Application of a new crossover treatment to a generalized cubic equation of state

Fluid Phase Equilibria 302 (2011) 241–248

Contents lists available at ScienceDirect

Fluid Phase Equilibria journal homepage: www.elsevier.com/locate/fluid

Application of a new crossover treatment to a generalized cubic equation of state Moussa Dicko, Christophe Coquelet ∗ Mines ParisTech, Laboratoire CEP/TEP 35, rue Saint Honoré 77305 Fontainebleau cedex, France

a r t i c l e

i n f o

Article history: Received 12 May 2010 Received in revised form 26 October 2010 Accepted 30 October 2010 Available online 9 November 2010 Keywords: Supercritical fluid Critical point Modeling Crossover

a b s t r a c t In spite of recent progress in the field of equations of state, the use of cubic equations of state (EoS) such as Redlich–Kwong EoS remains common in the industry. It has already been proved that such classical EoS which follow a mean field approach fail to represent physical properties in the vicinity of the critical point. Moreover, it is well known that in this area the representation of thermodynamic properties must rely on scale invariance. In this work, a new Landau-crossover treatment for pure fluids with two parameters has been applied to a generalized form of cubic EoS. The model calculations have been confronted with experimental data. The crossover treatment allows a correct representation of PVT values in the critical region while keeping classical results at distance. Discussion on possible improvements is also provided. © 2010 Elsevier B.V. All rights reserved.

1. Introduction In a context of growing concern about environmental issues, supercritical processes are becoming more and more attractive. They are particularly used in the pharmaceutical and food industry where the innocuousness and the physical properties of fluids like supercritical CO2 are appealing. Besides, supercritical fluids are not only used as extraction solvents but also as chromatography eluents, reaction media, etc. Up to now, two main different approaches have been used to represent fluids properties in a wide range of PVT values. The first one is the Landau-crossover method. Successfully employed by Sengers and coworkers and also by Kiselev in a simplified way [1], this method consists in rewriting the free energy with the inclusion of a crossover function. This function allows the renormalization of temperature and volume variables as the region of concern gets closer to the critical point. The phenomenological method of Kiselev uses a simple Padé approximant of the crossover free-energy obtained from the numerical solution of the renormalization-group equations. It is supposed to deviate from theoretical solutions [2] but its relative simplicity is fully compatible with chemical engineering issues. However, many authors privileged the second approach: White’s recursive procedure [3,4]. The reason claimed is systematically that this model requires a reduced number of parameters [5–9]. In this article, it is therefore our aim to reduce the number of parameters of the Landaucrossover fitted on experimental data. We shall compare the results

∗ Corresponding author. Tel.: +33 164694962; fax: +33 164694968. E-mail address: [email protected] (C. Coquelet). 0378-3812/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2010.10.026

obtained with those of White’s recursive procedure in a next publication. 2. Theory In the classical Landau theory of critical phenomena [10], it is assumed that the critical part of the free energy, A¯ can be represented by a Taylor expansion in powers of the order parameter ϕ: A¯ =



aij  i ϕj

(1)

i=0 j=1

with  = (T/Tc ) − 1, ϕ = (V/Vc ) − 1 (if the same order parameter as Kiselev is chosen) and aij system dependent parameters. Therefore, in the critical region, A¯ ≈ a12 ϕ2 + a04 ϕ4

(2)

since ϕ  1 and   1. Chen et al. [11] constructed the following crossover expression for the thermodynamic potential of a system in the critical region: A¯ = a12 Y (2−

−1 )/ω

Y −/ω ϕ2 + a04 ϕ4 Y 1/ω Y −2/ω − K( 2 )

(3)

K( 2 ) is the so-called kernel term which provides the correct scaling behaviour of the isochoric specific heat asymptotically close to the critical point. It will not be used thereafter since derivative properties will not be calculated. The expression (3) is based on the renormalization-group calculations of Nicoll and Albright [12,13] and can be rewritten thanks to scaling laws in the form: A¯ = a12 ϕ2 Y (−2ˇ−˛)/2 1 + a04 ϕ4 Y (−2ˇ)/ 1 − K( 2 )

(4)

242

M. Dicko, C. Coquelet / Fluid Phase Equilibria 302 (2011) 241–248

Table 1 Values of parameters c and d as a function of the EoS. EoS

Parameter c

Parameter d

Van der Waals [22] Redlich–Kwong [19] Peng–Robinson [21]

0 0 √ b(1 − 2)

0 b √ b(1 + 2)

Harmens–Knapp [35]

1−c ∗ 2

b+

Patel–Teja [24]

b+c ∗ 2

+





 1−c∗ 2 b

2

c∗ b +

 − c ∗ b2

 b+c∗ 2  2

In all these equations , ω, , ˇ, ˛,  and 1 are critical exponents. The values from [14] have been used for the calculations: ˛ = 0.110, ˇ = 0.325,  = 1.239 and 1 = 0.51 Y is the crossover function and depends on the distance to the critical point. It is then easy to verify that, following the proposition of Kiselev, the renormalization of the variables  and ϕ into (Eq. (5)) allows the crossover from (4) to (2) to be asymptotically verified. ¯ = Y −(˛/2 1 ) and ϕ ¯ = ϕY (−2ˇ)/2 1

(5)

Starting from a Wegner expansion [15], Kiselev et al. represented the Helmholtz free energy in a parametric form (6) [16] which is physically equivalent to the six-term crossover model [2,17].



¯ A(r,

) = kr 2−˛ R˛ (q) a0 ( ) + r(1 − b2 2 )

= ϕ = kr ˇ R−ˇ+1/2 (q) + d1 

5 



ci r i R− i (q)i ( )

i=1

r G

Y (q) =

q2

We take here into account the so-called root-square corrections for the isochoric specific heat and because of the limits: q→∞

q→0

q→∞

q→0

Y (q) −→ 1 and Y (q)−→q2 1



c∗ b +

b

 − c ∗ b2

 b+c∗ 2  2

to a (mean field) Landau theory next to the critical point. Moreover, they require only the knowledge of critical properties (Tc and Pc ). Process optimization needs accurate thermodynamics models which can predict densities and phase equilibria in a large range of temperatures. The classical equations of state have been modified several times during the last 30 years particularly in order to improve the densities determination around the critical point. Peneloux et al. [23] enhanced the representation of liquid densities thanks to a volume translation but failed to represent the densities close to the critical point. Patel and Teja [24] introduced a third parameter (dependent on critical compressibility) and modified the attractive part of classical EoS. They improved the representation of the densities but not satisfactory enough in the critical region. In this work, the critical part A¯ has been extracted from a generalized equation of state following Zielke and Lempe [25] concept. Their generalization presents the advantage to provide a single form for the calculation of physical properties which is valid for every cubic EoS. Indeed, a classical cubic EoS can be written as: RT

a(T ) (v + c)(v + d)

(7)

(8)

R(q) −→ q2 and R(q)−→1

−

2

where P is the pressure, v is the molar volume, R is the gas constant and T is the temperature. As shown in Table 1, the two parameters c and d can be expressed from the parameters b and c* of the original form of the cubic EoS. a(T) depends on the choice of the alpha function but also on Tc and Pc values as it is explained in Ji and Lempe paper [26]. The determination of the critical parameters ˝a , ˝b and ˝c* which are linked to a, b and c* in Eq. (12) is done by solving Eq. (11) developed in volume at the critical point.

 1

R(q)

b+c ∗ 2

 1−c∗ 2

P=

It has been proved that g is proportional to the inverse of the Ginzburg number which evaluates the size of the fluctuation zone. The crossover function R(q) was introduced in order to suppress the singularities in the expansion (6) when q tends to infinity. It is possible to show (see [18]) that (6) can be written in the form of (4). Then, considering the link between Y and R(q):



b−

(6)

q is related to the parametric variable r by: q2 = rg =



1−c ∗ 2

(9) (10)

this crossover function allows to switch progressively from Chen et al. expression (4) close to the critical point to the Landau expression (2) as q increases. 3. Classical equation of state The Landau-crossover has been applied to a cubic equation of state. Classical equations of state, like RKS [19,20] and Peng–Robinson [21], are mainly used in chemical engineering to represent the behaviour of pure component and mixtures. They belong to Van der Waals equation’s [22] family which is equivalent

v−b

−

a(T ) = ˝a

RTc2 Pc2

(11)

˛(T )

RTc Pc RTc ∗ c = ˝c Pc b = ˝b

(12)

In this paper, we have selected the RKS EoS because it is one of the mostly used equations in the industry and this equation fails strongly in the representation of densities. In a near future, we will also compare our results with the ones obtained following White’s procedure applied to RKS EoS [5]. Finally, c = 0 and b = d. An important remark must be made for programming purpose. If RKS (or PR) EoS is selected, Zc is supposed to be constant for any fluid. However, our experience proved that if ˝a and ˝b are not expressed with enough digits, the critical compressibility is not the one expected. This implies deviations in the calculation of the critical shift described below. As a consequence, we recommend to use (Eq. (13)) instead of their evaluation as in the original publication [19]. ˝a =

1 9(21/3 − 1)

21/3 − 1 ˝b = 3

(13)

M. Dicko, C. Coquelet / Fluid Phase Equilibria 302 (2011) 241–248

243

Fig. 1. d1 as a function of the acentric factor ω. Solid line: linear regression for n-alkanes.

4. Crossover equation of state

This equation is obtained by the combination of Eqs. (7), (15) and (16) which convert  and ϕ into parametric variables [28]:



4.1. Calculation framework We followed the method developed by Kiselev [27] to transform the generalized cubic EoS Eq. (11) into a generalized cubic crossover EoS. The method is detailed in Appendix A. In the end,  and ϕ are renormalized into: ¯ = ϕY (−2ˇ)/2 1 + (ϕ0c − ϕ)Y (2−˛)/2 1 ¯ = Y −(˛/2 1 ) and ϕ where ϕ0c = 1 − (V/V0c ) and ϕ = (V/Vc ) − 1ϕ0c is introduced to take into account the shift between the real critical volume, Vc and the one determined by the classical EoS, V0c [27]. The pressure is calculated from the dimensionless renormalized critical part of the free ¯ energy A. The first form of this crossover method leads to very good results in the critical region. However, three problems remained unsolved: - The calculation issues in the metastable region. - The number of crossover parameters to be fitted on experimental data. - The return to the classical equation of state was not effective (Y(q) did not reach 1) The first point has been solved by Fisher’s sine model [28]. The second one still remains and the third one has been mentioned by few authors. We found three of them in the literature [29–31] who proposed different solutions.

4.2. Reduction of the number of parameters The normalized distance from the critical point q is calculated by solving the equation:



 q − G 2



p2 1− 4b2



 1− 2 q G



=b

2

ϕ m0 Gˇ

2 Y (1−2ˇ)/2 1 (14)



 = rk( ) = r 1 − 2b2

1 − cos(p ) p2

ϕ = r ˇ m( ) = m0 r ˇ R−ˇ+1/2 (q)



(15)

sin(p ) p

(16)

We proceeded as Lee et al. in [30] and retrieved the original publication of Fisher. We saw that the sine model is indeed constructed with M = ( / c ) − 1. That is why Kiselev and Friend had to replace ϕ by empirical expressions such as ϕ = ϕ(1 + v1 ϕ2 exp(−ı1 ϕ)) in [32] to include asymmetry. The value of ı1 is completely empirical and changes with the publications. Since our goal is to reduce the number of parameters, we also used M instead of ϕ for the calculation of q. This allows to introduce asymmetry with the law of rectilinear diameter and simply replace M by M = ( / c ) − 1 + d1 . d1 has been calculated for 7 n-alkanes and CO2 in the temperature range 0.7Tc –Tc with Refprop 8.0. Our results are similar with those of Lee et al. [30]. Moreover, d1 seems to be linear with respect to the acentric factor (see Fig. 1) and its value is close to one for most of the fluids (see Table 2). In order to determine the diameter above Tc , we used the Ising value of the universal amplitude ratio A+ /A− = 0.523 0 0 which results in approximately half d1 above Tc (see [1], p. 398). Fisher et al. mentioned also that “m0 serves merely as a metrical factor and is irrelevant to the universal properties”. Then, we fixed m0 = 1. Fixing b2 = p2 = b2LM = 1.359, only one parameter is left: G. Table 2 Classical and crossover parameters for the equations of state. Component

Tc (K)

Pc (MPa)

ω

GN

d1

Carbon dioxide Methane Ethane Propane Butane Pentane Hexane Octane

304.1282 190.564 305.322 369.89 425.125 469.7 507.82 569.32

7.3773 4.5992 4.8722 4.2512 3.796 3.37 3.034 2.497

0.22394 0.01142 0.0995 0.1521 0.201 0.251 0.299 0.393

1.89 1.19 1.53 1.69 1.92 2.15 2.34 2.70

0.96 0.71 0.80 0.84 0.90 0.94 0.99 1.04

The source for Tc , Pc , ω is Refprop 8.0.

244

M. Dicko, C. Coquelet / Fluid Phase Equilibria 302 (2011) 241–248

Fig. 2. Plot of the functions R1 (q), R2 (q), R5 (q) and R(q) = q. Solid line: R1 (q). Dashed line: R2 (q). Dotted line: R5 (q).

The major interest of having this single parameter in the fitting procedure is that we can now look for direct correlations between G and the studied fluid. Our sine model is different from the one of Lee et al. [30] in two aspects. First, they decided to remove R−ˇ+1/2 (q) in (Eq. (15)). We keep it because this function is necessary in order to recover the classical exponent 1/2 far away from the critical point. Second, the crossover function is different. 4.3. Return to the classical EoS behaviour A very surprising effect occur when the crossover equation is programmed with the Y(q) function Y(q) = (q/(1 + q))2 1 . It should be expected that the isotherms get back to the ones calculated with the classical EoS. In fact, the convergence of the function Y(q) is

too slow to observe it. The same phenomenon occurred with the 2 parametric linear model and Y (q) = ((q(1 + q))/(1 + q + q2 )) 1 as noticed by Kudelkova et al. [29]. Therefore, she proposed to replace 2 Y(q) by Y (q) = ((q(1 + q))/(k + q + q2 )) 1 but we found that this modification is not satisfactory enough. Feyzi et al. [31] proposed the following correction: q = q exp(12.2 − (12.2T/Tc )). They claim that the value 12.2 has been obtained by trial and error. Lee et al. [30] chose the correction q = qe0.1ϕ . This correction is only effective for low densities and the value 0.1 was chosen because “it worked”. In the beginning of our investigations, we also looked for such type of correction. The transformation q = q exp(coefficient × (q − qc )p ) with qc standing for the normalized distance calculated at M = 0 led to good results (same

Fig. 3. G as a function of the acentric factor ω. Solid line: linear regression for n-alkanes.

M. Dicko, C. Coquelet / Fluid Phase Equilibria 302 (2011) 241–248

245

Fig. 4. Plot of three isotherms, pressure against density, for methane at T = 240 K; T = 195 K and T = 160 K. (–) Crossover RKS (CR-RKS); (. . .) classical RKS; (×××) calculated with Refprop 8.0.

Fig. 5. Plot of the coexistence curves for n-alkanes. (–) Crossover RKS (CR-RKS); (—) classical RKS; (×××) experimental data from [37] for C1, from [38,39] for C2, from [40–42] for C3, from [43] for C4, from [44–47] for C6, from [47–50] for C8.

Table 3 Calculated absolute average deviation AADa for VLE densities. Component [source]

AAD for L RKS (%)

AAD for L CR-RKS (%)

Carbon dioxide [36] Methane [36] Ethane [38,39] Propane [40–42] Butane [43] Hexane [44–47] Octane [47–50]

16.9 10.9 6.1 8.8 15.4 15.7 17.9

2.2 2.6 0.9 0.7 1.4 1.5 3.9

a

AAD =

1 Nexp

  exp − calc  exp

.

Range of temperature (K) 217–304.1 105–190.54 102.6–304 200–360 325–422.3 353–507 403–569.3

AAD for V RKS (%)

AAD for V CR-RKS (%)

Range of temperature (K)

2.7 3.1 2.9 0.7 1.2 – 3.6

2.4 2.1 2.1 2.4 2.9 – 3.6

217–303.8 105–190.54 248–305 230–363 325–422.3 – 463–563

246

M. Dicko, C. Coquelet / Fluid Phase Equilibria 302 (2011) 241–248

Fig. 6. Plot of five isotherms, pressure against density, for propane at T = 330 K; T = 350 K; T = 369.89 K; T = 410 K and T = 440 K. (–) Crossover RKS (CR-RKS); (. . .) classical RKS; (×××) calculated with Refprop 8.0.

shape of Y(q) in the critical region and increased convergence to one). However this method whether adds one parameter or requires to fix arbitrary limits for the unknown fluctuation zone. To avoid it, direct changes on the form of the function have been tried. Originally, the function R(q) = ((1 + q + q2 )/(1 + q)) was designed to converge to q for q values near 5 [33]. At that time, R(q) was the square root of R(q) defined in Eq. (7) because the isochoric heat correction was not added yet. So R(q) was supposed to tend to q when q tends to infinity. The family of function Rn (q) = (1 + q + q2 + . . . + qn )/(1 + q + . . .+ qn−1 ) has been studied. This form of crossover functions allows to keep the limits of Eqs. (9) and (10), and equivalently the crossover behaviour. By increasing n, the convergence to 1 is accelerated (see Fig. 2) while keeping the same behaviour in the critical region where q is very small. The obtained values of G are higher than with

usual R1 (q) or R2 (q) because of this enhanced convergence. If n is integer, the best form of Rn (q) is R5 (q) with the previously described sine model. R6 (q) would converge too fast leading to over evaluation of the pressure (because of the derivative of Y(q)). R1 (q) has also been plotted in Fig. 2 to show the slowness of the convergence in that case. Finally, our crossover function takes the form:

Y (q) =

q2 R5 (q)2

 1

=

q(1 + q + q2 + q3 + q4 ) 1 + q + q2 + q3 + q4 + q5

2 1 (17)

An additional correction has been added in order to recover the behaviour of ideal gases at low pressure. We calculated an exponential form to replace q using q (ϕ > 0) = q(ϕ > 0)e0.04ϕ so that the modification is strictly localized. The value 0.04 guarantees the return (Y(q) = 1) for V = 100 Vc .

Fig. 7. Plot of the coexistence curve of CO2 . (–) Crossover RKS (CR-RKS); (—) classical RKS; (×××) experimental data from [36].

M. Dicko, C. Coquelet / Fluid Phase Equilibria 302 (2011) 241–248

247

Fig. 8. Plot of three isotherms, pressure against density, for CO2 . T = 330 K; T = 304.13 K and T = 280 K. (–) Crossover RKS (CR-RKS); (. . .) classical RKS; (×××) calculated with Refprop 8.0.

5. Results

6. Discussion and conclusions

The results published in Lee et al. [34] have been reproduced using the former sine model and Patel–Teja EoS. The data are indeed well represented but no correlation exists between the crossover parameters. For instance, G is not monotonous with respect to the length of the carbon chain. This randomness of the parameters prevents the model from any predictive ability. In this work, a correlation between G and different n-alkanes has been searched. G has been fitted on Refprop 8.0 calculations for five isotherms in each case including the critical one. The data were selected in the range 0.9Tc and 1.2Tc for the temperatures and 0.5 and 1.5 c for the densities. The obtained parameters are in Table 2. It has been noticed that, as long as the same area of fluctuation (defined by the choice of the experimental data) is considered for fitting, G is linear with the acentric factor (see Fig. 3). With the chosen range of densities and temperatures, the results seem to be in good agreement with Refprop 8.0 curves for methane (Fig. 4). The coexistence curves (Fig. 5) are well rendered with a particular improvement for liquid densities. Table 3 presents the absolute average deviations obtained for the different components. Attention should be paid to the range of temperature in each case. Uncertainties on experimental data increase next to the critical point and some data are old (from 1900 and 1910 in the case of octane). In the future, we will modify our method of fitting because the calculated isotherms above Tc appeared to be more sensitive to G values than the coexistence curves. In fact, the isotherms for other hydrocarbons than methane confirmed that G has been over evaluated. The crossover effect is too strong so the CR-RKS curves are under the experimental ones at lower reduced temperatures than in the case of methane (see Fig. 6). For CO2 , the original RKS EoS gives poor results that are greatly improved by the crossover treatment for both the coexistence curve in Fig. 7 and the isotherms in Fig. 8. A huge deviation persists on the isotherms as the density increases but this is expected considering the classical results of RKS. An extensive study will be performed with different EoS and different fluids. In this work, our aim was simply to observe if a correlation appears when only one parameter is fitted.

A new crossover sine model including a different crossover function has been presented. This model is designed to improve the representation of fluid properties without adding many parameters. It greatly improves densities calculations especially in the critical region. The important question of the zone in which the data can be fitted has been raised. Many curves presented in the literature fit perfectly to experimental data even far away from the critical point. Previous models needed several parameters and no restriction was performed during the selection of experimental data. It was then possible to obtain crossover EoS following the experimental points instead of the classical EoS far away from the critical point (see Ref. [34] for instance). The choice of ϕ = (V/Vc ) − 1 as order parameter led to modifications of the model. It would be interesting to keep the same order parameter M = ( / c ) − 1 for both parts: the parametric model and classical EoS. The proposed corrections can surely be refined. Minor defects still persist far from the critical point (but maybe the original EoS should be used instead. . .). A correlation between G and the size of the component has been highlighted. For the moment, the correlation leads to overestimated values of G and should not be used if the acentric factor is above 0.3 (vapor pressure calculations inconsistencies might appear). In order to precise this relationship, we shall then test different families of fluids and reduce the area chosen to fit the data in a near future. Acknowledgments The authors would like to thank Dr. Pierre Carlès, Dr. Patrice Paricaud and Dr. Paolo Stringari for fruitful discussions. Appendix A. Calculation of the dimensionless free energy critical part from a generalized cubic EoS Using a Taylor expansion around the critical point leads to the following equation: ¯ r , Vr ) = A(Tr , Vr ) = A(, ) + A(T ¯ r , 1) − P¯ 0 (Tr )(Vr − 1) A(T RT

(A1)

248

M. Dicko, C. Coquelet / Fluid Phase Equilibria 302 (2011) 241–248

With Vr and Tr the reduced volume and temperature respectively. A cubic equation of state can be expressed with (10). This provides the expression of the residual free energy: Ares (T, v) = −RT ln(v − b) +

a ln d−c

v+c v+d

+ RT ln(v)

(A2)

In the dimensionless form and using the parameters ϕ = (v/vc ) − 1 and  = (T/Tc ) − 1, one obtains: A(T, v) A¯ res (, ϕ) = = ln RT +

A¯ res (, ϕ) = ln

 v (1 + ϕ)  c vc (1 + ϕ) − b

a() ln (d − c)RTc (1 + )

ϕ + 1 ϕ+B

+

 v (1 + ϕ) + c  c vc (1 + ϕ) + d

ϕ +C 

a() ln (d − c)RTc (1 + )

ϕ+D

(A3)

(A4)

with B = 1 − (b/vc ), C = 1 + (c/vc ) and D = 1 + (d/vc ). The dimensionless pressure is defined as follows: P¯ =

1 1 a() − ϕ+B RT v0c (ϕ + C)(ϕ + D)

(A5)

So, when ϕ = 0, the expression for the isochoric critical pressure used in Eq. (A1) takes the form: P¯ 0 =

1 a() 1 − B RT v0c CD

(A6)

Finally the critical term can be evaluated with: ¯ ¯ ¯ A(, ϕ) = A(, ϕ) − A(, 0) + P¯ 0 ()ϕ ¯ A(, ϕ) =

ϕ a()ϕ − ln − B CDRv0c Tc (1 + ) +

Finally,



P=−

∂A ∂V

 T

a() ln (d − c)RTc (1 + )

RT =− Vc



(A7)

ϕ B



+1

 (ϕ/C) + 1 

¯ , ¯ ϕ) ∂A( ¯ ∂ϕ

(ϕ/D) + 1

 + T

RT P¯ 0 (T ) V0c

(A8)

[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

[23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]

(A9)

References [1] J.V. Sengers, Equations of State for Fluids and Fluid Mixtures Part 1, Elsevier Science Ltd., Amsterdam, 2000, pp. 381–434.

[46] [47] [48] [49] [50]

M.A. Anisimov, S.B. Kiselev, J.V. Sengers, S. Tang, Physica A 188 (1992) 487–525. J. White, Fluid Phase Equilib. 75 (1992) 53–64. L.W. Salvino, J.A. White, J. Chem. Phys. 96 (1992) 4559–4568. J. Cai, J.M. Prausnitz, Fluid Phase Equilib. 219 (2004) 205–217. X.H. Xu, Y.Y. Duan, Fluid Phase Equilib. 290 (2010) 148–152. A. Bymaster, C. Emborsky, A. Dominik, W.G. Chapman, Ind. Eng. Chem. Res. 47 (2008) 6264–6274. J. Mi, C. Zhong, Y.G. Li, J. Chen, Chem. Phys. 305 (2004) 37–45. F. Llovell, L.F. Vega, D. Seiltgens, A. Mejia, H. Segura, Fluid Phase Equilib. 264 (2008) 201–210. L.D. Landau, E.M. Lifshitz, Statistical Physics, 3rd ed., Pergamon, New York, 1980. Z.Y. Chen, P.C. Albright, J.V. Sengers, Phys. Rev. A 41 (1990) 3161–3177. J.F. Nicoll, P.C. Albright, Phys. Rev. B 31 (1985) 4576–4589. J.F. Nicoll, P.C. Albright, Phys. Rev. B 34 (1986) 1991–1996. J.V. Sengers, J.M.H. Levelt, Sengers Ann. Rev. Phys. Chem. 37 (1986) 189–222. F. Wegner, Phys. Rev. B 5 (1972) 4529–4536. S.B. Kiselev, I. Kostyukova, A. Povodyrev, Int. J. Thermophys. 12 (1991) 877–895. S.B. Kiselev, J.V. Sengers, Int. J. Thermophys. 14 (1993) 1–32. S.B. Kiselev, J.C. Rainwater, J. Chem. Phys. 109 (1998) 643–657. O. Redlich, J.N.S. Kwong, Chem. Rev. 44 (1949) 233–244. G. Soave, Chem. Eng. Sci. 4 (1972) 1197–1203. D.Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. 15 (1976) 59–64. J.D. Van der Waals, Over de Continuiteit van den Gas-en Vloestoftoestand (On the Continuity of the Gaseous and Liquid States), 1873, Dissertation, Leiden University, Netherlands. A. Peneloux, E. Rauzy, R. Freze, Fluid Phase Equilib. 8 (1982) 7–23. N.C. Patel, A.S. Teja, Chem. Eng. Sci. 37 (1982) 463–473. F. Zielke, D.A. Lempe, Fluid Phase Equilib. 141 (1997) 63–85. W.R. Ji, D.A. Lempe, Fluid Phase Equilib. 147 (1998) 85–103. S.B. Kiselev, Fluid Phase Equilib. 147 (1998) 7–23. M.E. Fisher, S.Y. Zinn, P.J. Upton, Phys. Rev. B 59 (1999) 14533–14545. L. Kudelkova, J. Lovland, P. Vonka, Fluid Phase Equilib. 218 (2004) 103–112. Y. Lee, M.S. Shin, H. Kim, J. Chem. Phys. 129 (2008) 234503. F. Feyzi, M. Seydi, F. Alavi, Fluid Phase Equilib. 293 (2010) 251–260. S.B. Kiselev, D.G. Friend, Fluid Phase Equilib. 155 (1999) 33–55. S.B. Kiselev, High Temp. 28 (1990) 42–49. Y. Lee, M.S. Shin, J.K. Yeo, H. Kim, J. Chem. Thermodyn. 39 (2007) 1257–1263. A. Harmens, H. Knapp, Ind. Eng. Chem. Fundam. 19 (1980) 291–294. W. Duschek, R. Kleinrahm, W. Wagner, J. Chem. Thermodyn. 22 (1990) 841–864. R. Kleinrahm, W.J. Wagner, Chem. Thermodyn. 18 (1986) 739–760. J.E. Orrit, J.M. Laupretre, Adv. Cryog. Eng. 23 (1977) 573–579. D.R. Douslin, R.H. Harrison, J. Chem. Thermodyn. 5 (1973) 491–512. Y. Kayukawa, M. Hasumoto, Y. Kano, K. Watanabe, J. Chem. Eng. Data 50 (2005) 556–564. S. Glos, R. Kleinrahm, W. Wagner, J. Chem. Thermodyn. 36 (2004) 1037–1059. P. Sliwinski, Z. Phys. Chem. 63 (1969) 263–279. W.B. Kay, Ind. Eng. Chem. 32 (1940) 358–360. W.B. Nichols, H.H. Reamer, B.H. Sage, AIChE J. 3 (1957) 262–267. G.A. Mel’nikov, V.N. Vervenko, N.F. Otpuschennikov, Zh. Fiz. Khim. 62 (1988) 798–800. S.A. Beg, N.M. Tukur, D.K. Al-Harbi, E.Z. Hamad, J. Chem. Eng. Data 38 (1993) 461–464. S. Young, Sci. Proc. R. Dublin Soc. 12 (1910) 374–443. S. Young, J. Chem. Soc. 77 (1900) 1145–1151. J.H. McMicking, W.B. Kay, Proc. Am. Pet. Inst. 45 (1965) 75–90. J.F. Connolly, G.A. Kandalic, J. Chem. Eng. Data 7 (1962) 137–139.