A comparison of mixing rules for the combination of COSMO-RS and the Peng–Robinson equation of state

Fluid Phase Equilibria 275 (2009) 105–115

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A comparison of mixing rules for the combination of COSMO-RS and the Peng–Robinson equation of state Kai Leonhard a,b,∗ , Jan Veverka a , Klaus Lucas a a b

Lehrstuhl für Technische Thermodynamik, RWTH Aachen University, Schinkelstr. 8, 52062 Aachen, Germany Engineering Thermodynamics, Delft University of Technology, Leeghwaterstraat 44, 2628 CA Delft, The Netherlands

a r t i c l e

i n f o

Article history: Received 14 February 2008 Received in revised form 14 July 2008 Accepted 19 September 2008 Available online 8 October 2008 Keywords: Equation of state Peng–Robinson Mixing rule COSMO-RS Predictive thermodynamics

a b s t r a c t In this work, the COSMO-RS model is combined with a volume-translated Peng–Robinson equation of state (EOS) via a GE -based mixing rule. The performance of several mixing rules previously published for this purpose is compared and semi-empirical modifications to one of them are introduced to improve its performance in our application. The new mixing rule contains three internal parameters that are adjusted to achieve consistency between the mixing rule and COSMO-RS. No experimental binary data is needed for our EOS. The new COSMO-RS-based, predictive EOS introduces a density dependence into COSMO-RS and extends its applicability to higher pressures and to mixtures containing supercritical components. © 2008 Elsevier B.V. All rights reserved.

1. Introduction For the design of chemical separation processes knowledge of phase equilibria of the chemical compounds involved is essential. Presently, correlation equations based on accurate measurements are required to design the final apparatus installed in a plant. However, reliable estimates of phase equilibrium data are desirable in the early design phase, e.g. for choosing the best process, or for the screening of solvents and auxiliary compounds, etc. Today, group contribution methods like UNIFAC and modUNIFAC [1–3] are the standard for the computation of activity coefficients in the liquid phase and often an ideal vapor is assumed for vapor–liquid equilibrium (VLE) calculations. Such group contribution methods can be used to inter- and extrapolate experimental data to conditions and even compounds not previously measured and therefore are often called “predictive”. However, they are not generally predictive, because they cannot be used if a molecule of interest cannot

Abbreviations: AG, Ahlers–Gmehling; CEOS, cubic equation of state; COSMO, conductor-like screening model; COSMO-RS, COSMO for real solvation; COSMO-SAC, COSMOsegment activity coefficient model; EOS, equation of state; exp, experimental; HV, Huron–Vidal; LLE, liquid–liquid equilibrium; MHV1, Michelsen–Huron–Vidal; MR, mixing rule; PR, Peng–Robinson; SRK, Soave–Redlich–Kwong; UNIFAC, UNIQUAC functional group activity coefficient; UNIQUAC, universal quasi chemical; VLE, vapor–liquid equilibrium; VT, volume translated; VTPR, volume translated Peng–Robinson; WS, Wong–Sandler. ∗ Corresponding author. Tel.: +49 241 8098350; fax: +49 241 8092255. E-mail address: [email protected] (K. Leonhard). 0378-3812/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2008.09.016

be composed of the existing groups. Recently, other approaches have emerged that do not have this restriction, e.g. the conductorlike screening model for real solvents (COSMO-RS) developed by Klamt [4] and modifications of that model by other groups [5,6]. The weaknesses and strengths of group contribution methods and COSMO-RS-based models have been discussed, e.g., in [7] and [8]. An activity coefficient or excess Gibbs energy (GE ) model like COSMO-RS is, however, not sufficient for many applications. When a wide range of densities has to be considered in VLE calculations, or when supercritical compounds are present in a mixture, an equation of state (EOS) has to be employed. In the last decade, many studies have dealt with the combination of an EOS and a GE model by means of a suitable mixing rule (MR). Though an exact coupling of an EOS and a GE model is formally possible under certain conditions, it is of limited practical use. Therefore, a large number of MRs is described in the literature that differ in their assumptions used to simplify the exact relation. Many of them deal with the combination of a group contribution method with an EOS model, e.g. [9–14]. A first step towards a combination of COSMO-type models with an EOS has been performed by Constantinescu et al. [12]. They computed infinite dilution activity coefficients predicted with COSMO-RS, COSMO-SAC and UNIFAC at zero pressure, converted them to infinite pressure, and adjusted the parameters of a UNIQUAC model to them. Then they used a UNIQUAC+Huron–Vidal (HV)+Soave–Redlich–Kwong (SRK) EOS (GE +MR+EOS) model to calculate vapor–liquid equilibria (VLE) of water–alcohol (methanol, ethanol and 2-propanol), water–acetone, methanol–acetone, and

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K. Leonhard et al. / Fluid Phase Equilibria 275 (2009) 105–115

methanol–benzene mixtures. Since the number of classes of systems investigated in this study is quite limited, the authors note that a more extensive validation of their method is required. They combine UNIQUAC and COSMO-RS which is required here because the conversion from a zero pressure activity coefficient to one at infinite pressure is done at infinite dilution. The infinite pressure activity coefficients are in turn needed by the HV MR but the procedure is in principle unnecessary if a different MR is used. In this procedure, the inaccuracies of both models, COSMO-RS and UNIQUAC, may add up. A similar indirect approach was chosen by Shimoyama et al. who combined COSMO-RS with a SRK EOS model via a Michelson–Huron–Vidal (MHV1) MR modified for higher temperatures [15] and UNIQUAC as an intermediate GE model. Binary and ternary phase equilibria of systems containing water, hydrocarbons, and alcohols were computed and compared with experimental data. Lee and Lin [16] performed a direct combination of COSMOSAC and the Peng–Robinson EOS, using yet another MR, the Wong–Sandler (WS) MR. They studied VLE of n-alkane–n-alkane, n-alkane–alcohol, n-alkane–acetone, and water–alkane mixtures. In this work, a direct combination of COSMO-RS with the Peng–Robinson EOS model is performed. The effects of a variety of MRs are studied systematically to find out strengths and weaknesses of the models. This study is based on a set of systems covering a broader range of functional groups than earlier studies to gather experience on the range of applicability of the present approach. 2. Theory 2.1. The volume-translated Peng–Robinson equation of state Cubic EOSs (CEOSs) are widely used for calculations of phase equilibria in industrial applications. This is because of their simplicity and reliability, because of the global experience gained with such EOSs and because of the large data bases available for pure compound EOS parameters. Of the CEOS, the volume-translated (VT) Peng–Robinson (PR) EOS has the advantage of a better description of liquid densities compared to most other CEOSs for a wide range of molecules. The VTPR EOS reads p=

RT

v+c−b

−

a (v + c)(v + c + b) + b(v + c − b)

(1)

where p is the pressure, R is the gas constant, T is the temperature,

v is the molar volume, a is the attraction parameter as a function of temperature and composition, b is the co-volume as a function of composition and c is a constant for the volume translation. The VTPR model is based on a concept by Martin [17] and was first used by Péneloux et al. [18,19] in a group contribution EOS and later also by Ahlers and Gmehling [20]. Though the VT has no effect on the phase equilibrium calculations performed in this study, it is included here for later use since a more accurate description of liquid densities has been found for a VT CEOS compared to an untranslated one [17–20]. The VTPR EOS has been used with different functions for the temperature dependence of the attraction parameter a (˛-function) in the literature. In this work, a generalized ˛-function that was suggested by Twu et al.[21], which has only one compound-specific parameter, the acentric factor ω [11], with more recent general, i.e. substance-independent, parameters determined by Ahlers and Gmehling [20] is used. Tabulated values for ω can be used and no regression of pure compound vapor pressure is necessary in this version. Hence it is a four-parameter EOS.

2.2. The COSMO-RS GE model The COSMO-RS GE model [4] is based on surface charges that appear when a molecule is embedded in a cavity in a conductor. They are computed with the COSMO model [22] in combination with a quantum chemical DFT calculation. COSMO-RS is the bridge from single molecule properties (the screening charge densities) to macroscopic properties of pure systems and mixtures. It is based on assumptions on the interactions between the charged surfaces of molecules, the independence of such surface segments, and statistical thermodynamics. Note that no binary interaction parameters have to be determined by experiment or quantum-chemical calculations. Our investigations are built on the COSMOtherm sub-routine, version C1.2 [23], which has been implemented together with the GE -based MRs and the VTPR EOS in the ThermoC package for EOS calculations [24] during this study. The package is used for all computations using EOSs. COSMOtherm is used with its (older) element specific dispersion interactions instead of its element-pair specific interactions [23] since the former turned out to yield more accurate results for most small molecules. For mixtures, this procedure is equivalent to using the keyword “NOVDW” in the COSMOtherm input file.

2.3. The mixing rules studied 2.3.1. The Huron–Vidal mixing rule Huron and Vidal based their mixing rule on GE at infinite pressure [25]. This requires the assumption of a zero excess volume E → ∞, which in turn requires (vE = 0) at p → ∞ to prevent G∞ bm = x1 b11 + x2 b22 . Huron and Vidal find for the Redlich–Kwong equation of state

am = bm

 a

GE x − ∞ bii i ln 2



ii

i

(2)

where x stands for the mole fraction, the subscript m denotes the parameters of the mixture and ii those of pure component i. The assumption of vE = 0 in combination with the proposed MR for parameter a leads to a violation of the requirement of quadratic composition dependence of the second virial coefficient [26] and prevents the use of any other more sophisticated mixing rule for bm . Further, since the excess Gibbs energy at infinite pressure is required, existing predictive GE models cannot be used directly because they are optimized to yield GE at low pressure. This is why Constantinescu et al. [12] need to convert the infinite dilution activity coefficients to infinite pressure and need to use a second, intermediate activity coefficient model, namely UNIQUAC.

2.3.2. The Wong–Sandler mixing rule Wong and Sandler [26] start their derivation from the requirements for the composition dependence of the 2nd virial coefficient for the van der Waals equation of state, though the method can be applied to any EOS [26]. bm −



 am a = xi xj b − RT RT i

j

 ij

(3)

They suggest the following physical form of equations to compute bm and am and to fulfill Eq. (3) for an arbitrary function F(x)

K. Leonhard et al. / Fluid Phase Equilibria 275 (2009) 105–115

 i

bm =



xi xj b −

a RT

 ij

j

(4)

F(x) RT

1−

am = bm F(x)

(5)

and choose



b−

a RT



12

1 2

=



b−

a RT





11

+ b−

a RT



22

(1 − k12 )

(6)

By setting the excess Helmholtz energy of the van der Waals EOS equal to that of an AE (excess Helmholtz energy) model at infinite pressure, one arrives at AE∞ (x) =

am  aii + xi bm bii

(7)

i

which corresponds to F(x) =

 a ii xi

i

bii

(8)

Hence, the EOS reproduces the incorporated AE model exactly at infinite pressure and the EOS provides a switching mechanism to switch over to the behavior dictated by the 2nd virial coefficient at low pressure. It is assumed that AE at infinite pressure is approximately the same as GE at low pressure. Therefore, existing GE models can be used in conjunction with the WS MR. In their paper, Wong and Sandler give an example for demonstrating their assumption. Alternatively, the kij parameter can be used to adjust GE at a finite pressure to that of a GE model at the same pressure, while the mixing rule makes sure that AE of the EOS at infinite pressure approaches that of the GE / AE model used. In this case, kij should not be considered an additional parameter [27] and the MR essentially has the same number of adjustable parameters as the underlying GE model. This kind of parameter is called “internal parameter” in this work. Lee and Lin chose a simpler approach and set kij = 0 throughout their study [16]. Both Approaches will be evaluated in this work, i.e. (1) kij is fitted to GE (x1 = 0.5) computed with COSMO-RS at 0.85Tc and (2) kij = 0. Contrary to the good performance of the WS MR in many applications, several studies report problems that may occur when the WS MR is used. Ioannidis and Knox [28] point out that the WS MR contains a temperature-dependent size parameter, which may lead to inconsistencies for derived EOS properties. More importantly, Coutsikos et al. [29] find that the reproduction of the GE model at low pressure deteriorates as the asymmetry of the mixture increases. A perfect agreement can be obtained when kij is considered a function of composition. However, in this case, unphysical values of kij may result for very asymmetric mixtures. The authors show that, even at infinite pressure, AE of the AE model used is not recovered with the WS MR for very asymmetric mixtures where asymmetric can mean a large ratio b1 / b2 and also a large ratio 1∞ /2∞ . Hence, the advantage of fulfilling the 2nd virial coefficient constraint is paid for with another theoretical inconsistency. 2.3.3. Michelsen–Huron–Vidal-1 mixing rule In several MRs, the connection between the EOS and the GE model is made at zero pressure. In these cases, an estimate for the volume of the liquid phase at zero pressure, v0 , is necessary. With the assumption that v0 /b is a constant, the MHV1 MR is obtained [30] am = bm

  a ii i

C is a constant depending on the EOS used and the assumed inverse packing fraction r = v0 /b as long as the EOS is not volume translated. In the case it is there are several possible generalizations of the MR depending on the definition of the inverse packing fraction (e.g. v /b, v/b or v/b). Choosing r = v /b = (v + c)/b leads to results for C independent of c, i.e. identical to those of the corresponding EOS without VT. √ r+1− 2 1 (10) C = √ ln √ 2 2 r+1+ 2 2.3.4. Ahlers–Gmehling mixing rule (AG) Ahlers and Gmehling [11] designed a new MR for use in the Group Contribution Equation of State (GCEOS). Instead of the most often employed combination rule b12 = 12 (b11 + b22 ) taken from hard sphere theory, the following empirical combination rule is suggested for b 3/4

− AE∞ (x)

RT  bm GE (p = 0) xi + xi ln + C C bii bii i



(9)

107

b12 =

1 3/4 3/4 + b22 ) (b 2 11

(11)

This combination rule performs better on mixtures which are strongly asymmetric in terms of co-volumes than the original arithmetic-mean rule. For a, the following MR is used

 a GE am = xi ii + att ; C bm bii

C = −0.53087

(12)

i

E E is the attractive excess Gibbs energy. Gatt is comwhere Gatt puted without the combinatorial (hard core) contribution usually present in GE models, since that combinatorial contribution plus the ln bm /bii terms should yield zero, but in practice it is often does not because of accumulations of errors, especially for systems asymmetric in size. A similar approach has been proposed by Soave et al. [31] for use of a zero-pressure GE model with the infinite pressure HV MR.

2.3.5. A new mixing rule Kalospiros et al. [32] have found that problems with the MHV1like MRs arise when the mixture is asymmetric in terms of critical temperatures, not necessarily co-volumes. When the critical temperatures differ, the reduced temperatures differ as well and therefore it is impossible to find a reduced volume r that is appropriate for both pure components and the mixture. In such a case, the studied MRs do approximately conserve GE because of cancellation of errors, but deviations between ln  of the EOS and the GE model are generally larger. In an interesting review on GE /EOS MRs, Heidemann [33] cites an oral presentation by Boutrouille et al. [34] with the 1st idea to treat C in the Huron–Vidal MR as an adjustable parameter. Tochigi [35] suggested a mixing rule of r linear in x to overcome the problems described in the previous paragraph. In our new MR, the two previously discussed improvements are combined: (I) the combinatorial contribution in the GE model and in the MR are omitted as done by [31] as well as [11] and (II) the reduced molar volume, or equivalently the variable C, is treated as an internally adjustable parameter, as suggested by [36,33,37,35]. The question may arise why C is adjusted in our EOS model, though Ahlers and Gmehling obtain accurate results with a constant value of C. According to Orbey and Sandler [38], zero-pressure MRs have a similar performance for most systems independent of the value of C when the parameters of the underlying GE model are adjusted to experimental data. This may be the reason why Ahlers and Gmehling use the modUNIFAC structure for their GCEOS model, but re-fitted the parameters (apparently as a function of their choice for C) [11]. In an other study [14], accurate results were

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K. Leonhard et al. / Fluid Phase Equilibria 275 (2009) 105–115

Table 1 Temperatures used to fit parameter C to the COSMO-RS+ideal gas model and results of the fits. System

Tc,1 /K

Tc,2 /K

Tfit /K

Methane–n-hexane n-Hexane–n-octane CO2 –ethane CO2 –propane CO2 –n-butane CO2 –iso-butane CO2 –neo-pentane CO2 –n-octane CO2 –R22 CO2 –methanol CO2 –ethanol CO2 –acetone CO2 –CS2 CO2 –quinoline CO2 –n-eicosane R23 – CO2 R23–CS2 R22–CS2 R22–ethanol Ethanol–H2 O H2 O–1-butanol Methanol – benzene Methanol–acetone Acetone–n-hexane Ethane–methanol Propane–methanol Propane–ethanol R134a–propane R22–DME Methane–quinoline H2 S–methanol

190.6 507.6 304.13 304.13 304.13 304.13 304.13 304.13 304.13 304.13 304.13 304.13 304.13 304.13 304.13 299 299 369.3 369.3 513.9 647.1 512.6 512.6 508.1 305.3 369.8 369.8 374.3 369.3 190.6 373.4

507.6 568.7 305.3 369.8 425 407.7 433.8 568.7 369.3 512.6 513.9 508.1 552 782 767 304.13 552 552 513.9 647.1 563.1 562.1 508.1 507.6 512.6 512.6 513.9 369.8 400.1 782 512.6

162 432 258 258 258 258 258 258 258 258 258 258 258 258 258 254 254 314 314 437 478 436 432 432 260 314 314 314 314 162 317

C (1) −89.753 −19.993 −0.471 −0.317 −0.298 −0.285 −0.270 −0.352 −343.31 −0.755 −0.790 −1.137 −0.636 −0.016 −0.438 −0.791 −0.874 −0.473 −0.2754 −0.411 −0.412 −0.551 −0.414 −0.492 −0.838 −0.600 −0.594 −0.577 −0.508 –b −0.476

(3)

(3)

(3)

C12

C11

C22

GE a /(J/mol)

kij

82.367 73.755 0.5205 0.3306 0.3652 0.3381 0.3345 1.4621 0.00002 0.9903 1.0615 0.2841 0.8343 0.0222 1.5559 0.5147 1.0106 0.6126 0.4658 0.4053 0.3892 0.3873 1.0312 1.2227 0.8319 0.6072 1.2819 0.5462 0.5211 –b 0.635

−66.945 −187.41 −0.4516 −0.2936 −0.2456 −0.2391 −0.2226 −0.2909 −37.4175 −0.6596 −0.698 −2.2049 −0.4856 −0.0107 −0.3132 −0.952 −0.7042 −0.2554 −0.2131 −0.4078 −0.4272 −0.6358 −0.4146 −0.4954 −0.8635 −0.596 −0.5474 −0.5265 −0.5114 –b −0.4

−66.713 −112.77 −0.3697 −0.3435 −0.3197 −0.3085 −0.2949 −0.3379 −0.1961 −0.4868 −0.533 −1.3043 −0.5246 −0.0218 −0.3280 −1.2501 −0.6145 −0.461 −0.3147 −0.4289 −0.3656 −0.5029 −0.3997 −0.3572 −0.7362 −0.6138 −0.5507 −0.5977 −0.5075 –b −0.4228

−44 −9 319 335 336 321 295 302 −116 775 563 −191 334 −139 172 −133 332 234 −242 878 1292 1001 179 800 1218 1490 1086 368 −1253 264 413

0.350 0.0326 0.207 0.327 0.436 0.437 0.529 −0.490 −0.080 0.036 0.140 0.223 0.0923 0.368 0.976 −0.038 −0.134 −0.092 −0.254 0.222 0.5677 0.332 0.1045 −0.130 −0.110 0.1025 0.053 0.156 −0.542 −0.170 0.0814

a

GE (x = 0.5, T = Tfit ) computed with COSMO-RS to determine kij for the WS MR.

b

The value of C suggested by Ahlers and Gmehling has been taken because a fit is not possible due to the low vapor pressure of component 2.

presented for a combination of the PR EOS with a UNIFAC version with unchanged parameters without an internal adjustable parameter. However, all systems presented are relatively symmetric in terms of critical temperatures, the most asymmetric system is ethanol–n-butane with a ratio of critical temperatures of 1.21 while e.g. for CO2 –methanol and ethane–methanol, where our approach provides a real improvement, the ratio is about 1.68 (the effect is E is almost zero for this syssmall for methane–hexane because Gres tem). For the system CO2 –n-eicosane (see below) the ratio is even 2.52. Since it is intended to study asymmetric systems but not to change any parameters of the COSMO-RS GE model in this study, C has to be adjusted to the results of our GE model. In principal, the packing fraction is temperature dependent and so should be C. However, to avoid an excessive parameterization and unreliable extrapolations to supercritical temperatures, we decided to use a temperature-independent constant C. Preliminary studies for some systems showed that within a certain temperature range, from about 70 to 85% of the critical temperature of the component with the lower critical temperature, the chosen temperature has little influence on the result. While a low temperature is desirable in principle because of lower pressures, it turned out that for very asymmetric systems, the volatility of the less volatile compound may be so low that no reasonable calculation is possible. If we increase the temperature to more than 90% of one of the critical temperatures, real gas effects become too important to use our approach. Therefore, to determine C, we use the following empirical but universal procedure. A phase diagram is predicted with the COSMO-RS+ideal gas model at 85% of the critical temperature of the component with the lower critical temperature, where the ideal gas law is a reasonable model for the vapor phase for nonassociating molecules. The pure compound vapor pressures are

taken from the EOS. C is determined by a least square fit of the phase diagram computed with the EOS as a function of C to the phase diagram predicted with the COSMO-RS+ideal gas model. Then C is kept constant for other computations at different temperatures. No experimental data are used to determine C. For results of the internal parameter C see column C (1) in Table 1. This procedure leads to a considerable improvement of the predicted phase diagrams, especially qualitatively wrong predictions observed with the original AG MR for systems strongly asymmetric in terms of critical temperatures, e.g. CO2 –methanol, CO2 –ethanol, and propane–methanol are reduced to a large extent. Note however, that sometimes the values obtained for C are not physical, an extreme example being the system CO2 –R22 (the upper limit for C is reached at v/b = 1 which gives C = −0.623 for the VTPR EOS with our definition of the reduced volume). Exceeding this limit is a violation of the assumptions made in the derivation of the MR, but does not cause any practical problems [36]. Also note that, especially for systems that are only moderately non-ideal, the excess Gibbs energy computed from the EOS at 0.85Tc,min is not in very good agreement with that computed directly from COSMO-RS. The disagreement between GE model and EOS is, however, usually much larger for the MHV1 and AG MRs (see Section 3). The working equations for our model are 3/4

3/4

b12 = bm =

b1



3/4

+ b2 2

xi xj bij

(13) (14)

ij

 a GE am = xi ii + att C bm bii

(15)

cm = x1 c1 + x2 c2

(16)

i

K. Leonhard et al. / Fluid Phase Equilibria 275 (2009) 105–115 Table 2 A selection of representative systems for comparison of several MRs. System

Class

b2 / b1

Ethanol–propane n-Octane–carbon dioxide n-Eicosane–carbon dioxide n-Butanol–water

Unpolar + self–associating Unpolar Unpolar Self- + cross-associating

1.04 0.18 0.06 0.23

109

changed to a linear (two parameter) and quadratic (three parameter) mole fraction dependent approach. In total, four different assumptions for C were investigated: C = C (0) = C Ahlers–Gmehling

(17)

C = C (1)

(18)

C= C=

(2) x1 C1



(2) + x2 C2 (3)

xi xj Cij ,

(19) (3)

C12 =

ij

 1 2

(3)

(3)

C11 + C22



(3)

∗ C12

(20)

In all cases where C was not the constant suggested by Ahlers and Gmehling C was fitted to data of the vapor pressure generated with COSMO-RS at 85% of the lower one of the two pure compound critical temperatures as described above and a possible temperature dependence was neglected. Because of the small differences between the results for eqns. (19) and (20), only results for the constant (one internal parameter) and the quadratic (three internal parameters) model for C are shown in Table 1 and the results section. 3. Results and discussion 3.1. Comparison of several mixing rules for characteristic systems Fig. 1. Comparison of the excess Gibbs energy at 314 K and 2 MPa of the system ethanol(1)–propane(2) computed with the COSMO-RS model (open circles) and with the VTPR EOS using the COSMO-RS model via different MRs. The dotted line is computed with the AG MR, the dash-dotted lines with the 1P MR, the solid line with the 3P MR, the dashed line with the MHV1 MR (virtually identical to the AG MR on the scale of the diagram), the dash-dot-dotted line with the WS MR, and the thin solid line with dots with the 1P MR adjusted to GE instead of VLE data. The values for the internal parameters of the 1P, the 3P and the WS MR are given in Table 1.

E is the attractive (or residual) excess Gibbs energy. GE where Gatt att is computed without the combinatorial (hard core) contribution E usually present in G models. To test the importance of the mole fraction dependence of C, the constant (one parameter) approximation to parameter C was

For a detailed study of the most interesting MRs four systems consisting of unpolar+unpolar, unpolar+self-associating, and self-+ cross-associating components were chosen. One of these mixtures is symmetric in terms of b1 / b2 and three are not, see Table. 2. 3.1.1. Ethanol–propane Fig. 1 shows the excess Gibbs energy for the system ethanol(1)–propane(2) at 314 K and 2 MPa computed with the original COSMO-RS model and with the VTPR EOS using several MRs plus COSMO-RS with or without its combinatorial contribution as indicated for each MR in Section 2. Critical data and the VTPR parameters used in this work are listed in Table 3. For the system ethanol–propane, the WS MR with kij adjusted to the excess

Table 3 VTPR parameters used in this work and critical data computed from these parameters. a(Tc ) and b have been determined by Tc and pc , ω is taken from [39] and c is fitted to liquid densities over a range of temperatures when such data are available. Otherwise, c = 0 is used. “Source of exp. data” indicates the source of (pseudo)experimental data for critical data and liquid volumes. Name

CO2 Methane Ethane Propane n-Butane iso-Butane neo-Pentane n-Hexane n-Octane n-Eicosane Methanol H2 O Ethanol Acetone CS2 Quinoline R22 (CHClF2 ) R23 (CF3 H) 1-Butanol Benzene R134a H2 S

Critical values

VTPR EOS parameters (cm3 /mol)

pc (MPa)

Tc (K)

vc

7.3733 4.5990 4.8720 4.2480 3.7960 3.6400 3.1990 3.0250 2.4900 1.1010 8.0970 22.0640 6.1480 4.7000 7.9000 4.5000 4.9860 4.8394 4.4230 4.8950 4.0590 8.9630

304.1079 190.5600 305.3200 369.8299 425.1200 407.8500 433.7500 507.600 568.7002 766.998 512.6398 647.1401 513.9199 508.1000 552.0000 782.0000 369.2800 299.0181 563.0501 562.0501 374.2601 373.4000

105.8333 109.6018 163.7268 225.8555 286.2383 286.3792 346.5514 427.9435 576.5744 1766.115 156.4200 73.0768 207.8306 276.3082 178.5889 444.1571 189.6283 157.9227 312.7482 294.3799 233.5407 106.4788

a(Tc )

(J cm3 /mol2 )

396465 249582 604807 1017731 1504909 1444484 1858994 2692348 4105636 16752670 1025921 599965 1357908 1736256 1219170 4295506 864515 584000 2265637 2039908 1090790 491709

b

(cm3 /mol)

26.6783 26.8017 40.5361 56.3134 72.4402 72.4759 87.7040 108.5402 147.7332 446.9624 40.9527 18.9718 54.0700 69.9271 45.1967 112.4058 47.9069 39.9665 82.3426 74.2705 59.6415 26.9473

c

ω

Source of exp. data

0.225 0.011 0.099 0.152 0.200 0.186 0.197 0.300 0.399 0.907 0.565 0.344 0.649 0.307 0.109 0.315 0.221 0.267 0.590 0.210 0.326 0.090

[40] [41] [42] [43] [39] [39] [39] [44] [39,45] [39] [39,46] [47] [46] [39] [48] [39] [49] [50] [39,46] [39,45] [39,51] [52]

(cm3 /mol)

−0.4173 −3.6982 −3.5534 −3.3401 0 0 0 0.9393 7.1746 0 5.3995 1.8879 5.8199 0 0 0 −0.3303 0 12.6180 −0.9093 2.1252 0

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Fig. 2. Computation of the residual chemical potentials in the liquid phase at 314 K in the system ethanol(1)–propane(2) computed with the COSMO-RS model and with the VTPR EOS using the COSMO-RS model via different MRs. The values for the internal parameters of the 1P, the 3P and the WS MR are given in Table 1. The legend is the same as in Fig. 1 except that the Lin version of the WS MR is added and shown by a dot-dash-dashed line and the thin lines at the bottom show the residual chemical potentials in the gas phase for the 1P MR. The data have been calculated at 0.8 MPa because at this pressure a (partly metastable) solution exists for the liquid and for the vapor phase for the whole composition range.

Gibbs energy of the equimolar mixture can reproduce the underlying GE model best, while a closer look at the activity coefficients (or residual chemical potentials, i,res = i − i,id ) shows that that of propane is overestimated slightly and that of ethanol is underestimated compared to the COSMO-RS results (Fig. 2). The MHV1 and the AG MR, which are virtually identical because the combination rule of b differs significantly only for asymmetric systems and because the coefficient C is almost identical in both MRs, perform poorest for this system. Our new MRs have an intermediate performance, but the value for C, adjusted to reproduce the COSMO-RS+ideal gas VLE best, yields a GE that is about 10% too high, which is caused by an accurate reproduction of the ethanol activity coefficients for most compositions but too large propane activity coefficients, especially at low propane concentrations. By using three internal parameters instead of one, this deviation can be reduced slightly. When the internal parameter C in our 1P (parameter) MR is adjusted to GE data, one obtains a fit of the excess Gibbs energy of a quality similar to that obtained by the WS MR, but more accurate activity coefficients. However, in that case the VLE computed by our MR deviates more from the COSMO-RS+ideal gas predictions than it does when the parameter is fitted to VLE data (Fig. 3). The VLE predicted by the WS MR deviates similarly from the COSMO-RS+ideal gas result. When the resulting VLE is compared to experimental data (Fig. 4), the WS MR performs best for this system. This is due to a cancellation of errors, because the COSMO-RS+ideal gas model overestimates the non-ideality at subcritical conditions, i.e. the activity coefficient of ethanol which is reduced by the WS MR compared to the COSMO-RS prediction leads to an increased solubility of ethanol in propane, which agrees with the experimental data better because COSMO-RS predicts the solubility too low. From the isotherm at 350 K one can see that the non-idealities in the vapor phase, that are already present at 314 K (see Fig. 3 and the thin lines at the bottom of Fig. 2), are modeled quite well by the VTPR EOS in combination with any of the MRs studied. 3.1.2. Long alkanes–CO2 The systems C8 H18 (1)–CO2 (2) and C20 H42 (1)–CO2 (2) are the most asymmetric mixtures in terms of co-volume ratios studied by us in this work. To demonstrate the effect of the composition

Fig. 3. Comparison of the VLE at 314 K of the system ethanol(1)–propane(2) computed with the COSMO-RS model and with the VTPR EOS using the COSMO-RS model via different MRs. The values for the internal parameters of the 1P, the 3P and the WS MR are given in Table 1. The legend is the same as in Fig. 1.

dependence of coefficient C, Fig. 5 presents results for four MRs that differ only in this dependence for the system C8 H18 (1)–CO2 (2). Here it can be seen clearly that the use of a system dependent parameter C yields great improvement compared to a system independent constant and smaller further improvements are obtained when additional parameters are added. In Fig. 6 the capability of the MRs to reproduce the COSMORS GE data is shown. When kij in the WS MR is adjusted to match GE (x = 0.5) the shape of GE is wrong. As before, our MRs need to exaggerate GE to reproduce the VLE, but not as much as the MHV1 MR does. Also note that our new 3P MR matches the skewness for the GE curve best. The AG MR reproduces GE best, but at the expense of a bad reproduction of the VLE, see Fig. 5. In Fig. 7 we show a comparison to experimental data. Here it can be seen that the WS MR performs poorly for this strongly asymmetric system as does the MHV1 MR. The AG MR is better and the new MRs are still better for this system. The comparison also shows a

Fig. 4. Comparison of the VLE at 350 K and 450 K of the system ethanol(1)–propane(2) (squares [78]) computed with the VTPR EOS using the COSMO-RS model via different MRs with experimental data. The values for the internal parameters of the 1P, the 3P and the WS MR are given in Table 1. The dash-dotted is computed with the 1P MR, the solid line with the 3P MR, the dashed line with the MHV1 MR, the dash-dot-dotted line with the WS MR, and the open circles with the COSMO-RS+ideal gas model.

K. Leonhard et al. / Fluid Phase Equilibria 275 (2009) 105–115

Fig. 5. Comparison of the ability of several MRs to reproduce COSMO-RS+ ideal gas law results at 255 K for the system C8 H18 (1)–CO2 (2) dependent on the number of internal parameters for the packing fraction-dependent coefficient C. Open circles are used for COSMO-RS+ideal gas and lines for COSMO-RS+VTPR calculations with differen choices of C. The dotted line is for the zero parameter (=AG) version, the dash-dotted one is for the 1 parameter MR, the dashed one for the 2 parameter MR and the solid line for the 3 parameter MR.

slightly better agreement between experimental results and the one-parameter model than with the three-parameter model. An examination of the deviations introduced by COSMO-RS and by the MRs indicates, however, that the better performance of the 1P MR can clearly be attributed to a cancellation of errors. A comparison between experimental data and predictions for the system C20 H42 (1)–CO2 (2) is presented in Fig. 8. This very asymmetric system shows very different results for the various MRs. From the mixtures with short alkanes, e.g. with ethane, one finds that COSMO-RS understimates the non-ideality of the attraction in alkane–CO2 systems. Nevertheless, the MHV1 and Lin’s version of the Wong–Sandler MR overestimate the two-phase region strongly. The WS MR shows a reasonable high-pressure behavior at low temperature (a two-phase region at x2 = 0.85) but fails to predict a one-phase region at 473.2 K above 40 MPa. The AG MR and our new 1P and 3P MRs show very similar results among each other. All three underestimate the non-ideality of the system which leads to a qualitatively wrong prediction at low temperature and to a

Fig. 6. Comparison of the excess Gibbs energy at 255 K and 3 MPa of the system C8 H18 (1)–CO2 (2) computed with the COSMO-RS model and with the VTPR EOS using the COSMO-RS model via different MRs. The values for the internal parameters of the 1P, the 3P and the WS MR are given in Table 1. The legend is the same as in Fig. 1, except that the line with dots is not shown.

111

Fig. 7. Comparison of the VLE at 298.2 K (squares [53]) and 383.15 K(circles [54]) of the system C8 H18 (1)–CO2 (2) computed with the VTPR EOS using the COSMO-RS model via different MRs with experimental data. The values for the internal parameters of the New 1P, the New 3P and WS MR are given in Table 1. The legend is the same as in Fig. 1, except that the line with dots is not shown.

correct one at high T. We attribute this behavior to the underlying COSMO-RS model because the AG MR togehter with the VTPR and UNIFAC gives accurate results for the system C15 H32 (1)–CO2 (2) [11]. 3.1.3. 1-Butanol–water The system 1-butanol(1)–water(2) is an example for a crossand self-associating system that is very asymmetric in size. The excess Gibbs energy of the COSMO-RS model can be reproduced approximately with the WS MR, but again the skewness cannot be matched exactly (Fig. 9). Accidentally, the MHV1 MR and the new MR yield similar curves for GE but the AG MR yields only about 50% of the GE of COSMO-RS which corresponds to an underestimation of the non-ideality in the VLE, see Fig. 10. Our new MRs predict a somewhat too large liquid–liquid equilibrium (LLE) gap, as does the WS MR. However, the VLE part of the phase diagram is predicted quite well with COSMO-RS and the new MRs.

Fig. 8. Comparison of the VLE at 323.2 K and 473.2 K(circles [63,64]) of the system C20 H42 (1)–CO2 (2) computed with the VTPR EOS using the COSMO-RS model via different MRs with experimental data. The values for the internal parameters of the New 1P, the New 3P and WS MR are given in Table 1. The legend is the same as in Fig. 1, except that the line with dots is not shown and Lin’s version of the Wong–Sandler MR is shown (dot-dash-dashed).

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Fig. 9. Comparison of the excess Gibbs energy at 478 K and 3 MPa of the system 1butanol(1)–water(2) computed with the COSMO-RS model and with the VTPR EOS using the COSMO-RS model via different MRs. The values for the internal parameters of the 1P, the 3P and the WS MR are given in Table 1. The legend is the same as in Fig. 1, except that the line with dots is not shown.

3.2. Quantitative comparison of several mixing rules For comparison of the MRs, calculations have been performed to determine |p|/pexp , the mean absolute deviation in pressure relative to the experimental pressure, where p = pcalculated − pexperimental , and |y| is the mean absolute deviation in vapor phase composition. Both, pressure and vapor phase composition, are computed for the experimentally given temperature and liquid phase composition. Comparisons between the results of COSMO-RS+AG MR and COSMO-RS+ideal gas at a low pressure indicate that the results for most systems deteriorate significantly because of the mixing rule. Therefore, we investigated the suitability of the MHV1 MR, the WS MR in two versions, and finally developed our own modifications discussed above. The performance of those mixing rules with the VTPR EOS and COSMO-RS is shown in Table 4. Among the various proposed ways to determine kij for the WS MR the method proposed by Huang and Sandler [83] has been used in this work, i.e.

Fig. 10. Comparison of the VLE and LLE at 363.15 K of the system 1butanol(1)–water(2) (circles [72]) computed with the VTPR EOS using the COSMO-RS model via different MRs with experimental data. The values for the internal parameters of the New 1P, the New 3P and WS MR are given in Table 1. The experimentally determined water content of the coexisting liquid phases at low pressure is 98%/64%. Because of the very low reduced temperature of this plot, we used a specific set of pure compound parameters for this system to get accurate pure compound vapor pressures. The legend is the same as in Fig. 1, except that the line with dots is not shown.

E E GEOS = GCOSMO-RS at x1 = x2 = 0.5. Using this MR, the agreement of E G between the GE model and the EOS is the best, by definition. With respect to the AG MR the WS MR yields improved results for some systems, e.g. CO2 –C2 H6 , methanol–benzene and propane–ethanol, but the deviations are greatly enlarged for systems which are asymmetric in terms of the co-volumes b1 and b2 of both compounds like CO2 –C8 H18 , CO2 –acetone, CO2 –quinoline and quinoline–CH4 . This finding is in accordance with previous findings in the literature (see section 2.3.2). The average error in the pressure increases from 19.1 to 23.6% and from 3.4 to 4.4% in the mole fraction. Therefore, the use of the WS MR for the combination of COSMO-RS and the VTPR EOS is not recommended, at least not for asymmetric systems. Additionally, the Lee and Lin approach (kij = 0) [16] performs poorest and cannot be recommended at all. With our new one-parameter MR, the agreement between predicted and experimental data is improved for most systems, with only a few exceptions. In the system water–1-butanol, there is a LLE that is predicted roughly correct by COSMO-RS. However, our MR predicts a slightly too large LLE and some of the experimental VLE points lying close to the LLE are inside of the predicted LLE. This leads to an extremely large error for these points. The use of three internal parameters in the MR allows for a more accurate reproduction of the LLE predicted with COSMO-RS at low pressure. As a consequence of this, the error in VLE pressure and gas phase composition is reduced considerably for this system. For propane–methanol, the AG and MHV1 MRs produce results closer to the experiment, but our MRs reproduce the low-pressure COSMO-RS results better, as for the already discussed system propane–ethanol. Therefore, the better agreement for the AG and MHV1 MR for these systems is attributed to cancellations of errors. Comparing our 3P MR to the 1P MR for asymmetric systems, the results are better for CO2 –neo-C5 H12 and methanol–benzene but a slight deterioration between calculated and experimental data compared to the one-parameter MR is observed for CO2 –C8 H18 . However, this is due to a cancellation of errors as noted above in section 3.1.2. To determine to what extent the remaining inaccuracies are due to COSMO-RS and to what due to the mixing rule, calculations with CSOMO-RS+ideal gas and with CSOMO-RS+VTPR have been performed for 18 systems of our study for which at least one isotherm was available at less than 85% of the critical temperatures of both compounds. If the system 1-butanol–water is excluded from the comparison for the reasons described in the previous paragraph, the mean absolute deviation of the COSMO-RS+ideal gas model is 10.8% and that of COSMO − RS + VTPR + 3P mixing rule is 12.0%. Hence, about 90% of the deviation is due to COSMO-RS and only an additional 10% is introduced by the mixing rule. Though the main focus of this work is on systems previously not studied with such a model, some systems are included in our study that have already been investigated with other COSMO-RS+EOS approaches. With the work of Constantinescu et al. [12], the systems ethanol–water, methanol–benzene, and methanol–acetone are in common. Constantinescu et al. report mean deviations in the vapor pressure of 1.8, 4.0, and 2.3%, respectively, for their combination of COSMO-RS with the SRK EOS which are of similar quality as our results (1.8, 5.6, and 1.1%) obtained with our approach with the 3P MR. Shimoyama et al. combined COSMO-RS with a SRK model via a modified MHV1 MR and obtained a mean deviation averaged over pressure and composition of 3.2% [15] for the only system that can be compared with our results, water–ethanol. Their error is slightly larger than ours (1.6%). Compared to these two studies, our results are of similar accuracy but the methodological detour of using an additional intermediate GE model is avoided in the present work. The systems methane–n-hexane, n-hexane–noctane, and acetone–n-hexane were investigated by Lee and Lin

Table 4 Comparison of VLE predictions using COSMO-RS with and without dispersion interactions and with five different MRs. System

COSMO-RS AG

COSMO-RS MHV1

COSMO-RS WS

COSMO-RS WS, kij = 0

COSMO-RS 1P MR, no disp

COSMO-RS 3P MR, no disp

|p|/pexp in%

|y| in%

|p|/pexp in%

|y| in%

|p|/pexp in%

|y| in%

|p|/pexp in%

|p|/pexp in%

|y|

|p|/pexp

|y|

[55,56] [57] [58] [59] [60] [60] [61] [54,62,53] [63,64] [65] [66] [66] [66] [67] [68] [69] [69] [69] [70] [71] [72] [73,74] [75] [57] [76] [77] [78] [79] [80] [81] [82]

8.15 1.00 11.19 10.93 17.44 21.00 13.40 17.97 26.00 6.36 48.61 18.14 4.71 31.66 30.95 16.95 64.95 25.86 15.88 6.21 15.16 7.12 2.39 10.26 84.17 9.84 20.98 19.53 9.49 16.17 5.20 19.28

3.69 0.87 3.37 2.67 2.30 4.99 1.72 0.44 0.37 2.36 1.11 0.53 0.67 3.51 0.55 7.51 11.89 8.14 1.70 3.02 7.08 3.48 1.48 3.08 4.81 1.24 4.35 7.82 3.08 1.62 2.89 3.30

23.72 1.67 9.95 7.16 7.24 11.93 3.67 22.65 461.83 3.28 61.74 27.16 6.33 29.68 12.27 16.27 65.01 26.24 15.79 1.59 8.81 6.23 1.41 9.18 87.04 9.76 21.25 19.48 9.65 16.88 3.48 32.53

3.47 1.08 2.96 1.83 0.85 2.81 1.19 0.92 0.13 1.16 1.18 0.71 0.66 3.22 0.52 7.47 11.82 8.13 1.69 1.35 10.76 3.26 1.09 2.63 4.81 1.24 4.37 7.80 3.11 1.57 2.64 3.11

42.52 0.50 7.96 6.24 12.02 14.71 6.87 80.20 31.46 12.86 17.15 26.36 15.33 43.03 57.01 16.87 69.75 32.30 30.15 1.68 9.46 4.75 1.09 23.04 42.24 17.39 4.70 18.20 16.01 64.70 12.63 23.84

4.25 0.72 2.11 2.08 2.29 4.57 1.63 1.63 0.16 3.17 0.42 0.57 0.59 5.50 1.97 7.49 15.96 10.68 3.41 0.73 11.15 3.11 0.92 8.88 2.31 1.92 2.50 7.07 3.78 16.75 3.92 4.47

57.26 0.93 14.38 20.02 37.25 37.32 86.78 63.75 67.70 9.41 18.38 36.42 28.73 45.32 67.77 15.67 66.74 26.94 16.82 6.86 19.97 16.26 3.54 18.14 32.96 23.19 3.48 23.13 3.24 56.51 16.52 30.38

8.18 1.00 9.87 4.03 3.47 5.20 4.89 9.46 23.33 2.94 13.59 16.37 3.00 33.47 35.21 16.20 66.81 25.57 17.55 2.10 21.43 8.19 1.08 7.87 38.42 15.36 14.31 20.40 10.47 16.17 4.04 14.84

8.18 0.87 2.93 1.17 0.51 1.50 1.31 0.43 0.43 0.99 0.33 0.57 0.66 3.82 0.72 7.48 13.48 7.81 2.18 1.69 16.71 3.85 1.01 2.09 1.25 1.59 3.50 8.21 3.30 1.62 2.75 3.18

8.18 1.00 9.41 4.90 4.03 6.92 2.98 10.10 20.00 2.94 16.04 15.58 2.91 33.77 38.09 16.23 66.22 25.56 16.69 1.75 14.25 5.60 1.09 6.54 34.18 15.98 12.62 20.06 10.22 16.17 4.42 14.34

8.18 0.87 3.00 1.68 0.67 2.08 0.93 0.44 0.55 1.23 0.38 0.56 0.68 3.66 0.65 7.66 12.74 8.16 1.89 1.45 13.30 3.48 1.03 2.99 1.34 1.60 3.33 7.90 3.21 1.62 2.74 3.08

|y| in% 10.31 0.39 4,72 4.03 6.42 10.13 6.23 0.78 0.59 2.19 0.45 0.65 0.91 6.14 3.95 7.75 13.66 8.51 1.77 3.09 12.94 7.61 1.92 6.59 1.45 2.34 2.46 20.40 1.73 14.91 4.67 5.18

K. Leonhard et al. / Fluid Phase Equilibria 275 (2009) 105–115

Methane–n-hexane n-Hexane–n-octane CO2 –ethane CO2 –propane CO2 –n-butane CO2 –iso-butane CO2 –neo-pentane CO2 –n-octane CO2 –n-eicosane CO2 –R22 CO2 –methanol CO2 –ethanol CO2 –acetone CO2 –CS2 CO2 –quinoline R23–CO2 R23–CS2 R22–CS2 R22–ethanol Ethanol–H2 O H2 O–1-butanol Methanol–benzene Methanol–acetone Acetone–n-Hexane Ethane–methanol Propane–methanol Propane–ethanol R134a–propane R22–DME Quinoline–methane H2 S–methanol Average

Exp. source in%

113

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K. Leonhard et al. / Fluid Phase Equilibria 275 (2009) 105–115

[16] and in this work. Lee and Lin obtained averaged pressure deviations of 23.7, 1.7, and 13.8%, respectively, for these systems. Our 3P MR with COSMO-RS yields 8.2, 1.0, and 6.5%, respectively, for the same systems and same temperatures, but we have not been able to reproduce results similar to their COSMO-SAC+WS+PR approach (kij = 0) with our COSMO-RS+WS+VTPR EOS and obtained much larger deviations for two of the systems instead (57.3, 0.9, 18.1%). Hence the approach is similar in principle, but a combination of a COSMO based GE model and a MR that yields more accurate predictions for the investigated systems has been used in the present study. As a result of this comparison we summarize that the new MRs are suited better for the combination of COSMO-RS with a VTPR EOS than the MRs that were used for comparison. For systems not too asymmetric in terms of co-volume (0.5 < b1 / b2 < 2), the results obtained with the one-parameter and the three-parameter MR proposed in this study do not differ much. Since the parameter regression of both MRs is neither difficult nor time-consuming, the use of the three-parameter version is suggested because this version performs better for more asymmetric mixtures. In addition to some of those systems studied previously [12,15,16], typical gas molecules, such as CO2 , CS2 , and some refrigerants have been included in our comparison. Therefore, we can discuss, on the basis of our and previous work, for which classes of molecules the combination of COSMO-RS with an EOS can be recommended and for which it cannot. With a defintion of a mean absolute deviation in the vapor pressure of up to 10% as “good” for a prediction, 10–20% as “acceptable” and more than 20% as “poor”, the model works well for mixtures of hydrocarbons and of CO2 with hydrocarbons, with the exception CO2 –quinoline, for which the non-ideality is poorly predicted, but the reason for that deviation is not clear to us. It further yields generally good results for mixtures of CO2 with polar, non-hydrogen-bonding compounds, while the accuracy for systems with hydrogen-bonding is reduced to “acceptable”. When both molecules can participate in hydrogen bonds, the predictions are generally good (water–alcohol, H2 S–alcohol, alcohol-ketone, and ketone–water). The accuracy of the predictions for alkane–alcohol systems varies. While poor to acceptable results were obtained for mixtures of small alcohols (methanol and ethanol) with short alkanes (ethane and propane) other authors obtained acceptable to good agreement with experimental data for longer-chain alkanes (n-pentane to n-heptane) and/or alcohols (ethanol to butanol). Hence a clear trend of better predictions with longer chains can be seen. Shimoyama et al. [15] found good to poor results for mixtures of water with alkanes, without any identifiable trends. Especially poor are the predictions for systems consisting of CS2 mixed with CO2 , R22, or R23. In these systems, the nonideality which is due to dispersion non-idealities to a large extend [84], is poorly computed, and therefore the deviation from Raoult’s law, too. Interestingly, when the COSMO-RS element pair-specific dispersion model is turned on the deviations remain the same or increase even more. In the systems CS2 –R23 the pressure of the binary critical point is very sensitive to the temperature which is not predicted by the present model and leads to the poorest predictions of all systems in this study with most MRs. 4. Conclusions Finally we conclude this comparison with our finding that the combination of COSMO-RS with the VTPR EOS and the new MRs extends the range of applicability of the COSMO-RS model to higher pressures and temperatures by incorporation of non-idealities into the gas phase for systems for which it performs well at low pressure. For such systems it has been tested successfully up to both critical points. However, one should be aware that the present approach

does not extend the applicability of COSMO-RS to new systems for which the GE model alone performs not satisfactorily for the liquid phase. List of symbols A Helmholtz energy a attraction parameter b co-volume c constant for the volume translation C constant for the mixing rule c index ‘critical’ E superscript ‘excess’ G Gibbs energy  activity coefficient index ‘component i’ i index ‘components i and j’ ij adjustable parameter in a mixing rule Kij index ‘mixture’ m p pressure R universal gas constant T temperature v molar volume x mole fraction ∞ infinity Acknowledgments The authors wish to thank A. Klamt for helpful discussions, J. Gmehling for providing experimental data sets in electronic form. Financial support from the German research council (Deutsche Forschungsgemeinschaft, DFG) under the priority program SP 1155 is further acknowledged. References [1] A. Fredenslund, R.L. Jones, J.M. Prausnitz, AIChE J. 21 (1975) 1086–1099. [2] U. Weidlich, J. Gmehling, Ind. Eng. Chem. Res. 26 (1987) 1372–1381. [3] B.L. Larsen, P. Rasmussen, A. Fredenslund, Ind. Eng. Chem. Res. 26 (1987) 2274–2286. [4] A. Klamt, J. Phys. Chem. 99 (1995) 2224–2235. [5] S.-T. Lin, S.I. Sandler, Ind. Eng. Chem. Res. 41 (2002) 899–913, 2330–2334. [6] H. Grensemann, J. Gmehling, Ind. Eng. Chem. Res. 44 (2005) 1610–1624. [7] A. Klamt, F. Eckert, Fluid Phase Equilibr. 172 (2000) 43–72. [8] T. Mu, J. Rarey, J. Gmehling, Ind. Eng. Chem. Res. 46 (2007) 6612–6629. [9] T. Holderbaum, J. Gmehling, Fluid Phase Equilibr. 70 (1991) 251–265. [10] K. Tochigi, Fluid Phase Equilibr. 144 (1998) 59–68. [11] J. Ahlers, J. Gmehling, Ind. Eng. Chem. Res. 41 (2002) 3489–3498. [12] D. Constantinescu, A. Klamt, D. Gean˘a, Fluid Phase Equilibr. 231 (2005) 231–238. [13] E. Voutsas, K. Magoulas, D. Tassios, Ind. Eng. Chem. Res. 43 (2004) 6238–6246. [14] E. Voutsas, V. Louli, K. Boukouvalas, D. Magoulas, Tassios, Fluid Phase Equilibr. 241 (2006) 216–228. [15] Y. Shimoyama, Y. Iwai, S. Takada, Y. Arai, T. Tsuji, T. Hiaki, Fluid Phase Equilibr. 243 (2006) 183–192. [16] M.-T. Lee, S.-T. Lin, Fluid Phase Equilibr. 254 (2007) 28–34. [17] J. Martin, Ind. Eng. Chem. Fundam. 18 (1979) 81–97. [18] A. Péneloux, E. Rauzy, R. Fréze, Fluid Phase Equilibr. 8 (1982) 7–23. [19] W. Abdoul, E. Rauzy, A. Péneloux, Fluid Phase Equilibr. 68 (1991) 47–102. [20] J. Ahlers, J. Gmehling, Fluid Phase Equilibr. 191 (2001) 177–188. [21] C.H. Twu, D. Bluck, J.R. Cunningham, J.E. Coon, Fluid Phase Equilibr. 105 (1995) 49–59. [22] A. Klamt, G. Schüürmann, J. Chem. Soc., Perkin Trans. II (1993) 799–805. [23] F. Eckert, A. Klamt, COSMOtherm, version C1.2, release 01.03, 2003. [24] U.K. Deiters, Chem. Eng. Technol. 23 (2000) 581–584. [25] M.-J. Huron, J. Vidal, Fluid Phase Equilibr. 3 (1979) 255–271. [26] D.S.H. Wong, S.I. Sandler, AIChE J. 38 (1992) 671–680. [27] D.S.H. Wong, H. Orbey, S.I. Sandler, Ind. Eng. Chem. Res. 31 (1992) 2033–2039. [28] S. Ioannidis, D.E. Knox, Fluid Phase Equil. 187 (2001) 1–14. [29] P. Coutsikos, N.S. Kalospiros, D.P. Tassios, Fluid Phase Equilibr. 108 (1995) 59–78. [30] M.L. Michelsen, Fluid Phase Equilibr. 60 (1990) 213–219. [31] G.S. Soave, A. Bertucco, L. Vecchiato, Ind. Eng. Chem. Res. 33 (1994) 975–980. [32] N.S. Kalospiros, N. Tzouvaras, P. Coutsikos, D. Tassios, AIChE J. 41 (1995) 928–937. [33] R.A. Heidemann, Fluid Phase Equilibr. 116 (1996) 454–464.

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