simultaneous prediction of the critical and sub-critical phase behavior in mixtures using equations...

Chemical Engineering Science 58 (2003) 2529–2550www.elsevier.com/locate/ces

Simultaneous prediction of the critical and sub-critical phase behavior inmixtures using equations of state II. Carbon dioxide–heavy n-alkanes

Ilya Polishuka ;∗, Jaime Wisniaka, Hugo Segurab

aDepartment of Chemical Engineering, Ben-Gurion University of the Negey PO BOX 653, Beer-Sheva 84105, IsraelbDepartment of Chemical Engineering, Universidad de Concepci)on, Concepci)on, Chile

Received 29 July 2002; received in revised form 27 January 2003; accepted 13 February 2003

Abstract

In the present study we present the 3nal development of the Global Phase Diagram-based semi-predictive approach (GPDA), whichrequires only 2–3 key data points of one homologue to predict the complete phase behavior of the whole homologues series. The ability ofGPDA to predict phase equilibria in CO2–heavy n-alkanes is compared with the equations of state LCVM and PSRK. It is demonstratedthat both LCVM and PSRK are more correlative rather than predictive because their parameters are evaluated by the local 3t of aconsiderable amount of VLE experimental data. In addition, these models fail to predict accurately the VLE of systems, which havenot been considered in the evaluation of their parameters. They are also particularly inaccurate in predicting LLE and critical lines. Incontrast, GPDA is reliable in the entire temperature range and for all types of phase equilibria. It yields an accurate prediction of theglobal phase behavior in the homologues series and their critical lines. Moreover, increasing asymmetry does not a=ect the reliability ofGPDA; it predicts very accurately even the data of the heaviest homologues of the series.? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Phase equilibria; Supercritical >uid; Carbon dioxide; Alkanes; Parameter identi3cation; Equation of state

1. Introduction

During last decades a vast body of literature has beendevoted to development of models such as van der Waals(vdW)-type equations of state (EOS) and their mixingrules. However, the problem of predicting data withoutpreliminary resource to experimental results has not beensatisfactory solved yet. Indeed, the present developmentof molecular theory does not allow evaluation of entirelypredictive methods, which could be reliable in descrip-tion of experimental facts. Nevertheless, it is still possibleto develop approaches that use some experimental datain order to predict the missing ones. Such approachescan be classi3ed as semi-predictive. Thus, quality of thesemi-predictive approaches is de3ned not only by their ac-curacy, but also by the ratio of data input to data predicted.It is well known that the predictive ability of the conven-tional semi-predictive methods such as UNIFAC is basedon a large amount of experimental data. In this respect

∗ Corresponding author. Tel.: +972-7-646-1479;fax: +972-7-647-2916.

E-mail address: [email protected] (I. Polishuk).

the advantage of the quantitative Global Phase Diagram(klGPD)-based semi-predictive approach (GPDA), whichrequires no more than just 2-3 experimental critical pointsin order to predict the entire phase behavior in whole ho-mologues series of binary systems (Polishuk, Wisniak, &Segura, 2001), becomes evident.Since van Konynenburg and Scott (1980) 3rst introduced

the idea of the Global Phase Diagram (GPD), it has devel-oped signi3cantly (Boshkov, Deiters, Kraska, & Lichten-thaler, 2002). Recently, a quantitative form of GPD (kIGPD)has been proposed as a tool for proper evaluation of the EOSparameters (Polishuk, Wisniak, Segura, Yelash, & Kraska,2000a,b). Since then the kIGPD methodology has been im-plemented in numerous studies. For example, Imre, Kraska,and Yelash (2002) have employed it for investigating thephase behavior of polymer blends. It has been demon-strated that this methodology can be useful for investigatingthe e=ect of the chain length on the critical temperatureminimum. kIGPD is also a very e=ective tool for detectingfundamental regularities characteristic for phase behaviorgenerated by EOSs. In particular, implementation of kIGPDhas demonstrated that the accuracy in describing of VLE isdetermined by the exactness with which the models predict

0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0009-2509(03)00101-5

mailto:[email protected]

2530 I. Polishuk et al. / Chemical Engineering Science 58 (2003) 2529–2550

LLE (Polishuk et al., 2001). In other words, di=erent kindsof phase equilibria, such as VLE and LLE, are inter-relatednot only qualitatively but also quantitatively.Thus, it appears that an appropriate description of phase

equilibria in the system can be achieved by the accurate cor-relation of only two points that characterize VLE and LLEin the system, namely, the upper critical endpoint (UCEP)and the critical pressure maximum (CPM). In addition, it hasbeen demonstrated that the experimental values of those keypoints in di=erent homologues can be correlated by EOSsusing similar values of the binary adjustable parameters k12and l12. This means that each homologues series has its owncharacteristic balance between VLE and LLE. This obser-vation has allowed us to formulate a novel semi-predictiveapproach involving the estimation of the binary interactionparameters for a certain homologue using kIGPD, with theirfurther implementation for predicting data of other homo-logues. This approach can be implemented for those serieswhere at least one CPM point is available or can be extrap-olated and here we present its 3nal development.Here we begin by considering the homologues series car-

bon dioxide–alkanes. Yet we focus the attention on mixturesof homologues heavier than n-decane. The global phase be-havior of this part of the series is of major practical andtheoretical interest. Unfortunately, it has not been investi-gated with the same detail as in several other homologuesseries (see, for example, Raeissi, Gauter, & Peters, 1998).Hottovy, Lucks, and Kohn (1981) have established thatCO2–n-tetradecane exhibits phase behavior of the Type IIIaccording to the classi3cation of van Konynenburg and Scott(1980). In addition it is also known that CO2–n-tridecane ex-hibits a Type IV transitional phase behavior (Fall & Lucks,1985; Enick, Holder, & Morsi, 1985). Therefore, the dou-ble critical end point (DCEP) of the series must be locatedsomewhere between C13 and C14. The position of the tri-critical point (TCP) of the series is also still not known ex-actly. However, it is widely accepted that CO2–n-dodecanebelongs to Type II phase behavior (Gauter, Florusse, Peters,& de Swaan Arons, 1996). Hence, the TCP is probably lo-cated between C12 and C13.The importance of accurate prediction of phase equilib-

rium in carbon dioxide–paraLns is diLcult to overestimate.The data on solubility of CO2 in hydrocarbons is essentialnot only for the simulation of petroleum reservoirs, but alsofor the design of many industrial processes such as injec-tion of CO2 for enhanced oil recovery (Turek, Metcalfe, &Fishback, 1988) and production of coal liquids by Fisher–Tropsch syntheses. Accurate data on the solubility of hydro-carbons in CO2 vapor phase is also required for optimizingsupercritical gas extraction, which is rapidly becoming a use-ful tool for treating natural products like 3sh and plant oils,and in many other processes such as removal of wastes, pro-cessing of drugs, polymers (Levelt Sengers, 2000), regener-ation of poisoned catalysts (Vradman, Herskowitz, Korin, &Wisniak, 2001), and even production of ceramics (Bordet,Passarello, Chartier, Tufeu, & Baumard, 2001).

However, it is very diLcult to describe the phase behaviorof CO2–hydrocarbon mixtures using thermodynamic mod-els. Years ago, Jo=e and Zudkevitch (1966) explained thefailure of conventional methods to correlate these systemsby the presence of a large quadruple moment in the carbondioxide molecule, which had been previously discussed byMyers and Prausnitz (1965). Thus, the systems selected forthis study present a challenging task for the semi-predictiveapproaches.

2. Theory

The basic requirement given for a predictive thermody-namic model is consistency in the entire thermodynamicphase space. An example of such model is the vdW one,which expresses pressure as a contribution of a repulsiveand a cohesive term

P =RT

Vm − b− aV 2m; (1)

where a and b are the cohesion parameter and covolume, re-spectively. Additional engineering equations have proposedmore advanced expressions for the cohesive term in an at-tempt to improve the description of the volumetric propertiesof pure >uids. To obtain the required horizontal in>ection ofthe predicted critical isotherm it is necessary to determinethe value of the pure compound parameters aij and bij inany vdW-like equation by solving the following conditions:(@P@Vm

)Tc

=(@2P@V 2

m

)Tc

= 0: (2)

Eq. (1) may be extended to mixtures by stating mixing rulesfor the EOS parameters. The classical mixing rules accord-ing to the vdW one->uid theory are as follows:

a=∑ij

xixjaij;

b=∑ij

xixjbij: (3)

The cross-interaction parameters a12 and b12 are obtainedusing the combination rules:

a21 = a12 = (1− k12)√a11a22;

b12 = b21 = (1− l12)b11 + b22

2; (4)

where k12 and l12 are binary adjustable parameters, whichde3ne the accuracy with which the vdW model and its latersemi-empirical derivatives describe phase equilibria in mix-tures. A vast body of literature is devoted to this subject.Therefore, here we will only brie>y mention several refer-ences dealing with mixtures of carbon dioxide and n-alkanesheavier than n-decane.

I. Polishuk et al. / Chemical Engineering Science 58 (2003) 2529–2550 2531

Nung and Tsai (1997) have evaluated the values ofk12 for several cubic EOSs and numerous binary systems,including CO2–n-C16H34 and CO2–n-C20H42, at di=erenttemperatures. Although such practice cannot be consid-ered as predictive, it allows attaching a vdW-model by asemi-predictive ability by generalizing the values of binaryadjustable parameters obtained thus far. Usually, such gen-eralizations are performed in an entirely empirical manner.For example, Kato, Nagahama, and Hirata (1981) haveproposed to generalize the values of k12 as a function ofthe acentric factors of n-alkanes. Lately, by Kordas, Tsout-souras, Stamataki, and Tassios (1994) have proposed animproved temperature-dependent correlation valid up toCO2–n-C44H90.Alternatively, several investigators have developed their

correlations using molecular theory. For example, Trebbleand Sigmund (1990) have proposed a theoretically basedgeneralized expression for k12 as a function of two em-pirical constants, the volumes of the pure compounds andquadrupole moments. This correlation has been imple-mented for predicting of VLE in di=erent systems, includingCO2–heavy n-alkanes (Sigmund & Trebble, 1992; Trebble,Salim, & Sigmund, 1993).It should be noted that although Deiters and Schneider

(1976) have demonstrated the importance of the binary pa-rameter l12 for an appropriate description of phase behaviorin mixtures, its evaluation is neglected by all the correlationsabove, which consider only k12. An important contributionis that of Coutinho, Kontogeorgis, and Stenby (1994) thatdoes deal with both binary parameters. This paper providesnot only a comprehensive review of the di=erent attemptsto predict VLE in carbon dioxide–n-alkanes homologousseries, but also performs a fundamental analysis of the the-ory related to both binary interaction parameters. Coutinhoet al. (1994) have proposed theoretically based expres-sions for both binary parameters, which are valid for thesystems under consideration. In addition, they have dis-cussed several important topics, such as the temperaturedependency of k12 and the relation between the ionizationpotential and the molecular diameter of the Mie potentialfunction. Two additional important contributions should alsobe mentioned, that of Cassel, Matt, Rogalski, and Solimando(1997), who made a comprehensive comparison betweenthe results obtained by correlation and prediction methods;and that of Passarello, Benzaghou, and Tobaly (2000), whomodeled the mutual solubility of n-alkanes and CO2 usingthe SAFT EOS with two unique values of binary adjustableparameters.The main advantage of all the above models lies in their

handling of supercritical compounds. In contrast to activitycoeLcient models (ACM), such as UNIFAC (Fredenslund,Gmehling, & Rasmussen, 1977), Eqs. (1)–(4) do not re-quire a de3nition of hypothetical reference states, or the useof Henry’s law, and provide continuity in the critical region.At the same time, however, it is widely agreed that the gen-eralized expressions developed for Eq. (4) lack the powerful

estimation ability given for ACM. Hence, the major e=orthas been invested so far in the derivation of semi-predictivemethods that combine both ACM and vdW models.Several semi-predictive models have been developed

since Huron and Vidal (1979) proposed their idea of con-necting cubic EOS and ACM through the equivalence ofthe Gibbs energy obtained by both methods at a certain ref-erence state. In the present study we consider the two mostsuccessful GE-based equations that already have foundapplication in chemical engineering practice.

2.1. PSRK

The Predictive Soave–Redlich–Kwong (PSRK) modelwas proposed by Holderbaum and Gmehling (1991) andthen adjusted for a wide variety of systems by Fisher andGmehling (1996), Gmehling, Jiding, and Fischer (1997),and Horstmann, Fisher, and Gmehling (2000a). As a result,PSRK has the largest parameter matrix and its applicabilityexceeds all other GE-based equations. Hence, this modelhas become a conventional tool to predict phase behaviorwithout resource to experimental data of particular mix-tures and is available in commercial engineering software.PSRK is based on the Soave–Redlich–Kwong EOS (Soave,1972), given as follows:

P =RT

Vm − b− a�Vm(Vm + b)

; (5)

where

a= 0:42748R2T 2

c

Pc; b= 0:08664

RTcPc

; (6)

and � is the temperature functionality of Mathias and Cope-man (1983), given as follows:

�= [1 + c1(1−√Tr) + c2(1−

√Tr)2

+ c3(1−√Tr)3]2: (7)

Holderbaum and Gmehling (1991) have reported the val-ues of c1, c2, and c3 for a number of pure compounds. Thevalues for compounds that are absent in this reference arelisted in Table 1 and they have been evaluated by 3tting theempirical expression of vapor pressure given by the DIPPRdata bank (Daubert, Danner, Sibul, & Stebbins, 1989–2002).However, it should be realized that the experimental dataregarding vapor pressures of heavy hydrocarbons is usuallyscarce and sometimes non-reliable. Therefore, it is not al-ways possible to evaluate how accurate the pertinent empir-ical expressions are. Thus, a very accurate 3t of the vaporpressure data, as prescribed by PSRK, in many cases doesnot seem expedient. Moreover, it seems highly recommend-able to implement generalized expressions for the �-functionparameters. Although such expressions may not reproducethe empirical polynomials precisely, they attach an entirelypredictive character to the model. The critical data given byHolderbaum and Gmehling (1991) have been used here for


Table 1The values of c1, c2 and c3 in Eq. (7)

Compound c1 c2 c3

carbon dioxidea 0.8252 0.2515 −1:7039n-undecanea 1.3766 −0:9838 2.1446n-dodecanea 1.3026 −0:0059 0.1852n-tridecane 1.4495 −0:6000 1.2927n-tetradecanea 1.4596 −0:5074 1.4459n-pentadecane 1.5022 −0:2019 0.3771n-hexadecane 1.5502 −0:2796 0.6269n-heptadecane 1.6373 −0:3745 0.5726n-octadecane 1.7004 −0:4135 0.5520n-nonadecane 1 .7761 −0:5849 0.8344n-docosane 2.0585 −1:4461 1.8659n-tetracosane 1.9990 0.8293 −4:8972n-hexatriacontane 3.9498 −16:4926 47.1766

aHolderbaum and Gmehling (1991).

the PSRK model. The data for compounds absent in this ref-erence have also been obtained from the DIPPR data bank.The data given in this source has been used in all other mod-els considered in the present study for all pure compounds.At supercritical temperatures the values of c2 and c3 are

set to zero. This fact may lead to the result that certain prop-erties predicted by the model (such as heat capacities) willshow non-physical behavior (Deiters & de Reuck, 1997).In addition, Eq. (7) generates fundamental numerical pit-falls, such as non-physical breaking points and 3ctitiouscritical points of pure compounds. These pitfalls may leadto prediction of non-physical phase diagrams, as describedpreviously (Polishuk, Wisniak, & Segura, 2002a; Polishuk,Wisniak, Segura, & Kraska, 2002c).PSRK implements Eq. (3) for b and the modi3ed

Huron–Vidal (MHV1, Dahl & Michelsen, 1990) mixingrule for a, as follows:

a= b

[GE0APSRK +

∑i

xiaibi

+RTAPSRK

∑i

xi lnbbi

]: (8)

It should be realized that Eq. (8) does not produce the theo-retically required quadratic dependency of the mixture sec-ond virial coeLcient on the mole fraction at the limit ofP → 0 (Wong & Sandler, 1992). Nevertheless, the PSRKmodel equates the excess energy terms from ACM and fromEq. (5) at this limit. Holderbaum and Gmehling (1991)have de3ned the ratio of Vm=b equal to 1.1 and obtainedAPSRK =−0:64663.GE0 is the excess energy term obtained from the

original UNIFAC group contribution method (Hansen,Rasmussen, Fredenslund, Schiller, & Gmehling, 1991) ex-tended to include several gases (Holderbaum & Gmehling,1991; Fischer & Gmehling, 1996; Gmehling et al.,1997; Horstmann et al., 2000a). The original UNIFACgroup–group interaction parameters were determined forlow-pressure VLE and moderated temperatures. There-fore, to cover a wider temperature range, the two new

temperature-dependent interaction parameters for the newgroups proposed by Holderbaum and Gmehling were alsointroduced.Although the implementation of HV-type mixing rules

and additional temperature-dependent interaction parame-ters improves the >exibility of PSRK, it also complicatesconsideration of the overall picture of phase behavior. Al-though the parameters of PSRK are evaluated consideringthe widest variety of experimental data (Holderbaum &Gmehling, 1991; Fischer & Gmehling, 1996; Gmehling etal., 1997; Horstmann et al., 2000a; Gmehling & Lohmann,2001), this evaluation is performed by a way of the local3t. This approach tends to ignore the fact that di=erent re-gions of the thermodynamic phase space created by the EOSare closely inter-related. As a result, PSRK may not onlygenerate non-realistic phase diagrams, but it usually pre-dicts an incorrect balance between VLE and LLE (Polishuk,Stateva, Wisniak, & Segura, 2002a). Although this char-acteristic feature of PSRK might be insigni3cant for pre-dicting symmetric systems (which do not exhibit any LLE;see for example Horstmann, Fischer, & Gmehling, 1999;Horstmann, Fischer, Gmehling, & Kolar, 2000b; Guilbotet al., 2000; Horstmann, Fischer, & Gmehling, 2001), theresults usually become quite inaccurate in the case of asym-metric mixtures that exhibit LLE (Zhong & Masuoka, 1996;Li, Fischer, & Gmehling, 1998).Several attempts have been made to improve the results

of PSRK for asymmetric systems. Unfortunately all thesestudies have not succeeded in pointing out the main rea-son responsible for the inaccurate results yielded by PSRK,namely the local 3t of its parameters at the sub-critical VLE.Thus, it is not surprising that parameters obtained by suchway do not work outside the range of the 3t. In spite ofthis, several entirely empirical corrections for the existingPSRK parameters have been proposed so far. For example,Li et al. (1998) have developed the idea of the so-called ef-fective values for the relative van der Waals volumes andthe surface areas. Their proposed correction becomes sig-ni3cant only for heavy homologues. Zhong and Masuoka(1996) and Yang and Zhong (2001) have proposed di=er-ent corrections for APSRK. In what follows we will demon-strate the doubtful nature of these corrections, which are re-sponsible for prediction of the non-physical trend of globalphase behavior. However before doing so we will considersome alternative approaches for an accurate prediction ofphase behavior in mixtures.

2.2. LCVM

The linear combination of the Vidal and Michelsenmixing rules (LCVM) predictive model has been 3rst pro-posed by Boukouvalas, Spiliotis, Coutsikos, Tzouvaras, andTassios (1994), and its parameter matrix was extended bySpiliotis, Boukouvalas, Tzouvaras, and Tassios (1994a),Spiliotis, Magoulas, and Tassios (1994b), Voutsas,


Boukouvalas, Kalospiros, and Tassios (1996) and Yakoumiset al. (1996). The LCVM model is based on the Peng–Robinson EOS (1976), given as follows:

P =RT

Vm − b− a�V 2m + 2bVm − b2

; (9)

where

a= 0:45724R2T 2

c

Pc; b= 0:07780

RTcPc

: (10)

The relative inability of Eq. (9) to predict the volu-metric properties of pure compounds is improved here byimplementation of the volume-translation functionality ofMagoulas and Tassios (1990). This functionality is consis-tent, hence it does not in>uence the prediction of the phasebehavior of mixtures. However, it introduces a tempera-ture dependency in the repulsive term, which may lead tonon-physical phenomena such as intersection of isothermsat high pressure, as recently demonstrated by Pfohl (1999).Therefore, here we implement the LCVMmodel without thevolume-translation functionality of Magoulas and Tassios(1990).For non-polar compounds LCVM uses the 1-parameter

�-functionality of Soave (1972), given as follows:

�= [1 + (0:384401 + 1:52276!− 0:213808!2

+ 0:034616!3 − 0:001976!4)(1− T 0:5r )]2: (11)

Eq. (11) attaches an entirely predictive character to themodel. However, at the same time, it is characterized byseveral fundamental drawbacks, as described previously(Polishuk et al., 2002b, c). For example it yields that if!¿ 0:4283 then m¿ 1. This means that LCVM will gen-erate the second 3ctitious critical points for all n-alkanesheavier than n-octane and the temperatures of these pointswill be given by

Tr =(1 + m)2

(1− m)2: (12)

Obviously 3ctitious critical points of pure compoundswill generate non-physical phase behavior in mixtures. Suchresult may seriously a=ect the prediction of asymmetricsystems involving helium or hydrogen, information that isvaluable for modeling hydrogenation processes and as wellfor astrophysical research. However, in the particular case ofthe series considered here, it usually occurs outside the rangeof practical importance. Nevertheless, it should be realizedthat the GE-mixing rule may lead to a situation where thenon-physical phase equilibria predicted by Eq. (11) will alsoappear at ordinary conditions. Such possibility is strongerin the case of multi-compound systems.LCVM implements Eq. (3) for b and uses the following

mixing rule for a:

a=[(

�Av

+1− �AM

)GE

RT+

1− �AM

∑xi ln

(bbi

)

+∑

xiaibiRT

]bRT: (13)

Eq. (13) was obtained by the empirical combination of Eq.(8) with the original Huron–Vidal model (Tzouvaras, 1994),in spite of the fact that the 3rst model uses a zero-pressurereference state, and the second an in3nite pressure one.Thus, parameter � determines the relative contribution ofboth models above. Its value is equal to 0.36 and it has beenevaluated by 3tting the experimental data in numerous bi-nary systems. Am is set equal to −0:52 and Av =−0:62.

Although Orbey and Sandler (1997) have clearlydemonstrated the theoretical inconsistency of Eq. (13),Boukouvalas et al. (1994), Spiliotis et al. (1994a), Voutsaset al. (1996), Yakoumis et al. (1996), Pauly, Daridon,Coutinho, Lindelo=, and Andersen (2000) and Boukouvalas,Magoulas, Tassios, and Kikic (2001) have claimed thatLCVM is more accurate than any other semi-predictivemodel, especially for asymmetric systems.Kontogeorgis and Vlamos (2000) have explained this

observation using the argument that other models, suchas PSRK, generate a di=erence between the combinatorialterms of the ACM and the cubic EOS itself, although theo-retically these terms should be equal. Increasing asymmetryresults in increasing di=erence between both combinatorialterms, which according to Kontogeorgis and Vlamos (2000)strongly a=ects the accuracy of predicted data. In contrastto that, LCVM diminishes this di=erence in an empiricalmanner, resulting in more accurate predictions.It is our belief that the accuracy of thermodynamic mod-

els is de3ned not only by their theoretical correctness andstating this or other reference state, but also by the accuracywith which their parameters are evaluated. Analysis of thepublications dealing with prediction of phase equilibria in-dicates that attention is usually paid to development of noveland sophisticated models, while development of methodolo-gies for their appropriate implementation is in many timesneglected. As a result, the parameters in such models areusually evaluated in an arbitrary manner, mostly by a lo-cal 3t of sub-critical VLE. In this respect we do not noteany particular di=erence between PSRK and LCVM. Weexplain the relative success of LCVM for asymmetric sys-tems by the fact that its parameters have been 3tted usingdata for heavier homologues than the parameters of PSRK.For example, Holderbaum and Gmehling (1991) have 3ttedthe parameters of PSRK for the homologues series CO2–n-alkanes only up to n-decane and Boukouvalas et al. (1994)up to n-octacosane. However, it appears that such attemptdoes not remove the fundamental disadvantages of the localparameter 3t method. Actually, it can be demonstrated thatboth models fail to predict other (but not less important)than VLE data, such as LLE, LLV, and critical lines.It should also be realized that the complex and multidi-

mensional nature of GE-based models such as PSRK andLCVM, signi3cantly hinders the implementation of method-ologies that consider the complete phase behavior. As aresult, evaluation of parameters for such models should usu-ally be restricted to a local 3t. For this reason we advocateimplementation of the kIGPD methodology for the simplest


vdW mixing rules (Eq. (3)), for which binary interactionparameters have also a molecular meaning. This alternativeis considered in the following paragraphs.

2.3. GPDA

Appreciation of the fact that all parts of the thermo-dynamic phase space are closely inter-related encouragesdevelopment of models, which are simultaneously accu-rate for a maximum number of properties. It seems expe-dient to start the evaluation of the semi-predictive globalphase diagram-based approach (GPDA) developing an EOS,which will properly describe the properties of pure >uidsand their mixtures. Such a result can be achieved by statinga four-parameter equation (C4EOS), as explained below.

2.3.1. C4EOSRecently, we have proposed the following generalized

form of C4EOS (Polishuk, Wisniak, & Segura, 2000a):

P =RTVm

[1 + y(4− j)

1− yj

]− a�(T )

(Vm + c)(Vm + d); (14)

where

y =b

4Vm(15)

and j is a scaling factor that de3nes the accuracy with whichEq. (14) represents the virial coeLcients of the hard-spheretheory (Polishuk et al., 2000a). Since real molecules areusually not spherical it is not necessary that the value ofj, which is proper from the point of view of hard spheretheory, will be also the best descriptor of the experimentalfacts. We have found that di=erent values of j are optimalfor di=erent homologous series. For example, while j=3 isadequate for the series of CO2–alkanols, CF4–alkanes andmethanol–alkanes, j=4 yields better results for CH4–alkanesor acetone–alkanes. At the same time, the last choice willmake it impossible to use the EOS in the vicinity of thetriple point (TP) and to investigate compressed liquid states.Therefore in the present and the following studies we willuse one universal value of j=3:5 for all homologues seriesof systems.The values of the remaining parameters should be de3ned.

It is obvious that the proper values of the four parametersa, b, c and d may be obtained by 3tting the data of partic-ular pure compounds. However such practice violates thepredictive character of the model. Therefore the values ofthese parameters should be previously identi3ed. It is pos-sible to generalize these values in an empirical manner, assuggested thus far for other C4EOSs. However, experienceindicates that such practice a=ects the prediction of the be-havior of mixtures (see, for example, Polishuk, Wisniak,& Segura, 1999). Presently, our understanding of the rela-tion between the parameters of EOSs and the borders onthe kIGPD is incomplete. However, it can be said that thepure compound parameter b is a very signi3cant for theglobal phase diagram. Increasing b increases the region of

LLE on kIGPD and vice versa. The values of this param-eter, that correspond to an optimal balance between VLEand LLE are usually located around the value of the molarvolume at the TP and also depends on the value of the criti-cal compressibility used. Hence, we may guess that the un-successful results of the empirically generalized C4EOSs inpredicting mixtures are the result of using improper valuesof b. In addition, the evaluation of parameters for a particu-lar compound has a substantial advantage in predicting thedata of the heavy substances that form part of the asymmet-ric systems considered here.We have proposed (Polishuk et al., 2000a) that the values

of the four speci3c parameters for each pure compound, a,b, c and d, be obtained by solving four equations: the twoprescribed by Eq. (2) and the following additional ones:

b=4jVm;TP (16)

and

Vm;c;EOS = �Vm;c;EXPT; (17)

where � is a correlation factor for the critical volume. �=1is inaccurate for the gas densities, which has an immediateimpact on predicting the properties of mixtures (Polishuk etal., 2001). In particular, it is responsible for overestimatingthe composition of the heavy compound in the vapor phase.This feature seriously a=ects the accuracy of the model inpredicting important data such as solubility in supercritical>uids. Hence, we have considered the use of � values largerthan one, sacri3cing the experimental value of critical com-pressibility in order to keep the form of the EOS as simpleas possible.Since our main concern is not only the proper description

of properties of pure compounds, but also the accurate pre-diction of phase behavior in their mixtures, we have noticedthat � presents the following regularities:

1. The values of �, which are optimal for predicting thegas molar volume, are also appropriate for other prop-erties related to this phase, such as the Joule–Thomsoninversion and both the second and the third virial coeL-cients. They are also particularly accurate for describingthe high-temperature phase equilibria in mixtures, a re-sult which is not surprising since the reduced temperatureof the lighter compound may reach then relatively highvalues.

2. However, the values of �, which are optimal for predictingthe gas volume, are not so for predicting the liquid molarvolumes and, therefore, of the main property related tothis phase, namely the global phase behavior. This isdue to the fact that high values of � produce a >at trendfor the liquid densities near the critical points, both inpure compounds and in their mixtures, which may leadto the prediction of limited liquid–liquid splits. Hence,the proper values of � are located in halfway between 1and the optimal values for the gas phase.


Table 2The values of parameters in Eq. (14)

Compound a bar(g−mol=l)2 b L/g-mol c L/g-mol d L/g-mol

carbon dioxide 4.56886 0.03322 −0:02380 0.10151n-undecane 70.07878 0.19773 −0:09450 0.66389n-dodecane 79.58284 0.21645 −0:10256 0.73321n-tridecane 91.20757 0.22716 −0:10656 0.84812n-tetradecane 103.80541 0.24806 −0:12258 0.96608n-pentadecane 114.53455 0.26578 −0:12747 1.03695n-hexadecane 126.46827 0.28693 −0:13901 1.12622n-heptadecane 135.71891 0.30315 −0:13777 1.16032n-octadecane 146.82980 0.32138 −0:13983 1.22382p-nonadecane 155.54412 0.33713 −0:11935 1.20362n-docosane 191.71661 0.39005 −0:13617 1.44824n-hexatriacontane 335.34343 0.60293 0.32964 1.31729

3. There is no value of � appropriate for all compounds.However, we have detected that the values of � and b areinter-related. They both increase as the molecular weightincreases.

On the basis of these regularities we have evaluated thefollowing empirical expression for the value of dimension-less parameter �:

�= 1 + Vm;TP: (18)

In addition, we have found that if the ratio TTPVm;TPL=TcVm;c ¿ 0:117, then the solid molar volume at the TPshould be used in Eqs. (16) and (18). Usually this condi-tion is valid for light gases and substances having carbonnumber larger than 10. For other compounds the liquid mo-lar volumes at the TP should be used. These rules simplifyconsiderably the computation of the parameters in Eq. (14).The corresponding results are given in Table 2.

2.3.2. �-functionalityAnother very important issue of the present study is selec-

tion of the suitable temperature functionality for the cohe-sive parameter. Three basic requirements are given for suchfunctionality:

1. It should be >exible enough to 3t accurately all the ex-perimental vapor pressure data.

2. It should be consistent in the entire temperature rangeand should not generate numerical pitfalls.

3. It should have a simple form, which allows generalizationof its parameters to attach an entirely predictive characterto EOSs.

It is claimed that the �-functionality proposed byTwu, Bluck, Cunningham, and Coon (1991) andmodi3ed byTwu, Coon, and Cunningham (1995):

�(T ) = TN (M−1)r exp[L(1− TNMr )] (19)

meets these requirements. Recently this functionality hasbeen incorporated in the Ge-based semi-predictive modelproposed recently by Alhers and Gmehling (2001). How-ever, it should be pointed out that the form of Eq. (19) israther complex. Therefore it may easily lead to numericalpitfalls. In addition, our results indicate that the expressionproposed by Twu et al. (1991) does not contribute to theaccurate prediction of mixtures. Although our present un-derstanding of the in>uence of �-functionalities on kIGPDis incomplete, the inaccurate results of Eq. (19) may be ex-plained by a negative contribution of its exponential part. Atthe same time, if the 3rst part of Eq. (19) is taken alone, ityields quite accurate shapes of kIGPDs. This part of Eq. (19)is identical with the �-functionality proposed by Hederer,Peter, and Wenzel (1976):

�(T ) = Tmr : (20)

Although Eq. (20) is particularly accurate for mixturesand consistent in the entire temperature range, it has onlyone adjustable parameter, and, therefore, it is not >exibleenough. Thus, we have developed the following temperaturefunctionality for the cohesive parameter, which meets allour requirements:

�(T ) = Tm1Tm2r

r : (21)

Although Eq. (21) is very compact and has only two ad-justable parameters it is not less >exible than Eq. (19). Inaddition, the parameters of Eq. (21) can be easily general-ized. In particular, we have found that for all hydrocarbonsparameter m2 can be taken as 0.25. Then, for hydrocarbonshaving carbon number larger than 10, m1 (which should bealways negative) can be generalized as follows:

m1 =−0:4162 + 1:5447!2 − 2:5285!2 + 0:81466!3

b;

(22)

where the value of parameter b in the units of L/g-mol isused as a dimensionless number. At the same time, the highquality of data available for light gases makes evaluation of


both m1 and m2 easy. In particular, we have estimated thatfor carbon dioxide m1 =−0:33595 and m2 =−0:15187.

2.3.3. Combination rulesThe range of applicability of the kIGPD-based semi-

predictive approach can be substantially enlarged consider-ing the fact that the values of binary interaction parametersshould change proportionally to the values of the corre-sponding pure compound parameters. Such proportionalitycan be easily establish for the case of a linear combinationrule for the covolume, as follows:

l12 =b22 − b11b22 + b11

L12; (23)

where L12 is a value characteristic for a given homologuesseries.The proportional relation for the binary interaction pa-

rameter k12 is not so simple. Analysis of kIGPDs prescribesthat a12 should be dependent on the value of b12 (this obser-vation may have a theoretical background). In addition, theaccuracy of calculated data for asymmetric systems can besigni3cantly improved by introducing a temperature depen-dency for k12. There are three basic requirements for such adependency:

1. It should not violate the theoretical quadratic dependenceof the second virial coeLcient on the mole fraction.

2. It should not generate non-realistic values of the bi-nary parameter at very high temperatures outside therange of phase equilibria, as usually done by otherk12(T)-functionalities.

3. Its numerical contribution should decrease with a de-crease in the asymmetry of the system and it should bereducible to the form of Eq. (4).

In addition, many investigators (see for example Kato etal., 1981; Ohe, 1990; Kordas et al., 1994) have called atten-tion to the fact that for many very di=erent systems k12(T )should keep the U-type form. Since in the present study weattempt to develop a simple and universal temperature de-pendency for the cross energy parameter, we propose po-sitioning its minimum value at the critical temperature ofthe lighter compound in the system. Such practice will notonly avoid the undesirable prediction of limited liquid–liquidsplits near Tc1, but it will also improve the description of bothlow-temperature LLE and high-temperature VLE, which arestrongly a=ected by the imperfect crossover behavior gen-erated by EOSs. In other words, a U-type k12(T ) can signif-icantly improve the global phase behavior predicted by themodel. On the basis of these facts, we suggest adopting thefollowing combination rule:

k12 =(K11 − l12

Tc2Tc1

)(1− t) + K22t; (24)

where K11 and K22 are characteristic values for a given ho-mologues series and t is given by the following dimension-

less functionality:

t = tan h

[(T − Tc1T ∗c2 − Tc1

)2]: (25)

For homologues heavier than the reference homologue(for which K11, K22 and L12 are estimated), T ∗

c2 = Tc2. Forlighter homologues T ∗

c2 is equal to the Tc2 of the referencehomologue. For example for the series of CO2–alkanes,we have selected CO2–decane as the reference homologue.Therefore, T ∗

c2 of all lighter homologues will be taken as617:7 K, the Tc of n-decane (Daubert et al., 1989–2002).This will allow keeping the approximately same tempera-ture dependency of k12 along the homologues series.

In the 3rst version of the semi-predictive approach(Polishuk et al., 2001) the binary interaction parameterswere taken at the intersection of the loci that represent theexperimental values of the temperature UCEP and the pres-sure of CPM on the kIGPD of the reference homologue.Yet we have drown an additional locus that represents theexperimental temperature of CPM, which results in accu-rate correlation of the high-temperature data (additionaldetails see Polishuk, Wisniak, & Segura, 2003). It shouldbe noticed that two-parameter cubic EOSs usually do notallow intersection of all three key loci at one point of thekIGPD, even when implementing a temperature functional-ity for k12. However, such result can be achieved using Eq.(16) due to its better ability to represent the global phasebehavior in mixtures. For the series CO2–alkanes K11 =0:1,K22 = 0:35 and L12 = 0:02. The volumetric parameters cand d are calculated using linear combination rules withoutadjustable parameters.We will now proceed to analyze the results predicted by

PSRK, LCVM and GPDA.

3. Result

3.1. General

Fig. 1 presents the critical lines predicted by the four ap-proaches GPDA, LCVM, PSRK, and PSRK modi3ed by Liet al. (1998) for the systems under consideration. It is seenthat the GPDA has a clear advantage over all other modelsin predicting the data (Fig. 1a). It yields almost preciseresults for the vapor–liquid critical data of all homologuesup to CO2–n-heptadecane and slightly underestimates thecritical pressures of the systems CO2–n-nonadecane andCO2–n-docosane. Moreover, GPDA predicts correctly theglobal phase behavior of the homologues series (See Fig.2). TCP and DCEP have been calculated considering theternary systems of CO2–(n-dodecane+n-tridecane) andCO2–(n-tridecane+n-tetradecane). Fig. 2 shows a verygood agreement between the available experimental end-point data and those predicted by the GPDA. In additionGPDA gives a very accurate prediction of the K-points


4804003202400

100

200

300

400

T(K)

P(b

ar)

GPDA

C C C

C C C

C

C

22 19 17

13 14 15 16C

C

C

C

11

12

13

14

15

C16

4603803002200

200

400

600

T(K)

P(b

ar)

LCVM

C

C

C

C C C

C

C

17

19

22

13 14 15 16

C

C

C

C

11

12

13

14

15

C 16

4804003202400

200

400

600

800

T(K)

P(b

ar)

C 19

C 17 C C C16 15 14

PSRK

C 13

C 13

C 14

C 12

C 11

8007907800

500

1000

C 22

2 phases

1 phase

4804003202400

200

400

600

800

T(K)

P(b

ar)

PSRK(Li)

C

C

24

22

C24

C 16

C 16

C 19

(a) (b)

(c) (d)

Fig. 1. Critical curves of CO2–n-alkanes. Solid lines—predicted critical lines; gray line—vapor pressure line of CO2; —critical point of CO2;�—experimental UCEPs of CO2–n-dodecane and CO2–n-tridecane (Hottovy et al., 1981); x—predicted critical endpoints; +—critical points ofCO2–n-undecane extrapolated from the LLE data of Schneider et al. (1967) and Horvat (1965); ⊗—critical points of CO2–n-dodecane extrapolated fromthe LLE data of Meyer (1988); ⊕—critical points of CO2–n-tridecane of Enick et al. (1985) and those extrapolated from the LLE data of Schneider et al.(1967) and Meyer (1988); •—critical points of CO2–n-tetradecane of Scheidgen (1997); •—critical points of CO2–n-tetradecane extrapolated fromthe LLE data of Meyer (1988); ◦—critical points of CO2–n-pentadecane of Scheidgen (1994); 4—critical points of CO2–n-hexadecane of Scheidgen(1994); �—critical points of CO2–n-heptadecane of Scheidgen (1997); —critical points of CO2–n-nonadecane of Scheidgen (1997); �—criticalpoints of CO2–n-docosane of Scheidgen (1997).

(which are essential for the design of the supercritical ex-traction processes).However, Fig. 1 a shows that GPDA is somewhat less

accurate in predicting the liquid–liquid critical lines at highpressures due to the inaccurate critical slopes in the P–Tprojection. This disadvantage is common to all the modelsconsidered here. An important result is that the slopes ofthe liquid–liquid critical lines are de3ned by the values ofl12. The available experimental liquid–liquid critical datamay be 3t using large positive values of l12. However, suchvalues are inadequate for predicting VLE. In particular, theyresult in a signi3cant underestimation of the bubble-pointpressures.Since GPDA implements slightly positive values of l12, its

predictions of the liquid–liquid critical slopes are somewhatmore accurate than PSRK and, especially, LCVM.

Furthermore, it can be seen that the GE-based semi-predictive models fail in their estimation of both the globalphase behavior and the critical lines. In particular, LCVMpredicts an extremely large region of the transitional behav-ior of Type IV. Although such behavior is usually exhibitedby one homologue in the series, LCVM predicts it for sixhomologues: from CO2–n-undecane to CO2–n-hexadecane.As shown in Fig. 1b, these non-realistic results of theglobal phase behavior lead to the inaccurate prediction ofthe critical data.The results presented in the 3gure show that since LCVM

is a model based on the local 3t approach, it does not possessa high predictive ability. Therefore it is inaccurate not onlyin describing critical lines, but also LLE, K-points, and otherdata that have not been used for the 3t of its parameters(Fig. 1b).


21191715131197190

230

270

310

350

C

T(K

)

Type IIIType II Typ

eIV

UCEPs K-pointsDCEP

TCP

LCEPs

Liquid-Liquid Phase Split

Homogeneous Liquid Phase GPDA

21191715131197190

230

270

310

350

C

T(K

)

Type IIIType II Type IV

UCEPs

K-points

DCEP

TCP

LCEPs


Homogeneous Liquid Phase LCVM

21191715131197190

230

270

310

350

C

T(K

)

Type IIIType II Typ

eIV

UCEPsK-points

DCEP

TCP

LCEPs


Homogeneous Liquid Phase PSRK

Fig. 2. Global phase behavior in the homologous series of CO2–n-alkanes.Solid lines—predicted critical endpoint lines; 4—experimental UCEPsand LCEPs of Miller and Luks (1989); •—experimental K-points ofMiller and Luks (1989) and Iwade, Katsumura and Ohgaki (1993).

The same reasons explain why the results of PSRK areworse than the predictions of LCVM: the data of the systemsunder consideration have not been ever considered for 3ttingits parameters (Holderbaum&Gmehling, 1991). As a result,PSRK predicts Type III behavior already for the systemCO2–n-undecane (Fig. 2) and signi3cantly overestimates thedata of all the homologues up to the non-realistic predictionof CO2–n-docosane (Fig. 1c).Nevertheless, the predictions of the original PSRK give

qualitatively the correct patterns of phase behavior: a de-crease in solubility and increase of the critical pressure withincreasing carbon numbers of the homologues. This is not acase of the PSRK corrected by Li et al. (1998), which prop-agates the errors of the original local 3t by another local 3t.Such practice leads to the absurd result that increasing themolecular weight of the solute increases its solubility until areturn of Type II behavior for the system CO2–n-tetracosane(Fig. 1d). The same results are obtained by other corrections

of the PSRK model, which are probably di=erent in form,but are based on the similar principle (Zhong & Masuoka,1996; Yang & Zhong, 2001).The failure of GE-based semi-predictive models in de-

scribing the experimental data contradicts the widely ac-cepted opinion regarding their advantages in comparisonwith generalized correlations based on classical vdW mix-ing rules (Eqs. (3)–(4)). Fig. 3 presents typical predictionsyielded by such correlations. Although the correlations areless accurate than GPDA, they are usually better than PSRKand LCVM. In addition, they are much simpler and donot generate the non-realistic results characteristic for theGE-based models.

Blas and Galindo (2002) and Galindo and Blas (2002)have recently investigated the ability of SAFT-VR to pre-dict data in the series CO2–n-alkanes. The unique trans-ferable set of two binary adjustable parameters has beenobtained by 3tting the data of CO2–n-tridecane and thenused for predicting the critical lines of other homologues, ina way that may resemble the present GPDA. However, theresults of Blas and Galindo (2002) and Galindo and Blas(2002) are less accurate than the ones yielded by GPDA,presented on Figs. 1a and 2. In particular, they have pre-dicted a false limited liquid–liquid split (Type IV behavior)for CO2–n-dodecane. In addition, SAFT-VR underestimatesthe UCEPs of homologues heavier than C10. Accurate de-scription of UCEPs is essential for attaching a predictivecharacter to the model. Thus, Galindo and Blas (2002) havefailed to predict the critical curves of heavier homologuesusing the unique pair of binary parameters, which has re-quired re-3tting. Moreover, underestimation of UCEPs mayseriously a=ect the prediction of VLE; in particular it maylead to an underestimation of the bubble-points if the tem-perature independent binary parameters are used (Polishuket al., 2001).The fact that all regions of the thermodynamic phase space

are closely inter-related prescribes that robustness and relia-bility of the model can be achieved only by establishing theproper balance between VLE and LLE. We will now val-idate this statement by examining the accuracy of GPDA,LCVM and PSRK in predicting the available sub-criticaldata for the binary systems in question.

3.2. CO2–n-undecane

The experimental VLE data available for this importantsystem can be found only in the unpublished results ofHorvat (1965). The LLE data from this source have beenpartially published by Schneider, Alwani, Heim, Horvath,and Franck (1967). Fig. 4 compares the ability of the threemodels to predict the isopleth data of Horvat (1965). Again,the advantage of GPDA is clear since it predicts the datawith relatively high accuracy in the entire temperature range.This is not a case for the two other semi-predictive equationsconsidered here. They are relatively accurate only for VLE


44040036032028024050

100

150

200

250

300

T(K)

P(b

ar)

Kato et al. (1981)

Coutinho et al. (1994)

Kordas et al. (1994)

GPDA

Sigmund and Trebble (1990)

Fig. 3. Critical line of CO2(1)–tetradecane(2) predicted by generalizations based on classical mixing rules. •—critical points of CO2–n-tetradecane ofScheidgen (1997).

5304303302300

80

160

240

T(K)

P(b

ar)

GPDAGPDA

LCVM

PSRK

PSRKLCVM

Fig. 4. Phase equilibria in the system CO2(1)–n-undecane(2). Gray dotline—vapor pressure of CO2; —critical point of CO2; black solidline—data predicted by GPDA at x(1) = 0:88; gray solid line—datapredicted by LCVM at x(1) = 0:88; black dot line—data predicted byPSRK at x(1) = 0:88; ◦—experimental VLE data of Horvat (1965) atx(1)=0:88; •—experimental LLE data of Horvat (1965) at x(1)=0:88;+—experimental VLE data of Horvat (1965) at x(1) = 0:862.

at moderated temperatures, where the local 3t of their pa-rameters is typically performed. However, as expected, theGE-based models fail to predict data outside this range. Inparticular, LCVM signi3cantly underestimates the temper-ature of LLE and overestimates the high-temperature VLE.PSRK, in turn, overestimates both LLE and VLE. The shapeof the isopleths generated by PSRK leads to prediction of afalse absolute liquid–liquid split, a result, which is character-istic of this model (Polishuk et al., 2002c). LCVM predictsfor this system a false limited liquid–liquid split, which is

seen clearly in Fig. 2. GPDA predicts the correct topologyof phase behavior, without any false liquid–liquid split.

3.3. CO2–n-dodecane

Fig. 5 presents the results for this system. The availableexperimental data are located in a range of moderated tem-peratures, for which the predictions of the GE-based modelscan be relatively accurate. No VLE critical data is avail-able for this system. The 3gure demonstrates the doubtfulnature of the accuracy obtained by a local 3t. In particu-lar, although LCVM predicts the given experimental pointsalmost exactly and PSRK only slightly overestimates thebubble-point data, both models are unable to predict the cor-rect topology of phase behavior of the system. On the otherhand, GPDA gives an accurate prediction. It is somewhatless precise than LCVM for the bubble-point data but it isthe best model for predicting the solubility of n-dodecane insupercritical CO2. In addition, GPDA correctly predicts theUCEP and does not generate a false liquid–liquid split, asdo both GE-based models. In other words, GPDA shows abetter overall reliability of predictions.

3.4. CO2–n-tridecane

This homologue exhibits a transitional phase behavior ofType IV and, therefore, is among the most interesting mem-bers in the series. Not much experimental data are avail-able for this system, namely several critical points of Enicket al. (1985) and numerous data points of Schneider et al.(1967) in the very near vicinity of the critical lines. In ad-dition, Hottovy et al. (1981) and Fall and Lucks (1985) and


1.00.80.60.40.20.00

60

120

180

x(1)

P(b

ar)

y

318.15 K

353.15 K

393.15 K

GPDA

1.0000.9990.9980

20

40

60

80

363.15 K

343.15 K

318.15 K

1.00.80.60.40.20.00

60

120

180

x(1)

P(b

ar) 318.15 K

353.15 K

393.15 K

LCVM

1.0000.9990.9980

20

40

60

80

363.15 K

343.15 K

318.15 K

y

1.00.80.60.40.20.00

60

120

180

x(1)

P(b

ar)

LL318.15K

y

353.15 K

393.15K

PSRK

1.0000.9990.9980

20

40

60

80

318.15K

343.15K

363.15K

Fig. 5. Phase equilibria in the system CO2(1)–n-dodecane(2). Black solidlines—predicted VLE data; gray solid lines—predicted critical curves;x—predicted critical endpoints; 4—experimental data of Gardeler, Fischerand Gmehling (2002) at 318:15 K; experimental data of Henni, Ja=ar andMather (1996) at: ◦—313:15 K; �—353:15 K; �—393:15 K; experi-mental data of Ampueda Ramos (1986): ⊕—318:15 K; —343:15 K;⊗—363:15 K.

Enick et al. (1985) have reported detailed data regarding thethree-phase equilibria. Thus, the available data does not al-low investigation of VLE and restricts it to consideration ofthe critical lines and the three-phase equilibria.The P–T projection of the critical line is presented in

Fig. 1. It can be seen that only GPDA predicts the dataaccurately. LCVM overestimates the data at high tempera-tures and fails to predict the location of critical endpoints.The results of PSRK are inaccurate both qualitatively andquantitatively.

1.00.90.80.70

70

140

210

x(1)

P(b

ar)

K-point

LCEPUCEP

GPDA

V

L1L2

L2 L1

1.00.90.80.70

70

140

210

x(1)

P(b

ar) K-point

LCEP

UCEP

LCVM

V

L1

L2 L1

L2

1.00.90.80.70.60.50

60

120

180

240

300

x(1)

P(b

ar)

K-point

PSRK

V

L1L2

Fig. 6. Phase equilibria in the system CO2(1)–n-tridecane(2). Black solidlines—predicted three-phase lines; gray solid lines—predicted criticalcurves; �—predicted critical endpoints; 4—experimental critical data ofEnick et al. (1985); •—experimental three-phase data of Hottovy et al.(1981); ◦—experimental three-phase data of Fall and Lucks (1985).

Additional details are given on Fig. 6, which presentsthe P–x projection of the critical line and the three-phaseequilibria. It can be seen that although LCVM yields a qual-itatively correct prediction of Type IV, it is quantitativelyinaccurate because the critical composition is signi3cantlyunderestimated. We will show later that this underestima-tion of critical composition is not accidental; it is a resultof the attempt to achieve a precise local 3t of the availableexperimental bubble points. It should be realized that


neglecting parts of thermodynamic phase space always car-ries a price: underestimation of the critical compositionsresults in a very inaccurate description of the three-phaselines and leads to failure in predicting the global phasebehavior of whole homologues series. In other words, alocal 3t of bubble point data removes the robustness andreliability of the model.On the other hand, GPDA produces a reasonable distri-

bution of the inevitable deviations from experimental dataover the entire thermodynamic phase space. Figs. 1 and 6demonstrate that GPDA is accurate in predicting the criti-cal lines in both the P–T and P–x projections and generatesonly minor deviations for the three-phase lines. Once more,these results point to another advantage of the Global PhaseDiagram-based approach.

3.5. CO2–n-tetradecane

The only available sub-critical data for this system(Gasem, Dickson, Dulcamara, Nagarajan, & Robinson,1989; Danesh, Todd, Somerville, & Dandekar, 1990) are at344:3 K. The bubble points reported by Gasem et al. (1989)seem more accurate than those reported by Danesh et al.(1990). The data of Gasem et al. (1989) on the solubilityof n-tetradecane in supercritical carbon dioxide seem tobe overestimated not only when comparison with those ofDanesh et al. (1990), but also when compared to the avail-able data for other n-alkanes (see, for example, with Fig. 5).The system CO2–n-tetradecane is the 3rst in a series ex-

hibiting Type III behavior. Both GPDA and PSRK yield aqualitative prediction of the data but the results of GPDAare also quantitatively accurate. Although GPDA is slightlyless accurate than LCVM in predicting the bubble points, itis more exact in describing the solubility in supercritical car-bon dioxide. It addition, the critical line (Fig. 1) evidencesthe robustness of GPDA in the entire temperature range.On the other hand, LCVM fails to predict the topol-

ogy of phase behavior for the given system (see Fig. 1).Fig. 7 compares the predictions of the available VLE data ofCO2–n-tetradecane. It can be seen that LCVM predicts thebubble point data correctly but that the accuracy is achievednot by an overall robustness and reliability of the model, butby underestimating the solubility of n-tetradecane in the su-percritical carbon dioxide and the critical composition. In aprevious section we have already discussed this result andhave related it to the failure of LCVM to predict the globalphase behavior. In other words, although the local 3t of thegiven bubble-point data yields a very accurate result for theavailable experimental points, the LCVM model will fail topredict the data at other temperatures.

3.6. CO2–n-pentadecane

The latter statement may be validated by testing theability of the models to predict the available data of

1.00.90.80.750

100

150

200

250

x(1)

P(b

ar)

LCVMGPDA

PSRK

344.3 K

Fig. 7. Phase equilibria in the system CO2(1)–n-tetradecane(2) at 344:3 K.Black solid line—data predicted by GPDA; gray solid line—data predictedby LCVM; black dot line—data predicted by PSRK; �—experimentaldata of Danesh et al. (1990); �—experimental data of Gasem et al.(1989).

1.000.800.600.400.200.000

60

120

180

x(1)

P(b

ar)

292 K

LCVM

GPDAPSRK

LL

E

VLE

Fig. 8. Phase equilibria in the system CO2(1)–n-pentadecane(2) at 292 K.Black solid line—data predicted by GPDA; gray solid line—data predictedby LCVM; black dot line—data predicted by PSRK; •—experimentaldata of Scheidgen (1997).

CO2–n-pentadecane. The isotherm presented on Fig. 8 is ofpractical importance because it is typical of the temperaturesused in the injection of CO2 for the enhanced oil recovery.

The phase equilibria at the given temperature consist oftwo parts: the VLE and the LLE. However, the 3gure demon-strates that LCVM fails to predict any LLE at 292 K, atemperature located between the false LCEP and UCEP gen-erated by this model. Such result illustrates our previous con-clusion regarding the poor reliability characteristic of thismodel. In contrast, the predictions of GPDA appear to berobust, as usual. The data of Scheidgen (1997) at 292 K un-derestimates the CO2 content in the hydrocarbon-rich phasein comparison with the available three phase data sets ofHottovy et al. (1981) and van der Steen, de Loos, and deSwaan Arons (1989). Therefore, the accuracy of GPDA


1.00.80.60.40.20.00

50

100

150

200

250

x(1)

P(b

ar)

298 K

PSRK GPDA

LCVM

LL

E

VLE

1.00.80.60.40.20.00

50

100

150

200

x(1)

P(b

ar)

333.15 K

GPDA

LCVM

PSRK

1.00.80.60.40.20.00

90

180

270

360

x(1)

P(b

ar)

393.2 K

GPDA

LCVM

PSRK

1.00.80.60.40.20.00

50

100

150

x(1)

P(b

ar)

663.75 K

PSRK GPDA LCVM

Fig. 9. Phase equilibria in the system CO2(1)–n-hexadecane(2). Black solid line—data predicted by GPDA; gray solid line—data predicted by LCVM; blackdot line—data predicted by PSRK;•—experimental data of Scheidgen (1997); 4—experimental data of Schwarz and Prausnitz (1987); ⊗—experimentalhere-phase data of van der Steen et al. (1989); �—experimental data of King, Kassim, Bott, Sheldon, and Mahmud (1984); ◦—experimental data ofSpee and Schneider (1991); �—experimental data of Sebastian, Simnick, Lin, and Chao (1980).

seems somewhat better than it can be learned from Fig. 8.Although PSRK signi3cantly overestimates the immiscibil-ity, it is qualitatively correct.

3.7. CO2–n-hexadecane

Many data sets are available for this system in the en-tire temperature range, although some of them seem incon-sistent with the rest. For example, the results reported byCharoensombut-Amon, Martin, and Kobayashi (1986) dis-agree with the ones reported by Schneider et al. (1967) andScheidgen (1997). We have selected the data, which seemmore accurate and cover the entire temperature range from298 to 663.75 K.The pertinent results are presented in Fig. 9 and show

patterns of behavior similar to those discussed previously.Although the predictions of GPDA may somewhat deviatefrom the experimental points, their overall robustness is ev-ident. It is seen that although GPDA predicts accurately theLLE data at 298 K and above 150 bars, it is not a case ofLLE below this pressure. It can also be seen that the exper-

imental data of Scheidgen (1997) in this particular regionseems questionable, because they again do not agree withthe three-phase data of van der Steen et al. (1989). Thus,the predictions of GPDA are probably more accurate at 298K than it can be understood from the 3gure.PSRK, again, gives only a qualitative description of

the data in the entire temperature range due to signi3cantoverestimation of phase immiscibility. LCVM once morefails to predict even the topology of phase behavior at lowtemperature; it also underestimates the critical compositionat 333:15 K. However, this time such result does not im-prove the accuracy of the model for predicting the bubblepoints and it remains slightly less accurate than GPDA.Both LCVM and GPDA predict with similar accuracy thebubble point data at 393:2 K. However, now LCVM startsto overestimate the near-critical data, the result which canbe easily understood considering its inaccurate predictionof critical lines (Fig. 1b). LCVM underestimates the criticalcomposition at 333:15 K but overestimates it at 663:75 K.As a result, it is less accurate than GPDA in predicting thebubble points and somewhat better for the dew points.


At the high temperatures in question the inability ofanalytical EOS to describe the crossover critical behaviorincreases signi3cantly. Although both LCVM and GPDAprobably predict a reasonable value for the critical pres-sure, they fail to describe both sides of the phase envelope.On the other hand, although PSRK gives a rather accuratedescription of the available VLE data points at this temper-ature (663:75 K), such accuracy is just accidental becausePSRK predicts a non-realistic value for the critical pressure(234.5 bar).

3.8. CO2–n-heptadecane and–n-octadecane

The results for these two systems are presented inFigs. 10 and 11. It can be seen that the observations made forthe previous systems are also valid for the present ones. Themodels exhibit the same consistent regularities in predictingthe data for the homologues series, a fact that can also belearned from the kIGPD. It is seen that LCVM becomesless accurate as the molecular weight of the homologueincreases. This result might be explained by the procedureused to evaluate the parameters of the model. GPDA againpredicts the data accurately, showing the reliability of the

1.00.80.60.40.20.00

100

200

300

x(1)

P(b

ar)

P(b

ar)

P(b

ar)

LCVM

GPDA

PSRK353.2 K

1.00.80.60.40.20.00

100

200

300

400

x(1)

LCVM

GPDA

PSRK393.2 K

0.50.40.30.20.10.00

30

60

90

x(1)

P(b

ar)

LCVM

GPDA

PSRK

463.3 K

1.00.80.60.40.20.00

90

180

270

x(1)

LCVM

GPDA

PSRK

605.4 K

Fig. 11. Phase equilibria in the system CO2(1)–n-octadecane(2). Black solid line—data predicted by GPDA; gray solid line—data predicted by LCVM;black dot line—data predicted by PSRK; •;�—experimental data of PTohler (1994); ◦;�—experimental data of Kim, Lin, and Chao (1985).

1.00.80.60.40.20.00

100

200

300

400

500

x(1)

P(b

ar) LCVM

GPDA

PSRK

393.2 K

Fig. 10. Phase equilibria in the system CO2(1)–n-heptadecane(2)at 393:2 K. Black solid line—data predicted by GPDA; gray solidline—data predicted by LCVM; black dot line—data predicted by PSRK;•—experimental data of PTohler (1994).

model. Therefore, it is not surprising that GPDA has a clearsuperiority in predicting the solubility of n-octadecane insupercritical carbon dioxide (Fig. 12).


36034032030010-4

10-3

10-2

10-1

T(K)

y(2)

PSRK

LCVM

GPDA

100 bar

Fig. 12. Solubility of n-octadecane(2) in supercritical CO2(1). Blacksolid line—data predicted by GPDA; gray solid line—data predicted byLCVM; black dot line—data predicted by PSRK; •—experimental dataof Eustaquio-RincUon and Trejo (2001).

We will now analyze the behavior of the models withthe heaviest homologue, for which pure compound dataare available in the DIPPR data bank (Daubert et al.,1989–2002).

3.9. CO2–n-hexatriacontane

n-Hexatriacontane is a heavy asphaltene compound and,therefore, one may expect that prediction of its phase equi-libria with carbon dioxide without resource to experimen-tal data is questionable. Nevertheless, it can be seen thatGPDA yields a precise prediction of the data (Fig. 13),stressing once more the robustness and reliability of thismodel. Although LCVM is substantially less accurate thanGPDA, its predictions are not completely wrong. This is incontrast with the results of PSRK, which fails in predictingthe data. In particular, PSRK predicts a very deep liquid–liquid immiscibility, which it is not even in>uenced bytemperature.The bubble point data presented on the Fig. 13 are very

important but the dew-point data of systems like CO2–n-hexatriacontane are even more so because supercriticalextraction is performed by the vapor phase. The molecu-lar weight of n-hexatriacontane is similar to many essentialbiologically active compounds that are treated in the phar-maceutical and other industries by supercritical carbon diox-ide. However, the solubility of such compounds is extremelylow. This fact has a very negative impact not only on the re-liability of the available experimental data sets, which mayexhibit signi3cant deviations, but also on the eLciency ofthe extraction processes.It is not always realized that the solubility of even very

heavy compounds in the supercritical carbon dioxide canbe considerably enhanced in the vicinity of the critical line.Moreover, outside the heterogeneous region the solubility isunlimited and the extraction process is supposed to become

0.60.40.20.00

20

40

60

80

100

x(1)

P(b

ar) 373.2 K

423.2 K

473.35 K

573.25 K

GPDA

1.00.90.80.7180

220

260

300366.9 K

0.60.40.20.00

20

40

60

80

100

x(1)

P(b

ar)

373.2 K

423.2 K

473.35 K

573.25 K

LCVM

1.00.90.80.7180

220

260

300366.9 K

0.60.40.20.00

20

40

60

80

100

x(1)

P(b

ar)

373.2 K573.25 K

PSRK

1.000.750.500.25180

220

260

300366.9 K

Fig. 13. Phase equilibria in the system CO2(1)–n-hexatriacontane(2).Black solid lines—predicted VLE data; experimental data of Tsai, Huang,Lin, and Chao (1987): •—523:75 K; ◦—473:35 K; �—373:25 K;experimental data of Gasem and Robinson (1985): �—423:2 K;4—373:2 K; ⊕—experimental data of Nieuwoudt and du Rand (2002).

extremely e=ective. Therefore, it seems important to knowif the critical line of the system can be approached beforedecomposition of the heavy organic compound. Althoughthe pertinent experimental data are unavailable in literature


9507505503500

200

400

600

T(K)

P(b

ar)

GPDA

LCVM

PSRK

Fig. 14. Critical line of the system CO2(1)–n-hexatriacontane(2). Blacksolid line—data predicted by GPDA; gray solid line—data predicted byLCVM; black dot line—data predicted by PSRK.

36302418126220

280

340

400

460

C

T

(K

)c,

min

Possible location of T for CO - n-Cc, min 2 36

Fig. 15. UCST minima in the series CO2–n-alkanes. •—experimentaldata of Miller and Luks (1989), Meyer (1988) and Scheidgen (1997).

we may try to estimate it using the semi-predictive modelsdeveloped for this purpose. The answer to this question isprovided by the equations considered in this work, as shownin Fig. 14. It can be seen that the predictions of the criti-cal lines are very di=erent. In particular, PSRK shows thepresence of gas-gas immiscibility in the system and LCVMyields a minimal UCST of 648 K. It is very probable thatn-hexatriacontane and many of organic compounds of sim-ilar molecular weight will decompose before reaching thistemperature. In other words, both GE-based models predictthat we cannot reach the homogenous region in the systemunder consideration. In contrast, GPDA predicts that an ab-solute miscibility of CO2 and n-hexatriacontane is real (theminimal USCT is at 433 K).Let us try to analyze which prediction is more reliable.

Fig. 15 demonstrates that the minimal UCST changes overthe homologues series in a continuous manner. For the sys-

tem in question the minimal UCST can be estimated tobe between 370 and 430 K. Thus, the answer provided byGPDA seems correct and both GE-based models are prob-ably wrong. These results demonstrate again the advantageof an approach, which does not involve the local 3t of VLE,but considers the complete picture of phase behavior.

4. Conclusions

Modern thermodynamic models allow a precise correla-tion of experimental data. However, the problem of theirreliable prediction without resource to experimental resultsfor particular systems is still open. An optimal solution tothis problem seems extrapolation of the 3t of certain sys-tems to others using a semi-predictive procedure. However,development of the semi-predictive methodology so far hasnot received proper attention.In particular, it is usually assumed that models that are

characterized by high correlative ability should yield reli-able predictions as well. Therefore it is widely agreed thatcubic EOSs combined with Huron-Vidal type mixing ruleshave good predictive ability. The most successful modelsof the kind are PSRK and LCVM. These models have manyadjustable parameters, which attach to them an excellent>exibility and correlative ability. In addition, the ACM in-corporated by the Huron–Vidal type mixing rules have largeparameter matrixes, which are believed to contribute to thepredictive ability. However, the multidimensional nature ofthe GE-based models substantially hinders consideration ofthe complete thermodynamic phase space, which includesVLE in the entire temperature range, LLE, LLV, and criti-cal lines as well. As a result, the parameters of these modelsare usually evaluated by the way of local 3t of VLE, whichalso requires large bases of experimental data for manysystems.In the present study we complete the development of

an alternative semi-predictive approach based on accu-rate C4EOS and classical vdW mixing rules. The sim-plicity of the proposed model allows implementation ofthe global phase diagram methodology in order to con-sider the complete phase behavior. GPDA predicts thedata assuming that a basic property such as the balancebetween VLE and LLE (the relation between the UCEPand CPM) remains approximately constant over the homo-logues series. Therefore, in contrast to PSRK and LCVM,GPDA requires no more that 2–3 experimental criticalpoints of one homologue in order to predict the data ofthe whole series. In other words, the predictive characterof GPDA is much more evident. Comparison of the resultsfor the homologues series carbon dioxide–n-alkanes pre-dicted by GPDA, PSRK and LCVM leads to the followingconclusions:

• The fact that GE-based models require large database ofmany homologues is not accidental. The present study


shows that the extrapolative ability of these models is nothigh. They are accurate mostly in describing the data usedfor evaluation of their parameters. However, theGE-basedmodels hardly can estimate other data in a reliable man-ner. In particular, PSRK, whose parameters have been3tted to the homologues up to n-decane, considerablyoverestimates immiscibility and fails to predict quanti-tatively almost all the data of the homologues consid-ered in the present study. On the other hand, LCVM,whose parameters have been 3tted to homologues upto n-octacosane, can accurately describe the pertinentsub-critical VLE. Nevertheless, it looses accuracy in pre-dicting VLE for homologues heavier that n-octacosane.Therefore, both PSRK and LCVM can hardly be im-plemented for predicting asymmetric systems, whichhave not been considered in the evaluation of theirparameters.

• The local 3t approach tends to ignore the fact that allregions of the thermodynamic phase space are closelyinter-related. In other words, the accuracy of correlatingsome data, such as VLE, is achieved by the improperdescription of other data, such as LLE or critical lines.In particular, LCVM neglects the description of criti-cal points in order to yield a precise 3t of the availablebubble-point data. Its prediction of the critical lines iseven less accurate than those estimated using the empir-ical generalizations of vdW classical combining rules.As a result, LCVM predicts a non-realistic global phasebehavior over the entire homologues series and fails todescribe LLE and LLV. The attempts to correct the ACMparameters of PSRK in an arti3cial manner propagatethe errors of the original local data 3t by another local3t and lead to prediction of non-physical results. Thus,GE-based models can hardly yield robust estimations ofdata other than VLE, even for systems that have beenincluded in the evaluation of their parameters.

• The results of the present study indicate that both thePSRK and LCVM models appear to be more correlativerather than predictive. In contrast, the robust predictivecharacter of GPDA is evident. In particular, the accuratefour-parameters EOS used by GPDA is free of numer-ical pitfalls and contrary to other EOSs it allows theintersection of all key loci on the kIGPD. Thus, GPDAproduces a reasonable distribution of the inevitabledeviations from the experimental data over the wholethermodynamic phase space. Therefore, it is more re-liable in the entire temperature range and for all typesof phase equilibria and yields an accurate predictionof the global phase behavior in the homologues se-ries. Moreover, increasing asymmetry does not a=ectthe accuracy of GPDA. In other words, using onlya few key experimental points of one homologue,GPDA can accurately predict the data for the entirehomologues series, including its heaviest members.Hence, this approach is highly recommended for predic-tion of data, not available experimentally.

Notation

a cohesion parameterb covolumec; d attraction density dependence parameters in Eq.

(14)Cx carbon numberGE excess Gibbs energyP pressureR universal gas constantT temperaturex mole fraction of the lighter compound in the

liquid phasey mole fraction of the lighter compound in the

vapor phaseV volume

Greek letters

� temperature dependence of the attractionparameter

� correction factor of the critical volume! acentric factor

Subscripts

c critical statem molar property

Abbreviations

ACM activity coeLcient modelC4EOS four-parameter equation of stateCPM critical pressure maximumDCEP double critical end pointEOS equation of stateGPD global phase diagramGPDA global phase diagram approachklGPD global phase diagram in the k12–l12 projectionLCEP lower critical end pointLCST lower critical solution temperatureLCVM linear combination of the Vidal and Michelsen

mixing rulesLLE liquid–liquid equilibriaLLV liquid–liquid–vapor equilibriaPSRK predictive-Soave–Redlich–Kwong group

contribution EOSSAFT statistical associating >uid theory EOSTCP tri-critical pointsTP triple pointsUCEP upper critical end point (K-points)UCST upper critical solution temperaturevdW van der Waals modelVLE vapor–liquid equilibria


Acknowledgements

This work was 3nanced by the Israel Science Foundation,grant number 340/00 and by FONDECYT, Santiago, Chile(Project 1020340). We acknowledge the help of Prof. G.Schneider from the Ruhr-UniversitTat Bochum, for providinghis unpublished experimental data for some systems studiedhere.

Appendix A. The fugacity coe"cient of compound in amixture for the general C4EOS

Let us assume the most general form of C4EOS given asfollows:

P =RTVm

[1 + K1y1− K2y

]− a

(Vm + c)(Vm + d): (A.1)

Eq. (A.1) can be reduced to Eq. (14) by taking K1 = 4− jand K2 = j. It will then generate the second repulsivevirial coeLcient equal to 4, the value prescribed by thehard-sphere theory. However, in the most general case,one may also consider other options. The fugacity coeL-cient of compound in a mixture from Eq. (A.1) is given asfollows:

ln (i = (K1 + K2)b′=(4Vm − bK2)− ln[PV=RT ]− (K1 + K2)ln[1− (K2b=4Vm)]K2

+a(d′=(Vm + d)− c′=(Vm + c) + (1 + a′=a− (c′ − d′)=(c − d))ln[(Vm + d)=(Vm + c)])

(c − d)RT; (A.2)

where

x′ =(@Nx@Ni

)T;Nj �=1

(A.3)

and Ni—is number of moles of compound i.

References

Alhers, J., & Gmehling, J. (2001). Development of an universal groupcontribution equation of state I. Prediction of liquid densities for purecompounds with a volume translated Peng-Robinson equation of state.Fluid Phase Equilibria, 191, 177–188.

Ampueda Ramos, F. (1986). Phase diagrams of hydrocarbons(C12)—supercritical ?uids (N2, CO2, N2–CO2). (in French). Ph.D.dissertation. Lyon: UniversitUe de Lyon.

Blas, F. J., & Galindo, A. (2002). Study of the high pressure phasebehaviour of CO2 + n-alkane mixtures using the SAFT–VR approachwith transferable parameters. Fluid Phase Equilibria, 194–197,501–509.

Bordet, F., Passarello, J. -P., Chartier, T., Tufeu, R., & Baumard, J. F.(2001). Modelling solutions of hydrocarbons in dense CO2 gas. Journalof the European Ceramic Society, 21, 1219–1227.

Boshkov, L. Z., Deiters, U. K., Kraska, T., & Lichtenthaler, R. N.(2002). Global phase diagrams. Physical Chemistry Chemical Physics,4, V.

Boukouvalas, C. J., Magoulas, K. G., Tassios, D. P., & Kikic, I. (2001).Comparison of the performance of the LCVM model (an EoS=GE

model) and the PHCT EoS (the perturbed hard chain theory equationof state) in the prediction of the vapor-liquid equilibria of binarysystems containing light gases. The Journal of Supercritical Fluids,19, 123–132.

Boukouvalas, C., Spiliotis, N., Coutsikos, P., Tzouvaras, N., & Tassios,D. (1994). Prediction of vapor–liquid equilibrium with the LCVMmodel: a linear combination of the Vidal and Michelsen mixing rulescoupled with the original UNIFAC and the t-mPR equation of state.Fluid Phase Equilibria, 92, 75–106.

Cassel, E., Matt, M., Rogalski, M., & Solimando, R. (1997). Phaseequilibria modeling for binary systems that contain CO2. Fluid PhaseEquilibria, 134, 63–75.

Charoensombut-Amon, T, Martin, R. J., & Kobayashi, R. (1986).Application of a generalized multiproperty apparatus to measure phaseequilibrium and vapor phase densities of supercritical carbon dioxidein n-hexadecane system up to 26 MPa. Fluid Phase Equilibria, 31,89–104.

Coutinho, J. A. P., Kontogeorgis, G. M., & Stenby, E. H. (1994). Binaryinteraction parameters for nonpolar systems with cubic equations ofstate: a theoretical approach. 1. CO2=hydrocarbons using SRK equationof state. Fluid Phase Equilibria, 102, 31–60.

Dahl, S., & Michelsen, M. L. (1990). High-pressure vapor-liquidequilibrium with a UNIFAC-based equation of state. A.I.Ch.E.Journal, 36, 1829–1836.

Danesh, A., Todd, A. C., Somerville, J., & Dandekar, A. (1990). Directmeasurement of interfacial tension, density, volume, and compositionsof gas–condensate system. Chemical Engineering Research andDesign, 68, 325–330.

Daubert, T. E., Danner, R. P., Sibul, H. M., & Stebbins, C. C. (1989).Physical and thermodynamic properties of pure chemicals. DataCompilations, Bristol, PA: Taylor & Francis.

Deiters, U. K., & de Reuck, K. M. (1997). Guidelines for publicationof equations of state-I. Pure >uids. Pure and Applied Chemistry, 69,1237–1249.

Deiters, U. K., & Schneider, G. (1976). Fluid mixtures at high pressures.Computer calculations of the phase equilibriums and the criticalphenomena in >uid binary mixtures from the Redlich-Kwong equationof state. Berichte der Bunsengesellschaft fBur Physikalische Chemie,80, 1316–1321.

Enick, R., Holder, G. F., & Morsi, B. I. (1985). Critical and threephase behavior in the carbon dioxide/tridecane system. Fluid PhaseEquilibria, 22, 209–234.

Eustaquio-RincUon, R., & Trejo, A. (2001). Solubility of n-octadecane insupercritical carbon dioxide at 310, 313, 333, and 353 K, in range10–20 MPa. Fluid Phase Equilibria, 185, 231–239.

Fall, D. J., & Lucks, K. D. (1985). Liquid–liquid–vapor phase equilibriaof the binary system carbon dioxide+n-tridecane. Journal of Chemicaland Engineering Data, 30, 276–279.

Fischer, K., & Gmehling, J. (1996). Further development, status andresults of the PSRK method for the prediction of vapor–liquid equilibriaand gas solubilities. Fluid Phase Equilibria, 121, 185–206.

Fredenslund, A., Gmehling, J., & Rasmussen, P. (1977). Vapor–liquidequilibria using UNIFAC, a group contribution method. Amsterdam:Elsevier.

Galindo, A., & Blas, F. J. (2002). Theoretical examination of the global>uid phase behavior and critical phenomena in carbon dioxide +


n-alkane binary mixtures. Journal of Physical Chemistry, B 106,4503–4515.

Gardeler, H., Fischer, K., & Gmehling, J. (2002). Experimentaldetermination of vapor–liquid equilibrium data for asymmetricsystems. Industrial and Engineering Chemistry Research, 41, 1051–1056.

Gasem, K. A. M., Dickson, K. B., Dulcamara, P. B., Nagarajan, N., &Robinson Jr., R. L. (1989). Equilibrium phase compositions, phasedensities, and interfacial tensions for carbon dioxide + hydrocarbonsystems. 5. Carbon dioxide + n-tetradecane. Journal of Chemical andEngineering Data, 34, 191–195.

Gasem, K. A. M., & Robinson Jr., R. L. (1985). Solubilities of carbondioxide in heavy normal paraLns (C20–C44) at pressures to 9:6 MPaand temperatures from 323 to 423 K. Journal of Chemical andEngineering Data, 30, 53–56.

Gauter, K., Florusse, L. J., Peters, C. J., & de Swaan Arons, J.(1996). Classi3cation and transformations between types of >uid phasebehavior in selected ternary systems. Fluid Phase Equilibria, 116,445–453.

Gmehling, J., Jiding, L., & Fischer, K. (1997). Further development ofthe PSRK model for the prediction of vapor–liquid–equilibria at lowand high pressures II. Fluid Phase Equilibria, 141, 113–127.

Gmehling, J., & Lohmann, J. (2001). About analysis of the molecularinteractions of organic anhydride + alkane binary mixtures using theNitta–Chao model by L. Lugo, E. R. LUopez, J. GarcUWa, M. J. P.Comunas, J. FernUandez [Fluid Phase Equilib. 170 (2000) 69–85]. FluidPhase Equilibria, 189, 193–195.

Guilbot, P., ThUeveneau, P., Baba-Ahmed, A., Horstmann, S., Fischer, K.,& Richon, D. (2000). Vapor–liquid equilibrium data and critical pointsfor the system H2S+COS. Extension of the PSRK group contributionequation of state. Fluid Phase Equilibria, 170, 193–202.

Hansen, H. K., Rasmussen, P., Fredenslund, A., Schiller, M., & Gmehling,J. (1991). Vapor–liquid equilibria by UNIFAC group contribution.5. Revision and extension. Industrial and Engineering ChemistryResearch, 30, 2352–2355.

Hederer, H., Peter, S., & Wenzel, H. (1976). Calculation ofthermodynamic properties from a modi3ed Redlich–Kwong equationof state. The Chemical Engineering Journal, 11, 183–190.

Henni, A., Ja=er, S., & Mather, A. E. (1996). Solubility of N2O and CO2in n-dodecane. The Canadian Journal of Chemical Engineering, 74,554–557.

Holderbaum, T., & Gmehling, J. (1991). PSRK: a group contributionequation of state based on UNIFAC. Fluid Phase Equilibria, 70,251–265.

Horstmann, S., Fischer, K., & Gmehling, J. (1999). Experimentaldetermination of critical points of pure compounds and binary mixturesusing a >ow apparatus. Chemical Engineering Technology, 22,839–842.

Horstmann, S., Fischer, K., & Gmehling, J. (2000a). PSRK groupcontribution equation of state: revision and extension III. Fluid PhaseEquilibria, 167, 173–186.

Horstmann, S., Fischer, K., Gmehling, J., & Kolar, P. (2000b).Experimental determination of the critical line for (carbon dioxide+ ethane) and calculation of various thermodynamic properties for(carbon dioxide + n-alkane) using the PSRK model. The Journal ofChemical Thermodynamics, 32, 451–464.

Horstmann, S., Fischer, K., & Gmehling, J. (2001). Experimentaldetermination of the critical data of mixtures and their relevance forthe development of thermodynamic models. Chemical EngineeringScience, 56, 6905–6913.

Horvat, E. (1965). Phase equilibira in binary systems of carbon dioxide–n-undecane, carbon dioxide–n-dodecane and ethane–squalane.(in German). M.Sc. dissertation under direction of G. Schneider.Karlsruhe: Technische Hochschule Karlsruhe.

Hottovy, J. D., Lucks, K. D., & Kohn, J. P. (1981). Three-phaseliquid–liquid–vapor equilibria behavior of certain binary

CO2-n-paraLn systems. Journal of Chemical and Engineering Data,26, 256–258.

Huron, M. J., & Vidal, J. (1979). New mixing rules in simple equationsof state for representing vapour–liquid equilibria of strongly non-idealmixtures. Fluid Phase Equilibria, 3, 255–271.

Imre, A. R., Kraska, T., & Yelash, L. V. (2002). The e=ect ofpressure on the liquid-liquid phase equilibrium of two polydispersepolyalkylsiloxane blends. Physical Chemistry Chemical Physics, 4,992–1001.

Iwade, S., Katsumura, Y., & Ohgaki, K. (1993). High-pressure phasebehavior of binary systems of heptadecane and octadecane with carbondioxide or ethane. Journal of Chemical Engineering of Japan, 26,74–79.

Jo=e, J., & Zudkevitch, D. (1966). Fugacity coeLcients in gas mixturescontaining light hydrocarbons and carbon dioxide. Industrial andEngineering Chemistry Fundamentals, 5, 455–462.

Kato, H., Nagahama, K., & Hirata, M. (1981). Generalized interactionparameters for the Peng-Robinson equation of state: carbondioxide-n-paraLn binary systems. Fluid Phase Equilibria, 4,219–231.

Kim, H., Lin, H. -M., & Chao, K. -C. (1985). Vapor–liquid equilibrium inbinary mixtures of carbon dioxide + n-propylcyclohexane and carbondioxide + n-octadecane. AIChE Symposium Series, 244, 96–101.

King, M. B., Kassim, K., Bott, T. R., Sheldon, J. R., & Mahmud,R. S. (1984). Prediction of mutual solubilities of heavy compoundswith super-critical and slightly sub-critical solvents: the role ofequation of state and some applications of a simple expanded latticemodel at subcritical temperatures. Berichte der Bunsengesellschaft fBurPhysikalische Chemie, 88, 812–820.

Kontogeorgis, G. M., & Vlamos, P. M. (2000). An interpretation ofthe behavior of EoS=GE models for asymmetric systems. ChemicalEngineering Science, 55, 2351–2358.

Kordas, K., Tsoutsouras, K., Stamataki, S., & Tassios, D. (1994).A generalized correlation for the interaction coeLcients ofCO2–hydrocarbon binary mixtures. Fluid Phase Equilibria, 93,141–166.

Levelt Sengers, J. M. H. (2000). Supercritical >uids: Their propertiesand applications. In E. Kiran, P. G. Debenedetti, & C. J. Peters(Eds.), Supercritical ?uids. Fundamentals and applications, NATOscience series E, Vol. 366 (p. 1). Dordrecht: Kluwer AcademicPublishers.

Li, J., Fisher, K., & Gmehling, J. (1998). Prediction of vapor–liquidequilibria for asymmetric systems at low and high pressures with thePSRK model. Fluid Phase Equilibria, 143, 71–82.

Magoulas, K., & Tassios, D. (1990). Thermophysical properties ofn-alkanes from C1 to C20 and their prediction for higher ones. FluidPhase Equilibria, 56, 119–140.

Mathias, P. M., & Copeman, T. W. (1983). Extension of thePeng–Robinson equation of state to complex mixtures: Evaluationof the various forms of the local composition concept. Fluid PhaseEquilibria, 13, 91–108.

Meyer, C. (1988). Experimental investigation of phase equilibria inbinary CO2-mixtures at the pressures up to 140 MPa (in German).M.Sc. dissertation. Bochum: Ruhr-UniversitTat Bochum.

Miller, M. M., & Luks, K. D. (1989). Observations on the multiphaseequilibria behavior of CO2-rich and ethane-rich mixtures. Fluid PhaseEquilibria, 44, 295–304.

Myers, A. L., & Prausnitz, J. M. (1965). Thermodynamics of solid carbondioxide solubility in liquid solvents at low temperatures. Industrialand Engineering Chemistry Fundamentals, 4, 209–212.

Nieuwoudt, I., & du Rand, M. (2002). Measurement of phase equilibriaof supercritical carbon dioxide and paraLns. Journal of SupercriticalFluids, 22, 185–199.

Nung, W.-C., & Tsai, F.-N. (1997). Correlation of vapor–liquidequilibrium for systems of carbon dioxide + hydrocarbon by thecorresponding-states principle. Chemical Engineering Journal, 66,217–221.


Ohe, S. (1990). Physical sciences data 42. Vapor–liquid equilibriumdata at high pressure. Amsterdam: Elsevier.

Orbey, H., & Sandler, S. I. (1997). A comparison of Huron–Vidal typemixing rules of mixtures of compounds with large size di=erences,and a new mixing rule. Fluid Phase Equilibria, 132, 1–14.

Passarello, J. P., Benzaghou, S., & Tobaly, P. (2000). Modeling mutualsolubility of n-alkanes and CO2 using SAFT equation of state.Industrial and Engineering Chemistry Research, 39, 2578–2585.

Pauly, J., Daridon, J. -L., Coutinho, J. A. P., Lindelo=, N., & Andersen,S. I. (2000). Prediction of solid->uid phase diagrams of light gases–heavy paraLn systems up to 200 MPa using an equation of state-GE

model. Fluid Phase Equilibria, 167, 145–159.Peng, D. Y., & Robinson, D. B. (1976). A new two-constant equation

of state. Industrial and Engineering Chemistry Fundamentals, 15,59–64.

Pfohl, O. (1999). Evaluation of an improved volume translation forthe prediction of hydrocarbon volumetric properties. Fluid PhaseEquilibria, 163, 157–159.

PTohler, (1994). Fluid phase equilibria in binary and ternary mixtures ofcarbon dioxide with low-volatile organic substances at temperaturesfrom 303 up to 393 K and pressures from 10 MPa up to100 MPa (in German). Ph.D. dissertation. Bochum: Ruhr-UniversitTatBochum.

Polishuk, I., Stateva, R., Wisniak, J., & Segura, H. (2002a). Prediction ofhigh pressure phase equilibria using cubic EOS. What can be learned?The Canadian Journal of Chemical Engineering, in press.

Polishuk, I., Wisniak, J., & Segura, H. (2002b). Closed loops of liquid–liquid immiscibility predicted by semi-empirical cubic equations ofstate and classical mixing rules. Physical Chemistry Chemical Physics,4, 879–883.

Polishuk, I., Wisniak, J., Segura, H., & Kraska, T. (2002c). About therelation between the empirical and the theoretically based parts ofvan der Waals-like equations of state. Industrial and EngineeringChemistry Research, 41, 4414–4421.

Polishuk, I., Wisniak, J., & Segura, H. (1999). Prediction of the criticallocus in binary mixtures using equation of state I. Cubic equations ofstate, classical mixing rules, mixtures of methane-alkanes. Fluid PhaseEquilibria, 164, 13–47.

Polishuk, I., Wisniak, J., & Segura, H. (2001). Simultaneous prediction ofthe critical and sub-critical phase behavior in mixtures using equationof state I. Carbon dioxide-alkanols. Chemical Engineering Science,56, 6485–6510.

Polishuk, I., Wisniak, J., & Segura, H. (2003). Estimation of liquid–liquid–vapor equilibria using predictive EOS models. 1. Carbon dioxide–n-alkanes. Journal of Physical Chemistry, B 107, 1864–1874.

Polishuk, I., Wisniak, J., & Segura, H. (2000a). A novel approach forde3ning parameters in a four-parameter EOS. Chemical EngineeringScience, 55, 5855–5870.

Polishuk, I., Wisniak, J., Segura, H., Yelash, L. V., & Kraska, T. (2000b).Prediction of the critical locus in binary mixtures using equation ofstate II. Investigation of van der Waals-type and Carnahan-Starling-typeequations of state. Fluid Phase Equilibria, 172, 1–26.

Raeissi, S., Gauter, K., & Peters, C. J. (1998). Fluid multiphase behaviorin quasi-binary mixtures of carbon dioxide and certain 1-alkanols.Fluid Phase Equilibria, 147, 239–249.

Scheidgen, A. (1994). Fluid phase equilibria in CO2+1-nonanol+ pentadecane and CO2 + 1-nonanol + hexadecane to100 MPa (in German). M.Sc. dissertation. Bochum: Ruhr-UniversitTatBochum.

Scheidgen, A. (1997). Fluid phase equilibria in binary and ternarymixtures of carbon dioxide with low-volatile organic substances up to100 MPa (in German). Ph.D. dissertation. Bochum: Ruhr-UniversitTatBochum.

Schneider, G., Alwani, Z., Heim, W., Horvath, E., & Franck, E. U. (1967).Phase equilibriums and critical phenomena in binary mixed systemsto 1500 bars. Carbon dioxide with n-octane, n-undecane, n-tridecane,and n-hexadecane. Chemie Ingenieur Technik, 39, 649–656.

Schwarz, H. J., & Prausnitz, J. M. (1987). Solubilities of methane,ethane, and carbon dioxide in heavy fossil–fuel fractions. Industrialand Engineering Chemistry Research, 26, 2360–2366.

Sebastian, H. M., Simnick, J. J., Lin, H. -M., & Chao, K. -C. (1980).Vapor–liquid equilibrium in binary mixtures of carbon dioxide +n-decane and carbon dioxide + n-hexadecane. Journal of Chemicaland Engineering Data, 25, 138–140.

Sigmund, P. M., & Trebble, M. A. (1992). Phase behavior prediction insupercritical systems using generalized solute critical properties. TheCanadian Journal of Chemical Engineering, 70, 814–817.

Soave, G. (1972). Equilibrium constants from a modi3ed Redlich–Kwong equation of state. Chemical Engineering Science, 27, 1197–1203.

Spee, M., & Schneider, G. M. (1991). Fluid phase equilibriumstudies on binary and ternary mixtures of carbon dioxide withhexadecane, 1-dodecanol, 1,8-octanediol and dotriacontane at 393:2 Kand at pressures up to 100 MPa. Fluid Phase Equilibria, 65,263–274.

Spiliotis, N., Boukouvalas, C., Tzouvaras, N., & Tassios, D. (1994a).Application of the LCVMmodel to multicomponent systems: Extensionof the UNIFAC interaction parameter table and prediction of the phasebehavior of synthetic gas condensate and oil systems. Fluid PhaseEquilibria, 101, 187–210.

Spiliotis, N., Magoulas, K., & Tassios, D. (1994b). Prediction of thesolubility of aromatic hydrocarbons in supercritical CO2 with EoS=GE

models. Fluid Phase Equilibria, 102, 121–141.Trebble, M. A., Salim, P. H., & Sigmund, P. M. (1993). A generalized

approach to prediction of two and three phase CO2–hydrocarbonequilibria. Fluid Phase Equilibria, 82, 111–118.

Trebble, M. A., & Sigmund, P. M. (1990). A generalized correlationfor the prediction of phase behavior in supercritical systems. TheCanadian Journal of Chemical Engineering, 68, 1033–1039.

Tsai, F. -N., Huang, S. H., Lin, H. -M., & Chao, K. -C. (1987). Solubilityof methane, ethane, and carbon dioxide in n-hexatriacontane. Journalof Chemical and Engineering Data, 32, 467–469.

Turek, E. A., Metcalfe, R. S., & Fishback, R. E. (1988). Phasebehavior of several carbon dioxide/west-Texas-reservoir–oil systems.SPE Reservoir Engineering, 3, 505–516.

Twu, C. H., Bluck, D., Cunningham, J. R., & Coon, J. E. (1991).A cubic equation of state with a new alpha function and a new mixingrule. Fluid Phase Equilibria, 69, 33–50.

Twu, C. H., Coon, J. E., & Cunningham, J. R. (1995). A new generalizedalpha function for a cubic equation of state Part 1. Peng–Robinsonequation. Fluid Phase Equilibria, 105, 49–59.

Tzouvaras, N. P. (1994). On standard and reference states compatiblewith the LCVM model. Fluid Phase Equilibria, 97, 53–60.

van der Steen, J., de Loos, Th. W., & de Swaan Arons, J. (1989).The volumetric analysis and prediction of >uid liquid–liquid–vaporequilibria in certain carbon dioxide + n-alkane systems. Fluid PhaseEquilibria, 51, 353–367.

van Konynenburg, P. H., & Scott, R. L. (1980). Critical lines andphase equilibria in binary van der Waals mixtures. PhilosophicalTransactions of the Royal Society of London, 298, 495–540.

Voutsas, E. C., Boukouvalas, C. J., Kalospiros, N. S., & Tassios, D. P.(1996). The performance of EoS=GE models in the prediction ofvapor–liquid equilibria in asymmetric systems. Fluid Phase Equilibria,116, 480–487.

Vradman, L., Herskowitz, M., Korin, E., & Wisniak, J. (2001).Regeneration of poisoned nickel catalyst by supercritical CO2extraction. Industrial and Engineering Chemistry Research, 40,1589–1590.

Wong, S. H. W., & Sandler, S. I. (1992). A theoretically correct mixingrule for cubic equations of state. A.I.Ch.E. Journal, 38, 671–678.

Yakoumis, I. V., Vlachos, K., Kontogeorgis, G. M., Coutsikos, P.,Kalospiros, N. S., Tassios, D., & Kolisis, F. N. (1996). Applicationof the LCVM model to systems containing organic compounds and


supercritical carbon dioxide. The Journal of Supercritical Fluids, 9,88–98.

Yang, Q., & Zhong, C. (2001). A modi3ed PSRK model for the predictionof the vapor–liquid equilibria of asymmetric systems. Fluid PhaseEquilibria, 192, 103–120.

Zhong, C., & Masuoka, H. (1996). Mixing rules for accurate prediction ofvapor–liquid equilibria of gas/large alkane systems using SRK equationof state combined with UNIFAC. Journal of Chemical Engineeringof Japan, 29, 315–322.

simultaneous prediction of the critical and sub-critical phase behavior in mixtures using equations...

Documents