous and equivalent to the one proposed by Chapman et al. (1990). The corresponding expression for chemical potentials is given in the Appendix. (2) Mixing and Combining Rules. The application of the CPA EoS to mixtures requires explicit mixing rules only for the parameters of the physical part, while the extension of the association term to mixtures is straightforward. The classical van der Waals one-fluid mixing and combining rules are used for the parameters R and b:

R)

∑i ∑j xixjRij

b)

∑i ∑j xixjbij

(8)

bi + bj 2

(9)

where

Rij ) xRiRj(1 - kij)

bij )

and

Ri ) a0i(1 + c1i(1 - xTri))2

(10)

The interaction coefficient kij is the only adjustable parameter of the CPA EoS for a binary mixture containing a self-associating and an inert compound. As evident from eqs 4 and 5 no combining rules are needed for the association parameters AiBj and βAiBj for the case of mixtures containing only one associating compound. Estimation of Pure Compound Parameters. CPA, as shown by eqs 1-5, contains five parameters for each associating compound. These pure compound parameters (R0, c1, b, AB, and βAB) are estimated by fitting vapor pressure and saturated liquid volume experimental data for water in the temperature range 278-578 K (Tr range 0.43-0.9) using the following objective function: NDAT

F)

∑ i)1

(

) (

Pexp - Pcalc i i Pexp i

2

NDAT

+

∑ i)1

)

Vexp - Vcalc i i Vexp i

2

(11)

where Pexp and Pcalc are the experimental and the calculated vapor pressure, respectively, and Vexp and Vcalc are the experimental and calculated saturated liquid volume, respectively. With this procedure several sets of pure compound parameters are obtained with the two-, three-, and foursite models that give very good correlation of both vapor pressures and saturated liquid volumes (average percentage errors below 2.5%). The multiplicity of pure compound parameter sets with the CPA EoS has also been observed by Kontogoergis et al. (1996). Three typical sets of parameters for each model in terms of the magnitude of the association energy AiBj are presented in Table 1. For the inert (nonassociating) compounds, three parameters (R0, c1, and b) are required and are obtained again by fitting vapor pressure and saturated liquid volume data. One set of parameters is obtained in this case and is presented for the compounds of this study in Table 2. They can be also obtained by using the critical properties and acentric factor as done typically

Table 1. Typical Sets of Pure Compound Parameters for Watera set

b

R0

c1

AB

βAB

∆P (%)

∆V (%)

2b1 2b2 2b3

0.01466 0.01517 0.01576

2.7311 2.9013 3.4169

0.1752 0.5815 0.6896

269.26 200.27 155.43

0.0202 0.0652 0.0888

2.05 0.65 1.36

0.85 0.44 0.92

3b1 3b2 3b3

0.01552 0.01598 0.01599

3.5746 3.9456 3.8031

0.1528 0.3687 0.6758

233.49 183.27 122.39

0.0068 0.0125 0.0462

2.07 1.60 1.52

2.19 1.92 1.29

4c1 4c2 4c3

0.01435 0.01453 0.01481

1.0612 0.9894 1.0358

0.4333 1.0669 1.6098

180.74 162.01 137.83

0.0524 0.0787 0.1251

1.10 0.77 1.24

1.09 0.33 0.11

a In the first column of this table the name of each set is demonstrated. 2b, 3b, and 4c denote the model used for the correlation of vapor pressures and saturated liquid volumes; 2b is the 2-site model, 3b is the three-site model, and 4c is the foursite model. The notation 2b, 3c, and 4c is that used by Huang and Radosz (1991).

in the SRK EoS, which represents an advantage of CPA over SAFT in the case that supercritical compounds are involved. For vapor-liquid equilibrium the two approaches give practically the same results, while for liquid-liquid equilibrium calculations, as most of the ones performed in this work, the parameters obtained by fitting vapor pressure and liquid volume data give better results (Voutsas et al., 1997) and they will be used in this study. All the experimental vapor pressure and liquid volume data have been taken from the DIPPR data compilation (Daubert and Danner, 1989). Choice of the Pure Compound Parameters for Water for Each Association Model. Tsonopoulos and Wilson (1983) have measured the mutual solubilities at the three-phase equilibrium pressure of the water/n-hexane system. The three different sets of pure compound parameters for water for each association model (2-0, 3-0, and 4-0), which are presented in Table 1, have been used in describing this system with the CPA EoS. The prediction and correlation performance of all models depends on the set of pure compound parameter of water, which is more pronounced for the solubility of the n-hexane in the water-rich phase where it may vary up to 4 orders of magnitude, as shown with the typical prediction results for the two-site model in Figure 1. For the two-site model best results are obtained with the 2b1 set of pure compound parameters for water, for the three-site model with the 3b1 set, and for the four-site model with the 4c2 set. These sets of pure compound parameters of water are evaluated next in the correlation/prediction of phase equilibria for water/alkane systems. Comparison of the Three Different Association Models for Water. Figure 2 presents a comparison of the three different association models for water in the correlation of the liquid-liquid equilibria of the water/ n-hexane system using the best set for each model (2b1, 3b1, and 4C2). The 4-0 model performs satisfactorily in both phases, while the other two models (2-0 and 3-0) underestimate by several orders of magnitude the solubility of n-hexane in the water-rich phase. Furthermore, the 4-0 model requires a much smaller interaction coefficient than the 2-0 and 3-0 models. Similar results are obtained for the other water/ alkane systems leading to the conclusion that the four-

4178 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 Table 2. CPA EoS Parameters for the Inert Compounds of This Studya

a

compound

a0

c1

b

∆P (%)

∆V (%)

propane n-butane n-pentane n-hexane n-octane n-decane cyclohexane ethylcyclohexane n-butylcyclohexane 2-methylbutane 2,3-dimethylbutane 2,2,4-trimethylpentane 1-hexene 1-octene 1-decene

9.1187 13.143 18.198 23.681 34.875 47.389 21.257 32.297 43.900 17.621 25.625 32.141 22.176 33.672 46.299

0.63070 0.70771 0.79858 0.83130 0.99415 1.1324 0.74265 0.82438 0.99058 0.76463 0.74607 0.86993 0.83588 0.95785 1.0748

0.057834 0.072081 0.091008 0.10789 0.14244 0.17865 0.09038 0.12437 0.15974 0.090132 0.10519 0.13875 0.10288 0.13770 0.17258

0.93 0.19 0.54 0.45 0.27 0.42 0.37 0.85 1.45 0.16 0.84 0.16 0.20 0.32 1.50

1.86 0.97 0.93 0.54 0.50 0.56 1.11 0.78 0.59 1.17 2.04 0.97 0.69 0.60 0.89

Parameters are based on the representation of vapor pressures and saturated liquid volumes, in the Tr range 0.55-0.9.

Figure 1. Importance of pure compound parameters for water in the prediction of mutual solubilities of water/n-hexane with the 2-0.

site model for water is clearly superior to the other two for phase equilibria calculations. Thus, in the remainder of this paper the 4-0 model will be used in order to perform phase equilibrium calculations for water/ normal alkane, water/branched alkane, water/cycloalkane, and water/alkene binaries. Results and Discussion All water/hydrocarbon data considered in this study are correlated by use of a single temperature-independent binary interaction parameter (kij). Our preliminary study showed that the kij values can be very well correlated linearly with the carbon number of the hydrocarbon, in each homologous series:

kij ) a + bNc

(12)

Figure 2. Performance of the three association models of water in the correlation of the mutual solubilities of the water/n-hexane system. The interaction parameters were chosen so that they give about the same absolute percent error in the correlation of the upper (hydrocarbon-rich) phase. Table 3. Coefficients of Equation 12 water/

Nc range

a

b

n-alkanes branched alkanes cycloalkanes 1-alkenes

g3 g4 g5 g4

0.194 0.147 0.295 -0.040

-0.0280 -0.0333 -0.0325 -0.0100

The coefficients of eq 12 for each homologous series are given in Table 3 and they are valid for hydrocarbons with carbon numbers greater or equal to 3. Equation 12 thus allows prediction of mutual solubilities for these four types of hydrocarbons with water. The obtained prediction results, using the generalized expression for kij (eq 12), are presented in Table 4 and typical ones are shown graphically in Figures 3-8. For comparison purposes, published SAFT results (Economou and Tso-

Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4179 Table 4. Prediction Results with the CPA EoS and Correlation Ones with the SAFT EoSa CPA water/

refb

% AAD in xw

% AAD in yw

propane n-butane n-pentane n-hexane n-octane n-decane 2-methylbutane 2,3-dimethylbutane 2,2,4-trimethylpentane cyclohexane ethylcyclohexane n-butylcyclohexane 1-hexene 1-octene 1-decene

1 2 3 4 5 6 3 3 3 4 5 6 6 6 6

5.8 11.9 18.4 8.5 7.6 10.9 6.4 8.3 14.0 8.2 14.0 5.4 15.1 11.0 10.5

0.0004

0.0233

0.0090 0.0022 0.0095 0.0130

SAFT-vdW

SAFT-PHCT

OMD

% AAD in xw

OMD

% AAD in xw

OMD

=0.7 =0.4 =1.2 =0.2 =0.4 =1.1 =0.1 =0.7 =0.8 =0.7 =0.2 =0.7 =0.9 =1.1 =1.3

c c 5.3 5.3 6.7 c c c c c c 3.3 9.6 20.6

c c =8.5 =10.1 =13.7 c c c c c c =7.0 =8.1 =11.0

c c 6.3 6.2 8.2 c c c c c c 4.6 10.7 21.6

c c =2.7 =5.7 =10.2 c c c c c c =1.7 =2.1 =5.6

a Results were obtained by using temperature-independent binary interaction parameters and two different mixing rules. % AAD is the percent average absolute deviation in the water mole fraction in the hydrocarbon-rich phase (xw) and AAD is the average absolute deviation in the vapor phase (yw). OMD is the maximum absolute deviation in the hydrocarbon solubility in the water-rich phase expressed in orders of magnitude. Values of AAD in yw for the SAFT models are not available in the literature. The results for SAFT are from Economou and Tsonopoulos (1997) whenever available. b References of experimental data: 1, Kobayashi and Katz (1953); 2, Sage and Lacey (1955); 3, Sorensen and Arlt (1979); 4, Tsonopoulos and Wilson (1983); 5, Heidman et al. (1986); 6, Economou et al. (1997). c Results for these systems are not available in the literature.

Figure 3. Prediction of the vapor-liquid-liquid equilibrium for the water/propane system.

nopoulos, 1997) with water treated as a four-site molecule are presented where available. SAFT-vdW is the SAFT EoS in conjunction with the van der Waals one-fluid mixing rules, while SAFT PHCT is the SAFT EoS in conjunction with the asymmetric mixing rule originally developed for perturbed-hard-chain theory (PHCT) (Economou and Tsonopoulos, 1997). The performance of CPA is very satisfactory for both phases considering the uncertainty in the experimental data (especially in the water-rich phase, where reported solubilities may differ by an order of magnitude; Econo-

Figure 4. Mutual solubilities for the water/n-octane system at the three-phase equilibrium pressure. Prediction results were obtained with CPA, while for SAFT the interaction coefficient was obtained by fitting the water solubility data.

mou et al., 1997); both for lowsenvironmentally importantstemperatures (0-100 °C) and for the higher oness of interest to the petroleum industry. The predictions of water solubilities in the hydrocarbon-rich phase are comparable to the correlation ones for this phase with SAFT and orders of magnitude better for hydrocarbon solubility in the water-rich phase (Table 4 and Figures 4, 5, 7, and 8). The predicted vapor phase compositions presented in Table 4 also must be considered very

4180 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998

Figure 5. Mutual solubilities for the system water/n-decane at the three-phase equilibrium pressure. Prediction results were obtained with CPA, while for SAFT the interaction coefficient was obtained by fitting the water solubility data.

Figure 7. Mutual solubilities for the water/1-octene system at the three-phase equilibrium pressure. Prediction results were obtained with CPA, while for SAFT the interaction coefficient was obtained by fitting the water solubility data.

predictions are obtained with the CPA EoS, as can be seen in Figure 9 with the results for the water/propane, water/n-hexane, water/n-octane, and water/n-decane mixtures. We believe that a very important factor, which partially explains the improved performance of CPA over that of SAFT, is the choice of the pure-compound parameters for water, as demonstrated in Figure 1 for the case of the two-site model. The importance of the pure-compound parameters in correlating binary data with SAFT has been also suggested by Huang and Radosz (1991), while Suresh and Beckman (1994) revised the originally developed SAFT parameters (Huang and Radosz, 1990) of a series of pure associating compounds, including water, to be able to correlate satisfactory binary LLE data. Conclusions

Figure 6. Prediction of the mutual solubilities for the water/ cyclohexane system.

satisfactory for the systems for which data are available. Finally, excellent three-phase equilibrium pressure

Three different models for pure water, the two-, three-, and four-site ones, have been considered with the CPA EoS in the correlation of liquid-liquid equilibria of water-hydrocarbon systems. Several sets of parameters for water are possible for each model and have a pronounced effect on the obtained LLE results. Furthermore, of these three models only the four-site one gives successful results. One temperature-independent interaction parameters in the physical termsis sufficient and is very well correlated linearly with the molecular weight of the hydrocarbon in each homologous series, something that allows CPA EoS to be used as a predictive tool. Very satisfactory prediction results are obtained, considering the uncertainty in the experimental data, with typical

Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998 4181

Figure 8. Mutual solubilities for the water/1-decene system at the three-phase equilibrium pressure. Prediction results were obtained with CPA, while for SAFT the interaction coefficient was obtained by fitting the water solubility data.

errors of 5-15% in the water mole fraction in the hydrocarbon-rich phase and less than 1 order of magnitude for the hydrocarbon mole fraction in the waterrich phase. Furthermore, these results are similar to the correlation ones reported with the four-site SAFT model (Economou and Tsonopoulos, 1997) for the water solubility but are orders of magnitude better for the hydrocarbon solubilities even when SAFT employs the PHCT asymmetric mixing rules. This improved performance is attributed to the correct choice of parameters for water. Acknowledgment We acknowledge Shell Research and Technology Centre, Amsterdam, for financing part of this work; H. Meijer (Shell/SRTCA), R. A. S. Moorwood (Infochem Ltd., U.K.), and I. G. Economou (NRCPS “Democritos”) for useful discussions; and the latter for providing the results with the SAFT EoS. We also thank A. Kalezi and N. Tzinieris, seniors at NTUA, who performed some of the calculations. Nomenclature a ) energy parameter a0 ) parameter in the energy term b ) co-volume parameter b1 ) co-volume parameter of the associating compound b2 ) co-volume parameter of the inert compound c1 ) parameter in the energy term g ) radial distribution function kij ) adjustable binary interaction parameter P ) pressure Ps ) vapor pressure

Figure 9. Experimental and predicted results by the CPA EoS three-phase equilibrium pressures for the water/propane, water/ n-hexane, water/n-octane, and water/n-decane mixtures.

R ) gas constant T ) temperature Tr ) reduced temperature V ) molar volume Vl ) saturated liquid volume XA ) mole fraction of the compound not bonded at site A z ) compressibility factor Greek Letters β ) parameter in the association term of CPA ∆AB ) strength of interaction between sites A and B AB ) association energy of interaction between sites A and B F ) molar density ∑A ) summation over all the sites (starting with A) on the molecule Subscripts and Superscripts A, B ) for sites A and B on the molecule calc ) calculated exp ) experimental r ) reduced assoc ) association

Appendix: Derived Properties for the Association Term of the CPA EoS Under the assumptions made in the cluster partition function approach (Hendriks et al., 1997) and the additional assumption that association clusters mix ideally (this means that there are no cluster-cluster interactions), the contribution of the association term to the compressibility factor is given by eq 3. The corresponding contribution of the association term to the

4182 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998

chemical potential of the component i is given by

µassoc i

)

RT

N ∂ ln g

d(Ai) ln XA ∑ 2 A i

i

∂ni

xjd(Bj)(1 - XB ) ∑j ∑ B j

j

(A1)

where

∂g b 2.5 - y ) ∂F 4 (1 - y)4

(A2)

and

N

∂g

)

∂ni

Dg Dxi

-

Dg

∑j xjDx

+F

i

∂g ∂F

(A3)

The differential operator D/Dxi indicates differentiation with respect to xi where all other xj are formally held constant:

Dg 2.5 - y biF ) Dxi (1 - y)4 4

(A4)

Although no derivatives of XAi occur in eqs 3 and A1, these expressions are rigorous and applicable to every equation assuming radial distribution function density and composition-dependent and constant for the whole mixture. Furthermore, they are simpler than the corresponding expression proposed by Chapman et al. (1990). The derivation of eq A1 is based on the following idea. If in eq 5 g would be unity, the physical interactions are ideal and a closed expression for the Helmholtz free energy is available, and corresponding to this, eq A1 is valid without the second term. The trick is, now, to absorb g as a factor into the composition, so that the equations formally correspond to the ideal case. In this way, scaled composition variables are introduced, and by making use of the correspondence between the Helmholtz free energy and the reduced chemical potential in terms of these scaled compositions, eq A1 is derived by differentiation. In a completely similar way, eq 3 is derived by absorbing g (in eq 5) into density (F) (in eq 4), defining a new density variable, F′. Literature Cited Anderko, A. A simple equation of state incorporating association. Fluid Phase Equil. 1989a, 45, 39. Anderko, A. Extension of AEOS model to systems containing any number of associating and inert compounds. Fluid Phase Equil. 1989b, 50, 21. Anderko, A. Phase equilibria in aqueous systems from an equation of state based on the chemical approach. Fluid Phase Equil. 1991, 65, 89. Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New reference equation of state for associating fluids. Ind. Eng. Chem. Res. 1990, 29, 1709. Daubert, T. E.; Danner, R. P. Physical and thermodynamic properties of pure compounds: Data compilation; Hemisphere: New York, 1989. Economou, I. G.; Donohue, M. D. Equation of state with multiple association sites for water and water-hydrocarbon mixtures. Ind. Eng. Chem. Res. 1992, 31, 2388. Economou, I. G.; Donohue, M. D. Equations of state for hydrogen bonding systems. Fluid Phase Equil. 1996, 116, 518. Economou, I. G.; Heidman, J. L.; Tsonopoulos, C.; Wilson, G. M. High-temperature mutual solubilities of hydrocarbons and water. Part III: 1-hexene, 1-octene, C10-C12 hydrocarbons. AIChE J. 1997, 43 (2), 535.

Economou, I. G.; Tsonopoulos, C. Associating models and mixing rules in equations of state for water/hydrocarbon mixtures. Chem. Eng. Sci. 1997, 52, 511. Fu, Y.-H.; Sandler, S. I. A simplified SAFT equation of state for associating compounds and mixtures. Ind. Eng. Chem. Res. 1995, 34, 1897. Galindo, A.; Whitehead, P. J.; Jackson, G.; Burgess, A. N. Predicting the high-pressure phase equilibria of water + n-alkanes using a simplified SAFT theory with transferable intermolecular intermolecular parameters. J. Phys. Chem. 1996, 100, 6781. Goral, M. Cubic equation of state for calculation of phase equilibria in associating systems. Fluid Phase Equil. 1996, 188, 27. Heidemann, R. A.; Prausnitz, J. M. A van der Waals type equation of state for fluids for associating molecules. Proc. Natl. Acad. Sci. U.S.A. 1996, 73, 1173. Heidman, J. L.; Tsonopoulos, C.; Brady, C. J.; Wilson, G. M. Hightemperature mutual solubilities of hydrocarbons and water. Part II: Ethylbenzene, Ethylcyclohexane, and n-Octane. AIChE J. 1985, 31, 376. Hemptinne, J.-C. Importance of water-hydrocarbon phase equilibria for reservoir production and drilling operations. Production of reservoir fluids in frontier conditions, International Conference, 4-5 December 1997, Rueil-Malmaison, France. Hendriks, E. M.; Walsh, J.; van Bergen A. R. D. A general approach to association using cluster partition functions. J. Stat. Phys. 1997, 87, 1287. Huang, S. H.; Radosz, M. Equation of state for small, large, polydisperse and associating molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. Huang, S. H.; Radosz, M. Equation of state for small, large, polydisperse and associating molecules: Extension to fluid mixtures. Ind. Eng. Chem. Res. 1991, 30, 1994. Ikonomou, G. D.; Donohue, M. D. Extension of the associated perturbed anisotropic chain theory to mixtures with more than one associating component. Fluid Phase Equil. 1988, 39, 129. Kabadi, V. N.; Danner, R. P. A modified Soave-Redlich-Kwong equation of state for water-hydrocarbon phase equilibria. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 537. Kobayashi, R.; Katz, D. L. Vapor-liquid equilibria for binary hydrocarbon-water systems. Ind. Eng. Chem. 1953, 45, 440. Kontogeorgis, G. M.; Voutsas, E. C.; Yakoumis, I. V.; Tassios, D. P. An equation of state for associating fluids, Ind. Chem. Eng. Res. 1996, 35, 4310. Michel, S.; Hooper, H. H.; Prausnitz, J. M. Mutual solubilities of water and hydrocarbons from an equation of state. Need for an unconventional mixing rule. Fluid Phase Equil. 1989, 45, 173. Panayiotou, C.; Sanchez, I. C. Hydrogen bonding in fluids: equation of state approach. J. Phys. Chem. 1991, 95, 10090. Sage, B. H.; Lacey, W. N. Some properties of the lighter hydrocarbons, hydrogen sulfide and carbon dioxide; API Research Project 37; American Petroleum Institute: New York, 1955. Sorensen, J. M.; Arlt, W. Liquid-liquid equilibrium data collection. Binary systems. DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, Germany, 1979; Vol. V, Part 1. Suresh, J.; Beckman, E. J. Prediction of liquid-liquid equilibria in ternary systems from binary data. Fluid Phase Equil. 1994, 99, 219. Suresh, S. J.; Elliott, J. R. Multiphase equilibrium analysis via a generalized equation of state for associating mixtures. Ind. End. Chem. Res. 1992, 31, 2783. Tsonopoulos, C.; Wilson, G. M. High-temperature mutual solubilities of hydrocarbons and water. AIChE J. 1983, 29, 990. Tsonopoulos, C.; Heidman, J. L. High-pressure vapor-liquid equilibria with cubic equations of state. Fluid Phase Equil. 1986, 29, 391. Voutsas, E. C.; Kontogeorgis, G. M.; Yakoumis, I. V.; Tassios, D. P. Correlation of liquid-liquid equilibria for alcohol/hydrocarbon mixtures using the CPA EoS. Fluid Phase Equil. 1997, 132, 61. Yakoumis, I. V.; Kontogeorgis, G. M.; Voutsas E. C.; Tassios, D. P. Vapor-liquid equilibria for alcohol/hydrocarbon mixtures using the CPA EoS. Fluid Phase Equil. 1997, 130, 31.

Received for review December 30, 1997 Revised manuscript received June 3, 1998 Accepted June 16, 1998 IE970947I