- Email: [email protected]
Comments on the modeling of hydrogen and hydrogen-containing mixtures with cubic equations of state
Fluid Phase Equilibria 352 (2013) 93–96
Contents lists available at SciVerse ScienceDirect
Fluid Phase Equilibria journal homepage: www.elsevier.com/locate/fluid
Comments on the modeling of hydrogen and hydrogen-containing mixtures with cubic equations of state Ulrich K. Deiters ∗ Institute of Physical Chemistry, University of Cologne, Luxemburger Str. 116, 50939 Köln, Germany
a r t i c l e
i n f o
Article history: Received 4 December 2012 Received in revised form 25 February 2013 Accepted 28 May 2013 Available online 6 June 2013 Keywords: Equation of state Hydrogen Alpha function
a b s t r a c t Many cubic equations of state have attractive terms that are products of a density function and a temperature function, the so-called alpha function. By making the alpha function substance-specific – usually by letting it depend on the acentric factor – reasonably good representations of the thermodynamic properties can be achieved for many substances. This approach, however, does not always work well for hydrogen and hydrogen-containing mixtures. It is shown in this work that, for hydrogen, an alpha function can be reasonably defined in the temperature range 30–300 K only, provided that the densities are not too high. For higher temperatures or for modeling liquid hydrogen it is necessary to introduce corrections for soft repulsion into the hard-sphere terms of the equations of state. © 2013 Elsevier B.V. All rights reserved.
1. Introduction a b f kij p R T U V ˛ T
attraction parameter of an equation of state size parameter of an equation of state density-dependence of the attraction or repulsion term binary interaction parameter for species i and j pressure universal gas constant temperature internal energy volume temperature dependence of EOS attraction term internal pressure molar density
Subscripts att attraction critical property c rep repulsion high-temperature limit 0
∗ Tel.: +49 221 470 4543; fax: +49 221 470 4900. E-mail address: [email protected] 0378-3812/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fluid.2013.05.032
Cubic equations of state, but also some other theory-based equations, consist of a repulsion and an attraction term. In applied thermodynamics, it is frequently assumed that (a) the repulsion term is a hard-core term and thus strictly proportional to temperature, and (b) the attraction term can be split into a temperature-dependent and a density-dependent part,
p = RT frep () − fatt ()ac ˛(T ) ,
(1)
where denotes the molar density, T the temperature, and ac a substance-specific attraction parameter; the functions frep (), fatt (), and ˛(T) can of course contain more substance-specific parameters. For example, the combination frep () = /(1 − b),
2
fatt () = 2 /(1 + b), and ˛(T ) = [1 + m(1 − T/Tc )] gives Soave’s version of the Redlich–Kwong equation of state [1]. Eq. (1) constitutes a strong simplification, but appears to work well for most substances. Systematic studies of some popular ˛(T) functions were performed by Neau et al. [2,3] and – for Soave-type functions – by Segura et al. [4]. Now ˛(T) is not only important for the modeling of purefluid thermodynamic data. When thermodynamic properties of mixtures are calculated from cubic equations of state, usually an
94
U.K. Deiters / Fluid Phase Equilibria 352 (2013) 93–96
100 140
0.04 120
80
0.02 100
60
0.016
80
p/MPa
p/MPa
0.032
60
0.01
40
0.008
0.005
40
0
20
0.004
20
0
200
400
600
800
0
1000
T/K
adjustable1 binary interaction parameter kij has to be used in the combining rule of the attraction parameter:
aii ajj
(2)
Evidently, the optimal value of kij depends on the choice of ˛(T). This was recently pointed out by Jaubert et al. [5] in their analysis of the behaviour of kij in hydrogen-containing mixtures. The authors then proposed an elegant and useful method to convert kij parameters from one choice of ˛(T) to another. But is ˛(T) a meaningful function in connection with hydrogen?
600
800
Hydrogen, however, is different, as Fig. 2 shows. Here the experimental isochores are distorted and even intersect the hard-core pressure functions. The peculiarity of hydrogen becomes even more evident in Figs. 3–6, which show the dependence of the internal pressure, T = (∂U/∂V)T , on temperature. Here the behaviour of the reference equation of state by Leachman et al. [8] has been compared with that of four other equations of state, namely
1
0.5
UV /MPa
As the effect of temperature on fluid structure decreases with temperature (at constant density), ˛(T) can be expected to approach a limiting value at high temperatures. Consequently, p(T) isochores have a concavely curved low-temperature part,2 which reflects the ˛(T) function, and become increasingly linear at high temperatures. This is shown by Fig. 1, which contains isochores of argon, based on the argon reference equation of Tegeler et al. [6]. The limiting slopes of the isochores are of course equal to Rfrep (). A comparison of the experimental pressures with prep = RTfrep () shows that, except for really extreme thermodynamic conditions, the experimental pressures are always below the hard-core pressures and converge not against them, but a linear function slightly below them, i.e. the high-temperature behaviour of isochores is approximately given by
0
±0.5
(3)
where ˛0 is the high-temperature limiting value of ˛(T). It should be noted that ˛0 is evidently not zero [7]. Argon has been chosen here as an example merely because there exists a reference equation which is valid to very high pressures and temperatures. Most other gases behave similarly.
1
400
Fig. 2. Isochores of hydrogen. (—) Reference equation [8], (· · ·) hard-core pressure (estimated from the limiting slopes of the isochores). Parameter: /(mol cm−3 ).
2. Reconstruction of ˛(T) from reference equations
p → RT frep () − fatt ()ac ˛0 ,
200
T/K
Fig. 1. Isochores of argon. (—) Reference equation [6], (· · ·) hard-core pressure (estimated from the limiting slopes of the isochores). Parameter: /(mol cm−3 ).
aij = (1 − kij )
0
Whether kij is fitted directly to experimental data or estimated from group contribution scheme is not relevant for this work. 2 A closer inspection reveals that inflection points can occur in the critical region.
±1
0
200
400
600
800
T/K Fig. 3. Internal pressure of hydrogen as a function of temperature at a fixed molar density of 0.005 mol cm−3 , calculated with several equations of state. (—) Reference equation [8], (· · ·) Peng–Robinson [9], (−· −) SPHCT [10], (−·· −) PC-SAFT [11], (−−−) soft SPHCT [12].
U.K. Deiters / Fluid Phase Equilibria 352 (2013) 93–96
95
4 40 3 2
20
UV /MPa
UV /MPa
1 0
0
±1 ±20 ±2 ±3 ±4
±40
0
200
400
600
0
800
200
T/K Fig. 4. Internal pressure of hydrogen as a function of temperature at a fixed molar density of 0.01 mol cm−3 , calculated with several equations of state. See Fig. 3 for an explanation of the line types.
(1) the Peng–Robinson equation [9], as a representative of the class of cubic equations of state; it conforms to Eq. (1), (2) the simplified-perturbed-hard-chain equation (SPHCT) [10], which is a simple noncubic equation conforming to Eq. (1), too, (3) the PC-SAFT equation [11], an advanced noncubic equation of state, (4) and the “soft SPHCT” equation [12], an SPHCT equation with corrections for soft repulsion and thermodynamic quantum effects.
15
10
UV /MPa
5
0
±5
0
200
400
600
800
Fig. 6. Internal pressure of hydrogen as a function of temperature at a fixed molar density of 0.04 mol cm−3 , calculated with several equations of state. See Fig. 3 for an explanation of the line types. The curve of the reference equation has been terminated where the pressure exceeds 200 MPa.
For all equations, their substance-specific parameters were fitted to the critical point of hydrogen and, if possible, to the vapour pressure at 0.7 Tc . The calculations were carried out with the ThermoC program package [13]. Evidently, the Peng–Robinson, SPHCT, and PC-SAFT equations fail totally, which means that these equations cannot represent the cohesive energy of fluid hydrogen. Only the “soft SPHCT” equation can match the experimental data at least semiquantatively; it is, of course, too simple to achieve reference quality. It turns out that, for a good representation of the thermodynamic data of hydrogen, it is necessary to account for soft-repulsion effects. In fact, the Joule inversion temperature, i.e. the temperature at which T changes its sign, is at about 550 K and therefore well within the range of technical applications. A first conclusion from this might be that the PC-SAFT, SPHCT, and Peng–Robinson equation as well as a number of other cubic equations of state conforming to Eq. (1) should not be used for hydrogen or hydrogen-containing mixtures. But this conclusion is perhaps excessive. Alternatively (following the approach of Neau et al. [2,3]) we can ask what kind of ˛(T) function would be necessary to “rescue” the equations of state. For the Peng–Robinson equation, Eq. (1) can be rearranged to give
˛(T ) =
±10
±15
400
T/K
600
800
T/K Fig. 5. Internal pressure of hydrogen as a function of temperature at a fixed molar density of 0.02 mol cm−3 , calculated with several equations of state. See Fig. 3 for an explanation of the line types.
RT frep () − p . fatt ()ac
(4)
This function has been calculated for several molar densities and is shown in Fig. 7. The ˛(T) functions thus obtained are U-shaped, which is hard to reconcile with physical insight; the weird shape results from compensating an insufficient repulsion term by twisting the attraction term. Still, it should be noted that for the isochores with densities below 0.02 mol cm−3 , in the temperature range 30–300 K, the
96
U.K. Deiters / Fluid Phase Equilibria 352 (2013) 93–96
an ˛(T) function, and only there the kij parameter of binary mixtures can be properly defined. Incidentally, if the experimental acentric factor of hydrogen, −0.218, is used with the correlation of Peng and Robinson (Eq. (4) of [5]), the resulting ˛(T) function is far off the mark. It would be better to use an effective acentric factor of −0.05 or to employ the ˛(T) function of Twu et al. (see Fig. 7). For applications involving gaseous hydrogen above 300 K or liquid hydrogen, the approximation Eq. (1) breaks down. It is then necessary to use temperature-dependent size parameters after the fashion of the soft-SPHCT equation. It is known, however, that temperature-dependent size parameters in cubic equations of state can lead to isotherm crossing and similar artifacts [14,15]. Hence for such applications appropriate noncubic equations of state (containing softness corrections based on statistical thermodynamics) may be preferable; these would be valid for hydrogen as well as for ordinary fluids.
1.5
0.04
α(T)
1
PRorig Twu
0.5
0.02 0.01 0.005 0
PRfit 0
200
400
600
800
T/K Fig. 7. ˛(T) function for hydrogen. (—) Calculated from the reference equation [6] for the Peng–Robinson equation [9] from Eq. (4); parameter: /(mol cm−3 ). The region in which ˛(T) is approximately density-independent (for low to middle densities) is shaded grey. (· · ·) Correlation functions of Peng and Robinson (PR, using either the original or a fitted value of the acentric factor), and Twu et al.
˛(T) functions match reasonably well,3 so that it is indeed possible to split patt into a temperature-independent fatt () and a density-independent ˛(T). At liquid densities or at high temperatures, however, this is no longer true. 3. Conclusion Therefore it is in that temperature and density range only – 30–300 K and 0–0.02 mol cm−3 – where it makes sense to define
3
cf. critical density: 0.0155 mol cm−3 , normal liquid density: 0.0352 mol cm−3 .
References [1] G. Soave, Chem. Eng. Sci. 27 (1972) 1197–1203. [2] E. Neau, O. Hernández-Garduza, J. Escandell, Ch. Nicolas, I. Raspo, Fluid Phase Equilib. 276 (2009) 87–93. [3] E. Neau, I. Raspo, J. Escandell, Ch. Nicolas, O. Hernández-Garduza, Fluid Phase Equilib. 276 (2009) 156–164. [4] H. Segura, Th. Kraska, A. Mejía, J. Wisniak, I. Polishuk, Ind. Eng. Chem. Res. 42 (2003) 5662–5673. [5] J.-N. Jaubert, J. Qian, R. Privat, C.F. Leibovici, Fluid Phase Equilib. 338 (2013) 23–29. [6] Ch. Tegeler, R. Span, W. Wagner, J. Phys. Chem. Ref. Data 28 (1999) 779– 850. [7] U.K. Deiters, Chem. Eng. Sci. 36 (1981) 1139–1146. [8] J.W. Leachman, R.T. Jacobsen, S.G. Penoncello, E.W. Lemmon, J. Phys. Chem. Ref. Data 38 (2009) 721–748. [9] D.Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. 15 (1976) 59–64. [10] C.-H. Kim, P. Vimalchand, M.D. Donohue, S.I. Sandler, AIChE J. 32 (1986) 1726–1734. [11] J. Gross, G. Sadowski, Ind. Eng. Chem. Res. 40 (2001) 1244–1260. [12] O.L. Boshkova, U.K. Deiters, Int. J. Thermophys. 31 (2010) 227–252. [13] U.K. Deiters, ThermoC project homepage, http://thermoc.uni-koeln.de/index.html [14] M.A. Trebble, P.R. Bishnoi, Fluid Phase Equilib. 29 (1986) 465–474. [15] P.H. Salim, M.A. Trebble, Fluid Phase Equilib. 65 (1991) 59–71.
Contents lists available at SciVerse ScienceDirect
Fluid Phase Equilibria journal homepage: www.elsevier.com/locate/fluid
Comments on the modeling of hydrogen and hydrogen-containing mixtures with cubic equations of state Ulrich K. Deiters ∗ Institute of Physical Chemistry, University of Cologne, Luxemburger Str. 116, 50939 Köln, Germany
a r t i c l e
i n f o
Article history: Received 4 December 2012 Received in revised form 25 February 2013 Accepted 28 May 2013 Available online 6 June 2013 Keywords: Equation of state Hydrogen Alpha function
a b s t r a c t Many cubic equations of state have attractive terms that are products of a density function and a temperature function, the so-called alpha function. By making the alpha function substance-specific – usually by letting it depend on the acentric factor – reasonably good representations of the thermodynamic properties can be achieved for many substances. This approach, however, does not always work well for hydrogen and hydrogen-containing mixtures. It is shown in this work that, for hydrogen, an alpha function can be reasonably defined in the temperature range 30–300 K only, provided that the densities are not too high. For higher temperatures or for modeling liquid hydrogen it is necessary to introduce corrections for soft repulsion into the hard-sphere terms of the equations of state. © 2013 Elsevier B.V. All rights reserved.
1. Introduction a b f kij p R T U V ˛ T
attraction parameter of an equation of state size parameter of an equation of state density-dependence of the attraction or repulsion term binary interaction parameter for species i and j pressure universal gas constant temperature internal energy volume temperature dependence of EOS attraction term internal pressure molar density
Subscripts att attraction critical property c rep repulsion high-temperature limit 0
∗ Tel.: +49 221 470 4543; fax: +49 221 470 4900. E-mail address: [email protected] 0378-3812/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fluid.2013.05.032
Cubic equations of state, but also some other theory-based equations, consist of a repulsion and an attraction term. In applied thermodynamics, it is frequently assumed that (a) the repulsion term is a hard-core term and thus strictly proportional to temperature, and (b) the attraction term can be split into a temperature-dependent and a density-dependent part,
p = RT frep () − fatt ()ac ˛(T ) ,
(1)
where denotes the molar density, T the temperature, and ac a substance-specific attraction parameter; the functions frep (), fatt (), and ˛(T) can of course contain more substance-specific parameters. For example, the combination frep () = /(1 − b),
2
fatt () = 2 /(1 + b), and ˛(T ) = [1 + m(1 − T/Tc )] gives Soave’s version of the Redlich–Kwong equation of state [1]. Eq. (1) constitutes a strong simplification, but appears to work well for most substances. Systematic studies of some popular ˛(T) functions were performed by Neau et al. [2,3] and – for Soave-type functions – by Segura et al. [4]. Now ˛(T) is not only important for the modeling of purefluid thermodynamic data. When thermodynamic properties of mixtures are calculated from cubic equations of state, usually an
94
U.K. Deiters / Fluid Phase Equilibria 352 (2013) 93–96
100 140
0.04 120
80
0.02 100
60
0.016
80
p/MPa
p/MPa
0.032
60
0.01
40
0.008
0.005
40
0
20
0.004
20
0
200
400
600
800
0
1000
T/K
adjustable1 binary interaction parameter kij has to be used in the combining rule of the attraction parameter:
aii ajj
(2)
Evidently, the optimal value of kij depends on the choice of ˛(T). This was recently pointed out by Jaubert et al. [5] in their analysis of the behaviour of kij in hydrogen-containing mixtures. The authors then proposed an elegant and useful method to convert kij parameters from one choice of ˛(T) to another. But is ˛(T) a meaningful function in connection with hydrogen?
600
800
Hydrogen, however, is different, as Fig. 2 shows. Here the experimental isochores are distorted and even intersect the hard-core pressure functions. The peculiarity of hydrogen becomes even more evident in Figs. 3–6, which show the dependence of the internal pressure, T = (∂U/∂V)T , on temperature. Here the behaviour of the reference equation of state by Leachman et al. [8] has been compared with that of four other equations of state, namely
1
0.5
UV /MPa
As the effect of temperature on fluid structure decreases with temperature (at constant density), ˛(T) can be expected to approach a limiting value at high temperatures. Consequently, p(T) isochores have a concavely curved low-temperature part,2 which reflects the ˛(T) function, and become increasingly linear at high temperatures. This is shown by Fig. 1, which contains isochores of argon, based on the argon reference equation of Tegeler et al. [6]. The limiting slopes of the isochores are of course equal to Rfrep (). A comparison of the experimental pressures with prep = RTfrep () shows that, except for really extreme thermodynamic conditions, the experimental pressures are always below the hard-core pressures and converge not against them, but a linear function slightly below them, i.e. the high-temperature behaviour of isochores is approximately given by
0
±0.5
(3)
where ˛0 is the high-temperature limiting value of ˛(T). It should be noted that ˛0 is evidently not zero [7]. Argon has been chosen here as an example merely because there exists a reference equation which is valid to very high pressures and temperatures. Most other gases behave similarly.
1
400
Fig. 2. Isochores of hydrogen. (—) Reference equation [8], (· · ·) hard-core pressure (estimated from the limiting slopes of the isochores). Parameter: /(mol cm−3 ).
2. Reconstruction of ˛(T) from reference equations
p → RT frep () − fatt ()ac ˛0 ,
200
T/K
Fig. 1. Isochores of argon. (—) Reference equation [6], (· · ·) hard-core pressure (estimated from the limiting slopes of the isochores). Parameter: /(mol cm−3 ).
aij = (1 − kij )
0
Whether kij is fitted directly to experimental data or estimated from group contribution scheme is not relevant for this work. 2 A closer inspection reveals that inflection points can occur in the critical region.
±1
0
200
400
600
800
T/K Fig. 3. Internal pressure of hydrogen as a function of temperature at a fixed molar density of 0.005 mol cm−3 , calculated with several equations of state. (—) Reference equation [8], (· · ·) Peng–Robinson [9], (−· −) SPHCT [10], (−·· −) PC-SAFT [11], (−−−) soft SPHCT [12].
U.K. Deiters / Fluid Phase Equilibria 352 (2013) 93–96
95
4 40 3 2
20
UV /MPa
UV /MPa
1 0
0
±1 ±20 ±2 ±3 ±4
±40
0
200
400
600
0
800
200
T/K Fig. 4. Internal pressure of hydrogen as a function of temperature at a fixed molar density of 0.01 mol cm−3 , calculated with several equations of state. See Fig. 3 for an explanation of the line types.
(1) the Peng–Robinson equation [9], as a representative of the class of cubic equations of state; it conforms to Eq. (1), (2) the simplified-perturbed-hard-chain equation (SPHCT) [10], which is a simple noncubic equation conforming to Eq. (1), too, (3) the PC-SAFT equation [11], an advanced noncubic equation of state, (4) and the “soft SPHCT” equation [12], an SPHCT equation with corrections for soft repulsion and thermodynamic quantum effects.
15
10
UV /MPa
5
0
±5
0
200
400
600
800
Fig. 6. Internal pressure of hydrogen as a function of temperature at a fixed molar density of 0.04 mol cm−3 , calculated with several equations of state. See Fig. 3 for an explanation of the line types. The curve of the reference equation has been terminated where the pressure exceeds 200 MPa.
For all equations, their substance-specific parameters were fitted to the critical point of hydrogen and, if possible, to the vapour pressure at 0.7 Tc . The calculations were carried out with the ThermoC program package [13]. Evidently, the Peng–Robinson, SPHCT, and PC-SAFT equations fail totally, which means that these equations cannot represent the cohesive energy of fluid hydrogen. Only the “soft SPHCT” equation can match the experimental data at least semiquantatively; it is, of course, too simple to achieve reference quality. It turns out that, for a good representation of the thermodynamic data of hydrogen, it is necessary to account for soft-repulsion effects. In fact, the Joule inversion temperature, i.e. the temperature at which T changes its sign, is at about 550 K and therefore well within the range of technical applications. A first conclusion from this might be that the PC-SAFT, SPHCT, and Peng–Robinson equation as well as a number of other cubic equations of state conforming to Eq. (1) should not be used for hydrogen or hydrogen-containing mixtures. But this conclusion is perhaps excessive. Alternatively (following the approach of Neau et al. [2,3]) we can ask what kind of ˛(T) function would be necessary to “rescue” the equations of state. For the Peng–Robinson equation, Eq. (1) can be rearranged to give
˛(T ) =
±10
±15
400
T/K
600
800
T/K Fig. 5. Internal pressure of hydrogen as a function of temperature at a fixed molar density of 0.02 mol cm−3 , calculated with several equations of state. See Fig. 3 for an explanation of the line types.
RT frep () − p . fatt ()ac
(4)
This function has been calculated for several molar densities and is shown in Fig. 7. The ˛(T) functions thus obtained are U-shaped, which is hard to reconcile with physical insight; the weird shape results from compensating an insufficient repulsion term by twisting the attraction term. Still, it should be noted that for the isochores with densities below 0.02 mol cm−3 , in the temperature range 30–300 K, the
96
U.K. Deiters / Fluid Phase Equilibria 352 (2013) 93–96
an ˛(T) function, and only there the kij parameter of binary mixtures can be properly defined. Incidentally, if the experimental acentric factor of hydrogen, −0.218, is used with the correlation of Peng and Robinson (Eq. (4) of [5]), the resulting ˛(T) function is far off the mark. It would be better to use an effective acentric factor of −0.05 or to employ the ˛(T) function of Twu et al. (see Fig. 7). For applications involving gaseous hydrogen above 300 K or liquid hydrogen, the approximation Eq. (1) breaks down. It is then necessary to use temperature-dependent size parameters after the fashion of the soft-SPHCT equation. It is known, however, that temperature-dependent size parameters in cubic equations of state can lead to isotherm crossing and similar artifacts [14,15]. Hence for such applications appropriate noncubic equations of state (containing softness corrections based on statistical thermodynamics) may be preferable; these would be valid for hydrogen as well as for ordinary fluids.
1.5
0.04
α(T)
1
PRorig Twu
0.5
0.02 0.01 0.005 0
PRfit 0
200
400
600
800
T/K Fig. 7. ˛(T) function for hydrogen. (—) Calculated from the reference equation [6] for the Peng–Robinson equation [9] from Eq. (4); parameter: /(mol cm−3 ). The region in which ˛(T) is approximately density-independent (for low to middle densities) is shaded grey. (· · ·) Correlation functions of Peng and Robinson (PR, using either the original or a fitted value of the acentric factor), and Twu et al.
˛(T) functions match reasonably well,3 so that it is indeed possible to split patt into a temperature-independent fatt () and a density-independent ˛(T). At liquid densities or at high temperatures, however, this is no longer true. 3. Conclusion Therefore it is in that temperature and density range only – 30–300 K and 0–0.02 mol cm−3 – where it makes sense to define
3
cf. critical density: 0.0155 mol cm−3 , normal liquid density: 0.0352 mol cm−3 .
References [1] G. Soave, Chem. Eng. Sci. 27 (1972) 1197–1203. [2] E. Neau, O. Hernández-Garduza, J. Escandell, Ch. Nicolas, I. Raspo, Fluid Phase Equilib. 276 (2009) 87–93. [3] E. Neau, I. Raspo, J. Escandell, Ch. Nicolas, O. Hernández-Garduza, Fluid Phase Equilib. 276 (2009) 156–164. [4] H. Segura, Th. Kraska, A. Mejía, J. Wisniak, I. Polishuk, Ind. Eng. Chem. Res. 42 (2003) 5662–5673. [5] J.-N. Jaubert, J. Qian, R. Privat, C.F. Leibovici, Fluid Phase Equilib. 338 (2013) 23–29. [6] Ch. Tegeler, R. Span, W. Wagner, J. Phys. Chem. Ref. Data 28 (1999) 779– 850. [7] U.K. Deiters, Chem. Eng. Sci. 36 (1981) 1139–1146. [8] J.W. Leachman, R.T. Jacobsen, S.G. Penoncello, E.W. Lemmon, J. Phys. Chem. Ref. Data 38 (2009) 721–748. [9] D.Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. 15 (1976) 59–64. [10] C.-H. Kim, P. Vimalchand, M.D. Donohue, S.I. Sandler, AIChE J. 32 (1986) 1726–1734. [11] J. Gross, G. Sadowski, Ind. Eng. Chem. Res. 40 (2001) 1244–1260. [12] O.L. Boshkova, U.K. Deiters, Int. J. Thermophys. 31 (2010) 227–252. [13] U.K. Deiters, ThermoC project homepage, http://thermoc.uni-koeln.de/index.html [14] M.A. Trebble, P.R. Bishnoi, Fluid Phase Equilib. 29 (1986) 465–474. [15] P.H. Salim, M.A. Trebble, Fluid Phase Equilib. 65 (1991) 59–71.