Elliott−Suresh−Donohue and Simplified PC-SAFT - ACS Publications

Oct 31, 2006 - Two Wertheim-based association models, the simplified PC-SAFT and the Elliott-Suresh-Donohue (ESD) equation of state, are compared in t...
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8170

Ind. Eng. Chem. Res. 2006, 45, 8170-8179

Comparison of Two Association Models (Elliott-Suresh-Donohue and Simplified PC-SAFT) for Complex Phase Equilibria of Hydrocarbon-Water and Amine-Containing Mixtures Andreas Grenner,† Ju1 rgen Schmelzer,† Nicolas von Solms,‡ and Georgios M. Kontogeorgis*,‡ Department of Chemical Engineering, Hochschule fu¨r Technik und Wirtschaft DresdensUniVersity of Applied Sciences, Friedrich-List-Platz 1, 01069 Dresden, Germany, and Centre for Phase Equilibria and Separation Processes (IVC-SEP), Department of Chemical Engineering, Technical UniVersity of Denmark, DK-2800 Lyngby, Denmark

Two Wertheim-based association models, the simplified PC-SAFT and the Elliott-Suresh-Donohue (ESD) equation of state, are compared in this work for the description of vapor-liquid equilibria (VLE) and liquidliquid equilibria (LLE) in binary systems of aniline, cyclohexylamine (CHA), hydrocarbons, and water. Furthermore, the predictive capabilities of the models are investigated for four ternary systems composed of these components, which exhibit complex liquid-liquid(-liquid) equilibria (LLLE). Various aspects of association models which have an influence in the results are studied for the PC-SAFT equation of state, e.g., the choice of various association schemes for the amines and parametrization of water as well as different approaches for describing solvation. It is shown that simplified PC-SAFT using water parameters estimated in this work can describe successfully water-alkane LLE. In general, both models perform overall similarly for the binary systems, although ESD shows a remarkably good behavior despite its simplicity and the use of only the two-site scheme for all associating compounds. The prediction of the LLE in the ternary systems water + octane + aniline and water + CHA + aniline is satisfactory whereas larger deviations were obtained for the systems water + octane + CHA and octane + CHA + aniline. 1. Introduction Most advanced association models especially those belonging to the SAFT family have been extensively applied to mixtures containing polymers as well as several mixtures containing associating compounds. However, only few studies have been reported for complex solvating systems and aqueous mixtures as well as multicomponent mixtures exhibiting liquid-liquid equilibria. Thus, the capabilities of the models to such systems have not been thoroughly investigated. Complex associating mixtures are of importance both from an engineering and a scientific point of view. Industrially, they are met in the chemical industry as well as in oil and gas applications where methanol and glycols are used as gas-hydrate inhibitors. From scientific point of view, previous results1-4 for systems of phenol and cresols have shown interesting phase equilibria behavior such as three liquid phases, sigmoid excess molar volume curves, or azeotropes. Mixtures with amines are also of interest as amines are weaker hydrogen bonding compounds compared to alcohols, and still, they exhibit complex interactions with water and even three-liquid phase equilibria for ternary mixtures with hydrocarbons. Finally, aqueous mixtures with hydrocarbons are very complex due to the very low mutual solubilities and SAFTtype models typically have problems for such mixtures, especially the very low hydrocarbon solubility in the aqueous phase. Simultaneous description of both liquid phases with a single interaction parameter is very difficult. The models employed in this work are the Elliott-SureshDonohue (ESD) and the simplified PC-SAFT equation of state. * To whom correspondence should be addressed. Telephone: +45 45 25 28 59. Fax: +45 45 88 22 58. E-mail: [email protected]. † Hochschule fu¨r Technik und Wirtschaft DresdensUniversity of Applied Sciences. ‡ Technical University of Denmark.

Both have been described in the recent literature,5-13 and the main equations are presented in the Appendix. For our discussion, it suffices to say the following: (i) Both ESD and PC-SAFT use the Wertheim theory9-12 for the association contributions. ESD uses a cubic function for the physical interactions, while PC-SAFT uses a physically more realistic term for these effects, accounting for dispersive interactions between chains of hard-sphere molecules. (ii) Both models have three pure component parameters for nonassociating components and five in total for self-associating substances. (iii) The association scheme terminology of Huang and Radosz13 is employed. In accordance with the Suresh and Elliott6 recommendations, all ESD associating components are treated using the 2B scheme. (iv) The simplified PC-SAFT by von Solms et al.8 is used in the work. This equation is identical to the original PC-SAFT for pure compounds but uses simpler mixing rules taking into account that the segment diameters are similar for different segments belonging to different molecules. Tables 1 and 2 summarize the pure compound parameters used in this work for the two models. Some are taken from the literature; others have been estimated in this work based on vapor pressures and saturated liquid densities. The parameters were correlated to the DIPPR15 data compilation. The accuracy of the fit for the two amines is within the accuracy of the DIPPR correlations (about 5% for the vapor pressures and about 1-3% for the liquid densities of the two amines). For PC-SAFT, various association schemes are used for amines; both the simpler 2B and the more rigorous 3B scheme are examined. The PC-SAFT parameters for alkanes, alkenes, aniline 2B, and water 2B were taken from Gross and Sadowski.7,16 For water, several 4C parameters are tested: those presented by von Solms et al.17 and a new set obtained in this

10.1021/ie0605332 CCC: $33.50 © 2006 American Chemical Society Published on Web 10/31/2006

Ind. Eng. Chem. Res., Vol. 45, No. 24, 2006 8171 Table 1. ESD Pure Component Parametersa compound

ref

c

/k (K)

V* (cm3 mol-1)

HB/RTc

κAB/V*

Tr range

water aniline CHA pentane hexane heptane octane decane cyclohexane 1-hexene 1-octene

14 this workb this workb 14 14 14 14 14 14 14 14

1.0053 2.0205 1.1089 1.9117 2.1266 2.3002 2.4842 2.8366 1.7843 2.0316 2.3994

427.254 402.3207 460.2676 268.579 274.049 280.687 285.211 293.152 329.557 278.787 288.605

9.411 31.2260 45.8062 35.304 41.588 47.761 54.157 66.281 34.913 40.084 53.120

4.0000 6.21112 3.3200

0.100 9.79 × 10-7 0.0621

na 0.47-0.90 0.47-0.90 na na na na na na na na

a

∆P AAD (%)

∆F AAD (%)

na 1.11 1.32 na na na na na na na na

na 3.50 1.28 na na na na na na na na

Associating components are considered to be 2B molecules. The terminology refers to the work of Huang and Radosz.13

b

DIPPR15 data.

Table 2. PC-SAFT Pure Component Parametersa compound water

aniline CHA pentane hexane heptane octane decane cyclohexane 1-hexene 1-octene a

association scheme 2B 4C 4C 4C 4C 2B 3B 2B 3B

ref

σ (Å)

 (K)

m

AB(K)

κAB

Tr range

∆P AAD (%)

∆F AAD (%)

16 17 17 17 this workb 16 this workc this workc this workc 7 7 7 7 7 7 7 7

3.0007 2.3533 2.0794 1.9134 2.6273 3.7021 3.7050 3.7862 3.9001 3.7729 3.7983 3.8049 3.8373 3.8384 3.8499 3.7753 3.8133

366.62 207.84 183.61 199.88 180.30 335.47 331.58 291.18 298.09 231.20 236.77 238.40 242.78 243.87 278.11 236.81 243.02

1.0656 2.00 2.75 3.50 1.50 2.6607 2.6742 2.92438 2.71892 2.6896 3.0576 3.4831 3.8176 4.6627 2.5303 2.9853 3.7424

2500.7 1506.4 1354.1 839.0 1804.22 1351.6 1321.65 688.71 747.14

0.034868 0.1550 0.3374 0.7901 0.0942 0.074883 0.0392 0.0452 0.0376

0.42-1.00 0.50-0.90 0.50-0.90 0.50-0.90 0.50-0.90 0.38-1.00 0.50-0.90 0.50-0.90 0.50-0.90 0.30-1.00 0.35-1.00 0.33-ovcrit 0.38-1.00 0.39-ovcrit 0.50-1.00 0.26-1.00 0.30-1.00

1.88 0.70 0.44 0.73 0.93 0.79 1.11 1.72 1.97 1.45 0.31 0.34 0.77 0.24 0.53 0.42 0.79

6.83 1.66 1.05 1.24 2.62 1.09 1.03 0.68 0.85 0.78 0.76 2.10 1.59 1.18 3.12 1.23 0.75

Terminology refers to Huang and Radosz.13

b

NIST18 data. c DIPPR15 data.

work based on data of the National Institute of Standards and Technology (NIST).18 The range of temperature for which the equation of liquid density of the DIPPR data compilation is valid is only from 273 to 333 K. However, the modeling in this work extends to 373 K. 2. Water-Alkane Liquid-Liquid-Equilibrium: The Effect of Water Parameters The effect of the parametrization of water has been thoroughly studied in this work with the purpose of investigating the performance of simplified PC-SAFT on water-alkane liquidliquid equilibria. The following parameter sets have been considered: (1) The 2B water set presented in the literature by Gross and Sadowski.16 (2) Three of the various 4C sets recently presented by von Solms et al.17 These parameter sets were estimated with a fixed segment number from 2.00 to 3.50 in steps of 0.25. The sets chosen in this work correspond to m ) 2.00, 2.75, and 3.50. (3) A new 4C parameter set which is based on a number of physically justified arguments. (i) The number of segments for water should be small around unity, as water is a low molecular weight compound. (ii) The dispersion energy parameter should be low, as molecular simulations studied by Errington et al.19 give a dispersion energy for water between  ) 74-160 K. (iii) The association energy should be close to that reported by Koh et al.20 which is AB ) 1813 K. The water parameter set obtained from simultaneous fitting of vapor pressure and liquid density data results to the values of m ) 1.5,  ) 180.3 K, AB ) 1804.22 K which are in good agreement with the “expected” physically reasonable values of these parameters.

Figure 1. Calculated association strength of PC-SAFT for water. The different parameter sets are those of von Solms et al.,17 Gross and Sadowski,16 and this work.

The results are compared also with the ESD EoS (using a 2B water parameter set). Figure 1 shows a comparison of the calculated association strength ∆AB of PC-SAFT for the used water parameter sets.

[ ( ) ]

∆AB ) g(d)seg exp

AB - 1 κABσ3 kT

(1)

where g(d)seg is the radial distribution function of a reference fluid of hard spheres, d is the diameter of a chain segment on a molecule, AB is the association energy, κAB is the association volume, k is the Boltzmann constant, and T is the temperature. The parameter sets of von Solms et al.17 (4C; m ) 2.00, 2.75,

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Figure 2. LLE of octane + water. The experimental values are those of Tsonopoulos.22 The prediction of simplified PC-SAFT is shown with water parameters of von Solms et al.17 with 4C, Gross and Sadowski16 with 2B, and this work with 4C, and the prediction of ESD is shown with 2B.

and 3.50) have a lower association strength compared to the parameter set of this work. The 2B parameters of Gross and Sadowski16 show noticeable higher association strengths than the other sets. The 4C m ) 1.5 water parameter of this work gives an association strength which is between those of von Solms et al.17 and those of Gross and Sadowski.16 Perhaps a slightly high value for the association strength together with the physically correct association scheme, which is 4C for water as von Solms et al.17 pointed out, are in accord with the hydrophobic effect whereby entropic effects resulting from higher ordering of the water play an important role. Figure 2 shows the prediction (kij ) 0) results for the wateroctane LLE. The solubility of water in octane is predicted to be too high with all sets except the one developed in this work. The 2B water parameters of Gross and Sadowski16 give a too high solubility of octane in water. ESD strongly overpredicts the solubility in both phases. The prediction with simplified PCSAFT using the water parameters estimated in this work is very satisfactory. The octane-rich phase is predicted very well but the hydrocarbon solubility in the water-rich phase is predicted to be higher than the experimental value. Equally satisfactory results have been obtained for the other systems studied (LLE of water with pentane until octane, decane, cyclohexane, 1-hexene, and 1-octene). Some typical results are shown in Figures 3 and 4. Table 3 summarizes the results for all systems studied. For simplified PC-SAFT, it is necessary to mentioned that unfortunately within the good prediction of the hydrocarbon rich phase the results of the water rich phase are, without fitting of the kij, not satisfactory. However, for industrial application, the results for the hydrocarbon-rich phase are more important and that was the focus of this work. ESD performs with a fitted kij much better for the water-rich phase but is slightly worse for the hydrocarbon-rich phase. Suresh and Elliott6 proposed also a predictive equation for calculating the kij for the LLE of hydrocarbon + water which needs only two pure component parameters of each hydrocarbon and water. However, these kij diverge from those estimated in this work for about 10%. The results with the calculated kij are in comparison to the fitted one of this study are quiet similar for the hydrocarbon-rich phase but are about 20% better for the water-rich phase. In conclusion, simplified PC-SAFT predicts for all alkanes studied very satisfactorily the solubilities in the hydrocarbonrich phase, while the solubility of the hydrocarbon in water is

Figure 3. LLE in the system pentane + water. The experimental values are those of Tsonopoulos.22 PC-SAFT prediction; kij fit to water-rich phase (both with water parameters of this work) and ESD fit to pentane-rich phase.

Figure 4. LLE in the system decane + water. The experimental values are those of Economou et al.24 PC-SAFT prediction; kij fit to water-rich phase (both with water parameters of this work) and ESD fit to decanerich phase.

higher than the experimental values. The water-rich phase can be better described with simplified PC-SAFT using a kij ∼ 0.06 with the resulting consequence of underestimating the water solubility in the hydrocarbon-rich phase. LLE predictions for alkenes are less satisfactory, possibly due to ignoring the solvation between water and the olefins. This is in agreement with previous investigations with SAFT.25 ESD provides results which compare favorably to other SAFT variants, e.g., those shown by Voutsas et al.25 or Economou and Tsonopoulos.26 3. Amine-Containing Self-Associating Systems This section presents VLE and LLE results with simplified PC-SAFT and ESD for self-associating mixtures: a saturated amine (CHA) and an aromatic amine (aniline) with octane. The results are compared to the experimental data from Grenner et al.27,28 and Arlt and Onken.29 Some typical results are presented in Figures 5 and 6, while all results are summarized in Table 4. The following points summarize our conclusions: (1) For PC-SAFT, the 3B scheme is recommended for CHA and aniline, as this is the best overall choice. The differences between association schemes are small, though.

Ind. Eng. Chem. Res., Vol. 45, No. 24, 2006 8173 Table 3. Water + Hydrocarbon Liquid-Liquid Equilibria of PC-SAFT with Water 4C Parameters of This Work and ESD with Water 2B Parameters of Reference 14 PC-SAFT bias (%)a) pentane hexane heptane octane decane cyclohexane 1-hexene 1-octene a

ESD bias (%)a

water in HC

HC in water

kij

water in HC

HC in water

kij

3.88 0.64 -4.07 -10.91 -5.49 22.46 -40.48 -45.10

572.18 989.62 1461.05 2224.00 342.92 1514.38 226.72 645.43

0 0 0 0 0 0 0 0

-17.03 -16.70 -14.31 -12.45 -15.41 -20.36 -20.68 -26.01

-81.00 -82.48 -79.41 -76.16 -98.28 -89.13 -74.28 -85.83

0.24 0.23 0.23 0.23 0.24 0.25 0.19 0.20

Bias (%) ) 100/np∑(xcalcld - xexptl)/xexptl where np is the number of data points. Table 4. Deviations and kij Values for PC-SAFT and ESD in Self-Associating Systems

CHA 2B + octane aniline 2B + octane

data

kij

∆Pa (%)

∆yb

VLE 333.15 K

ESD -0.004

1.29

0.0161

VLE 363.15 K VLE 383.15 K

-0.0012 0.037

1.43 3.73

0.0130

LLE CHA 3B + octane aniline 3B + octane aniline 2B + octane Figure 5. VLE in the system CHA + octane, simplified PC-SAFT CHA parameter 3B, this work, and ESD results with fitted kij: (ESD) kij ) -0.005, (PC-SAFT) kij ) 0.001 (for both temperatures, respectively). Experimental data are taken from Grenner et al.27

0.021

∆xb

0.0378

VLE 333.15 K

PC-SAFT 0.001

0.79

0.0132

VLE 363.15 K VLE 383.15 K

0.001 0.012

1.02 0.78

0.0163

LLE VLE 383.15 K

0.015 0.016

0.76

LLE

0.025

0.0931 0.0862

∆P ) 100/np∑((|Pcalcld - Pexptl|)/Pexptl). b ∆Z ) 1/np∑(|Zcalcld - Zexptl|) where np is the number of data points and Z represents y or x. a

4. Solvating Systems Accurate description of the solvating systems, i.e., those that contain at least two associating compounds, is crucial for good multicomponent results, and such mixtures often represent a very challenging problem. It is not always the case that simple combining rules can describe the complex interactions in such mixtures. Such rules are often based on simplifications. For an example, see the Elliott combining rule (ECR)6 for the association strength used here.

∆AiBj ) x∆AiBi∆AjBj

Figure 6. LLE in the system aniline + octane, simplified PC-SAFT results with aniline parameter 2B of Gross and Sadowski,16 3B parameter of this work, and ESD results. Experimental data are taken from Arlt and Onken.29

(2) The VLE is satisfactorily calculated with both models. However, the pressure at the azeotropic point is underestimated by simplified PC-SAFT, whereas it is overestimated by ESD. The interaction parameter values are small, and the predictions (using kij ) 0) are satisfactory. (3) ESD performs better than simplified PC-SAFT for the LLE system (aniline + octane) at least with respect to the overall temperature dependency, but the solubility in the aniline-rich phase is overestimated.

(2)

It is assumed that, e.g., in the system CHA + water that the association strength is the same for O-H‚‚‚N bonds and for N-H‚‚‚O bonds. Although such assumptions are approximate, the ECR has been previously successfully applied30,31 and will be used for all solvating systems in this work. Table 5 summarizes all results for the systems considered here, while Figures 7-9 show a few characteristic results. We can observe the following: (1) Figure 7 shows the quasi-ideal VLE behavior of the system CHA + aniline, where solvation compensates the weak interactions between the saturated and the aromatic hydrocarbon. The ECR is a reasonable choice, and the performance of both models is very good with small interaction parameters. In the case of PC-SAFT, 3B performs slightly better than 2B. (2) VLE in the system CHA + water is of special interest. There is a bubble pressure maximum at high concentrations of water at 333.15 and 363.15 K. Another peculiarity is that the system is completely miscible, while cyclohexane + water and

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Table 5. Deviations and kij Values for PC-SAFT and ESD in Solvating Systems

CHA 2B + aniline 2B CHA 2B + water 2B water 2B + aniline 2B

CHA 3B + aniline 3B CHA 3B + aniline 2B CHA 3B + water 4Cd water 4Cd + aniline 3B water 4Cd + aniline 2B kij ) kij + represents y or a

data

kij

∆Pb (%)

∆yc

VLE 333.15 K VLE 363.15 K VLE 333.15 K VLE 363.15 K VLE 333.5 K/363.15 K VLE 373.15 K LLE

ESD -0.0004 0.0004 0.045 0.071 kij ) -0.028 kTij ) 0.00086a kij ) 0.088 kTij ) -0.0002a kij ) 0.088 kTij ) -0.0002a

2.16 0.88 2.71 3.65 3.21 8.93

0.0002 0.0001 0.0135 0.0180 0.0159 0.0099

VLE 333.15 K VLE 363.15 K VLE 333.15 K VLE 363.15 K VLE 333.15 K VLE 363.15 K VLE 373.15 K LLE VLE 373.15 K LLE

PC-SAFT -0.025 -0.02 -0.022 -0.018 1.9ECR kij ) -0.05 1.9ECR kij ) -0.03 0.085 0.055 0.03 0.00

kTij(T - 273.15 K). b ∆P ) 100/np∑((|Pcalcld x. d Water parameter of this work.

- Pexptl|)/Pexptl).

Figure 7. VLE in the system CHA + aniline, prediction of simplified PC-SAFT (both components 3B) and ESD. Experimental data are taken from Grenner et al.27

cyclohexanol + water are not. The system cyclohexane + water has a large miscibility gap (at 298.15 K water in cyclohexane 0.058 mol %, cyclohexane in water 0.0012 mol %),34 which is somewhat smaller for cyclohexanol + water (at 298.15 K water in cyclohexanol 43.3 mol %, cyclohexanol in water 0.708 mol %).34 All of these indicate strong interactions between the CHA and water which should occur through hydrogen bonding, but the hydrogen bonds between cyclohexanol and water are stronger (hydroxyl-hydroxyl) than those between CHA and water (amine-hydroxyl). However, a more detailed consideration of CHA provides a clue, as ionic interaction exists between CHA + watersCHA is a base having a pKa ) 10.64.35 The prediction of this system with simplified PC-SAFT is poor. The vapor pressure prediction is too high over the entire range of concentration, which indicates that the strength of interaction between CHA and water is underestimated. This could be attributed to the fact that PC-SAFT is not using a term for ionic interaction. One way to compensate for this is to increase the association strength by multiplying the solvation strength given by ECR by a factor of 1.9. Clearly better results can be obtained in this way. ESD also without a term for ionic interactions

c

∆xc

0.0296 3.55 3.54 3.45 2.74 3.02 3.50 3.26

0.0002 0.0001 0.0100 0.0086 0.0403 0.0289 0.0093

3.75

0.0094

0.0334 0.0402

∆Z ) 1/np∑(|Zcalcld - Zexptl|) where np is the number of data points and Z

Figure 8. VLE in the system water + aniline at 373.15 K, simplified PCSAFT results (water parameters of this work) aniline 2B kij ) 0.03, aniline 3B kij ) 0.085; ESD results kij ) 0.088, kijT ) -0.0002. Experimental data are taken from Griswold et al.32

predicts ideal behavior for CHA + water, and the correlation is very satisfactory using a single kij. (3) Figures 8 and 9 show the VLE and LLE results for water + aniline. The basicity of aniline is weak, pKa ) 4.87,35 because the phenyl group pulls electrons from the amine group and additionally the free electron pair is delocalized in the aromatic structure. The performance of both models is similar, with simplified PC-SAFT being better for VLE and ESD being better for LLE. Temperature-dependent parameters are used for ESD/ LLE. 5. Multicomponent Systems: Prediction of Ternary Liquid-Liquid(-Liquid) Equilibria The final target of this work was the prediction of complex ternary LLE and LLLE at 298.15 and 333.15 K of mixtures consisting of the compounds already discussed in the sections on binaries above. The experimental data of the system water + octane + CHA were taken from a work of Klauck et al.,36 and the data for the systems water + octane + aniline, water + CHA + aniline, and octane + CHA + aniline were taken from the recent work of Grenner et al.28

Ind. Eng. Chem. Res., Vol. 45, No. 24, 2006 8175 Table 6. Parameters (kij) Summarized from Tables 4 and 5 Which Were Used for the Predictions in Ternary Systems system

data

kij

water 2B + octane CHA 2B + octane aniline 2B + octane CHA 2B + water 2B CHA 2B + aniline 2B water 2B + aniline 2B

LLE VLE 333.15 LLE VLE 333.5/363.15 VLE 333.15 LLE

ESD 0.23 -0.004 0.021 kij ) -0.065 kTij ) 0.00086a -0.0004 kij ) 0.088 kTij ) -0.0002a

water 4Cd + octane CHA 3B + octane aniline 2B + octane CHA 3B + water 4Cd CHA 3B + aniline 2B water 4Ce + aniline 2B

LLE VLE 333.15 LLE VLE 333.15 VLE 333.15 LLE

PC-SAFT 0 0.001 0.025 1.9ECR kij ) -0.05 -0.022 0.00

kij ) kij + represents y or a

kTij(T - 273.15 K). b ∆P ) 100/np∑((|Pcalcld x. d Water parameter of this work.

- Pexptl|)/Pexptl).

Figure 9. LLE in the system water + aniline, simplified PC-SAFT results (water parameters of this work) aniline 2B kij ) 0, aniline 3B kij ) 0.055; ESD results kij ) 0.088, kijT ) -0.0002. Experimental results are taken from Sazonov et al.33 and Sørensen and Arlt.34

The observed behavior is rather complex, e.g., in the systems CHA + aniline + octane, the miscibility gap becomes smaller with increasing temperature, with the opposite being the case in the system CHA + aniline + water, and the temperature dependency is very small for water + octane + CHA (see Figure 10). For the latter system, there is a narrow temperature range of about 323-343 K where three liquid phases occur. This system has type I behavior, since there is only one binary pair of partial miscible liquids. Among such systems, LLLE is seldom seen. Three liquid phases also occur in the system anline + octane + water, but this is not surprising since all three binary border systems have large miscibility gaps. The predictions are based on the selected fitted binary kij parameter from Tables 4 and 5. Table 6 summarizes the parameters used for the prediction of the ternary systems. Aniline has been considered as a 2B molecule as in all cases; better results are obtained in this way. Aniline is a special case having the free electron pair delocalized in the aromatic structure. Additionally, the free electron pair of the slight pyramidal amine group can interact with the π-system of the aromatic ring.37 As one of the three association sites is restricted, it is reasonable to use the 2B scheme. For the system CHA + water, it was necessary to use a temperature-dependent kij parameter for ESD because with a single kij a miscibility gap occurs in this system.

c

∆Pb (%)

∆yc

1.29

0.0161

3.21 2.16

0.0159 0.0002

∆xc 18.33 3.78

2.96 0.1091 0.79

0.0132

3.02 3.45

0.0403 0.0100

0.0862 0.0402

∆Z ) 1/np∑(|Zcalcld - Zexptl|) where np is the number of data points and Z

Figure 10. LL(L)E in the system octane + CHA + water. Experimental data are taken from Klauck et al.36

Figure 11. LLE in the system octane + CHA + water at 298.15 K, prediction of ESD and simplified PC-SAFT based on binary kij parameter (Table 6 and Table 7). Experimental data are taken from Klauck et al.36

All results are presented graphically in Figures 11-18. Predictions for the system CHA + water + octane with ESD and simplified PC-SAFT are shown in Figures 11 and 12. For ESD, two results are shown: one result with parameters only fitted to one data set (either VLE or LLE) (ESD-1), similarly as for PC-SAFT. This approach is followed in all the other predictions presented in this work. The results of both models

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Figure 12. LL(L)E in the system octane + CHA + water at 333.15 K, prediction of ESD and simplified PC-SAFT based on binary kij parameter (Table 6 and Table 7). Experimental data are taken from Klauck et al.36

Figure 14. LL(L)E in the system octane + aniline + water at 333.15 K, prediction of ESD and simplified PC-SAFT based on binary kij parameter (Table 6). Experimental data are taken from Grenner et al.28

Figure 13. LL(L)E in the system octane + aniline + water at 298.15 K, prediction of ESD and simplified PC-SAFT based on binary kij parameter (Table 6). Experimental data are taken from Grenner et al.28

Figure 15. LLE in the system octane + CHA + aniline at 298.15 K, prediction of ESD and simplified PC-SAFT based on binary kij parameter (Table 6). Experimental data are taken from Grenner et al.28

are similar at both temperatures, both giving a larger miscibility gap than experiment. The temperature dependence is not very well reflected, and the occurrence of three liquid phases is not predicted. The other prediction, ESD-2, is a result of a simultaneous fit to several data sets. For the simultaneous fit, the following were used: for CHA + octane, two isothermal VLE data sets of Grenner et al.;27 for octane + water, the LLE regressions of Tsonopoulos22 and azeotropic data;38 for CHA + water, two isothermal VLE data sets of Grenner et al.,27 one isobaric data set of Tanaka et al.,39 and azeotropic data of Carswell and Morrill.40 The parameter and deviations are given in Table 7. The results can be improved somewhat for both temperatures but are still not really satisfactory. Figures 13 and 14 show results for the system water + octane + aniline. Each binary system has a large miscibility gap over an extended temperature range. At 298.15 K, both models predict the LLE and the LLLE satisfactorily. At the higher temperature of 333.15 K, simplified PC-SAFT gives better results for LLLE. The LLLE predicted with ESD gives an octane mole fraction for the octane-rich phase which is too low. Figures 15 and 16 show results for the system octane + CHA + aniline. A strong temperature dependence exists for the

Figure 16. LLE in the system octane + CHA + aniline at 333.15 K, prediction of ESD and simplified PC-SAFT based on binary kij parameter (Table 6). Experimental data are taken from Grenner et al.28

binodal curve. Both models give similarly poor results for the binodal curve at 298.15 K, which is too large. The tie line prediction with the two EoS is better since the slope of these is

Ind. Eng. Chem. Res., Vol. 45, No. 24, 2006 8177

Figure 17. LLE in the system water + CHA + aniline at 298.15 K, prediction of ESD and simplified PC-SAFT based on binary kij parameter (Table 6). Experimental data are taken from Grenner et al.28

ternary systems with water, octane, cyclohexylamine (CHA), and aniline. An extensive study of LLE in water + alkane systems was conducted; good results were obtained with both models but especially with simplified PC-SAFT using new water parameters obtained as part of this study. For binary selfassociating systems, both ESD and simplified PC-SAFT give similarly good VLE results, while for LLE, ESD presents better results. ESD performs overall somewhat better compared with simplified PC-SAFT for the solvating systems considered. However, when single systems are considered, the ranking is different; simplified PC-SAFT gives, e.g., in the system aniline + water, better results while ESD, e.g., in the system CHA + water. Finally, among the four ternary liquid-liquid(-liquid) equilibria, no model shows a clear advantage over the other and similar results were obtained. For two of the multicomponent systems (water + octane + aniline, water + CHA + aniline), the experimental data are well represented by both models, while larger deviations are obtained for the two others (water + octane + CHA, octane + CHA + aniline). Acknowledgment The authors are grateful to the Danish technical research council for financial support of this work as part of the project “Advanced Thermodynamic Tools for Computer-Aided Product Design”. We also thank Michael L. Michelsen for helpful discussions. Appendix: Applied Equations of State

Figure 18. LLE in the system water + CHA + aniline at 333.15 K, prediction of ESD and simplified PC-SAFT based on binary kij parameter (Table 6). Experimental data are taken from Grenner et al.28 Table 7. Parameter (kij) and Deviation of Simultaneous Fitting of VLE and LLE Data Used for ESD-2 Results in Figures 11 and 12 system

data

kij

kTij a

water + ref 22, 0.154728 0.0006638 octane 38 water + ref 27, -0.005326 0.0008409 CHA 39, 40 octane + ref 27 -0.030420 CHA

∆Pb (%)

∆yc

∆xc

5.97

0.0176 0.0014

2.28

0.0114

1.46

0.0154

a k ) k + kT(T - 273.15 K). b ∆P ) 100/n ∑((|P ij ij p calcld - Pexptl|)/ ij Pexptl). c ∆Z ) 1/np∑(|Zcalcld - Zexptl|) where np is the number of data points and Z represents y or x.

very similar to the experimental one. The LLE prediction at 333.15 K is challenging since the binodal is very flat and small. ESD performs better as a smaller two-phase area is predicted. Finally, Figures 17 and 18 show predictions for water + cyclohexylamine + aniline. Both models perform very well, with simplified PC-SAFT giving slightly better results than ESD at 298.15 K. Conclusions Two association models, the ESD and the simplified PCSAFT equations of state, were compared for six binary and four

The theory of Wertheim9-12 is the basis for the association term used in both models. We refer to the original papers of Suresh and Elliott6 and Huang and Radosz13 for more details about how the theory is employed. A. Elliott-Suresh-Donohue Equation of State (ESD). The ESD-EoS (ESD) was presented in the basic form by Elliott et al.5 and in the final version with regard to the thermodynamic perturbation of Wertheim by Suresh and Elliott.6 The ESD written as an expression of the compressibility factor is

4cη + 1 - 1.9η 9.49qηY -1 + XAi (A1) + 1 + 1.7745ηY 1 - 1.9η0

Z ) 1 + Zrep + Zatt + Zassoc ) 1 +

where Zrep is the contribution of repulsive forces, Zatt is the dispersion contribution, and Zassoc is the contribution due to association. The ESD has, equal to PC-SAFT, three pure component parameters for nonassociating components and five if the component is self-associating. The ESD approach to model a molecule is the following. Starting from a spherical molecule with the volume V* (V*, the so-called “size parameter”, is a pure component parameter of ESD) will give a characteristic shape by the pure component parameter c (c is the shape parameter for the repulsive term). Vapor pressure analysis of hydrocarbons was performed to determine a parameter which gives a relation between the shape parameter c of the repulsive term and the shape parameter q in the attractive term.

q ) 1 + 1.90476(c - 1)

(A2)

Furthermore, the reduced density η is given by

η≡

NV* V

(A3)

8178

Ind. Eng. Chem. Res., Vol. 45, No. 24, 2006

and the parameter Y which contains the pure component parameter /k is

Y ≡ exp

(kT ) - 1.0617

(A4)

The symbol XAi in the association term denotes the fraction of unbounded sites. B. Simplified PC-SAFT. The PC-SAFT presented by Gross and Sadowski7 was used in the full modification of von Solms et al.8 The compressibility factor for a mixture of associating molecules

Z ) 1 + Zhc + Zdisp + Zassoc

(A5)

where Zhc is the contribution of the hard-sphere chain reference system, the contributions of Zdisp and Zassoc are the contributions of dispersion forces and association. Zdisp and Zassoc were used as in the original PC-SAFT7 version. The two modifications of von Solms et al.8 affect only the hard-sphere chain term. For the first modification, it is defined that η ≡ ζ3 where η ) πFd3/ 6∑ximi and the “average” segment diameter is given by the following expression

d)

( ) ∑i ximidi3

1/3

∑i ximi

(A6)

Therewith will be obtained a very simple expression for the radial distribution function

ghs(d+ ) )

1 - η/2 (1 - η)3

(A7)

Using eq A7, the hard-sphere term given by the CarnahanStarling equation is reduced to

Zhs )

1 + η + η 2 - η3 (1 - η)3

(A8)

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ReceiVed for reView April 27, 2006 ReVised manuscript receiVed August 9, 2006 Accepted September 18, 2006 IE0605332