Generalized Cubic Equation of State Adjusted to ... - ACS Publications

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Ind. Eng. Chem. Res. 2009, 48, 10708–10717

Generalized Cubic Equation of State Adjusted to the Virial Coefficients of Real Gases and Its Prediction of Auxiliary Thermodynamic Properties Ilya Polishuk* Department of Chemical Engineering & Biotechnology, Ariel UniVersity Center of Samaria, 40700, Ariel, Israel

This study demonstrates that the overall reliability of a simplest cubic equation of state (EOS) might be improved by developing its accuracy in estimating the virial coefficients. In particular, it has been demonstrated that the progress can be achieved in predicting the sound velocities and, probably, the Joule-Thomson inversion curves. At the same time, the limitations of the simplest model are evident. In particular, it fails in predicting of the virial coefficients in the low temperature range. As a result, its predictions of the liquid phase properties are still imperfect. Further directions of development of the proposed EOS are discussed. The limitations of the proposed model will be discussed in order to set directions for further development of its theoretical base.

Introduction Development of a simple semiempirical engineering equation of state (EOS) model reliable for prediction of a large variety of thermodynamic properties is a challenging problem, which has not been satisfactorily solved yet. Although cubic EOS models are usually implemented for modeling phase equilibria in mixtures, so far their development has been mostly focused on the accurate description of certain pure compound properties, such as liquid molar volumes and vapor pressures. Considering that additional thermodynamic properties may increase the scientific value of the EOS models, which has been strongly outlined by IUPAC’s Guidelines for Publication of Equations of State.1 The recent developments of the cubic EOS models have been reviewed by Anderko2 and Valderrama.3 Nevertheless, several studies that have considered the accuracy of these models in predicting the auxiliary thermodynamic properties should be mentioned. Some of them (see for example refs 4-7) have examined the Joule-Thomson inversion and the ideal gas curves. Others (see for example refs 8-13) have considered properties such as the isochoric and the isobaric heat capacities, sound velocities, etc. It is widely recognized that the accurate description of virial coefficients is essential for the reliable prediction of different thermodynamic properties.1,14,15 Therefore it seems expedient to consider the virial coefficients in development of the EOS parameters. So far, several studies (see for example refs 16-19) have dealt with the fitting of the second virial coefficients by cubic EOS models. However, such practice usually affects the results for the vapor pressures. In addition, the attempts to improve the results for the third virial coefficients are still scarce. A potential disadvantage of approaches for developing overall accurate EOS models could be the need for introducing many compound-specific parameters. Such a practice creates an undesired complexity and hinders the implementation of the model for engineering calculations and tutorial purposes. Hence, at this preliminary stage of the research, attention should be given to the development of the optimal form of the simple cubic equation with analytical expressions for its parameters. * To whom correspondence should be addressed. E-mail: polishuk@ ariel.ac.il; [email protected]. Phone: +972-3-9066346. Fax: +972-3-9066323.

Theory Most cubic van der Waals-type equations can be generalized as follows: P)

a RT V-b (V + c)(V + d)

(1)

Recently, Privat et al.20 have considered the virial expansion of eq 1 and have proposed solutions for its several particular cases. However, after some algebra, the following simple and universal solution can be found for the analytical EOS models:

( )

∂Zi-1 Ff0 ∂i-1F ci ) (i - 1)! lim

(2)

where ci is the ith virial coefficient. Substitution of eq 1 into eq 2 yields a RT

(3)

a(c + d) RT

(4)

a(c2 + cd + d2) RT

(5)

B)bC ) b2 + D ) b3 -

It is a well-known fact that cubic equations of state that typically yield accurate description of the volumetric properties and vapor pressures are not always appropriate for the quantitative prediction of the experimental values of B and C. For example, as discussed elsewhere,1 cubic equations are unable to represent the behavior of the third virial coefficient C, namely its approaching negative values at low temperatures. However, inspection of eq 4 indicates that such behavior can be obtained if the sum (c + d) is a small negative number. Regrettably, an accurate description of the volumetric properties and B by eq 1 requires positive values of (c + d). (It should be pointed out that the situation can be different when considering noncubic van der Waals-like models). This result indicates that the relative success of cubic equations in describing volumetric properties should be explained by the error-compensation mechanism between its virial

10.1021/ie900905p CCC: $40.75  2009 American Chemical Society Published on Web 09/15/2009

Ind. Eng. Chem. Res., Vol. 48, No. 23, 2009

coefficients. However, this error-compensation mechanism may become less successful when attempting to predict auxiliary thermodynamic properties, such as the Joule-Thomson inversion curves or the sound velocity. Therefore, it is expedient to improve the accuracy of eq 1 in estimating of the experimentally available virial coefficients at least in the middle and the high temperature ranges. Let us consider two options for such an improvement. A. Temperature Dependency of the Covolume b. Equation 1 originates from the hard-sphere approach, which claims the temperature-independent covolume b. However, real molecules do not have hard repulsion potentials.1 This fact has been considered by several theoretically based models, including different modifications of statistical associating fluid theory (SAFT), which have introduced temperature dependencies for their size parameters. In fact, such dependencies may substantially increase the flexibility of eqs 1, 4, and 5 and, theoretically, allow a simultaneous fitting of the volumetric properties of real compounds, their vapor pressures, and the virial coefficients. Let us consider the consequences of making the covolume temperature-dependent for the isochoric heat capacities. It has been demonstrated1 that the repulsive term of eq 1 has a zero contribution to the residual isochoric heat capacities. Making the covolume temperature dependent as follows: Prep )

RT V - b[T]

measurements of the third virial coefficients are usually inexact, and there are significant disagreements between different data sources regarding the values of this property.14 In contrast, the values of the second virial coefficient are known with much higher certainty. At each relation between c and d, the parameters of eq 1 can be evaluated by solving a system of four equations, namely, by setting the first and the second derivatives of P with respect to V equal to zero, the critical pressure equal to the pertinent experimental value, and eq 4 equal to the experimental value of the second virial coefficient at the critical point (Bc). If the value of c approaches the value of d, the value of b decreases. The best results for the nonpolar compounds are obtained at c ) d, which reduces the number of the EOS parameters and makes their analytical expressions very simple, as follows: a)

27(RTc)2 64Pc

b ) Bc +

27RTc 64Pc

c ) d ) -Bc -

(6) Vc,EOS ) Bc +

and substituting eq 6 into the equation

CVR )

(

{ ∫ [T( ∂P ∂T ) V

∂

∞

∂T

V

] }

- P dV

)

V

(7)

yields CVR )

R(Tb′[T])2 2RTb′[T] + RT2b′′[T] V - b[T] (V - b[T])2

(8)

At very high pressures, V will approach b[T] and both denominators will become very small numbers. Since, in addition, R(Tb′[T])2 is always positive, it can be seen that R(Tb′[T])2 2RTb′[T] + RT2b′′[T] . 2 V - b[T] (V - b[T])

19RTc 64Pc

43RTc 64Pc

(10a)

(10b)

(10c)

(10d)

Regrettably, eqs 10 still overestimate the covolume’s values for some polar compounds. Moreover, the requirement of evaluating the experimental value of Bc may hinder implementation of the model by engineers. In addition, eq 10c usually results in significant overestimation of C. Thus, it seems more expedient to refer the EOS parameters to the experimental value of the critical volume, which is typically available together with other critical constants and which may also decrease the value of the parameter c, improving the predictions of C (see eq 4). Analysis of inter-relation between eqs 10b-10d and taking into account the experimental data of hundreds of pure compounds leads to definition of the following universal EOS:

(9)

Hence, at very high pressures, CV R will necessarily become a large negative number. Obviously, at the infinite pressure its value is indeterminate. Such a nonphysical result totally excludes the possibility of further consideration of eq 6. The advanced hard-sphere repulsive terms, such as the one of Carnahan and Starling21 attached by the temperaturedependent covolume, may not yield the indeterminate residual isochoric heat capacity at infinite pressure. Nevertheless, the positive values of b′[T] and b′′[T] still result in the nonphysical negative values of CVR. At the same time, the negative b′[T] results in another nonphysical phenomena, namely, the crossing of isotherms. In addition, the negative b′′[T] contradicts the theoretical studies of softly repulsive systems.1 As an alternative to the temperature-dependent covolume approach, the Guidelines for Publication of Equations of State1 recommend the development of the attractive term of the EOS. In what follows, we will consider this option. B. Development of the EOS’s Attractive Term. Simultaneous fitting of the experimental values of the virial coefficients and the volumetric properties by eq 1 is not an easy task. The

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P)

aR RT V-b (V + c)2

(11)

27(RTc)2 64Pc

(12a)

where a)

b)

c)

Vc,experimental 2√3

3RTc - 4√3PcVc,experimental 24Pc

Vc,EOS )

3RTc + 2√3PcVc,experimental 12Pc

(12b)

(12c)

(12d)

For most nonpolar compounds, eqs 10 and 12 usually yield very similar numerical values. In such cases, the RedlichKwong-Soave EOS (RKS)22 is often also accurate for the virial coefficients,18 being, however, less successful in describing the

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Figure 1. Second virial coefficients of light n-alkanes. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental data.26

volumetric and the caloric properties than eqs 12, or the popular EOS of Peng and Robinson (PR).23 Equations 12 describe the third virial coefficients and the properties of polar compounds more accurately than eqs 10. Unfortunately, all cubic EOS models, including eq 11, do not always allow the simultaneous fitting of the second virial coefficient and the vapor pressure curves. Therefore, the results for B may deviate from the experimental data at low temperatures, which correspond to liquids and diluted gases. In spite of the inaccurate description of the virial coefficients, diluted gases approaching the ideal behavior are predicted by cubic equations with sufficient accuracy. In liquids, a major role is played by the higher virial coefficients. As discussed previously, eqs 1 and 11 provide no robust solution for fitting the third virial coefficient at low temperatures. Thus, at the present stage of the research, it does not seem expedient to put more effort in improving the predictions of B at low temperature range. In contrast, the appropriate modeling of B and C in the supercritical range has significant practical value and it can be improved by the appropriate behavior of R at high temperatures. The exponential temperature dependencies may be superior for B;17 however, they typically require at least two parameters for the accurate fitting of the vapor pressures. Such practice hinders their generalization for different kinds of compounds. In fact, none of the currently available temperature dependencies has been found appropriate for the aims of the present study. After examining plenty of possible theoretically based and empirical expressions, a following simple one-parameter temperature

dependency that asymptotically approaches zero at high temperatures has been developed: R)

1 1 + m(√Tr - 1)2/3

(13)

where m is generalized with the acentric factor ω defined as follows: m ) 0.3671 + 0.7697ω

(14)

Equations 13 and 14 typically represent the experimental vapor pressure data with the accuracy similar to the popular wellknown temperature dependency proposed by Soave.22 In what follows, the performance of the proposed model will be compared with two cubic EOS models: PR and the equation of Patel-Teja-Valderrama (PTV).24 The last equation is similar to the proposed model because it also has three generalized parameters. In addition, according to the author’s experience, this model is one the most successful cubic equations in predicting the volumetric and the auxiliary properties (see also ref 25). Results Figures 1 and 2 present the results for B of several representative polar and nonpolar compounds. Since cubic equations of state are very simple models, they are unable to yield simultaneously accurate results for various properties. For example, the RKS’s universal value of B at the critical


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Figure 2. Second virial coefficients of miscellaneous compounds. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental data.26

Figure 3. Third virial coefficients of miscellaneous compounds. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental data.14

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Figure 4. Third virial coefficients of miscellaneous compounds. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental data.14

temperature is accurate for many nonpolar compounds.18 However, as shown elsewhere,23 there is a price to pay, namely, the poor prediction of the volumetric properties. Oppositely, the EOS of Trebble-Bishnoi-Salim estimates the volumetric properties better than RKS, but it usually yields poorer predictions of the second virial coefficients.8 Among the two-parameter cubic EOS models, PR seems to present an appropriate compromise in estimating the volumetric properties and B. It yields the universal value -0.37944 for the reduced second virial coefficient at the critical temperature, which typically improves the results in the supercritical range but underestimates the data around the critical temperatures. Such underestimation is increased as the difference between the covolume and the critical volume decreases. The parameters of the proposed model and of the PTV are dependent on the critical volume, and, therefore, these models have no universal value of the reduced second virial coefficient at the critical temperature. However, the PTV EOS does not aim at an accurate description of B. Its results for the nonpolar compounds are slightly better than those yielded by PR, and they are less accurate for the polar compounds. Figures 1 and 2 demonstrate the clear superiority of the proposed EOS in predicting the second virial coefficients. For some polar compounds, such as methanol, the proposed model may still underestimate B around the critical temperature. This is, however, a necessary compromise with the requirement of an accurate description of the volumetric properties. Obviously, the overall accuracy of the model could be increased by introducing additional parameters, which, however, contradicts the goals of the present study. Figures 3 and 4 show the results for the third virial coefficients. It should be pointed out that there are significant

deviations between different data sources of this property, and those, having relatively small estimated errors,14 are presented in the figures. A clear superiority of the proposed EOS over PR and PTV in predicting these data in the supercritical range can be seen. As discussed above, currently there is no appropriate solution for the low-temperature third virial coefficient data within the frame of eq 1. In fact, the desired behavior can be artificially reproduced by making the parameter c temperaturedependent. However, having two temperature-dependent parameters (R and c) in the cohesive term may hinder their mutual evaluation. In addition, overloading the model by empirical functionalities can affect its robustness. Thus, a more promising direction of further research seems development of a noncubic EOS, accurate for the volumetric properties and B with small negative (c + d), yielding positive C at the supercritical temperatures and negative at the subcritical ones. Figures 5 and 6 compare the accuracy of the proposed EOS with that of PR and PTV in predicting the vapor pressure, the enthalpy, and the entropy of phase transitions for methane and water. It can be seen that all the models yield relatively similar results. It is widely recognized that the accurate predictions of the Joule-Thomson inversion curves are the challenging task for EOS models. The inter-relations between the virial coefficients and the inversion data have been recently discussed.30-33 Figure 7 presents the predictions of the Joule-Thomson inversion curves for methane and carbon dioxide. It would be tempting to conclude that the relative success of the proposed EOS in predicting the data should be explained by its superiority in description of the virial coefficients. However, it is important to acknowledge the fact the available estimations of the Joule-Thomson inversion curves are often based on very old


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Figure 5. Equilibria properties of methane. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental data.27

Figure 6. Equilibria properties of water. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental data.28

and not always reliable data. Therefore, it seems expedient to challenge the equations of state by predictions of sound

velocities, which also combine the volumetric and the caloric properties, and for which the reliable data are available.

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Figure 7. Joule-Thomson inversion curves. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental data.29

Figure 8. PVT data of methane. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental saturated data.27 (0) Experimental single phase data.27

Figure 9. PVT data of nitrogen. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental saturated data.34 (0) Experimental single phase data.34

However, before doing so, it is important to consider the accuracy of the models under consideration in predicting the volumetric data. For this task, two nonpolar and three polar compounds with fully available “steam tables” have been selected (see Figures 8-12). Although the selected compounds are different, the patterns of behavior showed by the models are similar. In particular, it can be seen that the proposed model

Figure 10. PVT data of ammonia. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental saturated data.35 (0) Experimental single phase data.35

Figure 11. PVT data of methanol. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental saturated data.36 (0) Experimental single phase data.36

and PTV usually yield similar results. The proposed model may have some superiority at very high pressures (see for example water). PTV is more accurate for the saturated liquid volumetric properties near the critical points. Nevertheless, the issue of satisfying the critical point conditions (equality to zero of the first and the second derivatives of P with respect to V) probably requires additional discussion. PR is the less accurate equation for the volumetric properties at high pressures in the entire


Figure 12. PVT data of water. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental saturated data.28 (0) Experimental single phase data.28

Figure 13. Sound velocity data of methane. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental saturated data.27 (0) Experimental single phase data.27

temperature range and, as the polarity of the compound increases, the superiority of the proposed EOS increases as well. This result might be explained by the more appropriate selection of the covolume (eq 12b), which, in turn, has been made considering the virial coefficients (see discussion above). Such a result outlines the importance of inter-relating between different thermodynamic properties. The last statement may be supported by the results obtained for the sound velocities (Figures 13-17). While for the nonpolar compounds all the models yield similar results, the quality of predicting the polar compounds is different. In particular, it can be seen that PR and PTV are less successful in predicting the sound velocity data of methanol and water. In contrast, the proposed model yields relatively reliable estimations with the only exceptions of the low-temperature liquid-phase parts of the diagrams and around the critical points.

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Figure 14. Sound velocity data of nitrogen. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental saturated data.34 (0) Experimental single phase data.34

Figure 15. Sound velocity data of ammonia. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental saturated data.35 (0) Experimental single phase data.35

Conclusions and Discussion

Figure 16. Sound velocity data of methanol. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental saturated data.36 (0) Experimental single phase data.36

Considering the fact that different thermodynamic properties are inter-related might allow improvement of the overall reliability of the simplest cubic EOS model. This study has aimed to achieve such an improvement by adjusting the EOS parameters to the experimental values of the virial coefficients. It has been demonstrated that in certain cases some progress can be achieved in predictions of the sound velocities and, probably, the Joule-Thomson inversion curves. At the same time, the limitations of the simplest model are evident. In

particular, it fails in predictions of the virial coefficients in the low temperature range. As a result, its predictions of the liquid phase properties are still imperfect. Nevertheless, the proposed model should be considered as a good starting point for further development. One of the possible directions of its improvement could be replacing the van der Waals’s repulsive term by some theoretically advanced expression. However, such practice may create several unexpected problems. As discussed previously,37 approaching the theoretical

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(1) Correct predictions of the virial coefficients in the lowtemperature range. (2) Sufficient range of the absolute liquid-liquid immiscibility predicted with the zero values of binary parameters. Acknowledgment Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for support of this research, Grant 47338-B6. Literature Cited

Figure 17. Sound velocity data of water. Black solid lines-eqs 11-14. Dotted lines-PR.23 Red dashed lines-PTV.24 (O) Experimental saturated data.28 (0) Experimental single phase data.28

values of the hard-sphere virial coefficients reduces the numerical contribution of the required temperature dependency R, until at some point it can be completely vanished. The insufficient numerical contribution of R may affect the accuracy of the model in estimating the virial coefficients at high temperatures, which in turn could affect the entire thermodynamic phase-space. In addition, the analysis shows that R-functionality plays a very important role in predictions of phase equilibria in mixtures using the classical mixing rules. In particular, it has been demonstrated38 that the values of the binary parameter k12 required for obtaining the liquid-liquid critical line of type II by cubic EOS models and classical mixing rules must satisfy the following condition: k12 > 1 -

1 2

[ (

Pc2R2 Pc1R1 +1 Pc1R1 Pc2R2

)]

(15)

Similar expressions can be developed for several noncubic van der Waals-like equations. It can be demonstrated that as the difference between the R-functionalities of the pure compounds increases, the liquid-liquid phase equilibria can typically be obtained at lower values of k12. As observed elsewhere,39 van der Waals-like equations attached by the classical mixing rules with the zero values of binary parameters tend to underestimate the absolute liquid-liquid immiscibility of type II, which results in underestimation of the low-temperature bubble-point lines and overestimation of the limited liquid-liquid equilibria of type V. A correction of these phenomena may require high nonzero values of the binary parameters, which removes the predictive character of the models and may have negative consequences outside the fitting range. Analysis of eq 15 and the similar expressions for the noncubic models explains the poor performance of some theoretically based equations in predicting mixtures39 by smaller differences between the R-functionalities of pure compounds. In other words, not every improvement of theoretical base of the models will necessarily improve their overall accuracy. One of the promising directions of further research could be consideration of the existing soft-repulsion theories and their careful incorporation into the repulsive term in a way that will not introduce numerical pitfalls, such as the crossing of isotherms, and will improve the predictions of heat capacities at high pressures. The resulting EOS should satisfy the two following requirements:

(1) Deiters, U. K.; de Reuck, K. M. Guidelines for Publication of Equations of State - I. Pure Fluids. Pure Appl. Chem. 1997, 69, 1237. (2) Anderko, A. Cubic and Generalized van der Waals Equations. In Equations of State for Fluids and Fluid Mixtures. Part I; Elsevier: Amsterdam, 2000. (3) Valderrama, J. O. The State of the Cubic Equations of State. Ind. Eng. Chem. Res. 2003, 42, 1603. (4) Dilay, G. W.; Heidemann, R. A. Calculation of Joule-Thomson Inversion Curves from Equations of State. Ind. Eng. Chem. Fundam. 1986, 25, 152. (5) Matin, N. S.; Haghighi, B. Calculation of Joule-Thomson Inversion Curves from Cubic Equations of State. Fluid Phase Equilib. 2000, 175, 273. (6) Haghighi, B.; Laee, M. R.; Matin, N. S. A Comparison Among Five Equations of State in Predicting the Inversion Curve of Some Fluids. Cryogenics 2003, 43, 393. (7) Parsafar, G. A.; Izanloo, C. Deriving Analytical Expressions for Ideal Curves and Using the Curves to Obtain the Temperature Dependence of Equation-of-State Parameters. Int. J. Thermophys. 2006, 27, 1564. (8) Salim, P. H.; Trebble, M. A. Thermodynamic property Predictions from the Trebble-Bishnoi-Salim Equation of State. Fluid Phase Equilib. 1991, 65, 41. (9) Rogalski, M.; Carrier, B.; Pe´neloux, A. Correlation and Prediction of Physical Properties of Hydrocarbons with the Modified Peng-Robinson Equation of State. 3. Thermal Properties. A New Significant Characterizing Parameter m. Ind. Eng. Chem. Res. 1991, 30, 1613. (10) Gregorowicz, J.; O’Connel, J. P.; Peters, C. J. Some Characteristics of Pure Fluid Properties that Challenge Equation-of-State Models. Fluid Phase Equilib. 1996, 116, 94. (11) Coniglio, L.; Rauzy, E.; Pe´neloux, A.; Neau, E. Use of Heat Capacities for the Estimation of Cubic Equation-of-State Parameters Application to the Prediction of Very Low Vapor Pressures of Heavy Hydrocarbons. Fluid Phase Equilib. 2002, 200, 375. (12) Polishuk, I.; Burgin, I.; Cohen, N. Implementation of Mathematica for Development and Application of EOS models. III. Calculation of Heat Capacities, Enthalpy and Entropy of Evaporation and Speed of Sound. Chem. Eng. Commun. 2009, 196, 1291. (13) Polishuk, I.; Wisniak, J.; Segura, H. Some Observations Regarding the Prediction of Isochoric Heat Capacities by Engineering EOS Models. Fluid Phase Equilib. 2009, 277, 121. (14) Dymond, J. H.; Marsh, K. N.; Wilhoit, R. C.; Wong, K. C. Virial Coefficients of Pure Gases and Mixtures. Landolt-Bo¨rnstein Numerical Data and Functional Relationships in Science and Technology New Series Group IV: Physical Chemistry Vol. 21; Springer-Verlag: Berlin, 2002. (15) Larsen, E. R.; Prausnitz, J. M. High Pressure Phase Equilibria for the Water/Methane System. AIChE J. 1984, 30, 732. (16) Martin, J. J. Correlation of Second Virial Coefficients Using a Modified Cubic Equation of State. Ind. Eng. Chem. Res. 1984, 23, 454. (17) Mak, P. C. N.; Lielmezs, J. Correlation for the Second Virial Coefficient for Nonpolar Compounds Using Cubic Equation of State. Ind. Eng. Chem. Res. 1989, 28, 127. (18) Mathias, P. M. The Second Virial Coefficient and the RedlichKwong Equation. Ind. Eng. Chem. Res. 2004, 42, 7037. (19) Nasrifar, Kh.; Bolland, O. Square-Well Potential and a New R Function for the Soave-Redlich-Kwong Equation of State. Eng. Chem. Res 2004, 43, 6901. (20) Privat, R.; Privat, Y.; Jaubert, J.-N. Can Cubic Equations of State be Recast in the Virial Form. Fluid Phase Equilib. 2009, 282, 38. (21) Carnahan, N. F.; Starling, K. E. Equation of State for Nonattracting Rigid Spheres. J. Chem. Phys. 1969, 51, 635–636. (22) Soave, G. Equilibrium Constants Form a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197–1203. (23) Peng, D. Y.; Robinson, D. B. A New Two Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59–64.

Ind. Eng. Chem. Res., Vol. 48, No. 23, 2009 (24) Valderrama, J. O. A Generalized Patel-Teja Equation of State for Polar and Nonpolar Fluids and Their Mixtures. J. Chem. Eng. Jpn. 1990, 23, 87. (25) Valderrama, J. O.; Alfaro, M. Liquid Volumes from Generalized Cubic Equations of State: Take It with Care. Oil & Gas Sci. Tech.sReV. IFP. 2000, 55, 523. (26) Rowley, R. L.; Wilding, W. V.; Daubert, T. E.; Danner, R. P.; Adams, M. E. Physical and Thermodynamic Properties of Pure Chemicals: DIPPR: Data Compilation; Taylor & Francis: New-York, 2002. (27) Setzmann, U.; Wagner, W. A New Equation of State and Thermodynamic Properties for Methane Covering the Range from the Melting Line 625 K at Pressures up to 1000 MPa. J. Phys. Chem. Ref. Data 1991, 20, 1061. (28) Wagner, W.; Pruβ, A. The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use. J. Phys. Chem. Ref. Data 2002, 31, 387. (29) Perry, R. H.; Green D. W. Perry’s Chemical Engineers’ Handbook, 7th ed.; McGraw Hill: New York, 1997. (30) Castro-Marcano, F.; Olivera-Fuentes, C. G.; Colina, C. M. JouleThomson Inversion Curves and Third Virial Coefficients for Pure Fluids from Molecular-Based Models. Ind. Eng. Chem. Res. 2008, 47, 8894. (31) Colina, C. M.; Olivera-Fuentes, C. G. Predicted Inversion Curve and Third Virial Coefficients of Carbon Dioxide at High Temperatures. Ind. Eng. Chem. Res. 2002, 41, 1064. (32) Espinoza, D.; Segura, H.; Wisniak, J.; Polishuk, I. Comments on “Joule-Thomson Inversion Curves and Third Virial Coefficients for Pure Fluids from Molecular-Based Models” and “Predicted Inversion Curve and

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Third Virial Coefficients of Carbon Dioxide at High Temperatures”. Ind. Eng. Chem. Res. 2009, 48, 6901. (33) Colina, C. M.; Olivera-Fuentes, C. G. Reply to Comments by Espinoza et al. Ind. Eng. Chem. Res. 2009, 48, 6904. (34) Span, R.; Lemmon, E. W.; Jacobsen, R. T.; Wagner, W.; Yokozeki, A. A Reference Equation of State for the Thermodynamic Properties of Nitrogen for Temperatures from 63.151 to 1000 K and Pressures to 2200 MPa. J. Phys. Chem. Ref. Data 2000, 29, 1361. (35) Haar, L.; Gallagher, J. S. Thermodynamic Properties of Ammonia. J. Phys. Chem. Ref. Data 1978, 7, 635. (36) Goodwin, R. D. Methanol Thermodynamic Properties From 176 to 673 K at Pressures to 700 bar. J. Phys. Chem. Ref. Data 1987, 16, 799. (37) Polishuk, I. A Novel Four-Parameter EOS with TemperatureIndependent Parameters. Ind. Eng. Chem. Res. 2007, 46, 9248. (38) Polishuk, I.; Wisniak, J.; Segura, H. Transitional Behavior of Phase Diagrams Predicted by the Redlich-Kwong Equation of State and Classical Mixing Rules. Phys. Chem. Chem. Phys. 1999, 1, 4245. (39) Polishuk, I.; Wisniak, J.; Segura, H.; Yelash, L. V.; Kraska, T. Prediction of the Critical Llocus in Binary Mixtures Using Equation of State II Investigation of van der Waals-Type and Carnahan-Starling-Type Equations of State. Fluid Phase Equilib. 2000, 172, 1.

ReceiVed for reView June 2, 2009 ReVised manuscript receiVed July 27, 2009 Accepted August 31, 2009 IE900905P

Generalized Cubic Equation of State Adjusted to ... - ACS Publications

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