prediction of the critical locus in binary mixtures using equation of state: ii. investigation of...

Ž .Fluid Phase Equilibria 172 2000 1–26www.elsevier.nlrlocaterfluid

Prediction of the critical locus in binary mixtures using equationof state

II. Investigation of van der Waals-type and Carnahan–Starling-typeequations of state

Ilya Polishuk a,) , Jaime Wisniak a, Hugo Segura b, Leonid V. Yelash c, Thomas Kraska c,1

a Department of Chemical Engineering, Ben-Gurion UniÕersity of the NegeÕ, Beer-SheÕa, Israelb Department of Chemical Engineering, UniÕersidad de Concepcion, Concepcion, Chile´ ´

c Institute of Physical Chemistry, UniÕersity at Cologne, Cologne, Germany

Received 8 November 1999; accepted 11 April 2000

Abstract

The ability to predict critical lines of members of the methane–, perfluoromethane– and water–alkanesŽ . Ž .homologous series is compared for van der Waals vdW -type and Carnahan–Starling CS -type equations of

state. A temperature dependent combining rule for the binary attraction parameter is discussed and employed. Itis found that the appropriate choice of the adjustable parameters yields quite accurate results for both equations.A new application of global phase diagrams is proposed for the quantitative description of real mixtures. In thisdiagram, the boundaries of the different types of phase behavior are presented in the k –l plane. Analysis of12 12

this diagram has allowed us to reach conclusions that cannot be obtained by a simple fit of data points. Inparticular, it is demonstrated that the global phase diagram’s shape defines the correlative ability of theequations. It is found that CS-type equations tend to predict a larger region of liquid–liquid immiscibility, theaccuracy of the result depends on the particular experimental system. Changes in the density dependence of theattraction term of the two-parameter equations influence mostly the predicted critical volumes and not theirqualitative performance. In addition, the development of a CS-type equation suitable for engineering calcula-tions is discussed. q 2000 Elsevier Science B.V. All rights reserved.

Keywords: Vapor–liquid equilibria; Liquid–liquid equilibria; Global phase diagram; Method of calculation; Equation ofstate; Critical state

) Corresponding author.Ž . Ž .E-mail addresses: [email protected] I. Polishuk , [email protected] koelnode T. Kraska .–

1 Also corresponding author.

0378-3812r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0378-3812 00 00366-6

( )I. Polishuk et al.rFluid Phase Equilibria 172 2000 1–262

1. Introduction

Process design in the near-critical and critical regions has become an important problem in modernŽ .industry. Predictions in the critical region represent a severe test for different equations of state EOS

and mixing rules. However, not many of the publications available are devoted to the prediction ofcritical lines and their inspection shows that this problem has not been satisfactorily solved yet. In aprevious paper, a systematic investigation of the performance of different equations in the critical

w xregion was started 1 . Six of the most popular cubic equations of the van der Waals type withclassical mixing rules were used for predicting the critical lines of seven methane–alkane systems.Although each equation tested has a different attractive term, they yield very similar results. It wasfound that cubic equations are capable of quantitatively predicting the P–T projections of critical

Ž w x.lines of Types I and V systems according to the classification of van Konynenburg and Scott 2 .However, the values of the interaction parameters for the critical lines were different from those in thesubcritical region. In addition, deviations were found between the experimental and calculated data inthe P–x projections of type V systems. In the present study, we investigate some systems employinga simple linear temperature-dependent combining rule and show that it allows, simultaneously, a moreaccurate prediction of the gas–liquid critical lines in the P–T and the P–x projections.

The purpose of this work is to study the influence of the structure of the equation of state on itsw xpredictive ability, following a recent systematic investigation 3 discussing the general aspects of the

structure of the equation of state. Here we perform a quantitative comparison of the critical linespredicted by different types of EOS. An important goal of the present study is to analyze the results ofreplacing the van der Waals repulsion term with a more physically based equation. The hard-sphere

Ž .repulsion term of Carnahan–Starling CS has been chosen for this purpose. Here we focus on thecorrelation of experimental data with the CS-type equation and on the investigation of the shortcom-ings of the model. This approach should give directions for further improvements of hard sphere-typemodels.

The investigation of the performance of an equation can be accomplished systematically withŽ .global phase diagrams GPD . For this reason, another important topic of the present study is to

provide explanations based on GPD for the results obtained with the vdW-type and CS-typeequations. Global phase diagrams are maps of phase behavior in the molecular parameter space, they

w xwere introduced by van Konynenburg and Scott 2 and today have become a powerful tool of modernthermodynamics. The developments of the methodology and applications of GPD were recently

w xpresented at the First Workshop on Global Phase Diagrams 4 . Most of the research on GPD focuseson the general aspects of fluid phase equilibria, i.e., a qualitative description of the different types of

w xphase behavior. As discussed in Ref. 1 , representation of real binary systems by an EOS with onebinary interaction parameter and without temperature-dependent attraction parameters requires athree-dimensional GPD. Introduction of a second interaction parameter or a temperature dependentattraction parameter requires additional dimensions of the GPD, making difficult its application in theoriginal form for the quantitative description of particular real mixtures. Therefore, in the presentstudy we introduce a different intersection of the molecular parameter space, in which the boundariesof the different types of phase behavior are presented in the k –l plane. Here k and l are the12 12 12 12

binary interaction parameters for the attraction and the co-volume, respectively. It is shown that thisŽ .new k –l -global phase diagram klGPD is able to represent all possible types of phase behavior12 12

that a particular equation can predict for a particular binary system. The choice of this intersection

( )I. Polishuk et al.rFluid Phase Equilibria 172 2000 1–26 3

has practical reasons and does not reduce the generality of the GPD approach. In addition, klGPDallows the quantitative representation of binary systems and can be applied to the investigation of thepredictive abilities of equations of state. It must be realized that a simple fit of the parameters of anEOS will usually yield a good description in the fitted range only. It will represent a quantitativedescription of the experimental data in a given range of temperature and pressure. Proper overallrepresentation of phase behavior requires its analysis in a GPD, such as the klGPD. In addition, theGPD gives the direction for the optimal set of parameters globally. Comparison of the kl-global phasediagrams of different models is a very useful tool for investigating the characteristic features ofequations and the influence of different parameters. Such an investigation is necessary for improvingequation of state models.

2. Equations of state for pure components

The modeling of thermodynamic fluid phase properties of pure substances and mixtures is usuallyw xperformed with equations of state. The total number of EOS published thus far exceeds 2000 5 .

Most of them are based on the idea of expressing the pressure as the sum of a repulsive and anw xattractive term, as proposed by van der Waals 6 .

PsP R ,T ,V ,b yP V ,a,b 1Ž . Ž . Ž .rep m att m

where a is the attraction parameter and b the co-volume of the particular substance. The expressionw xfor the repulsive term proposed by van der Waals in 1873 6 continues to be the one most commonly

used

P sRTr V yb 2Ž . Ž .rep m

w xCarnahan and Starling 7 developed an expression for the repulsive term, which is an accuraterepresentation of the repulsion pressure of non-attracting hard-sphere molecules in the fluid phase

RT 1qyqy2 yy3

P s 3Ž .rep 3V 1yyŽ .m

where ysbr4V is the packing fraction.m

The Carnahan–Starling equation, as well as the repulsion term of the van der Waals equation, canbe combined with different attractive terms. Each combination of repulsion and attraction terms yieldsdifferent values of the parameters a, b and the critical compressibility factor Z . The attractionc

parameter a and the co-volume b can be calculated from the thermodynamic conditions of a criticalw xpoint of a pure fluid 8 .

EP E2Ps s0 4Ž .2ž / ž /EV EVm mT Tc c


In addition, the mechanic stability condition has to be fulfilled.

E3P-0 5Ž .3ž /EVm Tc

The values of a and b are related to the critical constants of the pure compounds

as V R2T 2rP a T bsV RT rP 6Ž . Ž .Ž .a c c b c c

Ž . Ž .where V and V are characteristic constants for a specific equation of state. In Eq. 6 , a T is aa b

temperature dependent function which is equal to 1 at the critical point. The following general formw xhas been proposed 3 to study the effect of the attraction term on the performance of vdW-type

equation

RT aPs y 7Ž .

V yb V V qcbŽ .m m m

where c is a parameter that scales between the density dependencies of different attraction terms. Forexample, cs0 yields the density dependence of the vdW equation and cs1 yields the density

Ž . w x Ž . w xdependence of the Redlich–Kwong RK 9 or the Redlich–Kwong–Soave RKS equation 10 . Inw x Ž . w xaddition, it was proposed 3 to combine the attraction term of Eq. 7 with the CS repulsion term 7 :

2 3RT 1qyqy yy aPs y 8Ž .3V 1yy V V qcbŽ .Ž .m m m

It was found that an increasing value of c decreases the value of the critical compressibility factor ofw xthe EOS 3 .

Accurate prediction of the vapor pressure line is essential for a quantitative description of criticallines and their end points in particular because the position of the three-phase line is often related tothe vapor pressure line of the more volatile substance. The analysis of van der Waals-like equations of

w xstate shows that these equations cannot yield a satisfactory quantitative fit to experimental results 11 .This problem is usually addressed by including empirical temperature dependencies of the attraction

w xparameter. De Santis et al. 12 have proposed combining the CS repulsion term with an attractionterm which, in addition to a temperature dependent attraction parameter, includes also a temperaturedependent co-volume. They have demonstrated that the resulting equation is capable of accuratelypredicting the vapor–liquid equilibria of pure compounds as well as that of several mixtures ofnon-polar substances.

w x Ž .Morrison and McLinden 13 have applied the Carnahan–Starling–De Santis equation CSD forcalculating the phase behavior of refrigerant gases and found that it yields an accurate representationof the volumetric properties along the bubble- and dew-point curves outside the critical region.Recently, the CSD equation has been used for correlating the thermodynamic properties for pure

w x w x w xrefrigerants 14 and their mixtures 15–17 . The results 17 indicate that although the CSD equationincludes a temperature dependence co-volume parameter b, it shows no significant advantage over the

w xvapor–liquid equilibria results predicted by the Peng–Robinson equation. Li et al. 18 have proposed


combining the repulsive term of Carnahan–Starling with the attractive term of the Patel–Tejaequation, making all three parameters of the resulting equation temperature-dependent. In addition, Li

w xand Zheng 19 have used their equation for correlating binary critical loci and concluded that in an

Fig. 1. Pressure–volume diagram of propane calculated using the CSD and CS-type equations. Solid lines – isotherms, thicksolid line – coexistence curve.


equation having temperature-dependent parameters, the Carnahan–Starling repulsive term is prefer-w xable over that of van der Waals. Recently, Al-Shafe’i and Mecarik 20 have also proposed improving´

the CSD equation by including a third temperature-dependent parameter into the attraction term.w xTrebble and Bishnoi 21 have shown that the introduction of a temperature dependent co-volume

improves the representation of the volumetric properties along the bubble- and dew-point curves alsow xfor the vdW-type equation. However, Salim and Trebble 22 have shown that a temperature

dependent co-volume leads to inconsistency in the thermodynamic properties predicted by a vdW-typeEOS. A similar inconsistency has also been observed for the temperature-dependent volume-transla-

w x w xtion technique 23 . So far, Salim and Trebble 22 have suggested to omit the temperaturew xdependence of the co-volume proposed before 21 . Fig. 1 compares the isotherms of propane

Ž .calculated with the CSD and CS-type equation where Eq. 8 has been used with the Soavetemperature functionality shown below. The parameters a and b of CSD have been taken from Ref.w x Ž .14 . It is seen that at high pressures, contrary to the CS-type equation CSD leads to a crossing ofthe isotherms. We can conclude that this equation should not be used outside the liquid–gascoexistence region and that its significance, in comparison with empirical correlations, is doubtful.Similar conclusions can be drawn also about other CS-type equations, which use a temperature-depen-dent parameter b.

In the present study, we have kept the co-volume temperature independent and introduced atemperature dependence for the attraction parameter a only. For both the vdW-type and the CS-type

w xequations, we have used the temperature dependence proposed by Soave 10

2as 1qm 1y T 9( Ž .ž /r

Here m is a constant characteristic for each pure compound. For non-polar substances, Soave hasdeveloped an empirical correlation for m as a function of the acentric factor. The same approach has

Table 1Values of constant m in the Soave temperature dependence

Substance vdW-type equation CS-type equation

Ž .methane cs1 0.491022 0.250674Ž .methane cs2 0.274696Ž .methane cs3 0.437264 0.276592Ž .methane cs5 0.392386

Ž .butane cs1 0.486522Ž .pentane cs1 0.861331 0.548088Ž .pentane cs2 0.580838Ž .pentane cs3 0.797584 0.585522Ž .pentane cs5 0.744388Ž .heptane cs1 1.006558 0.663698Ž .dodecane cs1 1.325560 0.915883

Ž .tetrafluoromethane cs1 0.750499 0.459526Ž .water cs1 0.981230 0.644282

Ž .2-butanol cs1 1.371387 0.948884


been adopted in other models such as the Peng–Robinson and the Patel–Teja equations. The value ofthe parameter m is characteristic for each pure compound for each specific equation of state. Table 1lists the values for m for the systems investigated here. The table shows that the m-values obtainedwith a CS-type equation are smaller than those obtained with a vdW-type equation. In other words, a

w xCS-type equation yields a better prediction of the vapor pressure with as1. It has been shown 1that although a temperature dependence with more than one parameter can give better correlations ofthe vapor pressure curves, its influence on the critical lines of mixtures is not significant.

3. Mixing rules

The extension of EOS to mixtures is usually accomplished by stating mixing rules for the equationparameters along the van der Waals one-fluid theory. With this approach a mixture is treated as apseudo-pure fluid and the attraction parameter a and the co-volume b are calculated using thefollowing quadratic mixing rules

as x x a 10Ž .Ý i j i jij

bs x x bÝ i j i jij

The cross-interaction parameters a and b are obtained using the combining rules12 12

a sa s 1yk a a 11(Ž . Ž .21 12 12 11 22

b qb11 22b sb s 1y lŽ .21 12 12 2

w xwhere k and l are interaction parameters. As shown before 1 , non-zero values of k can yield a12 12 12w xgood fit of the experimental data. Recently, Yelash and Kraska 24 have shown that l can improve12

the description of liquid–liquid immiscibility and described a phase diagram that cannot be obtainedwith a zero value of the parameter. Since l affects the co-volume and, therefore, the repulsive term,12

it becomes significant at the high densities that characterize critical states. Therefore, in the presentstudy the systems studied will be fit using also the second interaction parameter l . The relative12

influence of both parameters will be discussed.

4. Computation strategy

Having an equation of state and mixing rules, one can calculate phase diagrams of pure substancesand mixtures by solving the corresponding thermodynamic conditions. For a binary critical point, the

Žsecond and third derivatives of the Gibbs energy with respect to the mole fraction vanish G sG2x 3x. w x w xs0 8 . A previous publication 1 presented a simple numerical method for solving these conditions


for vdW-type equations. In this work, we propose a modified numerical method based on the sameidea of providing an initial value for the solution of critical conditions. In the case of gas–liquidcritical lines, the problem can be solved easily by starting the computation from the critical point of apure compound. The calculated critical point is then used as the initial value for computation of thefollowing one, in a point-by-point method. However, this method cannot be used for calculatingliquid–liquid critical lines because these do not start from the critical point of a pure substance. In the

w xcase of a vdW-type equations, we can apply the relations derived earlier 25 . These relations give azero-pressure solution for the liquid–liquid critical curves. For the CS-type equations, we use a

w xmethod similar to the one suggested by Deiters and Pegg 26 . Essentially, the procedure is as follows:for a given mole fraction, temperatures are obtained from the condition G s0 at 500 to 1000 values2x

of the molar volumes in the range where the EOS has physical validity. The second derivative G2x

has a relatively simple algebraic expression that converges well for arbitrary initial values of thetemperature. The values of the temperature and molar volume at which the derivative G changes3x

sign are chosen as initial guess for the calculation of the critical conditions. In some cases, oneobtains more than one solution for a critical point at the same temperature including liquid–vapor andliquid–liquid critical points. It is sufficient to find one point of the liquid–liquid critical line tocontinue the calculation with the point-by-point method. Each calculated critical point is checked for

w xglobal stability as described in Ref. 27 .The equations used in the present study were modified with the Soave-temperature dependence Eq.

Ž .9 . For this purpose, it was necessary to obtain the optimal value of m by fitting the experimentalvapor pressure lines. This parameter is specific for each pure compound and each equation asmentioned above. The computations of the critical lines and the value of the parameter m wereperformed with Mathematica 3.1.

5. Results

Investigation of the n-alkanes homologous series is important for understanding the fundamentalproblems of fluid phase equilibria because the critical properties of n-alkanes and their molecular size

w xchange in a regular manner 1 . The critical properties of the methane–alkanes series have alreadyw xbeen investigated using different approaches 1,28 . To analyze the correlation capability of the

CS-type and vdW-type equations, we have considered two members of the methane–alkane series, themethane–pentane and the methane–butane systems. The investigation has been performed with the

Ž .kl-global phase diagram klGPD using a vdW-type and a CS-type equation for the system methane–pentane. In addition, we have investigated the homologous series of perfluoromethane–alkanes andwater–alkanes and here we present the results for selected members of these series. The pure

w xcomponent properties have been taken from Daubert et al. 29 .

5.1. The klGPD for the methane-pentane system

Global phase diagrams are maps showing all the possible types of binary phase behavior generatedby a certain equation of state for mixtures. The axes of global phase diagrams are related to the


interaction parameters of the mixed substances and to the cross-interaction parameters. In the originalform of the global phase diagram, the cross attraction parameter is plotted against the difference ofthe attraction parameters of the pure substances for constant co-volumes. In such a diagram, eachbinary mixture is represented by one point. In the kl-global phase diagram proposed here, the binaryinteraction parameters are plotted at the axes while the parameters of the pure substances are keptconstant. Such a diagram shows all possible phase diagrams that can be obtained for a given binarymixture by variation of the cross-interaction parameters only. The regions of the different phase

Ž w x.diagram types according to van Konynenburg and Scott 2 are separated by boundary curves in thew xglobal phase diagram. Recent investigations 26,30–32 have shown the thermodynamic conditions

for the above boundary states.Because of solidification processes, real fluids cannot exhibit liquid phase equilibria at low

temperatures. So far, solid phases have been neglected in GPD calculations and for this reason thetransition between the phase diagram Types I and II or IV and V appears when the liquid–liquidcritical line coincides with the zero-temperature axis. This boundary is called zero KelÕin line.

ŽKnowledge of the lowest experimental solidification temperature of a particular system neglecting.possible eutectics allows estimating a new boundary on the klGPD that separates binary systems with

stable and meta-stable low-temperature liquid–liquid phase equilibria. So, the region between zeroKelÕin and solidifying critical lines contains meta-stable low-temperature liquid–liquid immiscibility.

Fig. 2 shows the klGPD of the methane–pentane system calculated using the vdW-type equation,Ž . Ž .Eq. 7 . The top part of the diagram large positive value of k corresponds to systems having weak12

cross-attraction. Such systems often present the Type III behavior that exhibits low miscibility in awide range of temperature and pressure. With the decrease of k the cross-attraction increases and12

the system becomes more miscible. In this direction, the phase behavior changes from Type III toType III by crossing the critical pressure step point boundary. In the top part of the klGPD, them

influence of the cross-co-volume parameter l on the qualitative phase behavior is very small. A12

further decrease of k increases the miscibility and leads to a variety of different phase diagram12

types. In this part of the klGPD, the cross-co-volume parameter l has a stronger influence on the12

types of phase behavior. In addition, it is seen that both zero KelÕin and solidifying critical lines arestrongly dependent on l . Therefore, one can conclude that this parameter plays a significant role12

whenever cross-attraction becomes significant. Similar results can be shown to be true for othersystems. Information of this nature is very important for a proper correlation of experimental data.

At negative k the klGPD is divided by the tricritical boundary curve into two major regions.12

Binary phase diagram types with a continuous liquid–gas critical curve are located on the left-handside of the tricritical boundary. On the right-hand side of this boundary, a liquid–liquid–gasthree-phase line interrupts the binary critical curve and a region of liquid–liquid immiscibility can befound. The experimental data suggest that the methane–pentane system is located close to a tricritical

w xboundary 33–37 , which is not the case of many other systems. This feature is important foranalyzing the predictive ability of equations using klGPD, because it allows definition of the values ofthe interaction parameters that will yield the correct typology of the calculated phase diagrams.

The form of the klGPD has additional advantages because it allows representation not only ofqualitative but also of quantitative information about the system. Experimental critical temperatures,pressures, volumes, and mole fractions can be predicted for different combinations of the binaryparameters. A point in the klGPD represents each of these combinations. It is also possible to trace acertain property of the binary phase diagram by a path in the klGPD. One can, for example, keep the


Fig. 2. kl-global phase diagram for the methane–pentane system calculated using the vdW-type equation. The bold lines areboundary lines: dashed line: critical pressure step point; solid line: tricritical line; thick solid line: double critical end point;

Ž .dot-dot-dashed line: solidifying critical line T s128 K, P s0 bar ; dot-dashed line: zero Kelvin line. The thin lines areŽ .lines of a constant property of the critical pressure maximum CPM-isobar, CPM-isotherm, CPM-isopleth . The thick arrow

shows the range of the temperature dependent k . The shaded area shows the region of the experimental data for12

temperature and mole fraction at the critical pressure maximum.

pressure of a maximum of a critical curve at the experimental value and calculate k for different12Ž .values of l . This procedure gives a curve of the critical pressure maximum CPM in the klGPD that12

is called CPM-isobar here. Similarly, one can calculate CPM-isotherms or CPM-isopleths. Theanalysis of such iso-lines allows the evaluation of the ability of the particular equation of state topredict the given experimental data.

So far, together with the boundary curves we have calculated quantitative curves in the klGPD. InŽ .these curves one property at a certain point of the experimental binary system is kept constant. Here

we have chosen the experimental data of the CPM as the characteristic property of this systemŽ .Ps169.25 bar, Ts210.9 K, xs0.8236 . In the klGPD three CPM-isotherms, twoCPM-isoplethsŽ .in order to take into account a possible uncertainty of the experimental data , and one CPM-isobarare plotted. The region in the global phase diagram where all three CPM-iso-lines intersect gives thevalue of the binary interaction parameters that exactly reproduce the experimental data at the CPM. Inaddition, as suggested by the experimental data, an accurate equation should locate this intersection

Ž .near the tricritical boundary. It is seen that Eq. 7 yields intersection of only two among threeCPM-iso-lines. One of these intersections is located where the CPM-isobar crosses the shaded region,which appears on the right-hand-side of the tricritical curve. A phase diagram calculated in the shaded


region on the left-hand side of the tricritical curve gives a continuous critical curve, as suggested bythe experimental data, but the critical pressure at the maximum differs substantially from theexperimental system. Fig. 3 represents phase diagrams along the CPM-isobar. In this region, the P–xdiagram points that there is a significant deviation between the calculated and the experimental data.It appears that a given model is unable to describe simultaneously both sides of the critical curve.

A klGPD allows understanding the relation between the tricritical boundaries and quantitativeŽ .iso-lines. Fig. 4 shows the klGPD for the CS-type equation Eq. 8 . Although the shape of the

boundary curves in klGPD of both equations look alike, a CS-type equation predicts the tricriticalboundary at lower values of l . Hence, there is no region of Type I on the klGPD of a CS-type12

equation and the region of Type II is small. This location of the tricritical boundary stronglyinfluences the iso-lines. It can be seen that iso-lines move away from one another and thatCPM-isotherms move to lower k and much lower l values. The CPM-isobar appears at larger12 12

values of l . We can conclude, therefore, that the performance of the equation is less accurate.12

Analysis of the relative location of tricritical boundaries and iso-lines on a klGPD indicates that whenthe tricritical boundary is moved to higher values of l and the region of Types I and II is increased,12

results in the iso-lines approaching each other. Hence, the quantitatiÕe performance of an equationŽ .i.e., location of the boundaries on a klGPD defines its correlating ability. It is remarkable that such afundamental conclusion cannot be reached by a simple fit of characteristic data points. In addition, itmust be realized that the shape of the klGPD is similar for many systems, therefore, parts of theconclusions obtained for the methane–pentane system are valid for other systems, especially thosewith similar values of the ratios a ra and b rb . We can conclude then that the klGPD is an11 22 11 22

important tool for analyzing and developing equation of state models.

5.2. Temperature-dependent combining rules

Ž . Ž .Investigation of the klGPD of the system methane–pentane has shown that Eqs. 7 and 8 ,Ž .together with the combining rules given by Eq. 11 , does not give an overall accurate description of

the critical line. From Fig. 3, we can learn that depending on the specific problem one can correlatedata at or on the other side of the critical curve that gives a binary interaction parameter located at oneor the other side of the tricritical curve. In other words, the correlation can be improved byintroducing a mole fraction- or a temperature-dependent k . Mixing rules derived from excess Gibbs12

w xenergy models include such a mole fraction-dependence 38 . In the present study, we have used thefollowing temperature dependence for the interaction parameter k along the critical line12

k sk tqk 1y t 12Ž . Ž .12 1 2

with

TyTc1ts 13Ž .

T yTc2 c1

Ž .When k sk the temperature dependence vanishes and Eq. 11 is recovered. Here we will use1 2

k sk for mixtures of substances having similar critical temperatures and for systems that present1 2Ž .liquid–liquid critical lines at very low temperatures T<T . It should be understood that thec1


Fig. 3. Critical lines for the methane–pentane system calculated using the vdW-type equation along the CPM-isobar. `Experimental data.


Fig. 4. kl-global phase diagram for the methane–pentane system calculated using the CS-type equation. For legend seeFig. 2.

Ž .temperature dependence of the interaction parameter, as given by Eq. 12 , keeps the quadraticŽ . Ž .dependence of the mixing rule on mole fraction expressed by Eqs. 10 and 11 . The second virial

) Ž . Ž .coefficient B of Eq. 8 and every vdW-type equation of state, such as Eq. 7 , is given by

B) sbyar RT 14Ž . Ž .

Consequently, the resulting mixing rule is able to generate the theoretical quadratic dependence of thesecond virial coefficient on the mole fraction. A similar temperature-dependent l may lead to the12

problems characteristic for temperature-dependent co-volumes discussed above, and thus it has notbeen used in the present study.

5.3. Accuracy of the models

5.3.1. Methane–pentaneOften the binary interaction parameters are correlated with the experimental critical data using the

w xinformation available for only one equimolar mixture 39–41 . The shortcomings of this method havew x w x w xalready been discussed 1 . Kolar and Kojima 42 and Englezos et al. 43 have proposed to minimize´

an objective function that includes all the experimental data available for a given system. Theadvantages of this method are clear, however, for a comparative investigation of the performance of


the different models it is more convenient to fit the temperature, pressure and mole fraction of acharacteristic point at the critical curve. One can, for example, use the extremum of the critical curve.

Ž .For the methane–pentane system the critical pressure maximum CPM was selected for this purpose.A comparison between the experimental data of this system and the critical lines calculated by a

Ž .generalized vdW-type and a CS-type equations Eqs. 7 and 8 , with the temperature-dependent k12Ž .Eqs. 12 and 13 is shown below.

Fig. 5 shows the results for the vdW-type equation. A large positive value of k is required for the12

prediction of the high-temperature portion of the critical line while a large negative value of k is12

necessary for the prediction of the low-temperature part.An important feature of the vdW-type equation with cs1, which corresponds to the RKS

equation, can be observed in Fig. 5a and b. The RKS equation is not accurate at the low temperaturepart of the critical line in both the P–x and P–T projections. In the P–T projection, the critical

Ž . Ž .pressures are over-predicted Fig. 5a while in the P–x projection Fig. 5b the calculated critical linecrosses the experimental points. Fig. 5c–f shows that increasing the value of c slightly improves theequation with respect to the simultaneous prediction in both the P–x and P–T projections.

Fig. 6 shows the critical lines of the methane–pentane system calculated using the CS-typeŽ .equation with a temperature-dependent k Eqs. 8, 12 and 13 , for different values of c. One can see12

that the CS-type equation gives results similar to those predicted by a vdW-type equation. However,the range of liquid–liquid immiscibility region is larger than that calculated using the vdW-type

Ž .equation as suggested by klGPD, Fig. 4 . A black arrow in Fig. 4 marks the range of the temperaturedependent cross-attraction parameter k . It is seen that the distance to the tricritical boundary curve is12

Ž .larger for the CS-type equation than for the vdW-type equation Fig. 2 . This distance is related to thelength of the liquid–liquid-gas three phase line. Here the influence of c is similar to that in theprevious case and can be explained by the fact that for both the vdW and CS-type equations, asignificant increase of c slightly moves the tricritical boundary on the klGPD to higher values of l ,12

which, as we have already shown, improves the results.The variation of the critical volume with composition for the vdW-type and the CS-type equation is

shown in Fig. 7 for different values of the parameter c. Since the fit was performed for the criticaltemperatures, pressures, and concentrations, the prediction of the critical volume shown in Fig. 7 isnot very accurate. Both equations give similar results in the V –x projection because increasing them

w xvalue of c decreases the critical compressibility predicted by the EOS 3 . Therefore, for larger valuesof c the critical lines move down in the V –x projection. We can summarize the influence of c asm

Ž .follows: a it slightly reduces the liquid–liquid immiscibility range and improves the performance ofboth models in the P–T and P–x projections. However, its influence is weaker than the structure of

Ž .the repulsive term; b increasing values of c decrease the values of the critical volumes. The possiblew xvalues of c for the critical lines can be those that yield a Z value equal to the experimental one 20 .c

However, due to the inaccurate predictions in the V –x projection, such values are not necessarily them

optimal for the critical volumes of mixtures. It seems reasonable to assume that the development of aproper form of repulsive term is more important than a search for the optimal values of c. Hence, allthe results that follow are restricted to the value cs1.

5.3.2. Methane–butaneFig. 8 presents a comparison between the experimental data for the methane–butane system

w x44–46 and the critical lines calculated using the vdW-type and the CS-type equations. It should be


Fig. 5. Critical lines for the methane–pentane system calculated using the vdW-type equation with a temperature dependentŽ . Žk parameter Eqs. 12 and 13 , and for different attraction density-dependence parameters c for cs1, k s0.04,12 1

.k sy0.12, l s0.01; for cs3, k s0.04, k sy0.11, l s0.01; for cs5, k s0.05, k sy0.11, l s0.01 . `2 12 1 2 12 1 2 12w x w x w xExperimental data of Berry and Sage 33 . v Experimental data of Sage et al. 34 . ^ Experimental data of Chu et al. 35

w xand Chen et al. 36 . I Critical points of pure compounds. q Critical endpoints. Solid lines – Calculated critical curves.Dotted lines – Vapor pressure curves of pure compounds.


Fig. 6. Critical lines for the methane–pentane system calculated using the CS-type equation with a temperature dependentŽ . Žk parameter Eqs. 14 and 15 , and for different attraction density dependence parameters c for cs1, k s0.12,12 1

.k sy0.18, l s0.05; for cs2, k s0.115, k sy0.18, l s0.05; for cs3, k s0.11, k sy0.17, l s0.05 . `2 12 1 2 12 1 2 12w x w x w xExperimental data of Berry and Sage 33 . v Experimental data of Sage et al. 34 . ^ Experimental data of Chu et al. 35

w xand Chen et al. 36 . I Critical points of pure compounds. q Critical endpoints. Solid lines – Calculated critical curves.Dotted lines – Vapor pressure curves of pure compounds.


Fig. 7. Volume-mole fraction projection of the critical lines for the methane–pentane system for different values of thew xattraction density dependence parameters c. ` Experimental data of Berry and Sage 33 . v Experimental data of Sage et

w xal. 34 . q Critical endpoints. Solid line – Calculated critical curves.

w x w xnoted that the data sets of Refs. 45,46 seem to be more accurate than those of Ref. 44 . The criticaltemperature of butane is closer to that of pentane than to the critical temperature of methane. Hence,the difference between k and k for both the vdW-type and CS-type EOS is smaller than for the1 2

methane–pentane system. One can see that both equations yield similar results and are in goodagreement with the experimental data. Both equations give Type I phase behavior for the systemmethane–butane in contrast to an investigation of this system with the SPHCT equation that gave

w xType V behavior 47 . This result shows that even the methane–butane system, which is the secondsystem in the methane–alkane series to the tricritical boundary, is non-trivial for correlation. Itappears that experimental systems that are close to boundary states are more difficult to correlate evenif the compounds happen to have a simple chemical structure.

5.3.3. Tetrafluoromethane–heptaneThe homologous series tetrafluoromethane–alkanes is of significant interest for the prediction by

equations of state because fluoroalkanes and alkanes have similar structure but different physical and


Fig. 8. Critical lines for the methane–butane system calculated with the vdW-type and the CS-type equation, with aŽ . Žtemperature dependent k parameter Eqs. 14 and 15 and with cs1 for CS-type EOS k s0.07, k sy0.05,12 1 2

. w xl sy0.03; for vdW-type EOS k s0.02, k s0, l sy0.07 . ^ Experimental data of Roberts et al. 44 . `12 1 2 12w x w xExperimental data of Elliot et al. 45 . v Experimental data of Kahre 46 . I Critical points of pure compounds. Solid line

– Calculated critical curves. Dotted line – Vapor pressure curves of pure compounds.

w xchemical properties. McCabe et al. 48 have recently calculated the critical lines of tetrafluo-romethane–alkanes systems using the SAFT-VR approach and obtained better results than those

w xreported by Sadus 49–51 . For our work, we have chosen the system tetrafluoromethane–heptanew xbecause it has been described as one that is difficult to predict 50,51 .

The system tetrafluoromethane–heptane exhibits Type III phase behavior. Critical lines of Type IIIsystems consist of two branches. A gas–liquid critical line starts at the critical point of the lessvolatile substance and goes through a critical temperature minimum to high pressure. This high

w xpressure part is called gas–gas immiscibility of the second kind in the classification of Schneider 52 .w xThis critical line goes first through a temperature minimum at 331.7 K 53 and continues then steeplyŽ .to high pressure. The other branch connects the upper critical endpoint UCEP of the liquid–liquid–

gas three-phase line with the critical point of the more volatile substance. The critical line ofw xtetrafluoromethane–heptane has also been calculated by Sadus 50 using the Guggenheim, Redlich–

Kwong, and Peng–Robinson equations. The combining rules were essentially the same as those in Eq.Ž . w x11 , including one interaction parameter. In a later publication, Sadus 51 calculated the critical line


of tetrafluoromethane–heptane using a CS-type equation and Boublik–Nezbeda equation with oneadditional molecular parameter, the non-sphericity parameter. The Boublik–Nezbeda equation is amodel for non-spherical molecules that gives the CS equation in the limit of spherical molecules.

w xAlthough this equation gave results that were less accurate than those of the SAFT-VR approach 48 ,w xits performance was better than that of the equations considered in Ref. 50 . Still, it is not clear

whether the additional interaction parameter or the theoretical correctness of the Boublik–Nezdedaequation is responsible for the improvement. To address this question, we have calculated the criticalline of this system using both the vdW-type and CS-type equation. We have used k without a12

temperature dependence because experimental data for this system is available only for the criticalw xpressures and temperatures 52 . In addition, this makes our calculation comparable with that of Sadus

w x51 . Fig. 9 indicates that both the CS-type and the vdW-type equations are capable of predicting theP–T projection of this system quantitatiÕely using two interaction parameters. Therefore, with anaccurate fit, both vdW-type and CS-type equations show a fair agreement with experimental data, as

w xthe SAFT-VR approach 48 . It is noteworthy that a CS-type equation yields a more accurate

Fig. 9. Pressure–temperature projection of the critical line of the system tetrafluoromethane–heptane for the vdW-type andŽ .CS-type equation for cs1 for CS-type EOS k s0.07, l sy0.01; for vdW-type EOS k s0.151, l sy0.047 . v12 12 12 12

w xExperimental data of Wirths and Schneider 53 . Solid line – Calculated critical curves. I Critical point of heptane.


correlation of the low-pressure gas–liquid part of the critical line. The high pressure branch of thecritical curve is predicted well by the CS-type and the vdW-type equation. It can be seen that similarto the case of methane–pentane, the CS-type equation tends to predict a larger region of liquid–liquidimmiscibility. Therefore, if both models are used to calculate liquid–liquid critical lines at the sametemperature, then the vdW-type equation predicts higher pressures for the gas–liquid critical line thanCS-type equation. Fig. 9 shows that in this particular case the CS-type equation agrees better with theexperimental data. We can conclude, so far, that increasing the region of Types I and II on the klGPDdoes not always improve the correlation ability of the equation. In other words, an EOS must beflexible enough in defining the tricritical boundaries on the klGPD for particular systems. Thiscondition imposes additional substance-dependent parameters in the repulsive term. A promisingdirection seems to be the introduction of molecular geometry into the repulsive term, as suggested byw x54 .

5.3.4. Tetrafluoromethane–butaneThe tetrafluoromethane–butane system belongs to the Type III phase behavior. The subscript ‘m’m

means that the critical line that starts from the less volatile compound has a pressure-maximum and apressure-minimum. The behavior of Type III systems is close to the transitional behavior to Typesm

III, II, or IV. In addition, in the P–x projection this type exhibits a characteristic loop with a molew xfraction-extremum 55 . In the homologous series tetrafluoromethane–n-alkanes, the tetrafluo-

romethane–butane system is the first member without a continuous critical line between the criticalpoints of the pure compounds. This system has been studied experimentally by Jeschke and Schneiderw x w x56 in the high pressure region and by de Loos et al. 57 up to 10 MPa. Fig. 10 shows a comparison

w xbetween the experimental data of de Loos et al. 57 and the calculation using the CS-type andvdW-type equation; both EOS give results that are qualitatively correct. Since Type III systems arem

difficult to predict accurately they are useful for comparing the performance of different models.Similar to the previous case, when both models are used to calculate the liquid–liquid critical lines atthe same temperature, the vdW-type equation predicts higher pressures for the gas–liquid critical linethan the CS-type equation. Again, this fact may be explained by the tendency of the vdW-typeequation to predict a tricritical boundary on the klGPD at higher values of k and therefore, a smaller12

region of the liquid–liquid immiscibility than a CS-type equation. However, in this particular case theŽ .vdW-type equation is in better agreement with experimental data see Fig. 10 . The calculations

shown in Fig. 10 were done using for both equations a temperature-dependent k . However, for the12Ž . Ž .vdW-type equation the value of k was almost identical to that of k so that Eqs. 12 and 131 2

Ž .reduced to Eq. 11 . In the P–x projection, both equations slightly over-predicted the gas–liquidbranch of the critical line that starts at the critical point of perfluoromethane. The vdW-type equationshows an overestimation of the size of the loop in the P–x projection while the CS-type equationunder-predicts the size of the loop but gives a good prediction of the liquid–liquid branch. It isinteresting to note that the CS-type equation gives a more accurate prediction of the concentration atthe extremum of the critical mole fraction than the vdW-type equation. In addition, one can see that inthe P–T projection both the CS-type and vdW-type equations over-predict the UCEP. McCabe et al.w x48 have detected a similar result with the SAFT-VR approach. For the methane–hexane system, it

w xwas also found that SAFT-VR and vdW-type equations gave similar results 1 . These facts show thatin the critical region the SAFT-VR approach has a similar performance to that of the semi-empiricalequations of state, developed for engineering purposes.


Fig. 10. Critical lines of the system tetrafluoromethane–butane calculated with the vdW-type and CS-type equation forŽ .cs1 for CS-type EOS k s0.085, k s0.065, l sy0.032; for vdW-type EOS k s0.101, l sy0.067 . ` Experi-1 2 12 12 12

w x w xmental data of de Loos et al. 57 . v Experimental data of de Loos et al. 57 . B Experimental critical curve starting fromtetrafluoromethane. I Critical points of pure compounds. q Critical endpoints. Solid lines– Calculated critical curves.Dotted lines – Vapor pressure curves of pure compounds.

5.3.5. Water–dodecaneThe water–alkane series is another important homologous alkane-series. The properties of water–

alkane systems are of fundamental importance in geology and mineralogy, i.e., in the geothermalw xprocesses, which form oil reservoirs, and in the oil and gas industries 58 . Furthermore, water–al-

kanes systems may serve as model systems that are useful for the development of processes such asw xthe hydrolysis of plastics, synthetic fibers or polycarbonates for recycling 59 , and the destruction of

w xhazardous wastes 60 .Ž w x.Water is characterized by a very high critical pressure 22.055 MPa, 29 that causes the critical

lines for several systems with water to show unusual shapes. The mutual solubility of alkanes andwater is very low and the three-phase equilibrium line occurs at significantly higher pressure than the

w xvapor pressure line of the pure alkanes. Brunner 61 has measured and analyzed the three-phaseregion and critical lines for 23 water–n-alkane systems and found that the series shows a transitionalbehavior between phase diagram types which differs from the other series. In these, the liquid–gas


critical curve that connects the critical end point of the three-phase line and the critical point of one ofthe substances is rather short. For the water–n-alkane series this critical line is very long. Systems of

Ž .water and alkanes exhibit mostly Type III hetero-azeotrope behavior III-H , which is difficult forw xquantitative correlation with simple equations of state 61 . Systems with carbon numbers between 26

and 28 show transitions through a tricritical boundary, a double critical end point boundary, and aw xType IV-H region 62 .

w x w xFor the systems water–methane and water–hexane, Shmonov et al. 63 and Heilig and Franck 64have correlated the critical lines that start from water using theoretically based EOS. Simultaneous

Žcorrelation of both the critical lines that start from water and from alkanes as done in the present.study is a challenging task. Our results show that both equations tested here have a similar ability to

correlate the critical lines of different members of this homologous series. The system water–dode-cane has been selected particularly because the concentration of some critical points is available in the

w x Ž .literature 65 Fig. 11 . The critical temperatures of water and dodecane are very close, therefore, the

Fig. 11. Critical curves of the water–dodecane system calculated using the vdW-type and the CS-type equations for cs1Ž . w xfor CS-type EOS k s0.18, l s0.29; for vdW-type EOS k s0.205, l s0.25 . ` Experimental of Brunner 61 . v12 12 12 12

w xExperimental of Stevenson et al. 65 . I Critical points of pure compounds. q Critical endpoints. Solid lines – Calculatedcritical lines. Dotted lines – Vapor pressure lines of pure compounds. Dot-dashed lines – Calculated three-phase line. =

w xExperimental three-phase points of Brunner 61 .


calculations have been performed assuming k to be temperature independent. Fig. 11 shows that the12

fit with two binary interaction parameters and classical mixing rules gives a relatively accurateprediction of the critical line. In the P–T and P–x projections, both the vdW-type and CS-typeequation yield very accurate predictions of the branch that starts from the critical point of water. Inthe P–T projection, there is a good agreement between the predicted branch of the critical curve

Ž .starting from the critical point of dodecane and the experimental data Fig. 11a and c . However, inŽ .the P–x projection the predicted critical curves lay below the experimental data Fig. 11b and d .

Both the vdW-type and the CS-type equation over-predict the pressure and temperature of the UCEP.In addition, it is seen that the vdW-type equation gives a slightly better prediction of the three-phaseline and that the CS-type equation gives a slightly better prediction of the critical line that starts fromwater. Similar to the previous cases, for water–dodecane the CS-type EOS tends to yield a largerregion of liquid–liquid immiscibility. However, this tendency has a minor influence on the accuracyof fitting because the system is located very far from the tricritical boundary. More complicated

w xapproaches have shown that this system is usually very difficult to model 66 . Although accurateresults were obtained for the critical lines, it is necessary also to investigate the performance of theCS- and vdW-type equations in the subcritical region.

6. Conclusions

In this work, we have compared the performance of vdW-type and CS-type equations in the criticalregion and discussed the development of a CS-type equation of state suitable for engineeringpurposes. A further development of the global phase diagram method that is suitable for chemicalengineering purposes is proposed. In this new diagram, the boundaries between the different types ofphase behavior are presented in the k –l plane. It is demonstrated that a klGPD offers the12 12

possibility to include quantitative data of a particular experimental mixture in the global phasediagram. In particular, we have investigated the klGPD diagram for the methane–pentane system witha vdW-type and a CS-type equation. Analysis of a klGPD has allowed us to reach conclusions thatcannot be obtained by a simple fit of characteristic data points. In particular, we have demonstratedthat the location of boundaries on a klGPD defines the correlative ability of equations.

It was found that an EOS with classical mixing rules and proper selection of the two interactionparameters gave a relatively good quantitative agreement with experimental data of the critical line,

Ž .even for systems that are difficult to correlate such as water–dodecane . In addition, a temperature-dependent combining rule for the binary attraction parameter k was used. It was found that this12

temperature dependence becomes stronger as the difference between the critical temperatures of purecompounds increases. The following specific results were obtained.

Ž .a The influence of the repulsive term of the EOS on the location of boundary states on the klGPDis stronger than that of the attractive one. In particular, changes in the density dependence of theattraction term of the two-parameter EOS influence mostly the predicted critical volume of mixturesand not its qualitative performance. Therefore, a proper form of the repulsive term is significant fordeveloping accurate EOS models.

Ž .b A CS-type equation tends to predict a larger region of liquid–liquid immiscibility than avdW-type equation. This tendency has minor influence on the accuracy of fitting for systems locatedfar from tricritical boundary like water–dodecane. On the other hand, for systems such as methane–


pentane it leads to results that are less accurate, while for others, such as tetrafluoromethane–heptane,to more accurate results. Therefore, it seems that an EOS must be flexible enough in defining thetricritical boundaries for particular systems. This condition imposes the inclusion of additionalsubstance-dependent parameters into the repulsive term. It is suggested that further improvement ofsimple equations of state is possible by considering the non-sphericity and chemical nature of themolecules.

Ž .c Parameter k has a strong influence on Type III phase diagrams with high immiscibility.12

Parameter l influences strongly Type V or I behavior that shows high miscibility. We have12

concluded that parameter l becomes important when cross-attraction becomes significant.12Ž .d Similar to vdW-type equations, CS-type equations should not be used with temperature-depen-

dent co-volumes.

List of symbolsa attraction parameterb co-volumeB) second virial coefficientc attraction density dependence parameterk binary attraction parameterl binary co-volume parameterP pressureR universal gas constantT temperature

Ž .x mole fraction of the more volatile compound compound 1V molar volumem

Z compressibility factor

Greek lettersŽ .a T temperature dependence of the attraction parameter

Ž .V , V characteristic constants of an equation of state, Eq. 6A B

Subscriptsc critical statem molar propertyr reduced value

AbbreÕiationsCPM critical pressure maximumCS Carnahan–Starling equationCSD Carnahan–Starling–DeSantis equationGPD global phase diagramklGPD global phase diagram in the k –l plane12 12


RK Redlich–Kwong equationRKS Redlich–Kwong–Soave equationUCEP upper critical endpointvdW van der Waals equation

7. Uncited reference

w x40

Acknowledgements

T.K. and L.V.Y. acknowledge the support of the Deutsche Forschungsgemeinschaft. Furthermore,T.K. would like to thank the Fonds der Chemischen Industrie for supporting this work. H.S. has beensponsored by the Fondap Program in Applied Mathematics, Chile.

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