Evaluation of the Truncated Perturbed Chain-Polar Statistical

Ind. Eng. Chem. Res. 2006, 45, 6063-6074

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Evaluation of the Truncated Perturbed Chain-Polar Statistical Associating Fluid Theory for Complex Mixture Fluid Phase Equilibria Eirini K. Karakatsani,† Georgios M. Kontogeorgis,*,‡ and Ioannis G. Economou*,† Molecular Thermodynamics and Modeling of Materials Laboratory, Institute of Physical Chemistry, National Centre of Scientific Research “Demokritos”, GR-15310 Aghia ParaskeVi Attikis, Greece, and Department of Chemical Engineering, Center for Phase Equilibria and Separation Processes (IVC-SEP), Technical UniVersity of Denmark, DK-2800 Lyngby, Denmark

Perturbed chain-statistical associating fluid theory (PC-SAFT) was extended rigorously to polar fluids based on the theory of Stell and co-workers [Mol. Phys. 1977, 33, 987]. The new PC-PSAFT was simplified to truncated PC-PSAFT (tPC-PSAFT) so that it can be practical for real polar fluid thermodynamic calculations. In this work, tPC-PSAFT is generalized to multicomponent mixtures and evaluated for a wide range of highly nonideal polar mixtures. Binary and ternary mixtures of dipolar, quadrupolar, and/or associating fluids are examined. Vapor-liquid and liquid-liquid equilibria at low and high pressures are calculated. Comparisons against PC-SAFT calculations are made. It is shown that tPC-PSAFT is an accurate model for polar fluid mixture phase equilibria. Introduction Equations of state (EoS) are used widely for the calculation of single phase thermodynamic properties and phase equilibria of pure components and mixtures.1 The unprecedented increase of computing power at relatively low cost over the last two decades allows the use of more accurate and, at the same time, relatively more complex thermodynamic models. Consequently, simple cubic EoS are systematically replaced by semitheoretical higher order EoS, both in academia and in industry. Furthermore, EoS are used today for the design of industrial processes that involve a wide range of fluids including highly polar fluids, polymers, etc. both for low and high pressure conditions.2-4 The major issue in all cases is the accuracy of the model over a wide range of conditions and the need for a minimum number of binary, preferably temperature-independent, adjustable parameters. The development of new accurate EoS is still a very active research area despite numerous models proposed in recent years. A very successful class of EoS is based on perturbation theory. Early perturbation models for real complex fluids were proposed in the 1970s and 1980s by Prausnitz, Donohue and coworkers.5-7 Perturbed anisotropic chain theory (PACT) is a very accurate model for polar fluids. Because of its relatively high complexity though, it never gained broad acceptance by the thermodynamic community. Today, the most widely used perturbation model is based on Wertheim’s first order thermodynamic perturbation theory (TPT-1)8-10 as developed into an engineering model,11-13 known as statistical associating fluid theory (SAFT). In its original formulation, SAFT’s reference term is the hard sphere fluid while weak attractions and association are treated as perturbations. SAFT attracted impressive attention in academia and in industry, and very many modifications and extensions of the model were proposed. One of the most successful SAFT extensions is the so-called * To whom correspondence should be addressed. E-mail: [email protected]. Tel.: ++ 45 45252859. Fax: ++ 45 45252800 (G.M.K.). E-mail: [email protected]. Tel.: ++ 30 2106503963. Fax: ++ 30 2106511766 (I.G.E.). † National Centre of Scientific Research “Demokritos”. ‡ Technical University of Denmark.

perturbed chain-SAFT (PC-SAFT) where the reference fluid is the hard chain fluid.14,15 A critical evaluation of SAFT-based models is given by Economou.16 Recently, Karakatsani et al.17,18 extended both SAFT and PCSAFT to polar fluids, namely dipolar, quadrupolar, and polarizable fluids, rigorously based on Stell and co-workers theory for simple polar fluids.19 To maintain a moderate model complexity, the new polar PC-SAFT (PC-PSAFT) was further simplified to truncated PC-PSAFT (tPC-PSAFT) using a truncated version of the multipolar perturbation expansion. Pure dipolar component and fluid mixture properties were calculated from the models and found to be in good agreement with experimental data. In this work, both models are further extended to mixtures of dipolar and quadrupolar fluids and fluids that exhibit both polar and hydrogen bonding interactions (e.g., water). Calculations are presented for binary and ternary mixtures with emphasis to vapor-liquid (VLE) and liquid-liquid equilibria (LLE) at low and high pressure. Comparisons are presented against the original SAFT and, mainly, PC-SAFT. In most cases, explicit inclusion of polar interactions improves model capabilities considerably. Model Description PC-PSAFT and tPC-PSAFT EoS proposed by Karakatsani et al.17,18 can be expressed in terms of reduced Helmholtz free energy as:

ares(T,F) ahs(T,F) achain(T,F) aassoc(T,F) adisp(T,F) ) + + + + RT RT RT RT RT apolar(T,F) aind(T,F) + (1) RT RT where T and F are the temperature and the molar density of the system, respectively, and R is the universal gas constant. The residual Helmholtz free energy is the difference between the Helmholtz free energy of the fluid, a, minus the Helmholtz free energy of an ideal gas, aideal, at the same T and F. All other thermodynamic properties (pressure, chemical potential, etc.) are calculated from the Helmholtz free energy of the fluid using standard thermodynamic relations.1

10.1021/ie060313o CCC: $33.50 © 2006 American Chemical Society Published on Web 07/15/2006

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Ind. Eng. Chem. Res., Vol. 45, No. 17, 2006

The hard-sphere, chain, association, and dispersion terms of eq 1 are the same as in PC-SAFT and are presented here briefly. The hard-sphere term is given by the Carnahan-Starling expression:

a 4η - 3η )m RT (1 - η)2 hs

()

adisp 1 1 2 3 ) -2πFI1(η,m)m2 σ3 - πFmC1I2(η,m)m2 σ (10) RT T˜ T˜ where C1 is the following compressibility expression

(

2

(2)

In this equation, m is the number of spherical segments per molecule and η is the reduced density (segment packing fraction):

η ) τFmV°

(3)

(

( ))

For the chain term in eq 1, the following expression is used based on Wertheim’s thermodynamic perturbation theory

(

)

M

(

ln XA ∑ A)1

)

RT

)

XA 2

+

M 2

I1(η,m) )

ai(m)ηi ∑ i)0

I2(η,m) )

bi(m)ηi ∑ i)0

m-1 m-1m-2 a + a m 1i m m 2i

(14)

bi(m) ) b0i +

m-1 m-1m-2 b + b m 1i m m 2i

(15)

The coefficients Rji and bji are given by Gross and Sadowski14 and are not repeated here. As for the polar and induced polar terms, PC-PSAFT and tPC-PSAFT use the simple Pade´ approximants proposed by Stell and co-workers:19

apolar 2 apolar )m polar polar RT 1 - a /a

(16)

aind 2 aind )m RT 1 - aind/aind

(17)

3

(7)

where M is the number of association sites per molecule and XA is the mole fraction of molecules not bonded at specific interaction site A. The summation is over all association sites on the molecule. The nonbonded fraction XA is calculated from the following:

1 M

1+

(8)

FXB∆AB ∑ B)1

where the summation is over all different types of sites and ∆AB is the association strength, given by

[ ( ) ]

AB 1 - 0.5η exp - 1 κAB 3 kT (1 - η)

∆AB ) x2V°°

(9)

In the latter, two new pure component parameters are introduced: the association energy, AB, and the association volume, κAB. In PC-SAFT, the dispersion contribution to the Helmholtz free energy is given from the expression:

(13)

ai(m) ) a0i +

2

and

3

XA )

(12)

where the coefficients ai and bi are functions of the chain length

(6)

while the association term is calculated from the expression:

aassoc

6

(4)

(5)

2-η achain ) (1 - m) ln RT 2(1 - η)3

(11)

6

where T˜ ) T/(u/k) and u/k is the dispersion energy parameter per segment. So far, three pure component parameters were introduced: the number of segments per molecule, m, the temperature-independent segment volume, V°°, and the dispersion energy per segment, u/k. In PC-SAFT, the segment diameter σ is often used rather than the segment volume V°° and the two parameters are related through the simple expression:

( )

-1

The integrals in the dispersion term are given by the following series expansions:

3

πNA 3 σ V°° ) 6τ

)

20η - 27η2 + 12η3 - 2η4 [(1 - η)(2 - η)]2

(1 - m)

where τ ) 0.74078 and V° is the temperature-dependent segment molar volume of the fluid, which can be calculated from the temperature-independent molar volume of the fluid, V°°, from the expression:

3 V° ) V°° 1 - 0.12 exp T˜

8η - 2η2 + (1 - η)4

C1 ) 1 + m

2

Subscripts 2 and 3 in eqs 16 and 17 denote the second and third-order terms in the perturbation expansion for polar and induced polar interactions, respectively. The third-order term for polar interactions consists of a two-body and a three-body term so that

apolar apolar apolar 3 3,2 3,3 ) + RT RT RT

(18)

tPC-PSAFT model uses a simplified, but yet accurate, version of the perturbation expansion proposed by Stell and co-workers19 for polar interactions. To account for the higher-order terms that are omitted, a new pure component parameter was introduced that accounts for the spatial extent of polar interactions compared to hard-core repulsive interactions. As a result, the polar terms in tPC-PSAFT assume the following form:

() [

]

apolar ˜4 2 1 2 η 4 4 12 µ˜ 2Q ˜ 2 12 Q + )µ˜ + 3 2 RT 5 K 5 K4 T˜ Κ 3

(19)

Ind. Eng. Chem. Res., Vol. 45, No. 17, 2006 6065

() [ () [

]

apolar ˜6 72 Q 3,2 1 3 η 6 4 2 144 µ˜ 2Q ˜4 Q ˜ + + ) µ ˜ RT 175 K2 245K4 T˜ K8 5

(20)

µ˜ )

]

µ/m

µ˜ ) 85.12

x(u/k)σ3 Q/m

Q ˜ ) 85.12

x(u/k)σ5

(22)

()

aind 3 1 2 η2 4 ) 10 µ˜ a˜ RT T˜ K3

ximi

(24)

(25)

(26)

2

i

Q ˜ )

∑i ∑j xixjmimjxQ˜ iQ˜ j

(∑ )

(27)

2

ximi

i

and

R˜ )

∑i ∑j xixjmimjxR˜ iR˜ j

(∑ ) ximi

i

As for the three-body terms, they are

2

Q ˜ )

∑i ∑j ∑k xixjxkmimjmk(Q˜ iQ˜ jQ˜ k)1/3 (30)

(∑ )

3

ximi

i

and

R˜ )

∑i ∑j ∑k xixjxkmimjmk(R˜ iR˜ jR˜ k)1/3 (31)

(∑ )

(23)

∑i ∑j xixjmimjxµ˜ iµ˜ j

(∑ )

i

ximi

where R˜ ) (R/m)/σ3 and R is the polarizability of the fluid (in cubic angstroms). tPC-PSAFT is extended to fluid mixtures in a straightforward way. The first four terms in eq 1 are identical to the terms used in PC-SAFT and are not repeated here. For the polar terms, mixing rules are needed in order to evaluate the reduced multipole moments and polarizability as a function of the multipole moments and polarizability of the pure components and the composition. For the case of two-body terms, they are

µ˜ )

3

3

i

where µ is the dipole moment of the fluid (in D) and Q is its quadrupole moment (in DÅ). The expressions for the dipoleinduced dipole two-body and three-body terms of tPC-PSAFT are

aind 2 8 η 2 )µ˜ a˜ RT T˜ K3

(29)

(∑ ) ximi

apolar ˜6 689 µ˜ 2Q 3,3 1 3 η2 10 6 159 µ˜ 4Q ˜2 ˜ 4 243 Q + + + ) µ ˜ RT 125 K2 1000 K4 800 K6 T˜ K3 9 (21) where K is a dimensionless quantity that accounts for the spatial range of polar interactions compared to hard-sphere interactions, K ) σp/σ. Consequently, σp (or Vp, equivalently) is the adjustable effective polar interactions segment diameter. The expressions for the reduced segment dipole moment and reduced segment quadrupole moment are

∑i ∑j ∑k xixjxkmimjmk(µ˜ iµ˜ jµ˜ k)1/3

(28)

The attractive potential between two dissimilar molecules is approximately given by the geometric mean of the potential between the like molecules at the same separation.1 This approximation is extended here to three-body interactions. This is obviously an oversimplification since the functional form that describes three-body interactions is not known exactly.1 Finally, additional mixing rules are necessary for the cross three-body dipole-dipole-quadrupole and dipole-quadrupolequadrupole terms (eq 21):

˜ )1/3 ) (µ˜ 2Q

∑i ∑j ∑k xixjxkmimjmk(µ˜ iµ˜ jQ˜ k)1/3

(∑ ) ximi

(32)

3

i

(µ˜ Q ˜ 2)1/3 )

∑i ∑j ∑k xixjxkmimjmk(µ˜ iQ˜ jQ˜ k)1/3

(∑ ) ximi

(33) 3

i

Results and Discussion In this work, tPC-PSAFT was applied to a wide range of mainly binary but also some ternary mixture fluid phase equilibria. Both VLE and LLE were considered. Results are presented and discussed for different classes of mixtures. Initially, a brief evaluation of the model for pure 1-alcohols is presented. Pure 1-Alcohols. Pure component parameters for methanol as well as other 1-alcohols are shown in Table 1 and are based on fitting to vapor pressure and saturated liquid density data taken from DIPPR.20 Hydrogen bonding interactions are calculated using a two-associating site per molecule model (known as 2B). Since quadrupolar interactions and dipole-induced dipole interactions for methanol are much lower than dipolar interactions, calculations were also made without the first two types of interactions. The accuracy of tPC-PSAFT for the correlation of vapor pressure and saturated liquid density from the two-parameter sets is similar. To justify the omission of quadrupolar and induced dipolar interactions, a comparison of the relative magnitude of intermolecular forces between two methanol molecules based on the numerator of the potential energy formula, Γii ) -B/r6 (the indices ii denote that the forces act between two identical

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Table 1. Pure 1-Alcohol and Water Parameters for tPC-PSAFT, Temperature Range used for Parameter Regression, and %AAD between Model Correlation and Experimental Data for Vapor Pressure and Saturated Liquid Densitya %ΑΑD m (-)

V°° (mL/mol)

u/k (K)

µb (D)

Q (DÅ)

1.647 1.743

13.2 12.5

178.48 180.45

1.70 1.70

4.13c

Methanol 3.23

2.697

11.8

182.78

1.69

5.60c

Ethanol 5.11

3.842 2.985

11.0 18.2

217.69 252.89

1.68 1.66

hb/k (K)

κhb

T (K)

Psat

Fliq

94.8 44.2

2836.7 2766.5

0.0459 0.0499

288-508 288-508

0.88 1.01

0.74 0.62

32.1

2549.4

0.0503

293-509

0.33

0.78

5.60

1-Propanol 6.74 65.6

1976.3

0.0196

333-531

0.51

0.79

5.60

1-Butanol 8.88

38.0

2565.2

0.0051

352-557

0.53

0.76

120.6

1575.2

0.3518

278-641

0.82

1.04

ad (Å3)

Vp (mL/mol)

Water 2.815

3.6

150.71

1.85

2.69c

1.49

a

For methanol, tPC-PSAFT parameters with dipolar interactions only are also shown. The two-site (2B) model is used for association in alcohols, and the four-site (4C) model, for association in water. b Reference 20. c Reference 65. d Reference 66.

Figure 1. Experimental data (points)22 and predictions for the second virial coefficient of methanol using PC-SAFT (dotted line),15 tPC-PSAFT with an explicit dipolar term (solid line), and tPC-PSAFT with explicit dipolar, quadrupolar, and induction terms (dashed line). The dotted line and the dashed line coincide.

molecules at distance r) is made.1 Note that, for quadrupolequadrupole interactions, this expression reads Γii ) -B/r10. To obtain comparable results, the potential energy was evaluated at distance r equal to two van der Waals radii for methanol21 and the quantity B′ ) B/(2rvdW)4 was calculated for quadrupolequadrupole interactions. At 0 °C, the dipolar interactions prevail (Bdd ) 147.6), followed by dispersion (BLJ ) 136.0), induction (Bd-indd ) 18.7), and finally quadrupolar interactions (Bqq ) 5.1) (all values are in 10-79 J/m6). An important test of model accuracy for pure fluids is presented in Figure 1 for the second virial coefficient of methanol. Accounting for dipolar interactions only results in the best agreement with experimental data.22 Additional explicit account for quadrupolar and induced dipolar interactions provides substantial deviation from experimental data, especially at lower temperatures. For comparison, PC-SAFT predictions are also shown. Consequently, in the rest of this paper, calculations for methanol systems are based on the parameter set with dipolar interactions only. An important thermodynamic property for pure fluids is liquid density. In Figures 2 and 3, isobaric liquid density at 50 MPa

Figure 2. Liquid density of methanol at 50 MPa: experimental data (points)48 and tPC-PSAFT predictions (lines).

Figure 3. Liquid density of methanol at 450, 510 and 540 K: experimental data (points)49 and tPC-PSAFT predictions (solid lines).

and isothermal liquid density for methanol at three different temperatures is shown. In all cases, tPC-PSAFT predictions are in excellent agreement with experimental data.

Ind. Eng. Chem. Res., Vol. 45, No. 17, 2006 6067 Table 2. Methanol-n-Alkane Mixture VLE Examined with tPC-PSAFTa n-alkane

T (K)

kij

∆P (%)

ref

methane ethane propane n-butane n-pentane n-hexane n-heptane

283.2-303.2 298.15 313-343 273-373 372.7-422.6 313-348 313

0.025 0.037 0.062 0.051 0.062 0.049 0.040

8.4 9.1 6.1 3.9 8.7 4.0 5.4

50 51 67 68 69 70, 71 72

a Values of k were regressed from experimental data. Bubble pressure ij deviation between experiment and model correlation is shown.

Figure 5. Phase equilibria for methanol-ethane mixture at 298.15 K: experimental data51 (points) and tPC-PSAFT correlation (lines).

Figure 4. Experimental pressures50 and calculated pressures with tPCPSAFT at 283.2, 293.2, and 303.2 K for methanol-methane.

Methanol-n-Alkane Mixtures. Methanol-n-alkane mixtures are important, both technologically and scientifically. Methanol is a widely used gas-hydrate inhibitor and the accurate knowledge of its content in oil is crucial since it affects, among others, the final price of the crude oil.23 From the scientific point of view, mixtures that contain a polar component and an alkane show a transitional behavior between nonassociating and associating mixtures.24 The specific interactions (polar and hydrogen bonding) exhibited between methanol molecules are strongly affected even by a single alkane molecule in their immediate proximity, because of the weak orientation of interacting polar molecules, that is weaker than the orientation of molecules associated by hydrogen bonds. On the other hand, at infinite dilution of the polar component (here methanol), the dipole-dipole interactions disappear and the dipole-induced dipole interactions between a methanol molecule and an alkane molecule appear. The induced polar interactions are much smaller than the dipole-dipole interactions, approximately by an order of magnitude.24 From structural studies in liquid alkanes, it is known that these molecules prefer to be surrounded by like molecules and intermolecular packing is observed, resulting in a quasiliquid-crystal arrangement.25 As a result, modeling of methanol-n-alkane mixtures should be based preferentially on theories with strong molecular/ microscopic origin that contain a quantitative account of the pertinent intermolecular forces between like and unlike molecules. In Table 2, a summary of the binary methanol-n-alkane mixtures examined is shown. In all cases, a temperatureindependent binary interaction parameter, kij, fitted to equilib-

rium VLE pressure is used. In Figure 4, experimental and calculated values for the methanol-methane mixture pressure at 283.2, 293.2, and 303.2 K are shown. The tPC-PSAFT correlation is in good agreement with experimental values. In the methanol-ethane mixture, complexity increases since this mixture exhibits VLE, LLE, and VLLE. In this case, kij was fitted to LLE data at 298.15 K and used subsequently for the prediction of VLE. Results are presented in Figure 5. The model is in good agreement with the experimental data in all cases. Calculations with SAFT for this mixture were reported by Li and Englezos.26 Results with tPC-PSAFT have a lower RMSD in pressure (11.9 vs 21.5), a similar RMDS in vapor composition of ethane (0.0102 vs 0.0110), and a lower kij value (0.037 vs 0.0641) compared to SAFT.26 The methanol-propane mixture is of immense technological interest because accurate prediction of the azeotrope formed at low methanol concentrations is crucial for the efficient design of propane-propene separation units.27 tPC-PSAFT provides accurate correlation of the experimental data at 313-343 K, although a slightly higher kij value is needed compared to PCSAFT correlation (0.062 compared to 0.052).28 In Figure 6, model predictions are shown concerning the azeotrope for this mixture at 298.15 K. Although no experimental data are available in the open literature, there is experimental industrial evidence for the occurrence of this azeotrope.27 Both tPCPSAFT and simplified PC-SAFT29 predict the azeotrope at 0.57 mol % methanol. For comparison, SAFT predicts the azeotrope also at about 0.6 mol % methanol.27 The azeotropic pressure is slightly higher for tPC-PSAFT compared to simplified PCSAFT. The LLE of methanol-n-octane mixture was examined, and results are shown in Figure 7 at two different pressures (0.1 and 100 MPa). A single kij value of 0.064 was used in both cases. Pressure has a relatively small effect on this phase diagram: an increase in pressure by 100 MPa results in an increase of the upper critical solution temperature (UCST) by ca. 25 K. tPC-PSAFT predicts accurately the UCST while it underpredicts the critical methanol concentration. In general, tPC-PSAFT provides a marginal improvement over PC-SAFT for the case of methanol-n-alkane mixtures. This may be attributed, among others, to the following

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Figure 6. Prediction of the azeotrope formed in the methanol-propane mixture at 298.15 K with tPC-PSAFT and simplified PC-SAFT EoS.

Figure 8. Water-n-hexane LLE at three-phase equilibrium pressure: experimental data (points)57 and model (tPC-PSAFT and PC-PSAFT) correlation (lines).

Figure 7. Methanol-1-octane LLE at 0.1 and 100 MPa: experimental data (points)52-56 and tPC-PSAFT correlation (lines).

reasons: (a) As the alkane chains become longer, the angledependent interactions break the symmetry of their structure and the molecular shapes become more anisotropic; therefore, rodlike geometries are favored and packing effects are important.30,31 As a consequence, the coordination numbers and the range of dipole-induced dipole interactions increase and so must, rigorously speaking, the effective polar diameters. (b) Hydrogen bonding cooperativity in 1-alcohol-n-alkane mixtures that is known to play an important role in 1-alcohol selfassociation32 is not accounted for by tPC-PSAFT. Aqueous Systems. Modeling of water is a challenging open problem that has attracted significant attention by physical chemists and chemical engineers. An ongoing debate is related to the number of hydrogen bonding sites per molecule that describes best the microscopic behavior of the fluid and results in the most accurate prediction of macroscopic thermodynamic properties. Clearly, hydrogen bonding in water is different than in alcohols. A four-site model for water is closer to reality compared to the two-site model used for alcohols. Interestingly, recent work revealed that at high pressure (30 MPa) the hydrogen bond distribution of water and alcohols becomes nearly identical, which is a totally different picture compared

to ambient conditions.33 Calculations presented here are based on a four-site model, and model parameters for water are given in Table 1. Water-Hydrocarbon Mixtures. Phase equilibria modeling of water-hydrocarbon mixtures is a challenging problem for EoS since these mixtures show highly nonideal behavior, resulting in very low mutual solubility over a broad range of conditions. The use of a four-site hydrogen bonding model over other models is justified based on previous work with SAFT and cubic plus association (CPA) models.34,35 The phase equilibria of binary aqueous mixtures with eleven hydrocarbons including n-alkanes of carbon number 5-10, 1-alkenes of carbon number 6, 8, and 10, and alkylcyclohexanes of carbon number 6 and 7 were correlated with tPC-PSAFT using of a single temperature-independent binary interaction parameter per mixture. In all cases, hydrocarbon parameters were taken from the original PC-SAFT parameter list.14 The objective function used for the regression of kij was in all cases the sum of the absolute average deviation of the K-factor:

F)

100

|

NcompNdata

∑∑ i)1

Ndata j)1

|

Kcalc - Kiexp i Kiexp

(34)

where Ndata is the number of experimental data, Ncomp is the number of components, and Ki ) xIi /xIIi is the equilibrium ratio of component i. Representative LLE results for mixtures with n-hexane and cyclohexane are shown in Figures 8 and 9 using PC-SAFT and tPC-PSAFT. PC-SAFT parameters for water were taken from ref 15, using the two-site associating model. In both cases, tPC-PSAFT correlates accurately the water solubility in the hydrocarbon and the hydrocarbon solubility in water at temperatures higher than the minimum solubility temperature. Results for the mixtures not shown here are of similar accuracy. Results presented here are also a clear improvement over SAFT predictions where hydrocarbon solubility was shown to be orders of magnitude lower than the experimental value35 and should

Ind. Eng. Chem. Res., Vol. 45, No. 17, 2006 6069 Table 3. 1-Alcohol(1)-Water(2) Mixtures VLE Examined with tPC-PSAFTa %AAD 1-alcohol

T (K)

kij

P

y1

ref

methanol ethanol 1-propanol 1-butanol

298.15-523.15 298.14-623.15 298.15-363.15 366.05-384.55

-0.023 0.0085 0.037 0.010

6.0 5.7 9.2 12.2

8.1 7.3 11.2 16.8

63,73,74 75,76 73,77,78 59

a Values of k regressed from experimental data and %AAD in bubble ij pressure and vapor phase composition between experiment and model correlation are shown.

Figure 9. Water-cyclohexane LLE at three-phase equilibrium pressure: experimental data (points)57 and model (tPC-PSAFT and PC-PSAFT) correlation (lines).

Figure 10. Values of kij for different water-n-alkane mixtures as a function of the n-alkane carbon number for tPC-PSAFT (squares) and CPA (diamonds).

values for CPA EoS are included.37 The binary interaction parameter shows a weak increase for higher hydrocarbons with the exception of n-hexane. This behavior should probably be correlated with the hydrophobic effect, arising from the strong hydrogen bonds between water molecules. Interestingly, the unfavorable Gibbs energy for solubilization of hexane in water is due entirely to the large entropy decrease that accompanies hexane dissolution in water (-32.4 kJ/mol), whereas the enthalpic contribution is zero at 25 °C.1 Molecular simulation results confirm that in aqueous solutions naturally occurring holes cluster together and they are filled by solute molecules. In this way, larger aggregates of solute molecules are created.38 Solute aggregation and large entropic mixing effects are not taken explicitely into account by SAFT models. Consequently, the features of aqueous solutions that contain highly hydrophobic solutes cannot be accurately described by such models.38 This can explain also the deficiency of the model to capture the minimum hydrocarbon solubility in water, presented above. As for 1-alkenes in water, an improvement in model prediction should be expected when the weak specific interaction between the π-bond of alkene and water molecules is included explicitly through a solvation model.39 Water-Alcohol Systems. Contrary to hydrocarbons, primary alcohols show less evidence of solute aggregation in water, mainly because of their polarity which leads to limited ability of the solute to penetrate the water lattice.38 Therefore, better correlation results are expected when using tPC-PSAFT pure component parameters which explicitly take into account all polar interactions for these highly nonideal mixtures. The 1-alcohol tPC-PSAFT parameters used in this study are presented in Table 1. As pointed out by Wolbach and Sandler,40 in binary mixtures of self-associating compounds, the mixing rules used to determine the cross-association parameters play an important role. In this work, the following combining rules were used:

AiBj ) be attributed to the explicit inclusion of polar interactions. A monotonic variation of the hydrocarbon solubility with temperature is predicted by the model, and the model inefficiency in capturing the minimum solubility remains. An empirical way to overcome this problem may be the inclusion of a second binary interaction parameter as proposed for the case of augmented BACK EoS for polar fluids.36 Addition of the second parameter is justified by the separation of nonpolar and polar forces. In the case of water-n-octane and water-1-octene mixtures, VLE calculations were also performed using the kij obtained from LLE and found in good agreement with experimental data. In Figure 10, the kij for different water-n-alkane mixtures as a function of carbon number is shown. For comparison, kij

A iB i + A jB j 2

(35)

and

κAiBj ) xκAiBiκAjBj

( ) xσiiσjj σij

3

(36)

which were already shown to be the most accurate.17,41 Table 3 summarizes the calculations for water-alcohol mixtures. For a methanol-water mixture, VLE calculations were performed at six different temperatures in the range 298.15523.15 K. A single temperature-independent kij value of -0.023 resulted in good correlation of the data, as shown in Figure 11. For aqueous solutions of ethanol, a structural model composed of three heterogeneous regions has been proposed on the basis

6070


Figure 11. VLE for methanol-water: experimental data (points)63,73,74 and tPC-PSAFT correlations (lines).

Figure 13. VLE (top) and LLE (bottom) for 1-butanol-water: experimental data (points)58-60 and tPC-PSAFT correlation (lines).

Figure 12. VLE for 1-propanol-water: experimental data (points)73,77,78 and tPC-PSAFT correlations (lines).

of infrared (IR) absorption spectroscopy, mass spectrometric analysis, and X-ray measurements.42 In this model, a hydrophobic core of ethyl groups is surrounded by an interfacing hydrogen-bonding water layer coupled with the hydroxyl groups of ethanols, whereas an outer “bulk” water area is highly prone to be damaged producing many nonlinear hydrogen bonding water pairs.42 This ordering increases with temperature, in accordance with the exothermic heat of mixing for this system.38 In the case of 1-propanol and 1-butanol, a problem encountered is that their quadrupole moment is not known. To overcome this problem, the reasonable assumption is made that the quadrupole moment of 1-propanol and of 1-butanol is the same as that of ethanol, i.e., it is assumed that the presence of one or two more methylene groups does not affect the value of the quadrupole moment. A similar assumption was made

by Donohue and co-workers for substituted aromatic molecules by assuming that the presence of aliphatic sidechain groups does not affect the value of the quadrupole moment of substituted aromatic molecules, and so, naphthalene and 1-methylnaphthalene have the same quadrupole moment.43,44 VLE calculations at three different temperatures, namely, 298.15, 333.15, and 363.15 K, were performed for a 1-propanol-water mixture and are presented in Figure 12. Results are in good agreement with experiment. The small kij values reported so far for the three alcohols are rather encouraging, considering the considerably higher values reported for these mixtures by Prausnitz and co-workers using the perturbed hard chain theory (PHCT) extended to include chemical dimerization equilibria45 and by Folas et al. using CPA.41 In the case of the more hydrophobic 1-butanol, both VLE and LLE are exhibited when mixed with water. The large entropic mixing effects that occur here are not considered fully by tPC-PSAFT and the use of a single kij adjusted to LLE experimental data results in unsatisfactory results, especially for the liquid organic phase. This is shown clearly in Figure 13, where both the VLE and LLE are shown. Nevertheless, it is encouraging that a single kij value is able to correlate both VLE and LLE.


Figure 15. VLE for acetone-decane mixture at 281.95, 313.15, and 333.15 K: experimental data (points)61,62 and tPC-PSAFT correlations (lines, kij ) 0.017).

Figure 14. Density of acetone-n-hexane at 278.15 and 298.15 K (top) and acetone-n-dodecane at 288.15 and 308.15 K (bottom) mixtures: experimental data (points)46,47 and tPC-PSAFT predictions (kij ) 0.0).

Acetone Binary Mixtures. When a nonpolar aliphatic alkane is mixed with the highly polar acetone, polar interactions among carbonyl groups weaken and highly positive excess molar volumes are observed, that increase with temperature and aliphatic chain length. In all cases, a maximum in VE is observed for equimolar composition.46,47 Here again, acetonealkane mixtures fall in the range between associating and nonassociating mixtures as far as it concerns excess thermodynamic properties.24 In the present case, tPC-PSAFT prediction capability was tested and representative results for acetone-nhexane and acetone-n-dodecane mixture densities are shown in Figure 14. Acetone tPC-PSAFT parameters can be found in ref 17, and PC-SAFT alkane parameters, in ref 14. tPC-PSAFT predictions are in satisfactory agreement with the experimental data.46,47 Furthermore, acetone mixture VLEs are correlated accurately with tPC-PSAFT. In Figure 15, representative results are shown for acetone-n-decane. On the basis of these, one can argue that tPC-PSAFT is more accurate for polar nonassociating components such as ketones, where properties are determined by long-range forces compared to polar associating liquids (water, alcohols, etc.).

Figure 16. Experimental data63 and tPC-PSAFT predictions for the equilibrium pressure of the water-methanol-ethanol mixture at temperatures of 323.15, 328.15, and 333.15 K.

Ternary Mixtures. Modeling of multicomponent phase equilibria is undoubtedly a strict test for macroscopic models. Here, the predictive performance of tPC-PSAFT EoS is tested for water-methanol-ethanol mixture in the temperature range 323.15-333.15 K. Calculations were based on pure component and binary parameters (for methanol-ethanol, kij ) -0.0232; experimental data were taken from ref 45), and results for the equilibrium bubble pressure are shown in Figure 16. The average deviation between experimental and calculated pressure is less than 9%. A second ternary mixture examined is methanol-water-nhexane (Figure 17). Accurate modeling of this mixture is important for the methanol synthesis process under supercritical conditions using n-hexane as solvent. Predictions are obtained again based solely on binary interaction parameters. The agreement between experiment and model is good although some deviations are observed in the high methanol concentration region.

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Figure 17. LLE of methanol-water-n-hexane mixture at 298.15 K (top) and 308.15 K (bottom): experimental data64 (short dotted lines and black squares) and tPC-PSAFT predictions (solid lines and open circles).

Conclusions In this work, the accuracy of tPC-PSAFT EoS to model several complex binary and representative ternary mixtures was examined. More specifically, tPC-PSAFT was applied to the VLE of methanol-n-alkane mixtures after carefully choosing the pure methanol parameters. For this purpose, prediction of pure fluid thermodynamic properties was examined. tPC-PSAFT was further applied to correlate water-1-alcohol VLE, wateraliphatic hydrocarbon LLE, and acetone-n-alkane mixture density and VLE. Finally, multicomponent phase equilibrium predictions were presented for water-methanol-ethanol and water-methanol-n-hexane mixtures. In all cases, model calculations were in good agreement with experimental data. The encouraging results obtained suggest that tPC-PSAFT is a reliable model for multicomponent, multiphase predictions under various conditions. Further extension of the model to highly asymmetric mixtures including polymers is underway. Acknowledgment The authors are grateful to the Danish Technical Research Council (STVF) for financial support of this work as part of a grant entitled “Advanced Thermodynamic Tools for ComputerAided Product Design”. E.K.K. gratefully acknowledges a shortterm visiting fellowship to DTU as part of this project. Dr. Oliver Pfohl of Bayer Technology Services GmbH is acknowledged for bringing the methanol-n-octane high pressure LLE data to our attention. Nomenclature Symbols R ) polarizability, Å3 R˜ ) reduced segment polarizability a ) molar Helmholtz free energy per mole

a2 ) second-order term of the Helmholtz free energy Pade´ approximant a3 ) third-order term of the Helmholtz free energy Pade´ approximant a3,2 ) two-body third-order term of the Helmholtz free energy Pade´ approximant a3,3 ) three-body third-order term of the Helmholtz free energy Pade´ approximant ai(m), bi(m) ) functions defined in eqs 14 and 15 aji, bji ) model constants defined in eqs 14 and 15 C1 ) compressibility expression of PC-SAFT, defined in eq 11 F ) objective function I1, I2 ) abbreviations defined in eqs 12 and 13 k ) Boltzmann constant, J/(molecule K) K ) σp/σ, effective polar interactions segment diameter reduced by the segment hard-sphere diameter Ki ) partition coefficient of component i kij ) binary interaction parameter of components i and j M ) number of association sites per molecule m ) number of segments per chain NA ) Avogadro’s number Q ) quadrupole moment, DÅ Q ˜ ) reduced segment quadrupole moment R ) gas constant, J/(mol K) r ) intermolecular distance rvdW ) van der Waals radius, Å T ) temperature, K T˜ ) reduced temperature u/k ) dispersion energy of interaction between segments, K XA ) monomer mole fraction (mole fraction of molecules not bonded at site A) Vp ) effective polar segment volume, mL/mol V° ) temperature-dependent segment volume, mL/mol V°° ) temperature-independent segment volume, mL/mol xi ) liquid mole fraction of component i Greek Letters Γij ) potential energy between molecules i and j, J/molecule ∆AiBj ) strength of association interaction between site A of component i and site B of component j, Å3 AiBj ) association energy of interaction between site A of component i and site B of component j, J/molecule η ) reduced density κAiBj ) volume of hydrogen bonding interaction between sites A of component i and site B of component j µ ) dipole moment, D µ˜ ) reduced segment dipole moment F ) molar density, mol/L σ ) temperature-independent segment diameter, Å σp ) effective polar interactions segment diameter, Å τ ) 0.74048 AbbreViations and Superscripts %AAD ) percent average absolute deviation assoc ) associating or due to association calc ) calculated chain ) residual contribution of hard-chain system dd ) dipole-dipole interactions d-indd ) dipole-induced dipole interactions disp ) dispersion exp ) experimental hs ) hard sphere ideal ) ideal gas ind ) induction


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ReceiVed for reView March 15, 2006 ReVised manuscript receiVed June 1, 2006 Accepted June 1, 2006 IE060313O

Evaluation of the Truncated Perturbed Chain-Polar Statistical

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