VLE calculation for non-polar fluid mixtures and polymer solutions using a SAFT-VR type equation of state

Fluid Phase Equilibria 232 (2005) 100–112

VLE calculation for non-polar fluid mixtures and polymer solutions using a SAFT-VR type equation of state Jung-Chin Tsai, Yan-Ping Chen ∗ Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan, ROC Received 13 October 2003; received in revised form 9 October 2004; accepted 24 February 2005 Available online 22 April 2005

Abstract An equation of state (EOS) using an improved expression of the coordination number model for square well (SW) fluids with variable well width was applied for vapor–liquid equilibrium (VLE) calculations of non-polar fluid mixtures and polymer solution. This modified coordination number model was compared with other equations in literature. Better agreement with molecular simulation data for fluid mixtures is observed from this study. Pure fluid parameters for non-polar low molecular weight compounds and polymers are reported. Satisfactory calculated results for the saturated properties are obtained. VLE calculations on non-polar fluid mixtures and polymer solutions were investigated using the van der Waals mixing rules. The equilibrium pressures, solvent solubility and activity of various binary systems were evaluated. This EOS presents comparably good results in comparison with those from other engineering models. © 2005 Elsevier B.V. All rights reserved. Keywords: Equation of state; VLE; Non-polar; Polymer solution

1. Introduction Equation of state (EOS) provides a useful tool for calculating the thermodynamic and phase equilibrium properties of fluids and their mixtures. Theoretically based equations developed from statistical mechanics usually agree well with related molecular simulated and experimental phase equilibrium results if optimal EOS parameters are available. In recent studies, EOS for chain-like polymer molecules receives much attention. For example, the statistical associating fluid theory (SAFT) EOS was established based on the thermodynamic perturbation theory of polymerization. Various versions of the SAFT type equations have been proposed in literature with modifications on either the repulsion or dispersion contributions. Generally, these EOS have been extended from hard sphere (HS) chains to square well (SW) chains with variable width. Dispersion contribution was derived from semi-empirical polynomial correlation [1], the Barker–Henderson perturbation theory [2,3], or the coordina∗

Corresponding author. Tel.: +886 2 2366 1661; fax: +886 2 2362 3040. E-mail address: [email protected] (Y.-P. Chen).

0378-3812/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2005.02.018

tion number equation [4,5]. Adidharma and Radosz [6] have compared various approximations of the SAFT type EOS for SW chains. Basically, different combinations of the expressions for the dispersion and chain formation terms give satisfactory correlation of the saturated properties of real nalkanes. A modified coordination number model for SW fluids was proposed in our previous study [5]. This modified coordination number expression gives correct results at low-density and close-packed regions. It has been tested with the molecular simulation data on pure SW fluids, and better results have been obtained in comparison with those from other coordination number models. A corresponding EOS was derived incorporating a modified chain term for the SW fluids with variable range (VR). In this study, this EOS was employed for real non-polar fluid mixture calculations. Pure fluid parameters were determined for non-polar components and polymers. Extension of our previous study to fluid mixtures and polymer solutions was examined. Comparisons with either the molecular simulation results or phase equilibrium data of real mixtures were investigated to test the applicability of this EOS.

J.-C. Tsai, Y.-P. Chen / Fluid Phase Equilibria 232 (2005) 100–112

2. Extension of the new equation of state to fluid mixtures A SW fluid is considered in this study to derive the coordination number model and the corresponding EOS. The potential function of a SW fluid is:    ∞, r ≤ σ Γ (r) = −ε, σ < r ≤ λσ (1)   0, r ≥ λσ where ε, σ and λ are the energy, the molecular size, and the characteristic well width parameter, respectively. In our previous work [5], the coordination number at any density was written as a product of the low-density limit expression and a correction term. The correction term was correlated using molecular simulation data of SW fluids. Extension to a mixture of unlike fluids i and j, the coordination number Nij is written as:   ε  4π 3 ij (λij − 1)xi ρσij3 exp Nij = 3 kT ε  ε  ij ij × 1 + 1.28 exp +f kT kT √ 

× 2 − ρσij 3 − 1 [1 − exp(1 − φij )] (2) The first part of Eq. (1) gives the exact low-density limit. The second part expresses the density effect for Nij that is different from various derivations in literature. From our correlation of the molecular simulation data, we have:  ε  ε   ε 2  ij ij ij f = 0.25112 − 0.58939 + 0.33225 kT kT kT  2εij × exp (3) kT √ 2 + ρσij 3 φij = √ 2 − ρσij 3

(4)

The cross-terms of the SW potential function parameters are simply defined in this study as: 1 (σi + σj ) 2 √ εij = εi εj σij =

(5)

101

and GWL-I have relatively larger errors. Among these six approaches, the model proposed in this study satisfies both the low-density limit and closed-packed limiting conditions [5], and is easily extended to multi-component expressions. It makes this coordination number model the most favorable one. Table 2 shows the AAD of calculated coordination numbers from six fluid mixture models for ternary SW fluids mixtures. It is indicated that the YC model and the model of this study show better results than those from other methods. Since Eq. (2) satisfactorily agrees with molecular simulation results from pure fluids to ternary mixtures, it is desirable to use this equation in deriving an EOS using the generalized van der Waals theory and the canonical partition function [11]. In this study, we apply the Carnahan–Starling equation [12] for the hard sphere term. The chain term is correlated using the molecular simulation data for the compressibility factors of SW chain fluids at various well widths and temperatures. A new SAFT type EOS for the SW chain molecules of variable range of well width was developed. In our previous study [5], this EOS has been tested for pure SW fluids and real n-alkanes with satisfactory results. This EOS is now extended for fluid mixtures: Z =1+

6 πNAV ρ

 3ξ23 ξ3 ξ23 ξ 0 ξ3 3ξ1 ξ2 × + + − 1 − ξ3 (1 − ξ3 )2 (1 − ξ3 )3 (1 − ξ3 )3  1.28εij εij /kT 2π  +m xi xj (λ3ij − 1)ρσij3 1+ −e 3 kT i

j

ε  1.28εij ij +f [(e1−φij − 1) kT kT

√ √ × ( 2 − 2ρσij3 ) + ( 2 − ρσij3 )e1−φij αij ]

  ξ3 1 + xi (1 − mi ) hs (1 − ξ3 )2 gii (σii ) i  2 2 2 3σjj (ξ2 + ξ2 ξ3 ) σjj (ξ2 + (1/2)ξ2 ξ3 ) + + 2(1 − ξ3 )3 (1 − ξ3 )4    (1 − 5ξ3,eff )(ξ3,eff + c2 ξ32 + 2c3 ξ33 ) corr εij + ×f kT 2(1 − ξ3,eff )3 × e1−φij (1 + αij ) −

(7)

(6)

We have compared the coordination numbers of binary SW fluids mixtures calculated from this model with those from Lee and Sandler (LS) [7], Lee and Chao (LC) [8], Guo et al. (GWL-I and GWL-II) [9], and Yu and Chen (YC) [10]. The molecular simulation data were taken from literature. The comparison results are shown in Table 1 where the calculated absolute average deviations (AAD) for coordination numbers of unlike pair (Nij ) and pure component (Nci ) are listed. It is observed that the coordination number models of LC and GWL-II give the best results while the models of LS

where ξk =

πNAV ρ  xi mi σiik , 6

k = 0, 1, 2, 3

(8)

i

f corr

ε  ij

kT

= 18.64084

ε  ij

 ε 2

+ 21.67648

kT  ε 3 ij − 4.19834 kT

ij

kT (9)

102

J.-C. Tsai, Y.-P. Chen / Fluid Phase Equilibria 232 (2005) 100–112

Table 1 Comparison of the calculated results for the coordination numbers of binary SW fluids from various models with λ = 1.5 Case

AAD (%)

Data sources

LS

LC

GWL-I

GWL-II

YC

This work

I

Nij Nci

15.22 12.04

5.41 3.76

9.40 7.13

5.41 3.98

7.37 3.92

7.41 4.57

[8] [8]

II

Nij Nci

11.64 10.81

4.04 3.49

8.76 6.97

4.65 4.12

3.96 3.58

4.11 3.54

[8] [8]

III

Nij Nci

10.29 8.07

6.67 5.31

10.76 8.54

7.78 6.55

7.41 5.18

7.35 5.36

[8] [8]

IV

Nij Nci

12.18 11.41

4.05 3.21

8.57 6.66

4.03 3.18

4.28 3.43

4.31 3.74

[8] [8]

V

Nij Nci

12.36 11.96

3.36 2.77

4.49 3.94

3.19 2.60

4.08 3.57

4.16 3.75

[20] [20]

VI

Nij Nci

9.93 8.80

7.61 7.01

9.73 9.62

8.97 8.42

8.06 7.78

7.48 6.71

[20] [20]

VII

Nij Nci

11.49 11.35

2.88 2.50

6.79 5.30

2.60 2.12

3.47 2.91

3.29 2.99

[7] [7]

VIII

Nij Nci

11.62 11.44

2.71 2.33

8.99 8.15

2.04 1.69

4.09 3.15

4.26 3.88

[7] [7]

Case I: ε11 /κT = 0.5, ε22 /κT = 1.0, ε12 /κT = 0.5, σ 22 /σ 11 = 1.0–2.0, x1 = 0.204–0.796, ρ* = 0.05–0.8. Case II: ε11 /κT = 0.5, ε22 /κT = 1.0, ε12 /κT = 0.71, σ 22 /σ 11 = 1.0–2.0, x1 = 0.204–0.796, ρ* = 0.05–0.8. Case III: ε11 /κT = 0.5, ε22 /κT = 1.0, ε12 /κT = 1.0, σ 22 /σ 11 = 1.0–2.0, x1 = 0.204–0.796, ρ* = 0.05–0.8. Case IV: ε11 /κT = 0.33, ε22 /κT = 1.0, ε12 /κT = 0.58, σ 22 /σ 11 = 1.0–2.0, x1 = 0.5, ρ* = 0.05–0.8. Case V: ε11 /κT = 0.4, ε22 /κT = 0.8, ε12 /κT = 0.57, σ 22 /σ 11 = 1.0, x1 = 0.25–0.75, ρ* = 0.1–0.7. Case VI: ε11 /κT = 0.6, ε22 /κT = 1.2, ε12 /κT = 0.85, σ 22 /σ 11 = 1.0, x1 = 0.25–0.75, ρ* = 0.1–0.7. Case VII: ε11 /κT = 0.6, ε22 /κT = 0.6, ε12 /κT = 0.6, σ 22 /σ 11 = 1.2–2.0, x1 = 0.25–0.75, ρ* = 0.1–0.7. * Case VIII: ε11 /κT  = 0.4, ε22 /κT = 0.4, ε12 /κT = 0.4, σ 22 /σ 11 = 1.2–2.0, x1 = 0.25–0.75, ρ = 0.1–0.7. AAD % = 100 [|N − N |/N ], n: number of data points. ij,k,MC ij,k,cal ij,k,MC k n

giihs (σii ) =

ξ22 σii2 1 3σii ξ2 + + 1 − ξ3 2 (1 − ξ3 )2 2 (1 − ξ3 )3

ξ3,eff = c1 ξ3 + c2 ξ32 + c3 ξ33    1.01217 1.41012 c1     c2  =  0.40337 3.47579 −4.27319 √ 2(1 − φij ) αij = √ 2 − ρσij3 c3

2.33210

(10) (11)



for a mixture of HS and SW fluid is depicted in Fig. 3. In general, the agreement between the calculated results from Eq. (7) and molecular simulation data is acceptable. This EOS is then extended for real fluid mixture calculations.



1 −1.42229 λ  −3.87916   ij  (12) λ2ij 1.94109 (13)

The new EOS shown in Eq. (7) has four pure fluid parameters: the number of segments m, the molar volume of a hard sphere segment V 0 = σ 3 , the characteristic well width parameter λ, and the interaction energy parameter ε/κ. In Fig. 1, we compare the vapor–liquid coexistence curve for a pure SW fluid from this EOS with the molecular simulation data. Generally, satisfactory match indicates the feasibility of employing this EOS on phase equilibrium calculations. Fig. 2 shows the graphical comparison of the compressibility factors for mixtures of HS chain fluids from the generalized Flory-Dimer (GFD) [13], TPT-D [14] and PHSC [15] EOS. This EOS again shows good agreement with molecular simulation data at various packing fractions ηmix . Comparison with the molecular simulation results for the temperature–composition relation

Fig. 1. The vapor–liquid coexistence curve for pure SW fluids calculated in the work (data Ref. [72]).

J.-C. Tsai, Y.-P. Chen / Fluid Phase Equilibria 232 (2005) 100–112

103

Table 2 Comparison of the calculated results for the coordination numbers of ternary SW fluids from various models with λ = 1.5 Case

AAD (%) LS

Data sources LC

GWL-I

GWL-II

YC

This work

I

Nij Nci

9.52 8.20

4.84 3.96

7.70 6.52

5.72 5.14

3.80 3.28

3.85 3.18

[21] [21]

II

Nij Nci

8.60 6.76

5.87 6.01

9.44 8.09

6.89 6.79

4.67 3.98

4.35 3.96

[21] [21]

III

Nij Nci

9.19 8.27

7.55 8.46

12.50 9.46

7.83 8.83

6.55 5.71

6.14 5.48

[21] [21]

IV

Nij Nci

13.78 11.15

12.55 13.23

17.35 12.92

11.96 12.61

11.06 9.70

11.17 9.32

[21] [21]

V

Nij Nci

8.84 7.52

5.44 4.96

9.59 7.81

5.84 5.68

4.41 3.44

4.26 3.20

[21] [21]

VI

Nij Nci

9.45 8.04

6.55 4.95

11.67 8.20

6.98 5.74

5.01 2.87

4.78 2.97

[21] [21]

VII

Nij Nci

21.14 19.46

18.67 18.15

18.15 20.09

18.65 18.07

17.38 16.07

16.94 15.84

[21] [21]

VIII

Nij Nci

9.56 7.65

9.80 8.77

15.78 13.81

10.16 9.52

7.95 6.46

8.98 6.81

[21] [21]

IX

Nij Nci

16.35 16.60

20.81 20.24

26.37 26.50

21.35 20.60

17.90 17.10

17.82 17.76

[21] [21]

Case I: ε11 /κT = 0.5, ε22 /κT = 1.0, ε33 /κT = 1.0, σ 22 /σ 11 = 1.5, σ 33 /σ 11 = 1.5, x1 = 0.204, x2 = 0.352, ρ* = 0.05–0.80. Case II: ε11 /κT = 0.5, ε22 /κT = 1.0, ε33 /κT = 1.0, σ 22 /σ 11 = 1.5, σ 33 /σ 11 = 2.0, x1 = 0.204, x2 = 0.352, ρ* = 0.05–0.80. Case III: ε11 /κT = 0.5, ε22 /κT = 1.0, ε33 /κT = 1.0, σ 22 /σ 11 = 1.5, σ 33 /σ 11 = 2.5, x1 = 0.204, x2 = 0.352, ρ* = 0.05–0.80. Case IV: ε11 /κT = 0.5, ε22 /κT = 1.0, ε33 /κT = 1.0, σ 22 /σ 11 = 1.5, σ 33 /σ 11 = 3.0, x1 = 0.204, x2 = 0.352, ρ* = 0.05–0.80. Case V: ε11 /κT = 0.5, ε22 /κT = 1.0, ε33 /κT = 1.0, σ 22 /σ 11 = 1.5, σ 33 /σ 11 = 2.0, x1 = 0.334, x2 = 0.333, ρ* = 0.05–0.80. Case VI: ε11 /κT = 0.5, ε22 /κT = 1.0, ε33 /κT = 1.0, σ 22 /σ 11 = 1.5, σ 33 /σ 11 = 2.0, x1 = 0.296, x2 = 0.602, ρ* = 0.05–0.80. Case VII: ε11 /κT = 0.5, ε22 /κT = 1.0, ε33 /κT = 0.5, σ 22 /σ 11 = 1.5, σ 33 /σ 11 = 2.0, x1 = 0.204, x2 = 0.352, ρ* = 0.05–0.80. CaseVIII: ε11 /κT = 0.5, ε22 /κT = 1.0, ε33 /κT = 1.25, σ 22 /σ 11 = 1.5, σ 33 /σ 11 = 2.0, x1 = 0.204, x2 = 0.352, ρ* = 0.05–0.80. = 0.5, ε22 /κT = 1.0, ε33 /κT = 1.5, σ 22 /σ 11 = 1.5, σ 33 /σ 11 = 2.0, x1 = 0.204, x2 = 0.352, ρ* = 0.05–0.80. Case IX: ε11 /κT AAD % = 100 k [|Nij,k,MC − Nij,k,cal |/Nij,k,MC ], n: number of data points. n

Fig. 2. Comparison of the compressibility factors for mixtures of hardsphere chain fluids from various EOS (data Ref. [73]).

Fig. 3. The temperature–composition relation for the mixture of hard spheres (1) and square wells (2) calculated in the work (data Ref. [74]).

104

J.-C. Tsai, Y.-P. Chen / Fluid Phase Equilibria 232 (2005) 100–112

The SAFT-VR type EOS shown in Eq. (7) has been applied for real n-alkane fluids with good accuracy. Extension of calculations to real fluid mixtures and polymer solutions is presented in this study. For the VLE calculations on mixtures, the equal fugacity criterion is employed for each component i: fˆ iV = fˆ iL

(14)

For polymer solutions, the amount of polymer in the vapor phase is close to zero. It is convenient to assume that there is no polymer in the vapor phase and Eq. (14) is only applied for the solvent molecule. To determine the EOS parameters for mixtures, the following simple van der Waals type mixing rules are used:  m= xi mi (15)  εij ε fi fj = kT kT i

(16)

j

Fig. 4. Calculated specific volumes for selected polymers at 1 bar and various temperatures (data Ref.: PE [75]; PCHMA, PS and PMMA [30]).

where fi is the volume fraction: xi mi Vi0 fi =  0 j xj mj V j

(17)

The following combining rules are employed: λi + λ j 2 σi + σj σij = 2 λij =

εij = (εi εj )0.5 (1 − kij )

(18) (19) (20)

One temperature-independent binary interaction parameter kij is required for each binary pair. Its optimal value is determined through data regression.

3. Results and discussion In this EOS, there are four parameters for each pure fluid that are regressed from experimental data. It is found that the molar volume of hard sphere and the binary interaction energy parameter are functions of temperature: V 0 = V 00 + 0.005T ε = kT

ε0 k

− 0.04 T

(21) (22)

These two equations are used for all pure fluids in this study. Besides n-alkane compounds shown in our previous work [5], other non-polar pure fluid parameters are presented in Table 3. Although five decimal points are reported for each parameter, it was observed that the numbers of decimal points are not very sensitive to the calculation accuracy and three decimal points are acceptable for further calculations. It is observed that the calculated saturated vapor pressures and

liquid densities are satisfactory and the overall AAD is about 1%. This EOS was further examined for polymer systems. In the regression of pure polymer parameters, the objective function was based on the minimization of calculated errors of liquid molar volumes. The results are presented in Table 4. For all polymers investigated, an optimal λ-value is found as 1.5. The calculated liquid molar volumes are again satisfactory as shown in Table 4. This EOS was also compared with the original SAFT EOS [1] and the GFD EOS [13]. As shown in Table 4, all equations yield good results and the SAFT-VR EOS in this study gives the minimum deviation. The calculated specific volumes at 1 bar for several polymers are shown graphically in Fig. 4 at various temperatures. Good agreement with experimental data confirms that this EOS is applicable for long-chain polymers. This SAFT-VR type EOS has been employed on the VLE calculations of binary mixtures. The calculated results using traditional van der Waals one-fluid mixing rules and the comparison with those from the GFD EOS are presented in Table 5. One temperature-independent binary interaction parameter was used for the energy parameter of each EOS. It is shown that both EOS models give satisfactory VLE calculation results. Table 6 presents a comparison of our VLE calculation results with the original SAFT EOS [1] and the recent perturbed-chain SAFT (PC-SAFT) EOS [16]. With the optimally fitted binary interaction parameters for each EOS, the calculated VLE constants (K-values) from this study yield the minimum absolute average deviation. It is also indicated that the binary interaction parameters are small. Even without using the binary interaction parameters, acceptable calculation results are still obtained from this study. Fig. 5 shows a graphical comparison between the experimental and calculated results from the SAFT-VR EOS in this study. It is

Table 3 Calculated results of saturated vapor pressure and liquid molar volumes for pure fluids using the equation of state in this study Compounds

Number of data points

PV

VL

PV

VL

200–401 236–447 257–433 289–497 283–504 236–488 259–499 284–524 283–529 300–532 286–512 260–529 290–525 291–535 284–543 301–546 303–630 302–616 296–616 297–616 203–397 203–285 229–510 279–535 283–604 120–281 163–364 215–413 221–463 283–353 217–401 212–282 217–302 183–365 89–227 220–503

213–369 236–440 261–425 289–480 273–494 174–463 275–489 284–572 288–518 300–523 286–503 297–522 273–513 291–527 284–543 282–546 312–612 302–582 296–491 297–487 197–386 203–273 229–496 279–535 297–575 128–277 176–364 215–388 221–443 283–334 217–387 242–282 217–298 203–357 107–183 212–414

32 29 45 25 43 41 45 41 44 28 28 37 41 30 41 27 38 37 42 33 35 41 40 42 45 41 33 40 37 45 40 36 42 45 45 39

25 28 42 22 42 31 39 40 41 26 27 30 42 29 41 29 35 33 26 20 32 35 38 42 39 37 30 35 34 33 37 21 38 38 25 28

λ

1.59242 1.59675 1.57731 1.58436 1.60134 1.57929 1.60049 1.59393 1.60192 1.62771 1.57903 1.59596 1.60021 1.59701 1.69522 1.65349 1.64106 1.62605 1.62880 1.63030 1.63727 1.66911 1.64084 1.63949 1.64830 1.62657 1.67468 1.74189 1.68279 1.65662 1.60355 1.63683 1.60375 1.93985 1.86895 1.64320

m

1.66783 1.74475 1.75413 1.94228 1.83720 1.90224 1.79400 2.14848 2.01801 1.92683 2.18118 2.10297 1.80204 1.95911 1.58386 1.73039 1.87555 1.95467 1.92257 1.90547 1.41086 1.41301 1.62389 1.65691 1.75005 1.32778 1.43735 1.45723 1.64858 1.80536 1.69118 1.80076 1.74674 1.43267 1.38833 1.64917

1030 V00 (m)3

43.85251 50.57464 50.27576 53.42933 56.18687 53.78994 57.39642 55.91995 58.59487 61.44723 54.08927 55.22687 64.95284 69.35194 46.06789 51.53922 55.55532 53.65989 54.72142 54.77212 32.70105 44.20858 45.91396 53.51279 58.37714 28.05806 36.67480 46.77521 51.00147 55.16348 26.60297 33.85187 15.42567 32.00159 29.48540 30.19893

ε0 /k (K)

215.80972 236.51768 235.11317 245.67580 250.18687 254.06947 250.95234 251.04540 255.28021 242.04331 255.08296 253.42128 266.04650 266.41433 228.81237 258.57663 275.72286 277.66231 277.48302 276.38762 200.68491 215.85423 236.49039 254.97878 263.89191 150.45695 163.90695 160.39594 191.48493 213.70577 205.26835 195.88717 151.30536 97.95842 68.76942 234.14923

This work

Data sources

AAD(P) %

AAD(VL ) %

1.04 0.73 1.09 1.29 0.91 0.92 0.98 1.47 1.24 0.63 1.40 1.01 0.93 1.12 0.69 0.59 1.21 0.81 0.93 1.11 0.79 1.02 0.87 0.98 0.72 0.90 0.69 0.72 0.86 0.12 0.99 1.06 0.23 1.50 1.36 1.26

0.86 1.32 1.03 1.09 1.08 1.35 1.09 1.08 1.01 1.27 1.07 1.04 1.04 1.18 1.07 1.32 1.38 0.66 0.57 0.54 0.99 0.46 1.04 0.91 1.30 0.89 0.77 1.23 0.86 0.26 0.96 0.38 1.28 1.01 0.89 1.26

[22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [22] [23] [22] [22] [22]

J.-C. Tsai, Y.-P. Chen / Fluid Phase Equilibria 232 (2005) 100–112

Isobutane 2-Methylbutane 2,2-Dimethylpropane 2-Methylpentane 3-Methylpentane 2,2-Dimethylbutane 2,3-Dimethylbutane 2-Methylhexane 3-Methylhexane 3-Ethylpentane 2,2-Dimethylpentane 2,3-Dimethylpentane 2,2,3-Trimethylbutane 2,2,4-Trimethylpentane Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Cyclopropane Cyclobutane Cyclopentane Cyclohexane Cycloheptane Ethene Propene 1-Butene 1-Pentene 1-Hexene Propyne 1-Butyne Carbon dioxide Chlorodifluoro methane Tetrafluoro methane Dichloro methane

Temperature range (K)

105

106

Table 3 (Continued). Compounds

Temperature range (K) V

Grand average

P

V

177–383 226–465 193–398 215–382 217–460 279–403 169–375 173–346 251–371 159–240 132–288 141–298 245–450 244–431

182–378 219–465 193–398 232–357 244–383 261–349 173–243 200–315 243–336 170–230 144–292 191–288 207–439 259–542

V

P

m

1030 V00 (m)3

ε0 /k (K)

L

V

40 37 40 39 45 35 45 45 42 11 38 45 38 27

38 38 39 31 19 25 14 26 33 8 34 27 42 38

1900

1602

2.02079 1.83058 1.89949 1.88789 2.18075 1.94109 1.76616 1.73303 1.85235 1.89770 1.77279 1.763575 1.64443 1.73197

1.31432 1.43385 1.26676 1.41972 1.17483 1.49036 1.44824 1.59073 1.48957 1.14548 1.32733 1.58036 1.63789 1.56667

46.38221 49.02094 30.19090 34.57423 43.28328 43.23966 31.33702 33.93456 54.54301 20.59342 22.48707 23.19527 34.02173 41.01033

94.04022 150.31105 128.76006 115.07701 101.23299 146.53395 138.23979 130.50482 163.52038 109.90392 124.19245 105.44668 206.35457 200.21793

This work

Data sources L

AAD(P) %

AAD(V ) %

1.49 1.12 0.82 1.51 1.46 0.12 1.57 0.82 0.11 0.17 0.56 1.24 0.73 0.52

1.24 1.48 1.59 1.64 1.67 0.80 1.25 0.58 0.84 0.93 1.08 1.80 1.04 1.29

0.95

1.06

[22] [22] [22] [22] [22] [22] [22] [22] [22] [24] [22] [22] [22] [22]

Fig. 5. VLE calculation results from this work for methane and m-xylene binary mixtures at 313.2 K (data Ref. [76]).

demonstrated that the calculated VLE data are satisfactory up to the critical region of the binary mixture. Table 7 lists the calculated results for the solubility of solvents in polymers. We compare our calculated results with those from Sanchez–Lacomb (SL) [17], GFD and the original SAFT EOS methods using van der Waals mixing rules and their optimally fitted binary parameters. Calculation results for the equilibrium pressures of polymer solutions from various EOS methods are presented in Table 8. Generally, this

Fig. 6. Calculated VLE results for heptane in polyethylene from this study (data Ref. [63]).

J.-C. Tsai, Y.-P. Chen / Fluid Phase Equilibria 232 (2005) 100–112

Dichloro difluoro methane Trichloro fluoro methane Chloro methane 1,1-Difluoro ethane Chloro ethane 1,2-Dichloro ethane Fluoro ethane 1,1,1-Trifluoro ethane 1,1,1-Trichloro ethane Hydrogen chloride Fluoro methane Trifluoro methane Dichlorofluoro methane Chloroform

L

λ

Number of data points

Table 4 Pure component parameters for polymers studied in this work and the comparison of calculated results with other EOS Polymer

Pressure range (MPa)

Temperature range (K)

BR HMDS i-PB i-PMMA i-PP LDPE LLDPE PAR PBMA PC PCHMA PDMPO&PS

0.1–283 10–90 49–196 0.1–200 49–196 10–100 0.1–200 29.4–177 0.01–20 29.4–177 0.1–200 29.4–177

277.15–328.35 298.15–343.15 406.99–514.08 328.45–463.35 446.66–571.63 413.15–473.15 409.85–472.65 465.92–588.43 295.15–472.65 453.26–610.01 382.75–472.05 428.42–597.0

PDMS PE PEG PIB PMMA PMP POM PoMS PPFE PS PT PTFE PVAC PVAL PVC

10–100 0.1–10 19.6–68.6 10–100 0.1–200 9.81–196 39.2–196 0.1–180 0.1–40 0.1–200 9.81–68.6 0.98–39.2 10–100 10–200 19.6–157

298.05–343.05 413.15–473.15 344.96–489.99 325.95–383.15 386.65–432.15 449.65–502.05 462.27–492.13 412.55–470.85 298.15–373.15 388.55–468.75 337.17–443.98 603.55–645.55 337.15–393.15 413.05–468.95 355.15–370.15

Grand average a

Molecular weight assigned; AAD V liq (%) =

Molecular weight, MW (g/mol)

m/MW (mol/g)

1030 V00 (m)3

ε0 /k (K)

156 36 36 96 36 38 61 63 168 89 90 79

100000a 100000a 1800000 100000a 570000 190000 100000a 35840a 100000a 254.2 100000a 120000

0.016492 0.015016 0.011292 0.016942 0.007340 0.010543 0.016907 0.010021 0.014713 0.014729 0.012233 0.008965

54.693 73.807 104.306 45.801 169.965 112.334 66.186 80.760 46.143 52.789 71.862 103.181

58 40 60 50 41 100 20 50 24 69 41 21 57 101 15

166000 25000 7500 36000 100000a 100000a 100000a 90700 16800 279000 40000 100000a 330000 100000a 100000a

0.010396 0.011945 0.014129 0.019466 0.014916 0.010035 0.010944 0.013130 0.006734 0.012938 0.013898 0.005415 0.015568 0.016391 0.017114

93.022 97 61.327 53.046 53.798 122.136 71.684 71.839 71.777 71.427 70.876 79.211 51.037 69.537 39.119

Number of data points

1695 100 n



liq liq |Vexp −Vcal | liq Vexp

AAD Vliq (%)

Data source

SAFT GFD EOS EOS

This work

673.581 293.596 595.686 542.993 660.895 574.179 480.019 705.439 593.255 739.158 592.823 647.813

0.086 0.228 0.260 0.289 0.124 0.096 0.104 0.068 0.178 0.087 0.123 0.117

0.049 0.182 0.228 0.173 0.332 0.091 0.047 0.063 0.221 0.099 0.117 0.086

0.014 0.021 0.118 0.047 0.102 0.099 0.054 0.099 0.013 0.028 0.017 0.106

[25] [26] [27] [28] [27] [29] [30] [31] [30] [31] [30] [32]

421.232 535.728 520.157 507.841 577.711 663.392 626.799 618.183 350.458 601.940 513.449 374.980 480.554 673.745 536.549

0.098 0.022 0.079 0.143 0.143 0.219 0.165 0.157 0.109 0.097 0.048 0.235 0.049 0.163 0.049

0.063 0.008 0.060 0.126 0.026 0.209 0.032 0.056 0.280 0.306 0.130 0.167 0.121 0.034 0.043

0.086 0.003 0.149 0.089 0.059 0.016 0.059 0.088 0.138 0.161 0.021 0.026 0.040 0.019 0.015

[29] [29] [33] [29] [30] [34] [35] [30] [36] [30] [33] [37] [29] [38] [39]

0.135

0.124

0.054

J.-C. Tsai, Y.-P. Chen / Fluid Phase Equilibria 232 (2005) 100–112

cis-1,4-Polybutadiene Hexamethyldisiloxane Isotactic poly(butene-1) Isotactic poly(methyl methacrylate) Isotactic polypropylene Low-density polyethylene Linear low-density polyethylene Polyarylate Poly(n-butyl methacrylate) Polycarbonate Poly(cyclohexyl methacrylate) Poly(2,6-dimethyl-1,4-phenylene-costyrene) Poly(dimethylsiloxane) Polyethylene Poly(ethylene glycol) Polyisobutylene Poly(methyl methacrylate) Poly(4-methyl pentene-1) Poly(oxymethylene) Poly(orthomethylstyrene) Poly(perfluoro ethers) Polystyrene Polytetrahydrofuran Poly(tetrafluoro ethylene) Poly(vinyl acetate) Poly(vinyl alcohol) Poly(vinyl chloride)

Symbol

.

107

108

J.-C. Tsai, Y.-P. Chen / Fluid Phase Equilibria 232 (2005) 100–112

Table 5 Comparison of the VLE calculation results of fluid mixtures from equations of state System (1) + (2) Ethane + propene Propene + propane n-Butane + n-hexane 1-Butene + propene n-Pentane + n-heptane n-Hexane + 2-meyhyl-pentane n-Hexane + 3-methyl-pentane n-Hexane + 2,3-dimethyl-butane n-Hexane + cyclohexane n-Hexane + n-heptane n-Hexane + benzene Dichloromethane + benzene n-Pentane + dichloromethane Toluene + n-decane Methane + n-butane Methane + n-pentane Methane + n-hexane Methane + octane Methane + n-decane Methane + n-eicosane Chloroform + benzene Benzene + cyclohexane CO2 + benzene CO2 + n-hexadecane CO2 + octane CO2 + ethane CO2 + n-pentane CO2 + n-hexane CO2 + ethene

Pressure range (kPa)

Temperature range (K)

689.12–2411.91 181.93–3097.99 5.44–190.90 2067.4–2756.46 18.08–105.37 10.68–49.52 10.35–45.27 21.26–30.20 77.31–100.79 10.36–87.12 13.33–105.51 14.68–294.80 63.30–1186.89 11.63–90.17 413.70–12410.3 137.9–16119.3 2028.5–25260.3 1013.3–7092.8 2756.0–23922.0 1008.2–5051.1 21.89–66.87 6.59–80.06 893.57–7748.13 2005.74–5065.0 2000.0–11350.0 1441.5–2134.39 227.41–9626.95 443.59–7655.55 1043.39–2618.61

260–277 244–344 253–293 277–344 293–343 283–313 283–313 298 343 303–340 293–343 298–348 298–398 373–383 144–394 191–444 464–543 298–423 511–543 373–573 308–323 283–343 298–313 463–664 313–348 250 278–377 298–313 232–253

Grand average

No. of data GFD EOS points Optimum AAD k12 P (%) 10 99 50 26 28 53 58 9 7 18 44 38 69 19 135 138 14 33 12 15 24 76 17 16 19 13 48 20 23

0.0005 0.0016 −0.0017 −0.0071 −0.0035 0.0018 0.0018 −0.0003 0.0013 −0.0030 0.0242 −0.0074 0.0574 0.0110 −0.0998 −0.0903 −0.1830 −0.0708 −0.1797 −0.1545 −0.0125 0.0190 −0.0328 −0.0895 0.0294 0.0985 0.0276 0.0245 0.0523

1131

This work

Data sources

y1 (%)

Optimum k12

AAD P (%)

y1 (%)

0.27 1.66 0.71 1.11 1.61 0.38 0.40 0.30 0.27 1.07 0.88 0.71 1.48 2.83 5.93 6.55 8.68 3.88 4.82 6.97 0.88 0.76 3.90 7.78 8.85 3.45 6.49 5.74 1.84

1.38 0.41 0.26 1.22 0.51 0.36 0.91 0.36 0.37 1.01 0.60 0.56 1.14 0.67 4.44 3.57 2.37 0.24 3.96 0.12 0.70 0.45 0.52 1.95 1.04 2.07 1.59 0.80 1.21

0.0008 0.0025 −0.0018 −0.0068 −0.0048 0.0021 0.0035 −0.0073 0.0021 −0.0025 0.0155 −0.0035 0.0257 0.0097 −0.0954 −0.0910 −0.0999 0.0048 0.0015 −0.0305 −0.0099 −0.0214 0.0015 −0.0198 0.0027 0.0875 0.0249 0.0202 0.0341

0.22 1.56 0.74 1.24 1.72 0.34 0.75 0.24 0.42 1.14 0.54 0.48 1.23 1.45 4.23 5.69 7.54 2.33 4.68 5.14 0.44 0.89 1.09 6.37 3.48 2.64 3.45 2.71 0.99

1.05 0.43 0.34 0.98 0.47 0.35 0.71 0.26 0.38 1.05 0.36 0.45 1.02 0.89 1.97 2.07 1.57 0.98 1.48 1.56 0.32 0.57 0.71 1.27 1.97 1.29 0.98 0.71 1.06

3.27

1.58

2.41

1.05

[40] [41,42] [43] [44] [43] [43,45,46] [43,45,46] [45] [43] [43] [43] [47] [48,49] [50] [51] [52] [53] [54] [55] [56] [43] [43] [57] [58] [59] [60] [61] [57] [62]

Table 6 Comparison of the VLE calculation results of fluid mixtures from various equations of state System

T range (◦ C)

(1) + (2)

Methane–butane Methane–pentane Methane–hexane Methane–heptane Methane–decane Methane–cyclohexane Methane–benzene Methane–toluene Nitrogen–hexane CO2 –methane CO2 –propane CO2 –butane CO2 –pentane CO2 –heptane Average

PC-SAFT k12

21–121 71–171 0.01–138 4–238 150–310 21–171 148–228 149–270 38–171 −43 to −3 −33 to 57 −45 to 145 4–104 37–204

0.022 0.024 0.021 0.016 0.056 0.045 0.037 0.052 0.119 0.065 0.109 0.120 0.143 0.129

SAFT AAD%

k12

K1

K2

1.64 1.56 3.62 7.28 1.58 4.18 5.19 3.34 7.73 3.04 3.92 5.91 6.99 8.68

2.60 1.65 1.44 3.06 2.11 2.26 2.85 3.47 3.97 2.61 5.45 5.56 5.04 2.65

4.62

3.19

0.074 0.077 0.076 0.053 0.121 0.076 0.091 0.111 0.127 0.091 0.108 0.116 0.149 0.129

AAD% AAD%

K1

This work K2

K1

K2

5.62 6.89 9.30 12.90 6.34 7.48 9.93 8.03 10.27 2.98 4.80 6.72 11.18 16.85

2.84 4.39 5.29 9.15 4.10 6.02 4.47 6.33 7.68 2.70 6.02 6.19 6.24 2.00

2.19 2.73 3.45 7.16 2.31 2.38 5.75 3.94 7.73 4.09 3.84 4.87 6.05 6.78

2.98 2.74 2.09 2.54 2.12 2.16 2.09 3.71 3.67 2.68 4.21 4.25 3.66 2.86

8.52

5.24

4.52

2.98

k12 −0.095 −0.091 −0.099 −0.031 0.001 0.029 0.017 0.079 0.104 0.113 0.073 0.044 0.025 0.013

AAD% K1

K2

1.43 1.69 2.14 6.31 1.83 1.14 5.49 3.74 6.83 3.41 3.19 4.41 5.83 5.94

2.17 2.01 1.54 1.73 1.57 1.21 1.97 3.59 3.02 2.14 3.83 3.89 3.19 2.12

3.82

2.43

Table 7 Comparison of the calculated solubilities of solvents in polymers from various EOS Pressure range (kPa)

Temperature range (K)

No. of data points

Hexane + PE Heptane + PE Octane + PE Benzene + PE Toluene + PE Cyclohexane + PE Ethylbenzene + PS Toluene + PS Benzene + PS m-Xylene + PS Butane + PIB Pentane + PIB Benzene + PVAC Ethyl benzene + BR Hexane + PP

29.99–1409.73 12.77–781.22 5.57–450.79 10.54–1104.54 7.30–587.28 15.81–1001.70 0.65–95.39 0.48–13.45 3.03–52.01 14.9–74.5 32.53–193.05 57.17–177.72 3.97–15.49 3.60–39.60 2.85–30.92

383–473 383–473 383–473 383–473 383–473 383–473 393–458 293–333 293–393 403–448 308–319 318–328 303 353–403 353

100 100 100 100 100 100 55 35 38 22 10 10 8 35 10

Grand average

SL EOS

GFD EOS

SAFT EOS

This work

Data sources

Optimum k12

Dev w1

Optimum k12

Dev w1

Optimum k12

Dev w1

Optimum k12

Dev w1

0.0140 0.0039 0.0036 0.0038 0.0004 −0.0091 0.0080 −0.0021 0.0017 0.0070 0.0207 0.0085 0.0107 0.0098 0.0045

0.0183 0.0144 0.0126 0.0112 0.0147 0.0154 0.0023 0.0217 0.0110 0.0021 0.0108 0.0178 0.0124 0.0196 0.0247

−0.0432 −0.0324 −0.0257 −0.0046 −0.0111 −0.0337 −0.0927 −0.1013 −0.1140 −0.0802 −0.1416 0.0283 −0.050 −0.0400 −0.0355

0.0221 0.0196 0.0159 0.0182 0.0115 0.0138 0.0051 0.0212 0.0288 0.0041 0.0215 0.0860 0.0098 0.0165 0.0198

−0.0066 −0.0052 −0.0046 0.0069 0.0006 −0.0052 0.0181 −0.3196 −0.3575 −0.2100 −0.0504 0.0787 −0.0272 −0.0125 −0.0148

0.0313 0.0288 0.0241 0.0299 0.0155 0.0385 0.0261 0.0342 0.0325 0.0104 0.0010 0.0910 0.0269 0.0102 0.0187

−0.0062 −0.0054 −0.0046 −0.0006 −0.0038 −0.0033 −0.0164 −0.1026 −0.1069 −0.1023 −0.0098 0.0874 −0.0313 −0.0169 −0.0127

0.0191 0.0154 0.0129 0.0143 0.0128 0.0116 0.0034 0.0156 0.0111 0.0018 0.0073 0.0142 0.0065 0.0152 0.0102

823

0.0138

0.0173

0.0274

[63] [63] [63] [63] [63] [63] [64] [65,66] [66] [67] [68] [68] [69] [68] [70]

0.0131

Table 8 Comparison of the results of vapor–liquid equilibrium calculations on polymer solutions using various equations of state System solvent (1) + polymer (2)

Pressure range (kPa)

Temperature range (K)

No. of data points

SL EOS Optimum k12

AAD P (%)

Optimum k12

AAD P (%)

Optimum k12

AAD P (%)

Optimum K12

AAD P (%)

Hexane + PE Heptane + PE Benzene + PE Toluene + PE Cyclohexane + PE Ethylbenzene + PS Toluene + PS Benzene + PS m-Xylene + PS Hexane + PIB Cyclohexane + PIB Benzene + PVAC Ethyl benzene + BR Hexane + PP Cyclohexane + PS

29.99–1409.73 12.77–781.22 10.54–1104.54 7.30–587.28 15.81–1001.70 0.65–95.39 0.48–13.45 3.03–52.01 14.9–74.5 53.48–410.05 29.67–281.27 3.97–15.49 3.60–39.60 2.85–30.92 8.50–19.21

383–473 383–473 383–473 383–473 383–473 393–458 293–333 293–393 403–448 398 398 303 353–403 353 297–307

100 100 100 100 100 55 35 38 22 20 20 8 35 10 17

0.0138 0.0051 0.0041 0.0022 0.0015 0.0051 −0.0018 0.0041 0.0070 0.0151 0.0117 0.0104 0.0082 0.0037 0.0292

5.32 5.95 5.36 3.74 4.32 5.71 14.45 7.03 6.76 6.03 4.88 3.48 9.30 4.46 6.71

−0.0412 −0.0353 −0.0111 −0.0136 −0.0157 −0.0851 −0.0842 −0.0837 −0.0795 −0.0673 −0.0428 −0.0496 −0.0437 0.1115 0.0137

6.40 5.82 5.75 3.98 4.01 14.25 11.93 8.77 6.62 4.06 4.52 4.41 10.22 1.57 5.46

−0.0068 −0.0067 0.0053 −0.0012 −0.0099 −0.0224 −0.2564 0.0162 −0.1949 −0.0261 −0.0176 −0.0280 −0.0985 −0.0013 0.0431

2.41 2.19 2.13 1.51 3.58 16.68 17.21 10.83 19.02 4.23 1.64 4.08 8.92 19.55 3.46

−0.0041 −0.0039 −0.0022 −0.0026 −0.0094 −0.0994 −0.0660 −0.0845 −0.0041 −0.0065 −0.0107 −0.0086 −0.0035 −0.0032 0.0036

2.04 1.98 2.23 2.04 2.15 7.89 10.14 7.24 3.41 4.56 2.13 3.17 9.29 2.17 3.78

760

5.84

SAFT EOS

6.51

This work

5.59

Data sources

4.12

[63] [63] [63] [63] [63] [64] [65,66] [66] [67] [63] [63] [69] [68] [70] [71]

109

Grand average

GFD EOS

J.-C. Tsai, Y.-P. Chen / Fluid Phase Equilibria 232 (2005) 100–112

System solvent (1) + polymer (2)

J.-C. Tsai, Y.-P. Chen / Fluid Phase Equilibria 232 (2005) 100–112

18.42 Grand average

920

11.66

6.12

4.78

3.43

[63] [63] [63] [63] [63] [64] [65,66] [66] [68] [63] [63] [68] [69] [70] [71] 2.57 3.01 3.24 2.71 3.01 4.57 8.76 3.14 5.27 2.71 3.59 4.21 4.12 3.97 4.23 −0.0038 −0.0021 −0.0020 −0.0064 −0.0016 0.0093 −0.0216 0.0020 0.0015 −0.0042 0.0069 −0.0027 0.0098 0.0102 0.0036 3.39 3.53 3.24 3.75 3.56 5.11 10.49 8.48 5.42 7.19 4.54 3.19 4.01 5.18 6.23 −0.0210 −0.0175 −0.0154 −0.0103 −0.0098 −0.0309 −0.0205 −0.0265 0.0021 −0.0015 0.0247 −0.0015 0.0074 0.0033 −0.0041 6.91 5.56 4.10 4.52 2.87 5.28 17.47 2.45 5.98 9.50 5.16 8.07 4.07 4.39 10.24 6.32 7.43 7.47 8.24 18.53 6.69 21.19 20.47 7.52 39.78 47.16 23.84 11.24 10.32 19.69 383–473 383–473 383–473 383–473 383–473 393–458 293–333 293–393 308–319 383–473 383–473 318–328 303 353 297–317 29.99–1409.73 12.77–781.22 5.57–450.79 15.81–1001.70 7.30–587.28 0.65–95.39 0.48–13.45 3.03–52.01 32.53–193.05 53.48–410.05 29.67–281.27 57.17–177.72 3.97–15.49 2.85–30.92 8.50–19.21 Hexane + PE Heptane + PE Octane + PE Cyclohexane + PE Toluene + PE Ethylbenzene + PS Toluene + PS Benzene + PS Butane + PIB Hexane + PIB Cyclohexane + PIB Pentane + PIB Benzene + PVAC Hexane + PP Cyclohexane + PS

100 100 100 100 100 55 35 38 10 100 100 10 8 10 54

15.78 14.81 14.19 10.57 4.11 12.99 16.37 10.48 5.43 13.01 8.59 2.88 3.66 8.26 13.75

−0.0080 −0.0064 −0.0050 −0.0062 −0.0004 0.0170 −0.2509 0.0263 0.0346 −0.0256 −0.0153 0.0715 −0.0291 0.0117 0.0434

AAD a1 (%) Optimum k12 AAD a1 (%) Optimum k12 AAD a1 (%) AAD a1 (%) AAD a1 (%)

Optimum k12

UNIFAC-FV GC-Flory

No. of data points Temperature range (K) Pressure range (kPa) System solvent (1) + polymer (2)

Table 9 Comparison of the calculated activity deviation for polymer solutions from various thermodynamic models

SAFT EOS

GFD EOS

This work

Data sources

110

SAFT-VR type EOS shows satisfactory accuracy and less peak error than those from other methods. A typical presentation for the equilibrium pressure calculation of a polymer solution is shown in Fig. 6. Table 9 presents the calculated activity of solvents in polymer solutions from various approaches. This SAFT-VR type EOS again shows less peak deviations from the experimental data. Two other group contribution methods from the GCFlory [18] and the UNIFAC-FV [19] equations are compared with this EOS. With one binary parameter, this EOS significantly improves the calculation error from predictive models.

4. Conclusion A new SAFT-VR type EOS based on a modified coordination number model was employed for real fluid mixture calculations in this study. Pure compound parameters from non-polar low molecular weight components to polymers are presented. This new equation gives satisfactory results for VLE calculations on non-polar organic systems and polymer solutions. It is demonstrated that this theoretically based EOS not only agrees well with molecular simulation data for binary and ternary mixtures, but also shows practical feasibility for engineering phase equilibrium calculations. List of symbols AAD absolute average deviation c parameter defined in Eq. (12) f fugacity, or volume fraction, or function defined in Eq. (9) g radial distribution function k Boltzmann constant kij binary interaction parameter m segment number Nij coordination number of molecule i around central molecule j P pressure r separation distance T temperature T* dimensionless temperature, kT/ε V volume V0 , V00 Molar volume parameters of hard spheres w weight fraction x, y mole fraction Z compressibility factor Greek letters α parameter defined in Eq. (13) ε energy parameter η packing fraction λ well width of the square well potential ξk functions defined in Eq. (8) π mathematical constant ρ number density ρ* dimensionless density, ρσ 3

J.-C. Tsai, Y.-P. Chen / Fluid Phase Equilibria 232 (2005) 100–112

σ Γ

temperature-independent molecular segment diameter potential function

Subscripts cal calculated property eff effective reduced variable exp experimental data i, j components i, j k data point mix mixture MC Monte Carlo simulation data Superscripts corr correlated property hs hard sphere L, liq liquid phase V vapor phase

Acknowledgement The authors are grateful to the National Science Council, Republic of China, for supporting this research. Jung-Chin Tsai is now with The Department of Chemical Engineering, Minchi University of Technology, Taipei, Taiwan, ROC.

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