A Comparative Study of the Perturbed-Chain Statistical Associating

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A Comparative Study of PC-SAFT EOS and Activity Coefficient Models in Phase Equilibria Calculations for Mixtures Containing Associating and Polar Components Ke zheng, Huashuai Wu, Chunyu Geng, Gang Wang, Yong Yang, and Yongwang Li Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b04758 • Publication Date (Web): 06 Feb 2018 Downloaded from http://pubs.acs.org on February 11, 2018

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Industrial & Engineering Chemistry Research

A Comparative Study of PC-SAFT EOS and Activity Coefficient Models in Phase Equilibria Calculations for Mixtures Containing Associating and Polar Components Ke Zheng,†,‡,§ Huashuai Wu,†,‡,§ Chunyu Geng,‡ Gang Wang,*,‡ Yong Yang,†,‡ and Yongwang Li,†,‡ † State Key Laboratory of Coal Conversion, Institute of Coal Chemistry, Chinese Academy of Sciences, Taiyuan 030001, PR China ‡ National Energy R & D Center for Coal to Liquid Fuels, Synfuels China Co., Ltd., Huairou District, Beijing 101400, PR China § University of Chinese Academy of Sciences, Beijing 100049, PR China *Corresponding Author. E-mail: [email protected]. Tel.: +86 010-60684022. Fax: +86 010-60684023.

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ABSTRACT

Vapor–liquid equilibria (VLE), liquid–liquid equilibria (LLE) and vapor–liquid–liquid equilibria (VLLE) for systems involving highly non-ideal components, namely, water, alcohols, alkanes, ketones, aldehydes, esters and ethers, were investigated to evaluate PC-SAFT EOS and two widely-used activity coefficient models, i.e., UNIQUAC and UNIFAC. Parameters used for PC-SAFT EOS were taken from literatures or estimated in this work, while those for UNIQUAC and UNIFAC were from commercial process simulator Aspen plus 8.4. It was found that all the three models yield reliable correlations/predictions for VLE calculations. However, UNIQUAC and UNIFAC were observed to be unreliable for LLE and VLLE calculations despite successful reproductions of experimental data in some cases. The calculated results deviate significantly from experimental data in many cases. Particularly, both models predict artificial liquid–liquid phase splitting for a number of miscible mixtures. Nonetheless, PC-SAFT EOS with the use of a single set of parameters reproduces experimental data quantitatively in most cases and provides reasonably accurate results in all other cases. This remarkable performance of PC-SAFT EOS potentially eliminates the need for various thermodynamic models and consequently the need for selecting a thermodynamic model when performing phase equilibria calculations using commercial software. This is important for practitioners, since (1) it remains unclear to select an appropriate model from the available models of a process simulator or thermodynamic package for a given phase equilibria calculation despite the presence of some rule of thumb and (2) it is also likely that none of the existing models is sufficiently accurate. In addition, it was shown that both pure-component parameters and binary interaction parameters for PC-SAFT EOS are well-behaved for a homologous series, which allows for parameterization for weakly characterized components by interpolation or extrapolation, and consequently, facilitates the development of a practical tool for phase equilibria calculations.


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1.


INTRODUCTION Complex mixtures containing non-polar (e.g., hydrocarbons), associating (e.g., water and alcohols)

and polar components (e.g., ketones, aldehydes, esters, and ethers) are basic feedstocks or intermediates in the chemical and process industries. These mixtures exhibit highly non-ideal phase behaviors due to various intermolecular forces, such as dispersion force, hydrogen bonding and dipole–dipole interactions. Reliable phase equilibria calculations are essential for the design, optimization and operation of chemical processes with these chemicals involved. Studies on modeling fluid phase equilibria of complex associating and polar mixtures with equations of state (e.g., Peng–Robinson and Soave– Redlich–Kwong) and activity coefficient models (e.g., UNIFAC and UNIQUAC) have been the subject of many investigations in the past few decades, as reviewed by literatures.1-3 Conveniently available in commercial process simulators, such as Aspen Plus®, these methods have become the workhorse and de facto industry standard for phase equilibria calculations. Nonetheless, there are also shortcomings for these established approaches. Firstly, one needs to choose different models based on the nature of the problem. For instance, Peng–Robinson or Soave–Redlich–Kwong EOS is often selected for oil and gas applications. Activity coefficient models (e.g., UNIFAC and UNIQUAC) are needed when highly polar components or those with hydrogen bonding are involved. Despite the presence of some rule of thumb, it could be confusing to select an appropriate model for a given mixture, e.g., gasoline or diesel with associating or highly polar additives. Secondly, distinct sets of parameter values are generally necessary for vapor–liquid equilibrium (VLE) and liquid–liquid equilibrium (LLE) calculations, respectively. As a result, dilemmas may arise in modeling of vapor–liquid–liquid equilibrium (VLLE). Last but not least, it is also a challenge for developers to develop and maintain various thermodynamic models and multiple sets of parameter values to meet increasing demands in different fields. Thus, it is desirable to develop a unified thermodynamic approach (i.e., a single model with a single set of parameter values) to perform different types of phase equilibira calculations for a large variety of mixtures. Statistical associating fluid theory (SAFT) family of EOS has emerged as the most promising



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candidate in this respect.4-7 A SAFT EOS was derived using Wertheim’s first-order thermodynamic perturbation theory,8-11 which models all intermolecular attraction forces as perturbations to the reference fluid that accounts for repulsion interactions. The resulting models have been fitted and validated using the data from molecular simulations.4,5 The main advantage of a SAFT EOS comes from its validated theoretical framework, which is capable of accurately describing various attractive intermolecular interactions, such as hydrogen bonding, van der waals force and dipole–dipole interactions.12 Numerous SAFT variants, such as SAFT-VR,13,14 soft-SAFT,15 and PC-SAFT,16,17 have been proposed following the initial success of the SAFT framework. Perturbed Chain SAFT (PC-SAFT) EOS by Gross and Sadowski16,17 has been the most successful variant among SAFT family.18-25 It has been shown that mixtures containing non-polar and associating components can be modeled with satisfactory results. Pure-component parameters for a homologous series are well-behaved, allowing for the estimation of parameter values by interpolation or extrapolation. Maximum of a single binary interaction parameter (BIP) is needed to be adjusted to fit binary phase equilibria data. The BIP was also observed to be well-behaved for a homologous series. It should be stressed that the value of the BIP is generally close to zero, implying that PC-SAFT EOS is a fully predicative approach since pure-component parameters can be easily derived either by data fitting, a group contribution method or by interpolation or extrapolation for a homologous series. The PC-SAFT framework has been extended to account for more intermolecular interactions, such as dipolar interactions. Gross and Vrabec added a dipolar term into the original PC-SAFT framework to enhance the description of dipolar interactions.26 It has been shown that a better fit can be obtained from regressions using VLE and LLE data without introducing additional adjustable parameters. Furthermore, the estimated value of the BIP for the polar PC-SAFT (PCP-SAFT) EOS is significantly reduced, implying its improved predicative capability. Jog and Chapman27 proposed a different dipolar term, leading to similar benefits. De Villiers et al.28 evaluated the two dipolar terms by comparing their performance in phase equilibria calculations for three series of binary mixtures, namely, Alkane/Alcohol, Alcohol/Alcohol and Water/Alcohol. It was found that similar results were delivered. Nevertheless, the formulation by Jog and Chapman27 needs one additional adjustable parameter, which is a disadvantage


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from both practical and theoretical point of view. Likewise, Gross29 added a quadrupolar term to account for quadrupolar interactions. Similar improvements have been achieved for mixtures containing quadrupolar components (e.g., CO2), showing the advantages of approaches with sound theoretical basis as compared to empirical methods. Kleiner and Sadowski30 proposed a straightforward approach to account for induced-association interactions in mixtures containing associating and polar components. Systematic improvements have been achieved in terms of the predicative capability and description of phase equilibria. Group contribution (GC) and critical point-based (CP) methods have been developed to deliver reliable estimates of parameter values for PC-SAFT EOS, leading to so-called GC-PC-SAFT and CP-PC-SAFT, respectively.31-40 GC-PC-SAFT and CP-PC-SAFT have been successfully used to model VLE and LLE for mixtures containing water, hydrocarbon and oxygenates. In addition, PC-SAFT framework has been successfully applied in the fields of petroleum, bio-fuels, polymers, electrolytes and ionic liquids, showing advantages over the classical cubic EOS frameworks and other traditional methods which are incapable of dealing with applications in such diverse areas.41-43 Despite significant developments in academia, PC-SAFT EOS has not been widely accepted and used in practice. To promote the use of PC-SAFT EOS in practical applications, this study is devoted to show that different types of phase equilibria calculations (e.g., VLE, LLE and VLLE) with different types of fluids (e.g., non-polar, polar and associating) involved can be performed accurately with PC-SAFT EOS using a single set of parameter values. The other objective is to show the advantages of PC-SAFT EOS by systematically comparing its performance to that of those well-established methods, namely, UNIQUAC44 and UNIFAC45. It should be stressed that the comparison was made over VLE, LLE and VLLE of a large variety of binary and ternary mixtures containing water, alcohols, ketones, aldehydes, ethers, esters, and hydrocarbons to reach unbiased conclusions. To the best of our knowledge, this has been the first attempt in this direction. The reason why only activity coefficient models were selected for the comparison is that cubic equations of state generally perform poorly for highly polar and associating fluids that are the focus of this study.41 Calculations using PC-SAFT EOS were performed with our in-house FORTRAN code, while those using the two activity models with commercial process simulator, Aspen plus 8.4. It is commonplace that more than one set of parameter values is available for a



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thermodynamic model. The net effect is that some improvements are made for some cases at the sacrifice of accuracy for some others. As an option available in Aspen, UNIFAC-LLE represents UNIFAC model with a different set of parameter values and was selected to show this effect. 2.

PARAMETERIZATION OF PC-SAFT EOS

2.1 Parameters Three pure-component parameters are needed for PC-SAFT EOS to describe non-associating fluids. The parameters are (1) segment diameter σ, (2) the depth of the potential ε k and (3) the number of segments per chain m. Two more pure-component parameters (the association energy effective association volume

κ

Ai B

j

ε

Ai B j

k

and the

) are needed for associating fluids and one more parameter (dipole

moment µ) for dipolar components. The dipolar term derived by Gross and Vrabec26 was used in this work to account for dipolar contribution, since measured dipole moments are used and no additional adjustable parameters are introduced. The cross-association parameters can be determined from pure-component association parameters by using simple combining rules suggested by Wolbach and Sandler46:

(

1 Ai Bi AB ε +ε j j 2

ε

Ai B j

=

κ

Ai B j

= κ Ai Bi κ

Aj B j

)

 2 σ iiσ jj   σ ii + σ jj 

(1) 3

   

(2) Cross-association occurs not only between associating components, but also between associating components and polar components, which is defined as the so-called induced-association.30 Cross-association parameters for induced-association interactions are obtained with the assumptions: (1) the association energy parameter ε A B k of the non-self-associating (polar) component is set to zero; i

j

ABj

(2) the association volume parameter κ i

of this component is assumed to be equal to the value of the

associating component in the pair. The equation of state can easily be extended to mixtures by applying van der Waals one-fluid mixing


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rules. The parameters for a pair of unlike segments are obtained by Berthelot-Lorentz combining rules:

εij = εiε j (1- kij ) σ ij =

(3)

1 (σ i + σ j )(1- lij ) 2

(4)

where one binary interaction parameter (BIP), kij , is introduced to correct segment–segment interactions of unlike chains. Though model correlations and predictions are generally satisfactory with the use of temperature independent kij , three additional measures has been taken to improve the results when necessary: (1) introduce additional BIP lij as defined in eq 4; (2) introduce an asymmetric BIP p ij as defined by Mathias et al.47; (3) use temperature dependent kij defined below: kij = a0 + a1

T T + a2 ln 1000 1000

(5)

where ai are constants and T is temperature in Kelvin. 2.2 Parameterization methodology Building a database of parameters is essential for practical use of PC-SAFT EOS. Generally, there are two types of approaches. One popular approach is to acquire all the parameter values by fitting experimental data, e.g., binary VLE data. However, this approach is very tedious given an extremely large number of fluid-specific constants. Moreover, this approach is also practically infeasible due to the lack of high quality data for many fluids, though a tremendous amount of data is available in literature. The other approach is the so-called group contribution method. Though there are thousands of fluids of interest in chemical and related industries, the number of functional groups that constitute these fluids is rather limited. If the activity coefficient of a fluid is assumed as the sum of contributions made by the molecule’s functional groups and one functional group makes the same contribution in all types of molecules, individual functional groups rather than individual fluids need to be characterized by parameters, significantly reducing the number of the parameters. Nevertheless, a functional group may make distinct contributions in different molecules. For instance, the contribution of a hydroxy group in phenol is not the same as that of a hydroxy group in methanol. With these understandings in mind, we parameterized PC-SAFT EOS by taking advantages of its well-behaved parameters: parameter values



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correlate well with the number of carbon atoms (Cn) or molecular weight (Mw) for each homologous series. This method consists of three steps: (1) estimate parameters for a few fluids with rich and high-quality experimental data for each homologous series by the standard data-fitting approach; (2) fit model parameters acquired from step 1 with respect to molecular weight or carbon number for each homologous series; (3) obtain both pure-component parameters and BIPs for all other fluids by interpolation or extrapolation based on the fitting functions acquired from step 2. On one hand, it is a practically feasible and efficient way to parameterize PC-SAFT EOS. The dependence of the parameterization on high-quality data is considerably reduced without the sacrifice of accuracy in model predictions on the other hand. Please note that this method may be not applicable to other equations of state or activity coefficient models, since model parameters are generally not well-behaved. It should be stressed that our approach is not correlative, but predictive. Our intention is to develop a predictive approach by correlating pure-component parameters and binary interaction parameters with the carbon number or molecular weight of each homologous series. Users can obtain all the parameter values using the fitting functions (e.g., eq 6 and the equation shown in Figure 2), and subsequently, perform their calculations. The fitting-function based parameterization strategy for PC-SAFT in this paper plays the same role as the group contribution method does for UNIFAC. Though the fluids we have considered are rather limited at this point, this investigation is just our first effort and we will include more components in the future. Please note that the first two steps in the above 3-step parameterization strategy should be carried out by the developers (e.g., the authors of this paper) of thermodynamic packages. To use our approach, an engineer just needs to obtain the parameter values using the established fitting functions and perform their calculations. In principle, this procedure is the same as that for UNIFAC.

Table 1. Pure-component parameters of associating components Mw

σ

ε k

m (g/mol)

ε

Ai B j

%AAD

κ

AB i j

[Å]

[K]

[K]

[-]

△P

b

T-range b

△ρ

[K]

Water

18.015

1.6619

2.4643

267.71

2500.57

0.12502

1.83

1.02

313-573

Methanol

32.042

1.8458

3.0530

193.94

2668.26

0.04647

1.12

0.68

220-500

46.069

2.4513

3.1478

198.15

2653.49

0.03030

0.75

0.67

220-500

60.096

2.9997

3.2522

233.4

2276.8

0.01526

Ethanol 1-propanol

a


240-537

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74.123

2.7452

3.6149

259.59

2544.6

1-pentanol

88.15

3.5762

3.4737

247.28

1-hexanol

102.177

3.6932

3.6104

256.34

1-heptanol

116.203

4.0966

3.6416

1-octanol

130.23

4.3738

3.7068

1-butanol

a

0.00654

0.95

1.48

240-560

2252.04

0.01177

1.14

0.37

260-550

2542.18

0.005435

1.55

0.41

300-600

261.51

2875.27

0.001263

1.56

0.73

330-625

262.75

2754.82

0.002048

1.53

0.43

325-640

144.257

4.6839

3.7292

263.64

2941.90

0.001427

1-decanol

158.284

4.8372

3.8136

269.14

2979.80

0.001206

1.41

1.06

350-680

1-undecanol

172.311

5.0115

3.9027

269.23

3017.69

0.002489

1.30

0.78

390-695

1-nonanol

368-680

1-dodecanol

186.338

5.1643

3.9791

274.49

3103.33

0.001466

1.11

0.91

370-700

1-tridecanol

200.365

5.1926

4.0657

287.30

3149.70

0.000766

0.55

0.81

380-700

1-tetradecanol

214.392

5.2074

4.1743

285.19

3160.03

0.002527

1.53

0.35

380-710

1-pentadecanol

228.419

5.2825

4.2553

397.24

3154.58

0.0012

1.85

1.33

400-710

1-hexadecanol

242.446

5.2398

4.3595

294.53

3158.08

0.004798

0.69

0.50

400-730

1-heptadecanol

256.473

5.2522

4.4376

306.08

3160.13

0.003869

0.62

1.02

420-710

0.90

1-octadecanol

270.5

5.2432

4.5528

301.29

3164.77

0.009686

0.71

1-nonadecanol

284.527

5.2693

4.6576

308.17

3165.53

0.009747

1.05

430-750

1-eicosanol

298.554

5.2953

4.7623

315.05

3166.30

0.009808

0.93

2-propanol

60.096

3.2276

3.1700

208.32

2253.92

0.02028

0.95

1.72

245-500

2-butanol

74.123

3.3686

3.3330

231.32

2246.41

0.00747

0.97

0.88

260-530

2-methyl-1-propanol

74.123

2.6805

3.6359

257.33

2545.60

0.00513

0.94

1.41

265-540

2-pentanol

88.15

3.4487

3.5221

244.31

2454.37

0.003379

0.79

0.81

295-540

440-720 460-740

3-pentanol

88.15

3.6975

3.3996

242.67

2461.57

0.001136

1.07

0.43

300-460

2-methyl-1-butanol

88.15

2.8268

3.7752

264.82

2619.47

0.005124

0.99

1.42

280-570

2-hexanol

102.177

3.1300

3.8413

256.81

2441.87

0.009024

0.62

1.15

260-560

cyclohexanol

102.177

2.5292

3.9664

327.48

2932.88

0.000889

0.90

0.77

320-630

2-heptanol

116.203

4.0415

3.6570

257.90

2873.38

0.0009

0.73

0.58

350-605

17

a.

The parameters for 1-propanol were taken from literature ;

b.

%AAD = 1 Np ∑1 Ωical Ωiexp − 1 × 100 , where Ω is vapor pressure or liquid density; Np is number of experimental points. Np

Table 2. Pure-component parameters of polar components %AAD Mw

σ

ε k

[Å]

[K]

μ

b

T-range

m (g/mol)

[D]

[K] △P

a

△ρ

58.08

2.7448

3.2742

232.99

2.88

200-508

a

72.107

2.9835

3.4239

244.99

2.78

186-536

a

86.134

3.3537

3.4942

246.66

2.7

234-561

2-hexanone

100.16

3.6441

3.5693

252.64

2.7

0.36

0.84

220-580

2-heptanone

114.19

3.9924

3.6284

254.56

2.59

0.44

0.84

240-605

1.29

0.56

Acetone

2-butanone

2-pentanone

2-octanone 2-nonanone

a

128.88

4.3312

3.6736

255.66

2.7

142.24

4.8255

3.6584

252.93

2.7

250-625 340-422

Butanal

72.107

2.8728

3.4416

247.16

2.72

1.20

0.32

220-500

pentanal

86.132

3.1205

3.5809

257.86

2.59

0.83

1.25

240-550

methyl acetate

74.079

3.1705

3.1870

231.63

1.72

0.90

1.18

230-500

ethyl acetate

88.106

3.5073

3.3070

230.24

1.78

0.55

0.92

190-520



propyl acetate

a

butyl acetate DEE

a,c

3.7658

3.4289

235.42

1.78

116.16

4.0486

3.5204

239.55

1.87

74.123

2.9726

3.5127

219.53

1.15

260-549 0.52

0.75

270-570 220-466

a,c

102.177

3.5753

3.6702

231.03

1.21

260-500

a,c

102.176

3.5458

3.6717

216.73

1.13

260-480

c

88.15

2.9444

3.7064

233.27

1.32

0.38

0.13

185-485

c

116.204

3.5693

3.8090

236.34

1.54

0.33

0.21

260-540

c

130.23

4.2131

3.7709

239.76

1.17

0.48

0.44

280-575

DNPE DIPE

102.13

Page 10 of 34

MTBE ETAE

DNBE

30

a.

The parameters for dipolar compounds were taken from reference ;

b.

Dipole moment values obtained from literature44;

c.

DEE: diethyl ether; DNPE: dipropyl ether; DIPE: diisopropyl ether; MTBE: methyl-tert-butyl ether; ETAE: ethyl tert-pentyl ether; DNBE: dibutyl ether.

In this study, pure-component parameters for the first few fluids of each homologous series were acquired either from literatures or by fitting vapor pressure and liquid density, which is a well-accepted procedure.16-20 Though multiple association schemes have been used in a few previous studies, there has been no consensus on which scheme is the best.6,17-19,43 Thus, the 2B scheme was used throughout in this study. Experimental data were used for the parameter of dipole moments as suggested by literatures.26,30 Tables 1 and 2 present regression results for alcohols and polar components. Given the wide temperature ranges, the average absolute deviations (AAD) of the saturated properties compared to experimental data48 are remarkably small. Since thermodynamic behavior near the critical point is drastically different, additional treatment has been used to improve model predictions.49,50 However, this issue is out of the scope of this study. Therefore, experimental data of vapor pressure and liquid density near the critical point was excluded in the parameter estimation.

Figure 1. Pure-component parameters of PC-SAFT for the series of 2-ketones as a function of molar mass.

Water has been well-known for its many abnormal phenomena.51 A large number of pure-component


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parameters have been published for water and no consensus has been reached on which set of parameter values is the best.17,19,24,40,52-55 In this study, the pure-component parameters of water and the BIPs for binary pairs of water and alcohols were obtained simultaneously by data-fitting. As shown in Table 1, the regression results are satisfactory. It will be shown later that this set of parameter values work well for a large variety of water-containing mixtures. It has been shown that pure-component parameters for normal alkanes can be fitted with high accuracy with respect to molecular weight or carbon number.16,21,56 Here, we show that this observation is also applicable to other homologous series, such as 2-ketones, as shown in Figure 1. The fitting functions are as follows:

m = 0.0253M w + 1.138 m ⋅ σ 3 100 = 0.0167M w − 0.00659

(6)

m ⋅ ( ε k ) 1000 = 0.00686M w + 0.234 The best estimates of BIPs were obtained in a similar fashion as those of pure-component parameters. The values of BIPs were acquired by fitting binary VLE data and/or LLE data. The regressions were performed by the least-square method with the residuals defined as the difference between the fugacity of the same component in different phases rather than the deviations of calculated values from experimental data. This is advantageous, since the latter approach requires solutions of phase equilibria formulations leading to a couple of practical difficulties: (1) the solution may not exist when the initial values of model parameters are not properly assigned; (2) the iterative solution process may not converge. The values of BIPs for PC-SAFT EOS for systems investigated in this paper are summarized in Tables S1, S3-5 in Supporting Information. Again, BIPs correlate with carbon number or molecular weight very well. BIPs for binary pairs of ethanol/n-alkanes are shown as one example in Figure 2.



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Figure 2. BIPs of PC-SAFT for binary ethanol/n-alkanes mixtures as a function of carbon number of n-alkanes.

Please note that experimental data of ternary mixtures were never used for parameter estimation in this study. On one hand, parameter estimation heavily depends on the quality and amount of experimental data. Experimental data of binary mixtures are ideal for meeting these needs. On the other hand, predicative capability is the most important indicator for the quality of a thermodynamic model. Validations against ternary data present an excellent opportunity to test its predicative capability. However, data for ternary mixtures may be used for parameter estimation for UNIQUAC or UNIFAC in order to improve model correlations with ternary data. This implies that model predictions with parameter values regressed using binary data only are inadequate for ternary mixtures and hence unreliable for more complex mixtures. 3.

RESULTS AND DISCUSSIONS A thorough evaluation of PC-SAFT, UNIQUAC and UNIFAC was presented by comparing model

correlations and predictions of VLE, LLE and VLLE for a large variety of binary and ternary mixtures containing water, alcohols, ketones, aldehydes, ethers, esters, and hydrocarbons to experimental data. The majority of binary VLE and LLE results is correlation and the ternary results are prediction for PC-SAFT. However, this is not clear for UNIFAC. Parameterization of the group contribution method for UNIFAC also uses a tremendous amount of experimental data. The calculations within the region where parameters were estimated are not predictions, but correlations. A clear distinction cannot be made between correlations and predictions, since it is vague what data were used to estimate a specific set of parameter values. However, the comparison is still meaningful, since a large variety of data sets


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were used to avoid biased results. As discussed earlier, PC-SAFT with a fitting-function based parameterization strategy is a predictive approach. Therefore, the comparison between UNIFAC and PC-SAFT is also fair. 3.1 VLE of binary mixtures

Figure 3. Boiling temperatures (Ta) and compositions of the azeotropes as function of carbon number (Cn) for (a) water/1-alcohols systems and (b) ethanol/n-alkanes systems at P = 1.013 bar. Comparison of PC-SAFT (solid lines), UNIQUAC (dashed lines) and UNIFAC (shot dotted lines) to experimental data57 (symbols, square for Ta and circle for compositions). The BIPs used for UNIQUAC can be found in VLE-IG and LLE-lit databanks of Aspen plus 8.4.

The VLE was calculated using the three models for 36 binary mixtures. The details of the results are shown in Table S1 in the Supporting Information. Azeotropes were also calculated with the three models for binary water/1-alcohols and ethanol/n-alkanes mixtures. The results are shown in Figure 3. Collectively, it was shown that model correlations/predictions are in close agreement with experimental data in all cases and the three models are neck to neck. Given the simplicity in model formulations, this remarkable performance justifies the popularity of UNIFAC and UNIQUAC in VLE calculations. Particularly, UNIFAC is even more appealing given the fact that all its model parameters are available via a group contribution method. Although the performance of PC-SAFT EOS is as good as that of its classical predecessors and all its model parameters are obtained by interpolation or extrapolation for each homologous series, there is no clear need for additional method for VLE calculations, since there is little room left for further improvements. 3.2 LLE of Water/hydrocarbon binary systems



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Figure 4. %AAD of mutual solubilities of (a) water in hydrocarbons and (b) hydrocarbons in water. The %AAD is set to 100% for visualization when actual values are greater than 100%.

Mutual solubilities of water and hydrocarbons including n-alkanes, branch-alkanes, cycloalkanes and 1-alkenes were calculated with the three models and were compared to experimental data. Typical results are summarized in Figure 4. More details of the results are given in Table S3 for brevity. Obviously, PC-SAFT EOS is superior in LLE calculations of water and hydrocarbons. Figure 5 presents the mutual solubilities of water and three n-alkanes, i.e., n-hexane, n-heptane and n-octane. As shown in Figure 5a, the solubilities of water in n-alkanes increase monotonically with an increase in temperature and are virtually independent of the carbon number of the n-alkanes. PC-SAFT EOS correlates well with experimental observations, whereas UNIQUAC and UNIFAC show considerable deviations at high temperatures. As shown in Figure 5b, the solubilities of n-alkanes in water exhibit a minimum as temperature increases; the solubilities decrease in orders of magnitude as the carbon number of the


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solute increases. When compared to experimental data, PC-SAFT EOS and UNIQUAC show negative deviations at low temperatures and positive deviations at high temperatures. UNIFAC predictions are a few orders of magnitude higher than the measured values. None of the three models is capable of capturing the dip in solubility at low temperatures. Surprisingly, the results with UNIFAC-LLE display negligible improvements as compared to those with default parameter values, as shown in Figure S2.

Figure 5. Mutual solubilities of (a) water in n-alkanes and (b) n-alkanes in water for C6 (black), C7 (red) and C8 (green). Comparison of PC-SAFT (solid lines), UNIQUAC (dashed lines) and UNIFAC (dash dotted lines) to experimental data58-60 (symbols).

Figure 6. BIPs of (a) PC-SAFT and (b) UNIQUAC as a function of carbon number of n-alkanes for binary water/n-alkanes mixtures. The BIPs for UNIQUAC can be found in VLE-lit databank of Aspen plus 8.4.



Figure 7. Predicted mutual solubilities with PC-SAFT of (a) water in n-alkanes and (b) n-alkanes in water for the series of n-alkanes: C13 C15 , C16 , C17 , C18 , C19 and C20 .

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, C14

,

As shown in Figure 6, the BIPs of PC-SAFT EOS are also well-behaved, whereas those of UNIQUAC cannot be well fitted with a function. Please note that this is not specific for binary water/n-alkanes mixtures, but a general observation. This is advantageous for PC-SAFT EOS from a practical point of view, since many BIPs related to weakly characterized components can be estimated by extrapolation. This feature greatly enhances the predictive capability and practical usability of PC-SAFT EOS. UNIFAC is a fully predictive approach and parameter values are available for all components by a group contribution method. However, its model correlation or predictions are generally less accurate than PC-SAFT EOS and UNIQUAC as shown in Figure 5. Figure 7 displays PC-SAFT predictions using parameter values acquired by extrapolation on mutual solubilities of water and n-alkanes with 13-20 carbon atoms. The prediction seems reasonable: the solubilities of water in n-alkanes continue to be independent of the carbon number of n-alkanes and the solubilities of n-alkanes in water continue to decrease in orders of magnitude as the carbon number of the solute increases. Similar calculations were carried out for binary water/1-alkenes mixtures and similar results were obtained. Refer to Figures S3 and S4 in Supporting Information for details.

3.3 LLE of Water/alcohol binary systems


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Figure 8. %AAD of mutual solubilities of (a) water in alcohols and (b) alcohols in water. The %AAD is set to 100% for visualization when actual values are greater than 100%.

Mutual solubilities of water and alcohols including 1-alcohol, 2-alcohol, 3-alcohol and methyl-1-alcohol were calculated with the three models and were compared to experimental data. Typical results are summarized in Figure 8. More details of the results are given in Table S4 and Figures S5-6 for brevity. Again, PC-SAFT EOS provides the best correlation results in all cases. UNIQUAC and UNIFAC are satisfactory for the solubilities of water in alcohols in most cases, but fail to describe the solubilities of alcohols in water accurately. Unfortunately, all the three models fail to capture the minimum in the solubilities of alcohols in water, as is the case for the solubilities of n-alkanes in water, as shown in Figures S5-6. The results with UNIFAC-LLE fail to present any improvement as compared to those with default parameter values, as shown in Figure S5. Again, the BIPs of PC-SAFT can be obtained by extrapolation for binary water/alcohols mixtures, whereas data regressions are necessary for



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each UNIQUAC parameter as shown in Figure 9. As shown in Figure 10, the mutual solubilities of water and 1-alcohols with 13-20 carbon atoms were predicted using PC-SAFT EOS. The parameter values needed were obtained using the fitting functions shown in Figure 9a. The predictions look plausible. Binary water and 2-alcohols mixtures were also investigated in this study and typical results are plotted in Figure S6 (Supporting Information).

Figure 9. BIPs of (a) PC-SAFT and (b) UNIQUAC as a function of carbon number of 1-alcohols for binary water/1-alcohols mixtures. The BIPs for UNIQUAC can be found in LLE-lit databank of Aspen plus 8.4.

Figure 10. Predicted mutual solubilities with PC-SAFT of (a) water in 1-alcohols and (b) 1-alcohols in water for the series of 1-alcohols: C13 C14

, C15

, C16

, C17

, C18

, C19

and C20

.

3.4 LLE of water/polar binary system


,

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Figure 11. %AAD of mutual solubilities of (a) water in polar compounds and (b) polar compounds in water. The %AAD is set to 100% for visualization when actual values are greater than 100%.

Figure 12. LLE of water and 2-butanone at 1.013 bar. Comparison of PC-SAFT (solid lines), UNIQUAC (dashed lines), UNIFAC (dash dotted lines) and UNIFAC-LLE (dash dot dotted lines) to experimental data61 (symbols: square for 2-butanone-rich phase, circle for water-rich phase).



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Figure 13. BIPs of (a) PC-SAFT and (b) UNIQUAC as a function of carbon number of 2-ketones for binary water/2-ketones mixtures. The BIPs for UNIQUAC can be found in LLE-Aspen databank of Aspen plus 8.4.

The mutual solubilities of water and polar compounds, namely, 2-ketones, aldehydes, esters as well as ethers, were calculated with the three models and compared to experimental data. The detailed results are given in Table S5 for brevity. As shown in Figure 11, PC-SAFT EOS provides satisfactory results in all cases. UNIQUAC also provides accurate calculation results for the binary mixtures of water/2-ketones and water/acetates, nonetheless, it delivers poor results for the binary mixtures of water/aldehydes and water/ethers. UNIFAC, though fully predictive with all parameters available, presents the worst results. Remarkably, both PC-SAFT EOS and UNIQUAC are able to capture the minimum of the solubility of water in 2-butanone as shown in Figure 12. The BIPs of PC-SAFT EOS are well fitted with respect to the carbon number of 2-ketones, whereas BIPs have to be estimated by data regression for UNIQUAC, as shown in Figure 13. Essentially no improvements can be made by replacing the default parameter values with UNIFAC-LLE as shown in Figure 12. More evidences can be found in Figures S7-10 (Supporting Information). Obviously, it is necessary to use PC-SAFT EOS for cases with a miscibility gap, in contrast to VLE calculations where all the three models are sufficient. It is not unexpected that UNIFAC and UNIQUAC models yield inaccurate results in LLE calculations despite their outstanding performance in VLE calculations. The reason is that LLE predictions are much more sensitive to the accuracy in activity coefficients than VLE predictions. Pure-component vapor pressures are the most important factor in VLE calculations, while activity coefficients are secondary. Pure-component vapor pressures calculated using other methods, e.g., Antoine equations, for UNIFAC and UNIQUAC are normally sufficiently


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accurate. Reliable predictions can be made without highly accurate estimates of activity coefficients by UNIFAC or UNIQUAC. The situation for LLE calculations is very different. Pure-component vapor pressures play no role in LLE calculations, whereas activity coefficients are the sole decisive factor. Small inaccuracy in activity coefficients by UNIFAC or UNIQUAC models can lead to considerable errors. Thus, validations against LLE data are more rigorous for thermodynamic models than those against VLE data. Rather than VLE data, LLE data should be preferred in the estimation of BIPs, since activity coefficients and hence BIPs are much more sensitive to LLE data than they are to VLE data. 3.5 VLE, LLE and VLLE of ternary systems The ultimate goal of establishing a thermodynamic model is to predict phase behaviors of complex multi-component fluids. Since binary data are often used for parameter estimation, it is necessary to use ternary data for validation purposes. VLE for 15 ternary systems were predicted with the three models and the details of the results are summarized in Table S2 in the Supporting Information for brevity. It was shown that all the three models work remarkably well. Given the fact that a large variety of mixtures are involved in these validations, all the three models can be used with confidence in VLE calculations. It should be noted that binary mixtures are limiting cases of ternary mixtures. Therefore, any inaccuracy in data regression for binary mixtures propagates into calculation results for ternary mixtures. This negative effect dominates in the vicinity of the boundaries of a triangle phase diagram and may extend all the way into the center. Accurate model correlations with binary VLE data partly explain why all the three models offer high precision predictions for ternary mixtures.

Figure 14. (a) LLE and (b) VLLE for ternary water/1-propanol/1-pentanol mixtures. Comparison of PC-SAFT (solid lines), UNIQUAC (dashed lines),



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UNIFAC (dash dotted lines) and UNIFAC-LLE (dash dot dotted lines) to experimental data62,63 (symbols) for organic phase (square), aqueous phase (circle) and vapor phase (triangle).

Figure 15. (a) LLE and (b) VLLE for ternary water/1-propanol/propyl acetate mixtures. Comparison of PC-SAFT (solid lines), UNIQUAC (dashed lines), UNIFAC (dash dotted lines) and UNIFAC-LLE (dash dot dotted lines) to experimental data64,65 (symbols) for organic phase (square), aqueous phase (circle) and vapor phase (triangle).

Figure 16. (a) LLE and (b) VLLE for ternary water/ethanol/n-hexane mixtures. Comparison of PC-SAFT (solid lines), UNIQUAC (dashed lines), UNIFAC (dash dotted lines) and UNIFAC-LLE (dash dot dotted lines) to experimental data66,67 (symbols) for organic phase (square), aqueous phase (circle) and vapor phase (triangle). The BIPs of ethanol/n-hexane and water/ethanol used for UNIQUAC can be found in VLE-IG databank of Aspen plus 8.4.

The three models were validated against LLE and VLLE data of a large variety of ternary mixtures and typical results are presented in Figures 14-18. Figure 19 replots the data in Figure 18b but with tie lines. Refer to Figures S11-14 (Supporting Information) for more results. As shown in Figures 14, 16 and 18d, UNIQUAC predicts artificial phase separations for the binary mixtures of ethanol/n-hexane, water/1-propanol and water/2-propanol. It can be seen that these artificial phase splitting leads to significant deviations from experimental measurements not only in the vicinity of the boundary of the


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triangle phase diagrams, but also in the areas far from the boundary. Moreover, as shown in Figures 14b, 16b and 18d, these artificial phase separations lead to artificial heterogeneous azeotropes for VLLE calculations and hence considerable errors in vapor compositions. It should be stressed that these artificial phase separations undermine predictions for any mixtures containing these binary pairs. UNIFAC also predicts artificial phase separations for the binary mixtures of ethanol/n-hexane, water/1-propanol and ethanol/isooctane as shown in Figures 14a, 16a and 18e. The inaccuracy in the calculations for binary mixtures propagates into predictions for ternary mixtures. Typical examples are shown in Figures 14-16, 17c,d, 18a,b,f, and 19. However, accurate correlations with binary data do not necessarily lead to accurate predictions for ternary mixtures. Examples are displayed in Figures 17b, 18b and 19. Please note that though large errors were observed in calculating the compositions of the aqueous phases with UNIFAC and UNIQUAC, these effects are indiscernible in the triangle phase diagrams, since all the points corresponding to the aqueous phase are very close to the vertex of the triangle phase diagrams due to trace amount of all the components other than water.



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Figure 17. LLE for various ternary mixtures. Comparison of PC-SAFT (solid lines), UNIQUAC (dashed lines), UNIFAC (dash dotted lines) and UNIFAC-LLE (dash dot dotted lines) to experimental data66,68-70 (symbols) for organic phase (square) and aqueous phase (circle). The BIPs of ethanol/1-hexanol, ethyl acetate and DIPE used for UNIQUAC can be found in VLE-IG databank of Aspen plus 8.4.


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Figure 18. VLLE at P = 1.013 bar for various ternary mixtures. Comparison of PC-SAFT (solid lines), UNIQUAC (dashed lines), UNIFAC (dash dotted lines) and UNIFAC-LLE (dash dot dotted lines) to experimental data70-74 (symbols) for organic phase (square), aqueous phase (circle) and vapor phase (triangle). The BIPs of used for UNIQUAC can be found in VLE-IG, LLE-lit and LLE-ASPEN databanks of Aspen plus 8.4.

Figure 19. VLLE with tie-lines for ternary water/acetone/2-butanone mixures at P = 1.013 bar. Comparison of (a) PC-SAFT (solid lines), (b) UNIQUAC (dashed lines), (c) UNIFAC (dash dotted lines) and (d) UNIFAC-LLE (dash dot dotted lines) to experimental data86 (symbols) for organic phase (square),



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aqueous phase (circle) and vapor phase (triangle). The BIPs of used for UNIQUAC can be found in VLE-ig, LLE-lit and LLE-ASPEN databanks of Aspen plus 8.4.

The artificial miscibility gap may be remedied using the so-called UNIFAC-LLE, as shown in Figures 14-16 and 18d,e. However, this solution comes at the expense of introducing more problems. As shown in Figures 14b and 15b, the predicted composition in vapor phase is less satisfactory. Furthermore, UNIFAC-LLE fails to correlate well the mutual solubilities of binary mixtures as discussed earlier. The inaccuracy propagates into the complex mixtures containing these binary pairs as shown in Figures 17b and 18a,d. As shown in Figure 16, the results with UNIFAC-LLE are in close agreement with the experimental data of the ternary mixtures. Nevertheless, it was shown in the previous section that the model fails to reproduce experimental data for the constituent binary mixtures, as shown in Figures S1 and S2. This is because the inaccuracy in binary interactions cancels out somehow in the calculations for ternary mixtures. However, this counteracting effect cannot be generally true, limiting the scope of its practical use. Currently, multiple thermodynamic models (e.g., UNIFAC and UNIQUAC) are necessary for properly handling applications in different areas due to the deficiencies of the models. Alternatively, distinct sets of parameter values are used to meet practical needs for accurate thermodynamic calculations in different fields and at various conditions. However, these measures fail to address this issue at a fundamental level. The use of a different model or the same model with a different set of parameter values leads to improved model correlations/predictions for some cases at the sacrifice of accuracy in some others. For instance, UNIFAC shows better performance than UNIQUAC, as shown in Figures 14b, 15b, 16b and 18d, while UNIQUAC is superior to UNIFAC as shown in Figures 5b and S7c-f. UNIFAC-LLE correctly predicts the absence of the artificial phase separations as shown in Figures 14-16, whereas UNIFAC fails to do so. However, UNIFAC-LLE delivers poor results for VLE calculations, as shown in Figure S1, while UNIFAC shows accurate and reliable predictive capability as summarized in supporting information. There is also a trade-off between accuracy in model correlations/predictions and convenience in parameterization. For instance, UNIQUAC performs better than UNIFAC when considering all the studied cases. However, UNIFAC parameters are easily obtained by a group contribution method, while UNIQUAC parameterization strategy depends heavily on the


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amount and quality of experimental data and is practically infeasible as discussed earlier. All in all, none of classical thermodynamic models is capable of yielding reliable and accurate correlations/predictions in different types of phase equilibria calculations, and simultaneously, offering a practically feasible strategy for parameterization due to their semi-empirical nature. As opposed to its predecessors, PC-SAFT EOS consistently yields accurate and reliable correlations/predictions in all studied cases. This outstanding performance of PC-SAFT EOS not only reflects substantial advance in theoretical aspect, but also greatly benefits practical users. Though some rule of thumb exists, many users (e.g., process engineers, analytical chemists and so on) struggle with selecting an option from the available thermodynamic models of a process simulator to obtain reliable results. Furthermore, it is likely that none of the available models give reliable predictions. For instance, none of thermodynamic models available in commercial software was found to be sufficiently accurate for applications in the Fischer-Tropsch synthesis.75 The advent of PC-SAFT EOS potentially eliminates the need for a large variety of thermodynamic models. As a result, it is possible to develop a process simulator or thermodynamic package with PC-SAFT as the only underlying model. Users of the thermodynamic package can obtain reliable results without the need to select a thermodynamic model. Remarkably, this excellent performance of PC-SAFT EOS was achieved with a single set of well-behaved parameters, allowing for parameterization for weakly characterized fluids by interpolation or extrapolation. 4.

CONCLUSION The performance of PC-SAFT EOS and two other widely-used activity coefficient models, namely,

UNIQUAC and UNIFAC, was evaluated by comparing model correlations/predictions to experimental data of various binary and ternary mixtures containing water, alcohols, alkanes, ketones, aldehydes, esters, and ethers. It was found that all the three models are sufficiently accurate for VLE calculations, whereas PC-SAFT EOS is superior to the other two models in terms of reliability and accuracy for cases with a miscibility gap. This difference in performance was attributed to the fact that LLE calculations are sensitive to small changes in activity coefficients and the most important factor for VLE calculations is pure-component vapor pressures. It was concluded that PC-SAFT EOS has become an ideal and practical tool for reliable



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thermodynamic calculations thanks to the two reasons. (1) PC-SAFT EOS with the use of a single set of parameters consistently gives better performance than its traditional predecessors in all studied cases, regardless of VLE, LLE and VLLE calculations. Potentially, PC-SAFT can replace all the models available in a thermodynamic package so that users do not need to select a model when performing their calculations. Since it is often vague on which model should be selected for a given calculation, this benefit can be substantial for many inexperienced users. This advance also relieves the developers of thermodynamic packages from the trouble and distraction caused by the development and maintenance of multiple thermodynamic models and multiple sets of parameter values. (2) The parameterization for PC-SAFT EOS is straightforward, since both pure-component parameters and BIPs are well fitted with respect to the carbon number or molecular weight for a homologous series, allowing for parameter estimation by interpolation or extrapolation.

SUPPORTING INFORMATION The deviations of VLE for binary systems with PC-SAFT, UNIQUAC and UNIFAC (Table S1); The deviations of VLE for ternary systems (Table S2); %AAD for the mutual solubility of water and hydrocarbons (Table S3); %AAD for the mutual solubility of water and alcohols (Table S4); %AAD for the mutual solubility of water and polar components (Table S5). Experimental and calculated results of VLE for binary systems with PC-SAFT and UNIFAC-LLE (Figure S1); The mutual solubilities of water and 1-alkanes with UNIFAC-LLE (Figure S2); The mutual solubilities of water and 1-alkenes with PC-SAFT, UNIQUAC, UNIFAC and UNIFAC-LLE (Figure S3); The prediction results of the mutual solubilities for water and 1-alkenes with 5-15 carbon atoms with PC-SAFT (Figure S4); The mutual solubilities calculated with PC-SAFT, UNIQUAC, UNIFAC and UNIFAC-LLE for water and 1-alcohols (Figure S5), for 2-alcohols (Figure S6), for water and 2-ketones (Figure S7), for water and acetates (Figure S8), for water and aldehydes (Figure S9), and for water and ethers (Figure S10); The prediction results of LLE with PC-SAFT, UNIQUAC, UNIFAC and UNIFAC-LLE for water/1-propanol/1-alcohols (Figure S11), for water/1-propanol/acetates (Figure S12), and for water/ethanol/n-alkanes (Figure S13); Tie-line prediction results of VLLE for ternary mixtures with


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PC-SAFT, UNIQUAC, UNIFAC and UNIFAC-LLE (Figure S14).

ACKNOWLEDGMENTS The authors acknowledge the financial support of the National Key Research and Development Program of China (No.2017YFB0602402). Ke Zheng thanks the financial support from Synfuels China Co. Ltd. REFERENCE (1) Wei, Y. S.; Sadus, R. J. Equations of state for the calculation of fluid-phase equilibria. AlChE J. 2000, 46, 169. (2) Valderrama, J. O. The state of the cubic equations of state. Ind. Eng. Chem. Res. 2003, 42, 1603. (3) Gmehling, J.; Constantinescu, D.; Schmid, B. Group contribution methods for phase equilibrium calculations. Annu. Rev. Chem. Biomol. Eng. 2015, 6, 267. (4) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. SAFT: Equation-of-state solution model for associating fluids. Fluid Phase Equilib. 1989, 52, 31. (5) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New reference equation of state for associating liquids. Ind. Eng. Chem. Res. 1990, 29, 1709. (6) Huang, S. H.; Radosz, M. Equation of state for small, large, polydisperse, and associating molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. (7) 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. (8) Wertheim, M. S. Fluids with highly directional attractive forces .1. statistical thermodynamics. J. Stat. Phys. 1984, 35, 19. (9) Wertheim, M. S. Fluids with highly directional attractive forces .2. thermodynamic perturbation-theory and integral-equations. J. Stat. Phys. 1984, 35, 35. (10) Wertheim, M. S. Fluids with highly directional attractive forces .3. multiple attraction sites. J. Stat. Phys. 1986, 42, 459. (11) Wertheim, M. S. Fluids with highly directional attractive forces .4. equilibrium polymerization. J. Stat. Phys. 1986, 42, 477. (12) Al-Saifi, N. M.; Hamad, E. Z.; Englezos, P. Prediction of vapor-liquid equilibrium in water-alcohol-hydrocarbon systems with the dipolar perturbed-chain SAFT equation of state. Fluid Phase Equilib. 2008, 271, 82. (13) Gilvillegas, A.; Galindo, A.; Whitehead, P. J.; Mills, S. J.; Jackson, G.; Burgess, A. N. Statistical associating fluid theory for chain molecules with attractive potentials of variable range. J. Chem. Phys. 1997, 106, 4168. (14) Gilvillegas, A.; Galindo, A.; Whitehead, P. J.; Mills, S. J.; Jackson, G.; Burgess, A. N. The thermodynamics of mixtures and the corresponding mixing rules in the SAFT-VR approach for potentials of variable range. Mol. Phys. 1998, 93, 241. (15) And, F. J. B.; Vega, L. F. Prediction of Binary and Ternary Diagrams Using the Statistical Associating Fluid Theory (SAFT) Equation of State. Ind. Eng. Chem. Res. 1998, 37, 660.



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A Comparative Study of the Perturbed-Chain Statistical Associating

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