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Ind. Eng. Chem. Res. 2002, 41, 4414-4421
About the Relation between the Empirical and the Theoretically Based Parts of van der Waals-like Equations of State Ilya Polishuk,*,† Jaime Wisniak,† Hugo Segura,‡ and Thomas Kraska§ Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel, Department of Chemical Engineering, Universidad de Concepcio´ n, Concepcio´ n, Chile, and Institute of Physical Chemistry, University at Cologne, Cologne, Germany
It is demonstrated that empirical temperature functionalities for the cohesive parameter in van der Waals-like equations can be responsible for prediction of nonphysical results such as fictitious critical points of pure compounds, that can result in prediction of unrealistic phase behavior in the entire thermodynamic phase space. Simple numerical tests for detecting this pitfall are proposed, and the roles played by the empirical and the theoretically based parts in semiempirical engineering equations are investigated. In particular, it is demonstrated that improvement of the accuracy with which equations of state approximate the hard sphere virial coefficients makes the theoretical part more self-sufficient for an accurate description of data. This significantly reduces the contribution of the corresponding empirical part and has a positive effect on the overall robustness and reliability of the model. In addition, an improved theoretical base contributes to the ability of the model so simultaneously and accurately predict both liquid and vapor densities. 1. Introduction Reliable prediction of thermodynamic properties and phase equilibria of pure compounds and their mixtures in the entire thermodynamic phase space is a critical problem in modern chemical engineering that has not been satisfactorily solved yet. Successful correlations exist for pure compound liquid and solid densities, vapor pressures, heats of vaporization, liquid and solid heat capacities, and many other industrially important properties.1 These correlations may describe the desired properties within experimental accuracy, yet they have no theoretical basis. As a result, their performance depends on the available experimental data and their implementation is restricted to particular temperature and pressure ranges. In addition, usually they have not been applied to mixtures, the prediction of which is more important than the description of pure substances. Phase behavior in mixtures is much more complex than that in pure compounds. While the latter may exhibit only vapor-liquid equilibria (VLE), mixtures may split into liquid-liquid equilibria (LLE) phases. Van Konynenburg and Scott2,3 have classified all possible relations between VLE and LLE in binary mixtures into five general types. They have also demonstrated that van der Waals’ (vdW) equation of state (EOS), which can be derived from the statistical mechanics, is capable of qualitative description of these types of binary phase behavior. Previous studies have established that EOSs with the improved approximation of the hard sphere model may also predict phenomena related to a phase diagram of the sixth type.4-7 These results clearly demonstrate the advantage of theoretically based EOSs over those entirely empirical. * Corresponding author. Telephone: +972-8-6461479. Fax: +972-8-6472916. E-mail address: polishyk@ bgumail.bgu.ac.il. † Ben-Gurion University of the Negev. ‡ Universidad de Concepcio´n. § University at Cologne.
Indeed, equations that do not include empirical functionalities can be consistent in the entire thermodynamic phase space. They do not generate the nonphysical predictions that empirical functionalities can produce outside the range of fitting. Nevertheless, the results of entirely theoretical equations are often quantitatively inaccurate. For example, even if such EOSs show good agreement with pure compound experimental data in a temperature-density projection, they can exhibit inaccuracies in a pressure-temperature diagram and vice versa.8 Phase behavior in real fluids is influenced not only by the repulsive-attractive intermolecular forces (considered by a hard-body molecular theory) but also by many different factors such as polarity, hydrogen bonds, association, and others. Existing models are often unable to treat these factors properly. As a result, even the theoretically based equations sometimes include crude approximations about the molecular picture. Thus, nowadays, it does not seem possible to get an accurate prediction of the thermodynamic properties and phase equilibria in real fluids without attaching empirical functionalities to theoretically based EOSs. Such equations can be defined as semiempirical. The conventional way to derive a semiempirical EOS is to make the parameters of the theoretically based equation temperature-dependent in an empirical manner. Although this practice may significantly improve the flexibility of the resulting models and, as a consequence, their ability to fit the experimental data, there is a price to pay: the empirical temperature functionalities may remove the robustness and reliability of EOSs. For example, it is common practice9 to introduce a temperature dependence of the covolume in a cubic EOS. However, it is known that such modification can result in nonphysical predictions such as intersection of isotherms.10 These pitfalls are characteristic not only for vdW-type cubic EOSs but also for noncubic ones,11 such as that of Carnahan-Starling-de Santis, which has become an accepted tool for describing phase
10.1021/ie020102t CCC: $22.00 © 2002 American Chemical Society Published on Web 07/19/2002
Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4415
equilibria in refrigerant systems.12 Another family of empirical modifications of cubic EOSs, namely the temperature-dependent volume-translation functionalities, also gives nonphysical results, as demonstrated recently.13 Attaching a temperature-dependent character to parameters other than covolume, such as the cohesion parameter, is dangerous too. It may generate multiple pure compound critical points and, as well, nondifferentiable and nonphysical breaking points,14 which strongly affect the predicted phase behavior. However, as opposed to the case of the covolume, the cohesive parameter can be temperature-dependent, without affecting the reliable and robust character of the equation. Hence, introduction of empirical functionalities into the vdW-like EOSs seems essential for an accurate presentation of experimental data; however, it should be performed in a careful manner. Although this problem is fundamental for modern chemical engineering thermodynamics, thus far it has not been studied systematically. Hence, it becomes important to investigate the roles played by the empirical and the theoretically based parts in semiempirical engineering EOSs. The aim of the present study is to consider the mutual relation between the quality with which a vdW-like EOS approximates the hard-sphere model and the corresponding numerical contributions of the empirical parts. Important regularities, which are established by empirical temperature functionalities, are investigated as well. 2. Theory All vdW-like equations (cubic and noncubic) present pressure as a contribution of the repulsive and attractive intermolecular forces, as follows:
P ) Prepulsive - Pattractive
(1)
In what follows we will consider these two terms in more detail. 2.1. Attractive Term. The original vdW attractive term is given as
Pattractive )
a Vm2
(2)
Here a is a cohesion parameter. Although eq 2 can be derived directly from statistical mechanics, it does not provide an accurate presentation of experimental data. Therefore, to attach the EOS with the ability to fit the vapor pressure curves of pure compounds, it is necessary to make the cohesion parameter temperature-dependent. In addition, the accuracy of calculated densities can be improved by addition of empirical volumetric parameters. Thus, the resulting semiempirical expression of the attractive term can be generalized as
Pattractive )
aR (Vm + c)(Vm + d)
far, such as the one of Soave,15 given as follows:
R ) [1 + m(1 - xTr)]2
where m is a substance-dependent parameter. The significance of eq 4 for modeling phase equilibria cannot be overestimated. It has been included into very important semiempirical EOSs, such as those of RedlichKwong-Soave,15 Peng-Robinson,16 and many others. In addition, eq 4 has been the basis for the development of many other R-functionalities such as that of MathiasCopeman,17 which have also found wide application in developing EOS models. However, eq 4 and all its recent modifications include some undesirable numerical pitfalls. Consider, for example, the following mixing rule for the cohesive parameter:
aR )
∑ij xixj(aijRij)
(5)
where
aijRij ) (1 - kij)xaiiRiiajjRjj
(6)
and kij is a binary adjustable parameter. One can then obtain
xRiRj ) |1 + mi[1 - xTr,i]||1 + mj[1 - xTr,j]|
(7)
Since the value of the parameter m is almost always positive, increasing the temperature results first in a decrease of the values inside the brackets, who intersect zero and become negative. Afterward, however, the absolute value changes direction and begins to increase again, and when 1 + mz[1 - xTr,z] ) 0, a nonphysical and nondifferentiable breaking point appears at the temperature Tr,z,breaking ) ((1 + mz)/mz)2, where z ) i, j. Although these breaking points usually occur at very high temperatures, they can strongly affect the phase behavior predicted by the model at ordinary conditions.14 Once again, such results prove the close interrelation between all parts of the thermodynamic phase space predicted by an EOS.18 There are additional numerical pitfalls that can be generated by empirical expressions such as that given by eq 4. The fact that these expressions may not decrease monotonically with temperature leads to prediction of a nonphysical trend for the second virial coefficient, which exhibits curvature changes. This behavior leads to the appearance of multiple JouleThompson inversion curves, as well as fictitious critical points of the pure compounds. In what follows, we propose a simple numerical test for detecting multiple pure compound critical points generated by thermodynamically wrong equations. The mechanical stability conditions at the critical point are given as follows:
( ) ( )
(3)
where c and d are volumetric parameters and R is a temperature functionality. While the temperatureindependent volumetric parameters are not supposed to affect the robustness and reliability of the EOS in the entire thermodynamic phase space, this is not the case of many empirical R-functionalities proposed thus
(4)
∂P ∂Vm
and
)
Tc
∂2 P ∂Vm2
( ) ∂3 P ∂Vm3
Tc
)0
(8)
Tc
1, the temperature of a fictitious critical point is given by
Tr )
(1 + m)2 (1 - m)2
RT
∞
)1+
Bm+1ym ∑ m)1
(13)
Here y is the packing fraction and B2, B3, ... are the virial coefficients, which are related to the molecular forces that exist between molecules.19 Thus, B2 represents interactions between two molecules, B3 represents interactions between three molecules, and so forth. For the repulsion, the hard sphere model is employed. The values of the virial coefficients in eq 13 for the hard sphere model are available in the literature.20 However, it is not practical to present the repulsive term of the EOS by a complex expression such as eq 13. For this reason, many engineering EOSs employ the simple vdW repulsion term:
Prepulsive )
RT Vm - b
weak theoretical basis because its virial expansion yields an inaccurate approximation of B3 as well as the subsequent hard sphere virial coefficients. To overcome this difficulty, several advanced expressions for the repulsion term that are capable of an accurate approximation of eq 13 have been proposed.21 Several of them do not keep the cubic form and cannot be reduced to eq 14, which still continues to be an integral part of a successful semiempirical EOS. Hence, the following more general expression for the repulsive term of the cubic EOS (which keeps the correct value of B2) has been recently developed:22
(12)
Examination of eqs 7 and 12 indicates that increasing the value of the parameter m promotes nonphysical predictions by decreasing the temperature of the fictitious critical point. That is, eq 4 outlines a fundamental regularity: an increase in the numerical contribution of the empirical part affects the robustness and reliability of any semiempirical EOS model. The same analysis can be done for other empirical temperature functionalities, usually with similar results. 2.2. Repulsive Term. Let us now consider the other term of a vdW-like EOS, the repulsion term. A common approach in the hard sphere theory is the virial equation
PrepulsiveVm
Figure 1. Influence of factor j on the accuracy of approximating hard sphere virial coefficients by eq 15.
(14)
Here b is the covolume. Although eq 14 is very important in chemical engineering thermodynamics, it has a
Prepulsive )
[
]
RT 1 + y(4 - j) Vm 1 - yj
(15)
where
y)
b 4Vm
(16)
and j is a scaling factor. Figure 1 presents the influence of the scaling factor j on the accuracy of approximating the theoretical values of hard-sphere virial coefficients by eq 15 in the plot of ln(Bi) against the number of the virial coefficient i, as suggested for such as an investigation recently.21 For j ) 2, eq 15 is equivalent to the expression proposed by Scott.23 Figure 1 demonstrates that the proposed equation predicts values that are close to the theoretical ones. Increasing the value of j to 2.5 yields the expression of Zhang et al.,24 which approximates an exact value of B3; however, it is less accurate than Scott’s expression for the subsequent hard sphere virial coefficients. A further increase of j affects the agreement between eq 15 and the hard sphere model. The agreement becomes poor for j ) 4, where eq 15 reduces to eq 14. In other words, increasing the scaling factor j has a similar effect to that of increasing the parameter m in eq 4: it reduces the theoretical base of the EOS and increases its empirical character. It seems, then, that the qualities with which vdW-like EOSs approximate the hard sphere virial coefficients and the corresponding numerical contribution of the empirical part (expressed by the values of j and m) are interrelated. We will now test the validity of this assumption and discuss some additional characteristic features of vdWlike EOSs.
Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4417 Table 1. Values of Parameters in Eq 17 compound
j
m
a
4 0.2996 2.6742 3.5 0.2203 2.7524 3 0.1242 2.8526 2.5 0.0042 2.9886 2 -0.1528 3.1901 carbon dioxide 4 0.4353 4.5547 3.5 0.3378 4.6636 3 0.2185 4.8049 2.5 0.0656 5.0020 2 -0.1592 5.3048 chlorobenzene 4 0.5843 30.931 3.5 0.4879 31.633 3 0.3681 32.519 2.5 0.2103 33.713 2 0.0278 35.481 n-heptane 4 0.6833 37.892 3.5 0.5731 38.700 3 0.4392 39.719 2.5 0.2616 41.100 2 -0.031 43.163
jb/4 (covolume)
methane
0.0355
0.0373
0.1023
0.1452
c
d
-0.0258 -0.0277 -0.0296 -0.0315 -0.0333 -0.0312 -0.0326 -0.0339 -0.0353 -0.0367 -0.0756 -0.0813 -0.0871 -0.0930 -0.0991 -0.1118 -0.1192 -0.1269 -0.1348 -0.1427
0.0805 0.0793 0.0770 0.0729 0.0667 0.1009 0.0970 0.0917 0.0844 0.0747 0.3249 0.3192 0.3089 0.2924 0.2673 0.4838 0.4729 0.4550 0.4279 0.3883
3. Results and Discussion Let us consider first the generalized vdW-like EOS obtained by combining eqs 3 and 15:
P)
[
]
a[1 + m(1 - xTr)] RT 1 + y(4 - j) (17) Vm 1 - yj (Vm + c)(Vm + d)
where the values of the parameters a, b, c, and d are calculated using the approach described previously.18,22 (The pertinent values appear in Table 1; pure compound data have been taken from ref 1.) In previous publications,18,22 the critical compressibility factor of the EOS has been set equal to the experimental value, giving an accurate representation of the liquid phase and an improvement of the ability of the model to predict global phase behavior in homologous series.18,22 However, the EOS has lost its accuracy to describe the behavior of the vapor phase. To solve this problem, it has been proposed to multiply the critical compressibility of the EOS by a number larger than one.10 Although this practice improves the prediction of the vapor phase, it may affect the topology of the predicted phase behavior in mixtures.18.22 The question may be asked: Is it possible for vdWlike EOSs to be simultaneously accurate for the liquid and vapor phases despite their inability to describe the density fluctuations in the critical region?25 To answer this question, we have multiplied the experimental value of the critical compressibility by a factor of 1.1 (to improve the description of the vapor phase) and studied the influence of factor j on the predicted liquid molar volumes of several pure compounds. We have selected methane and carbon dioxide to represent simple molecules; n-heptane, a chain molecule, and chlorobenzene, an aromatic compound with relatively complex geometry. All these substances do not exhibit aggregation, which may introduce undesired complexity and hinder analysis of the results yielded by eq 17. The pertinent results are presented in Figure 2. It is seen that although the selected compounds have different properties, factor j affects the prediction of their molar volumes in a similar manner. The results obtained using the classical value of j ) 4 are not surprising: they are accurate for the vapor phase and inaccurate for the liquid one. However, it can be seen that a decrease of j (which improves the theoretical base
of the EOS) has a positive effect on the predicted phase envelopes. In a previous study,8 a similar regularity has been found for a generalized vdW-like EOS and a generalized attracting hard-sphere EOS. In the case of simple molecules (Figure 2a and b), the relation between the quality of approximating the hard sphere virial coefficients and the accuracy of representing experimental volumetric data is evident. The best results correspond to j ) 2, which is a reasonable description of the hard sphere model. However, this is not the case for complex molecules (Figure 2c and d). It seems that values of j around 2.5-3 yield more accurate results than those for j ) 2. Figure 2d demonstrates that the latter value may even affect the prediction of the vapor phase. These results, however, do not contradict the theory, since chlorobenzene and n-heptane are not spherical molecules; their phase behavior cannot be described by the hard sphere theory properly. The development of a simple, accurate, and theoretically based EOSs for complex molecules is a very important task for modern chemical engineering thermodynamics. A promising approach is correcting the hard sphere theory by theoretically based shape factors. The following simple expression for the repulsive term has been recently proposed26 and successfully applied for investigating global phase behavior in binary fluid mixtures of chain molecules:27
Prepulsive )
(
)
RT 3 + Ay + By2 4y b (1 - y)(3 - 4y)
(18)
The parameters A and B represent different kinds of molecular shapes. Development of the appropriate mixing rules for these parameters and the semiempirical attractive term that could match eq 18 is a challenging aim for forthcoming research. However, at this stage it can be inferred that significant progress in solving the problem of the simultaneous prediction of pure compound liquid and vapor densities (which seems essential for the reliable description of the phase behavior of mixtures) may be achieved by proper combination of molecular theory and the mathematical structure of vdW-like EOSs. Another fundamental regularity can be detected by considering the data presented in Table 1: a reduction of the factor j causes a decrease of the parameter m. This result can be explained by analyzing Figure 3, which presents the influence of the factor j on the vapor pressure curves predicted by eq 17 without an empirical R-functionality. In this figure, the performance of the theoretically based part of the equation is analyzed separately. Once again, it can be seen that a decrease of the value of j significantly improves the results for both simple and complex molecules. For example, j ) 2.5 yields an accurate prediction of the vapor pressure of methane (Figure 3a). One may even think about the possibility of fitting vapor pressures not by empirical R-functionalities but by the values of factor j. The theoretical background of such a procedure is given by eq 18, in which the fit of the data may be performed with shape factors. It is obvious that this model would be free from undesired numerical pitfalls such as those generated by eq 4. Regarding eq 4, it can be concluded that the value of the parameter m reflects the inaccuracy with which the theoretically based part of vdW-like EOSs represents
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Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002
Figure 2. Influence of factor j on predicting pure compound molar volumes by eq 17: j ) 4, dotted line; j ) 2.5, solid line; j ) 2, dotdashed line; experimental points from ref 28, circles.
the vapor pressure lines of pure compounds. Accurate representation of these data requires low values of the parameter and, consequently, a less empirical character of the EOS. Therefore, the results presented here confirm the validity of the assumption that there is a close connection between the values of j and m, which measure the contributions of the theoretical and the empirical parts of eq 17. In other words, improvement of the theoretical basis of vdW-like EOSs makes the theoretical part more self-sufficient for an accurate description of vapor pressure data, which reduces the need for empirical corrections. We will now consider the practical consequences of this regularity. It has already been shown that an increase of the empirical part of the EOS decreases its robustness and reliability. Nonphysical phase diagrams generated by Soave’s temperature functionality15 and its derivatives17 have already been considered.14 However, the same numerical pitfalls also characterize the majority of empirical R-functionalities proposed thus far, including those not related to eq 4. For example, implementation of the test given by eq 11 to Melchem’s29 expression
ln R ) m[1 - Tr] + n[1 - xTr]2
(19)
indicates that it is thermodynamically wrong. Therefore, it is not surprising that several pure compounds listed in the ref 29 exhibit fictitious critical points at both low
Table 2. Values of Parameters in the EOS Combined from Eqs 15 and 19 and the Attractive Term of for sulfur trioxide Peng-Robinson j
m
n
4a
0.4072 0.4391 0.2978
6.1502 2.8008 2.0884
2.5 2 a
Values obtained from ref 29.
and high temperatures. Obviously, the best candidates for such nonphysical phase behavior are compounds having high values of the empirical parameters m and n (see discussion above). This is the case of sulfur trioxide (see Table 2), which, according to Melchem,29 exhibits, in addition to the normal critical point at 491 K, a fictitious one at 1005.95 K. Figure 4 provides an explanation for the high values of the empirical parameters: the Peng-Robinson EOS without temperature functionality is not a good descriptor of the SO3 vapor pressure curve. It is remarkable that the regularity present in Figure 3 is valid not only for eq 3 but also for every form of the attractive term. Therefore, it becomes evident why replacement of eq 14 by eq 15 in the EOS of Peng-Robinson,16 accompanied by a decrease in the value of j, improves considerably the performance of the theoretically based part (see Figure 4). When the value of j is assumed to be 2.5 the values of m and n are low enough (see Table 2) to avoid the nonphysical predictions yielded by j ) 4.
Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4419
Figure 3. Influence of factor j on predicting pure compound vapor pressures by eq 17 with m ) 0: j ) 4, dotted line; j ) 3.5, dashed line; j ) 3, dot-dot-dashed line; j ) 2.5, solid line; j ) 2, dot-dashed line; experimental points from ref 28, circles.
To emphasize this fact more, we have implemented Melchem’s modification of the Peng-Robinson EOS16 for correlating the critical data of the industrially important SO3-H2O system.31 The strong chemical interaction between SO3 and H2O makes the prediction difficult, although some attempts have been made in the past.31,32 The hard sphere approach does not consider such interactions; however, they can be represented empirically by the values of the binary adjustable parameters k12 in eq 6, and the values of l12 in the combining rule for covolume:
b11 + b22 2
b12 ) (1 - l12)
Figure 4. Influence of factor j on predicting the vapor pressure of SO3 by the EOS combined from eq 15 and the attractive term of Peng-Robinson without temperature functionality.
It can be argued that the numerical pitfalls take place outside the range of application of the model.30 Such an argument is invalid because it neglects the fundamental rule that all regions of the predicted thermodynamic phase space are closely interrelated. Although the fictitious critical point of SO3 appears at 1005.95 K (which is without any doubt outside the range of practical importance), still, it will also seriously affect the results inside the range of application of the model.
(20)
To simulate chemical interactions and their high excess enthalpy, it is necessary to consider strong attractive and weak repulsive cross-interactions by assuming high negative values for k12 and high positive values for l12. These values predict a strong negative azeotropy, which matches the experimental data. The critical lines have been calculated using the methods described previously.33-35 The pertinent results appear in Figure 5a and c, where it is seen that Melchem’s29 model predicts a critical curve, which does not connect the real critical points of pure compounds but the real and the fictitious critical points of sulfur trioxide. This result leads to the nonphysical prediction of the entire thermodynamic phase space. In contrast to this, using the value j ) 2.5 (Figure 5b and d) allows a qualitatively
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Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002
Figure 5. Critical lines in the system SO3- H3O calculated by the EOS combined from eqs 15-19 and the attractive term of PengRobinson: k12 ) -0.55; l12 ) 0.3; solid lines, calculated critical lines; circles, experimental points from the ref 31.
correct description of the data, showing once again the advantage of the theoretically based approach. IV. Conclusions An increasing request for modeling complex phase equilibria data encourages development of empirical functionalities that improve the flexibility of cubic EOS models. However, many of these functionalities can become inconsistent depending on the parameters obtained in the correlation. In particular, they may generate the prediction of fictitious critical points of the pure compounds or nondifferentiable breaking points. Although these pitfalls may take place outside the range of application of an EOS, they still can affect the results at ordinary conditions, because all regions of the predicted thermodynamic phase space are closely interrelated. A simple numerical test is proposed here for detecting these pitfalls. Although empirical functionalities have clear disadvantages, today it does not seem possible to get the simultaneous and accurate modeling of different thermodynamic properties without their assistance. This fact emphasizes the need for fundamental and systematic research of the roles played by the empirical and theoretically based parts in semiempirical engineering equations. In the present study we have considered the mutual relations exhibited by these two parts of vdWlike EOSs. In particular it was demonstrated that the improvement of the theoretical base of the EOS, namely the accuracy with which it approximates the hard sphere virial coefficients, presents the following advantages:
(1) It makes the theoretical part of the EOS more selfsufficient for an accurate description of vapor pressure data, which significantly reduces the numerical value of its corresponding empirical part and has a positive effect on the overall robustness and reliability of the model. (2) An improved theoretical basis also contributes to the ability of the model to yield accurate results for both liquid and vapor densities simultaneously. This capability is highly significant. The results presented here emphasize the advantage of replacing the theoretically inaccurate classical repulsive term of van der Waals in semiempirical engineering EOSs by one have a stronger theoretical basis. Acknowledgment This work was financed by the Israel Science Foundation, Grant Number 340/00. I.P. acknowledges a fellowship of Deutscher Akademischer Austauschdienst (DAAD) for a short-term stay at the University of Cologne. Literature Cited (1) Daubert, T. E.; Danner, R. P.; Sibul, H. M.; Stebbins, C. C. Physical and Thermodynamic Properties of Pure Chemicals. Data Compilations; Taylor & Francis: Bristol, PA, 1989-2001. (2) van Konynenburg, P. H. Critical Lines and Phase Equilibria in Binary Mixtures. Ph.D. Thesis, University of California, Los Angeles, 1968. (3) van Konynenburg, P. H.; Scott, R. L. Critical Lines and Phase Equilibria in Binary van der Waals Mixtures. Philos. Trans. R. Soc. London 1980, 298, 495.
Ind. Eng. Chem. Res., Vol. 41, No. 17, 2002 4421 (4) Yelash, L. V.; Kraska, T. Closed-Loops of Liquid-Liquid Immiscibility in Binary Mixtures of Equal Sized Molecules Predicted with a Simple Theoretical Equations of State. Ber. BunsenGes. Phys. Chem. 1998, 102, 213. (5) Yelash, L. V.; Kraska, T. On Closed-Loops of Liquid-Liquid Immiscibility. Phys. Chem. Chem. Phys. 1999, 1, 307. (6) Yelash, L. V.; Kraska, T.; Deiters, U. K. Closed-Loop Critical Curves in Simple Hard-Sphere van der Waals-Fluid Models Consistent With the Packing Fraction Limit. J. Chem. Phys. 1999, 110, 3079. (7) Wang, J.-L.; Wu, G.-W.; Sadus, R. J. Closed-Loop LiquidLiquid Equilibria and the Global Phase Behavior of Binary Mixtures Involving Hard-Sphere + van der Waals Interactions. Mol. Phys. 2000, 98, 715. (8) Yelash, L. V.; Kraska, T. Investigation of a Generalized Attraction Term of an Equation of State and its Influence on the Phase Behaviour. Fluid Phase Equilib. 1999, 162, 115. (9) Nasrifar, Kh.; Moshfeghian, M. A New Cubic Equation of State for Simple Fluids: Pure and Mixture. Fluid Phase Equilib. 2001, 190, 73. (10) Salim, P. H.; Trebble, M. A. A Modified Trebble-Bishnoi Equation of State: Thermodynamic Consistency Revisited. Fluid Phase Equilib. 1991, 65, 59. (11) Polishuk, I.; Wisniak, J.; Segura, H.; Yelash, L. V.; Kraska, T. Prediction of the Critical Locus 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. (12) Di Nicola, G.; Polonara, F.; Stryjek, R. P-V-T-x and VLE Properties of Difluoromethane (R32) + 1,1,1,2,3,3-Hexafluoropropane (R236ea) and Pentafluoroethane (R125) + R236ea Systems Derived from Isochoric Measurements. J. Chem. Eng. Data 2001, 46, 367. (13) Pfohl, O. Evaluation of an Improved Volume Translation for the Prediction of Hydrocarbon Volumetric Properties. Fluid Phase Equilib. 1999, 163, 157. (14) Polishuk, I.; Wisniak, J.; Segura, H. Closed Loops of Liquid-Liquid Immiscibility Predicted by Semiempirical Cubic Equations of State and Classical Mixing Rules. Phys. Chem. Chem. Phys. 2002, 4, 879. (15) Soave, G. Equilibrium Constants from a Modified RedlichKwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197. (16) Peng, D. Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. (17) Mathias, P. M.; Copeman, T. W. Extension of the PengRobinson Equation of State to Complex Mixtures: Evaluation of the Various Forms of the Local Composition Concept. Fluid Phase Equilib. 1983, 13, 91. (18) Polishuk, I.; Wisniak, J.; Segura, H. Simultaneous Prediction of the Critical and Sub-Critical Phase Behavior in Mixtures Using Equation of State I. Carbon Dioxide-Alkanols. Chem. Eng. Sci. 2001, 56, 6485. (19) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley & Sons: New York, 1964. (20) van Rensburg, J. Virial Coefficients for Hard Discs and Hard Spheres. J. Phys. A 1993, 26, 4805.
(21) Yelash, L. V.; Kraska, T. A Generic Equation of State for Hard-Sphere Fluid Incorporating the High-Density Limit. Phys. Chem. Chem. Phys. 2001, 3, 3114. (22) Polishuk, I.; Wisniak, J.; Segura, H. A Novel Approach for Defining Parameters in a Four-Parameter EOS. Chem. Eng. Sci. 2000, 55, 5705. (23) Scott, R. L. Physical Chemistry, an Advanced Treatise; Academic Press: New York, 1971; Vol. 8A. (24) Zhang, H.-L.; Sato, H.; Watanabe, K. A New ThreeParameter Cubic Equation of State for Refrigeration Engineering Calculations. Int. J. Refrig. 1997, 20, 421. (25) Sengers, J. V.; Levelt Sengers, J. M. H. Thermodynamic Behavior of Fluids Near the Critical Point. Annu. Rev. Phys. Chem. 1986, 37, 189. (26) Yelash, L. V.; Kraska, T.; Mu¨ller, E. A.; Carnahan, N. F. Simplified Equation of State for Non-Spherical Hard Particles: an Optimized Shape Factor Approach. Phys. Chem. Chem. Phys. 1999, 1, 4919. (27) Yelash, L. V.; Kraska, T. The Global Phase Behavior of Binary Mixtures of Chain Molecules: Theory and Application. Phys. Chem. Chem. Phys. 1999, 1, 4315. (28) Stephan, K.; Hildwein, H. Recommended Data of Selected Compounds and Binary Mixtures 1 + 2. Tables, Diagrams and Correlations; DECHEMA Chemistry Data Series; DECHEMA: Stuttgart, 1987. (29) Melchem, G. A. A Modified Peng-Robinson Equation of State. Fluid Phase Equilib. 1989, 47, 189. (30) Ungerer, P.; de Sant’Ana, H. B. Reply to the Letter to the Editor by O. Pfohl about the Paper ‘‘Evaluation of an Improved Volume Translation for the Prediction of Hydrocarbon Volumetric Properties”. Fluid Phase Equilib. 1999, 163, 161. (31) Stuckey, J. E.; Secoy, C. H. Critical Temperatures and Densities of the SO3-H2O System. J. Chem. Eng. Data 1963, 8, 386. (32) Engels, H.; Lieberman, A.; Olf, G. Thermodynamically Consistent Description of Vapor/Liquid-Phase Equilibria and Heat of Mixing of Water/Sulfur Trioxide (in German). Chem.-Ing. Tech. 1991, 63, 1247. (33) Polishuk, I.; Wisniak, J.; Segura, H. Prediction of the Critical Locus in Binary Mixtures Using Equation of State I. Cubic Equations of State, Classical Mixing Rules, Mixtures of MethaneAlkanes. Fluid Phase Equilib. 1999, 164, 13. (34) 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. (35) Segura, H.; Polishuk, I.; Wisniak, J. Phase Stability Analysis in Binary Systems. Phys. Chem. Liq. 2000, 38, 277.
Received for review February 5, 2002 Revised manuscript received June 5, 2002 Accepted June 20, 2002 IE020102T
