Thermodynamic consistency of high pressure ternary mixtures containing a compressed gas and solid solutes of different complexity

Fluid Phase Equilibria 242 (2006) 93–102

Thermodynamic consistency of high pressure ternary mixtures containing acompressed gas and solid solutes of different complexity

Jose O. Valderrama a,b,∗, Pedro A. Robles b

a Faculty of Engineering, Department of Mechanical Engineering, University of La Serena, Casilla 554, La Serena, Chileb Center for Technological Information (CIT), Casilla 724, La Serena, Chile

Received 12 November 2005; received in revised form 8 January 2006; accepted 12 January 2006

Abstract

A thermodynamic consistency test applicable to high pressure binary gas–solid mixtures is extended to ternary mixtures containing a compressedgas and two solid solutes. A high pressure mixture containing carbon dioxide as solvent and two chemically similar solutes (2,3 dimethylnaphthaleneand 2,6 dimethylnaphthalene) and a high pressure mixture containing carbon dioxide as solvent and two chemically different solutes (capsaicinand �-carotene), are considered in the study. Several sets of isothermal solubility data for binary and ternary mixtures are considered in the study.Tisc©

K

1

tolmsaprnf

daPcW

0d

he Peng–Robinson equation of state with the mixing rules of Wong and Sandler have been employed for modeling the solubility of the solidn the case of binary mixtures, while the classical van der Waals mixing rules were used for modeling the ternary mixtures containing two solidolutes. Then the proposed thermodynamic consistency test has been applied. The results show that the thermodynamic test for ternary mixturesan be applied with confidence determining consistency or inconsistency of the experimental data used.

2006 Elsevier B.V. All rights reserved.

eywords: Thermodynamic consistency; Supercritical fluids; Equations of state; Mixing rules; Solid–gas mixtures

. Introduction

Different methods to test inherent inaccuracies of experimen-al phase equilibrium data have been published in the literaturever the years. The important differences found in the data pub-ished by different researchers is one of the reasons that has

otivated the proposals of these so-called “thermodynamic con-istency tests”. Although it is difficult to be absolutely certainbout the correctness of a given set of experimental data, it isossible to check whether such data satisfy certain fundamentalelationships, establishing that the data is or is not thermody-amically consistent. The thermodynamic relationship that isrequently used to analyze thermodynamic consistency of exper-

Abbreviations: Aver, average value used for the area deviations; DMN,imethylnaphthalene; eq., equation; EoS, equation of state; GAs, geneticlgoritms; K, kelvin; Max, maximum value; NFC, not fully consistent; PR,eng–Robinson EoS; Ref, reference to the literature; TC, thermodynamicallyonsistent; TI, thermodynamically inconsistent; VL, van Laar; vdW, van deraals; WS, Wong–Sandler∗ Corresponding author. Tel.: +56 51 551158; fax: +56 51 551158.

imental phase equilibrium data is the Gibbs–Duhem equation.Depending on the way in which the Gibbs–Duhem equation ishandled, different consistency tests have been derived. Amongthese are the Slope Test, the Integral Test, the Differential Testand the Tangent–Intercept Test. If the Gibbs–Duhem equation isnot obeyed then the data is not consistent and can be consideredas incorrect. If the equation is obeyed, the data is thermodynam-ically consistent but not necessarily correct. Good reviews ofthese methods are found in the books by Raal and Muhlbauer[1] and Prausnitz et al. [2].

Valderrama and Alvarez [3] presented an interesting methodto test the thermodynamic consistency of phase equilibrium datain binary mixtures containing a liquid solute and a supercriti-cal fluid. They analyzed the difficulties normally found whenmodeling these type of mixtures and proposed a methodol-ogy to analyzed the experimental data and to conclude abouttheir thermodynamic consistency or inconsistency. In anothercommunication, the same authors discussed the correct formof analyzing the accuracy of a model when dealing with highpressure phase equilibrium data [4], aspects that are taken intoaccount in the proposed consistency method. More recently,

E-mail address: [email protected] (J.O. Valderrama). Valderrama and Zavaleta [5] presented a thermodynamic test

378-3812/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2006.01.012

94 J.O. Valderrama, P.A. Robles / Fluid Phase Equilibria 242 (2006) 93–102

Table 1Properties of the substances considered in this work

Compound Formula M M/MCO2 TC (K) PC (MPa) ω Ref. [1] V soli (cm3/gmol) Ref. [2]

2,3-DMN C12H12 156.20 3.6 785.0 3.13 0.424 Haselow et al. [40] 138.3 Haselow et al. [40]2,6-DMN C12H12 156.20 3.6 777.0 3.14 0.420 Haselow et al. [40] 168.2 Haselow et al. [40]Capsaicin C18H27NO3 305.42 6.9 1062.1 1.71 1.180 Skerget and Knez [19] 289.8 Vafai et al. [41]�-Carotene C40H56 536.85 12.2 1209.4 1.24 1.040 Subra et al. [15] 536.5 Vafai et al. [41]Carbon dioxide CO2 44.01 1.0 304.2 7.38 0.224 Daubert et al. [42] – –

Ref. [1] means literature sources for the critical properties and acentric factor and Ref. [2] means references for the solid molar volume.

to analyze high pressure binary gas–solid mixtures. So far, fewattempts have been done to treat ternary mixtures as presentedin this work.

The consistency method for ternary solid–solid–gas mixturesproposed in this work can be considered as a modeling procedureand can be easily extended to other multicomponent mixtures.In the method, a thermodynamic model that can accurately fitthe experimental data must be first used to apply the consistencytest. The fitting of the experimental data requires the calculationof some model parameters using a defined objective functionthat must be optimized.

Once a thermodynamic model (such an equation of state withappropriate mixing and combining rules) accurately fit the datafulfilling the equality of fugacities required by the fundamentalphase equilibrium equation, that model is used to check that theGibbs–Duhem equation is also fulfilled. Once should notice thatthese two steps, modeling of the data and the application of theGibbs–Duhem equation are independent so that good modelingdoes not guarantee consistency and that consistent data cannotnecessarily be well represented by a defined model.

The binary mixtures selected for this study present someinteresting peculiarities that make them appropriate for the ther-modynamic test for ternary mixtures that is presented here. Thesolutes themselves, 2,3 dimethylnaphthalene, 2,6 dimethylnaph-thalene, capsaicin, and �-carotene, present some very differentphysicochemical characteristics and properties that determinedi(hmlcpcd

FtifTCfcim

The solubility of 2,3DMN and 2,6DMN has been stud-ied by Kurnik et al. [6] for the binaries CO2 + 2,3DMN andCO2 + 2,6DMN at 308, 318 and 328 K. The range of pressurewas from 100 to 300 bars and the solubilities were all of theorder of 10−3. To the best of our knowledge, the only publishedset of data for the ternary mixture CO2 + 2,3DMN + 2,6DMN isthat of Kurnik and Reid [7]. They presented experimental datafor the ternary mixture at 308 and 318 K. The ranges of pressureand solubilities were about the same than those for the binarymixtures of Kurnik et al. [6]. Fig. 1 presents the data availableon this ternary mixture CO2 + 2,3DMN + 2,6DMN.

Solubility of capsaicin in high pressure CO2 has been mea-sured by Knez and Steiner [8], Hansen et al. [9], and morerecently by de la Fuente et al. [10]. These authors used differentexperimental methodologies and comparable results, at severaltemperatures from 298 to 333 K and over a pressure range from6 to 40 MPa were found. The solubility data for �-carotene inhigh pressure CO2 can be found in the literature at tempera-tures ranging from 288 to 353 K and pressures ranging from 5to 50 MPa [11–17]. The solubility data for this mixture reported

FD((×) 2,6-DMN at 318 K.

ifferent phase behavior (molecular size, molecular shape, polar-ty, chemical affinity). The dimethynaphthalene compounds2,3DMN and 2,6DMN) are non-polar polycyclic aromaticydrocarbons, chemical compounds that consist of fused aro-atic rings. On the other hand, capsaicin is a cytotoxic alka-

oid of the group known as capsaicinoid and �-carotene is aarotenoid made up of isoprene units as basic structures. Table 1resent some properties of these solutes included in the mixturesonsidered in this work. The literature sources from where theata were obtained are also given in the table.

The mixtures studied also have some special characteristics.or the binary mixtures, CO2 + capsaicin and CO2 + �-carotene,

he concentration of the solid solutes (capsaicin and �-carotene)n the supercritical solvent (CO2) is much lower than thoseor the binary mixtures CO2 + 2,3DMN and CO2 + 2,6DMN.he molecular weights ratio solute/solvent (M/MCO2 ) forO2 + capsaicin and CO2 + �-carotene is much higher than those

or CO2 + 2,3DMN and CO2 + 2,6DMN mixtures. Also, in allases, the concentration of the solvent in the compressed phases close to 1.0, but it cannot be considered as a pure gas, so

ixture properties must be determined.

ig. 1. Data for the ternary mixtures available in the literature: CO2 + 2,3-MN + 2,6-DMN at two temperatures. The data are from Kurnik and Reid [7]:�) 2,3-DMN at 308 K; (�) 2,6-DMN at 308 K; (�) 2,3-DMN at 318 K; and

J.O. Valderrama, P.A. Robles / Fluid Phase Equilibria 242 (2006) 93–102 95

Fig. 2. Data for the ternary mixtures available in the literature: CO2 + capsaicin+ �-carotene at two temperatures. The data are from Skerget and Knez [19]: (�)capsaicin at 298 K; (�) �-carotene at 298 K; (�) capsaicin at 313 K; and (×)�-carotene at 313 K.

by different researchers present large discrepancies, which canbe mainly attributed to the different experimental techniquesused to measure the solubility, to solute impurities, and alsoto solute degradation [18]. To the best of our knowledge, theonly set of data for the ternary mixture CO2 + capsaicin + �-carotene published in the literature is that of Skerget and Knez[19]. Fig. 2 present the data available on this ternary mixtureCO2 + capsaicin + �-carotene.

2. Thermodynamic modeling

Modeling mixtures containing supercritical components isfrequently presented in the literature, mainly for binary mixtures.Few papers have been dedicated to the modeling of ternary andmulticomponent mixtures, although this is a subject of specialimportance for analysis and design of supercritical extractionprocesses. Kurnik and Reid [7], presented experimental data forseven ternary mixtures containing supercritical carbon dioxideand two solutes (combinations of phenanthrene, naphthalene,2,3 DMN, 2,6DMN, and benzoic acid). Also, Kosal and Holder[20] studied the system CO2 + Phenanthrene + Anthracene. Forall these systems, the simple model of Peng–Robinson with clas-sical van der Waals mixing rules gave acceptable results forbinary and ternary mixtures. The reported absolute deviations forthe solid solubility are below 5%. However, for the more com-p[f

with simple van der Waals mixing rules and a Group Contribu-tion Equation of State. As the authors state, the results “showedpoor agreement with the experimental data”. Therefore, a betterthermodynamic model must be applied and studied, especiallyfor thermodynamic consistency purposes.

The theory of solid solubility in a compressed gas is found instandard books [22,2], so a summary only is given in what fol-lows. The fundamental equation of phase equilibria establishesthat at a given temperature and pressure, the fugacity of a com-ponent, for instance a solid solute, in the gas phase (f gas

i ) mustbe equal to the fugacity of the same component in the solid phase(f sol

i ). If subscripts 2 and 3 stand for the solid components, thenthe fundamental equation of phase equilibria reduces to:

f sol2 = f

gas2 (1)

f sol3 = f

gas3 (2)

Expressing the fugacities in terms of the fugacity coefficientsand considering the following appropriate assumptions: (i) lowsublimation pressure of the solid solutes; (ii) ideal gas behaviorfor the gas phase over the pure solid at the working temperature,so φs

2 ≈ 1; (iii) the solid phase is considered to be pure; and (iv)the volume of the solid is considered to be pressure independent,the mole fraction of the solute in the gas phase, or solubility, atthe temperature T for solute “i” (i = 2, 3), can be reduced to [2]:

y

sol s

http

d

R

cltafaaHIiirdt

tso

lex systems of interest in this work deviation are much higher21]. Recently, de la Fuente et al. [10] presented some resultsor the mixture CO2 + capsaicin using the Peng–Robinson EoS

i = P si e(V

i/RT )(P−P

i)

Pφi

(3)

ere, P si is the sublimation pressure of the pure solid “i”, V sol

i

he solid molar volume of “i”, all at the temperature T and φi ishe fugacity coefficient of “i” in the compressed gas phase at theressure P and temperature T.

The fugacity coefficient is calculated from standard thermo-ynamic relations as [22]:

T ln(φi) =∫ ∞

V

[(∂P

∂ni

)T,V,nj

− RT

V

]dV − RT ln Z (4)

Any equation of state could be used to evaluate the fugacityoefficients as long as low deviations are found between calcu-ated and experimental values of the solubility. Several combina-ions between cubic EoS and mixing rules have been employednd have been presented in the literature. These included dif-erent applications and modifications of the Peng–Robinsonnd Soave–Redlich–Kwong equations with mixing rules suchs van der Waals, Panagiotopoulos–Reid, Kwak–Mansoori,uron–Vidal, Kurihara et al., and Wong–Sandler, among others.

t is also known that for binary mixtures including a supercrit-cal component, such as those of interest here, more than onenteraction parameter in the mixing rules are needed if accu-ate correlation of the solute concentration in the gas phase isesired [23]. The classical VdW mixing rules with one interac-ion parameter do not work well for this type of mixtures [21].

In this work, the Peng–Robinson EoS was considered as thehermodynamic model to evaluate the fugacity coefficient of theolutes 2 and 3,φ2 andφ3: The modeling included the calculationf the sublimation pressures of the solid solutes as explained in

96 J.O. Valderrama, P.A. Robles / Fluid Phase Equilibria 242 (2006) 93–102

Table 2Mixing rules and combination rules used in this work

Equation of state Mixing rules

Peng–Robinson equation van der Waals (for consistency) Wong–Sandler (for determining the sublimation pressures)

P = RTV−bm

− amV (V+bm)+bm(V−bm) am =

∑i

∑j

yiyjaij bm =∑∑

yiyj (b−(a/RT ))ij

1−(∑

yia/biRTi)−((gE(y)/RT )/0.34657)am = bm

(Σyiai

bi+ gE(y)

0.34657

)a = 0.42748 R2T 2

cPc

α(T ) aij = √aiaj(1 − kij)

(b − a

RT

)ij

= 12 [bi + bj)] −

√aiaj

RT(1 − kij)

b = 0.08664 RTcPc

bm =∑

i

∑j

yiyjbijgE

RT=

N∑i

yi

∑N

jyjLij

1−yi

[1 −

yi

∑N

jyjLij

yi

∑N

jyjLij+(1−yi)yi

∑N

jyjLji

]2

α(Tr) = [1 + F (1 − T 0.5r )]

2bij = 1

2 (bi + bj)F = 0.48 + 1.574ω − 0.176ω2

the literature [24] using the mixing rule of Wong and Sandler[25], with the van Laar model for the excess Gibbs free energy.

Although this same model could be used to treat the ternarymixtures and to apply the consistency test method, there are tworeasons not to do it in this way: (i) for ternary mixtures, ninebinary parameters need to be calculated (three kij and six Lij);and (ii) usually, few experimental data points are available in theliterature for the ternary mixtures of interest in this work. Thus,for the ternary mixtures studied in this work, the equation ofPeng and Robinson [26] with the classical van der Waals mixingrules with all three parameters (k12, k13, and k23), calculatedfrom ternary mixture data were used. The model has shown towork reasonable well for most ternary mixtures [27]. Table 2describes the equation of state and the mixing rules used. ThePeng–Robinson EoS with Wong–Sandler mixing rule and thevan Laar equation for the excess Gibbs free energy is designatedas PR/WS/VL while the Peng–Robinson EoS with classical vander Waals mixing rules is designated as PR/vdW.

As known, the basic idea in the regression analysis usuallyemployed to determine the binary interaction parameters is toapply the EoS to the calculation of a particular property and thenminimize the differences between predicted and experimentalvalues of that property, according to a specified objective func-tion. The value of the interaction parameters, which minimize theobjective function, corresponds to the optimum values of thoseinteraction parameters. The objective function is arbitrarily butcisef

W

|

w

%

In the case of employing ternary mixture data (one solvent“1” and two solutes “2” and “3”), the objective function mustconsider the deviations in the correlation of both solutes: |%�y2|and |%�y3|. The objective function Wyy in this case is:

Wyy = 0.5[|%�y2| + |%�y3|] (8)

The global model accuracy is determined using the value ofthe objective function, although the individual deviations arealso analyzed.

The optimization procedure includes the use of genetic algo-rithms as the numerical optimization method. Genetic algo-rithms constitute a reliable method that uses biologically derivedtechniques such as inheritance, mutation, natural selection, andrecombination to find the optimum solution of the optimizationproblem. This technique has proved to give good, reliable resultsin applications to gas–solid equilibrium [24].

3. Consistency criteria

Similar to the Van Ness–Byer–Gibbs test [28], the consis-tency method proposed in this work can be considered as amodeling procedure. This is so because a thermodynamic modelthat can accurately fit the experimental data must be first used tothen apply the consistency test. Once the model parameters aredetermined and the calculated solubilities are within acceptablelehti

a

∑btfT

∫

onveniently defined and several criteria have been presentedn the literature [23]. For modelling, the solubility of a solidolute in a compressed gas the deviation between calculated andxperimental values of the solubility is defined as the objectiveunction Wy, that is:

y = |%�y| (5)

The average absolute deviation |%�y| is calculated as:

%�y| = 100

N

N∑i=1

[|ycal − yexp

yexp

]i

(6)

hile the average relative deviation %�y is determined as:

�y = 100

N

N∑i=1

[ycal − yexp

yexp

]i

(7)

imits of deviations, the Gibbs–Duhem equation is applied. Thequations defining the consistency criteria for binary mixturesave been presented by the authors [3,5]. The development forernary mixtures has not been yet presented in the literature, ands summarized in what follows.

For a ternary homogeneous gas mixture at constant temper-ture, the Gibbs–Duhem equation can be written as [29]:

yiRTd[(d ln ϕi)] = V R

RTdP (9)

eing yi the mole fraction of component “i” in the gas phase, Phe system pressure, VR the residual molar volume, and ϕi is theugacity coefficient of component “i” in the gas phase mixture.he equation can be conveniently arranged to get:

1

Pyj

dP =2∑

i=1

∫yi

(Z − 1)ϕiyj

d(ϕi) (10)

J.O. Valderrama, P.A. Robles / Fluid Phase Equilibria 242 (2006) 93–102 97

Fig. 3. Decision rules for the proposed consistency test method for ternary solid–solid–gas high pressure mixtures.

In this equation, yj is the gas phase solute mole fraction andϕi the fugacity coefficient of component “i” in the gas phasemixture. In Eq. (10), the left hand side is designated by APj andthe expression inside the summation term on the right hand sideby Aϕj, that is:

APj =∫

1

Pyj

dP (11)

Aϕj =2∑

i=1

[Aϕj ]i

(12)

Thus, if a set of data is considered to be consistent APj shouldbe equal to Aϕj for j = 2 and 3 within acceptable defined devi-ations. To set the margins of errors, an individual percent areadeviation [%�Aj|i between experimental and calculated valuesis defined as:

[%�Aj]i= 100

[Aϕ − AP

AP

]i

(13)

For binary mixtures, in which only one solute “2” is dissolvedin the compressed gas solvent “1” two areas are calculated: AP2

and Aϕ2. For ternary mixtures in which two solutes (2 and 3),are dissolved in the compressed gas solvent “1”, four areas mustbe calculated: AP2, AP3, Aϕ2, and Aϕ3.

To evaluate the integrals in Eq. (10) for a set of N experi-mental points, two consecutive data points were used and thetrapezoidal rule was applied, obtaining N − 1 values of the inte-grals. In another communication [21], the integration methodwas justified since a straight line between two consecutive pointscould be assumed without much error. To check the validity ofthe integration method, the authors fitted a third degree polyno-mial to the integrand functions and the integrals were calculatedusing the fitted polynomials. The deviations between the areascalculated using the trapezoidal rule and those determined usingthe polynomial functions were below 2%. These deviations wereconsidered acceptable for consistency analysis.

Fig. 3 shows a flow diagram that clarifies the different situa-tions that can be found when the proposed method is applied. Asobserved in the figure there are three exits, that is three possibleanswers for the consistency test: (i) the data are thermodynami-cally consistent (TC); (ii) the data are not fully consistent (NFC);and (iii) the data are thermodynamically inconsistent (TI). Andadditional answer is also included for cases in which some few

98 J.O. Valderrama, P.A. Robles / Fluid Phase Equilibria 242 (2006) 93–102

points could not be modeled within the established accuracy. Inthis case, when less than 25% of the points are not well mod-eled (for instance, 2 out of 10), these points are eliminated andthe remaining points are subjected to analysis. If these data passthe test, the conclusion is that the original data is NFC and theremaining data tested are TC.

As seen in the figure, the first requirement of the thermo-dynamic consistency test is that the model with the estimatedparameters allows calculating the solute concentration in thegas phase within defined maximum deviations. We request thatthe absolute deviation in the gas phase concentration for bothsolutes |%�y2| and |%�y3| are within (−20% to +20%). Afterthis, the individual areas are calculated. Similar to Valderramaand Alvarez [21], we define the maximum deviation for [%�Aj]i

(for both j = 2 and 3), to be in the range −20% to +20% and weset five as the minimum number of experimental points requiredfor the consistency criteria to have some reasonable meaning.

These percentages defined for consistency criteria are basedon information presented in the literature related to the accuracyof experimental data for this type of mixtures (solids dissolvedin a high pressure gas) and criteria used in other thermodynamicconsistency tests [30,1,21]. To confirm these “acceptable” devi-ation ranges, calculations of error propagation on the measuredexperimental data have been performed. This was done using thegeneral equation of error propagation [31], being the pressureP and the solid solubility y the independent measured variables.

4. Genetic algorithms

The genetic algorithms method (GAs) used in the correla-tion of the binary and ternary mixtures considered in this studyhas been described with detail elsewhere [5], so a brief descrip-tion only is summarized here. The optimization technique thatuses genetic algorithms is one of the so-called evolutionaryalgorithms that operate on entire populations of candidate solu-tions in parallel. This is fundamentally different from moretraditional numerical techniques, which iteratively refine a sin-gle solution vector as they search for the optimum solution ina multi-dimensional landscape. The parallel nature of a GAsstochastic search is one of the main strengths of the geneticapproach. This parallel nature implies that GAs is much morelikely to locate a global optimum than traditional techniques,because the GAs are much less likely to get stuck at localoptima. Therefore, the problem of finding a local optimumis greatly minimized since GAs can make hundreds of initialguesses.

Genetic algorithms use biologically derived techniques suchas inheritance, mutation, natural selection, and recombinationto evolve toward better solutions so the objective function isbest satisfy. At the beginning of the computation a number ofindividuals represented by chromosomes are randomly created,forming a set known as the population. Each chromosome con-sists of a number of “zeros” and “ones” and represents a valueomooa(twtrpd[[m

5

wfamaCceTtT

The calculated individual area Aϕ (evaluated using two consecu-tive points), is the dependent variable of interest. The error in thecalculated areas, EA and the percent error %EA are calculatedas:

EA =[∂Aϕj

∂P

]�P +

[∂Aϕj

∂y

]�y +

[∂Aϕj

∂T

]�T (14)

%EA = 100

[EA

Aϕj

]

We assumed maximum uncertainties of 0.5 bars for the exper-imental pressure, 0.5 K for the temperature, and 1% for theexperimental solubility. The partial derivatives in Eq. (14) werenumerically calculated, giving average estimated percent errors%EA of ±20%. Therefore, the range (−20% to +20%) is estab-lished as the maximum acceptable error for the individual areas[Aj]i.

The method proposed here is based on the hypothesis that trueexperimental data should be randomly distributed and thereforerandom distribution of errors in the testing method should befound. For this reason, we consider that the individual devi-ations in the calculated areas [%�Aj]i, as derived from theGibbs–Duhem equation, are the most important parameter todetermine the consistency of the data. Therefore, if an EoS modelproduces randomly distributed errors in the calculated solubil-ities and these values are within defined acceptable limits (asshown in Fig. 3), the proposed area test represents an acceptablecriterion to accept or reject a set of data from the thermodynamicconsistency point of view.

f the parameter to be calculated. The set of individuals form aatrix that represents the first/initial generation. After this, the

bjective function is evaluated for each of the individuals. If theptimization criteria are not met, the creation of a new gener-tion starts. Individuals are selected according to their fitnesshow close they fulfill the objective function), for the produc-ion of offspring (new individuals). All offspring are mutatedith a certain probability and the fitness of the offspring is

hen computed. The offspring are inserted into the populationeplacing the parents, producing a new generation. This cycle iserformed until the optimization criteria are reached. For furtheretails on genetic algorithms the works of Golberg [32], Davis33], Pohlheim [34], Chakraborty and Janikow [35], Whitley36], Rangaiah [37], Escudero [38], and Obitko [39], are recom-ended.

. Binary and Ternary Data used

The binary and ternary mixture data used in this studyere obtained from the literature. The sets of data selected

or the study are those for which ternary data are avail-ble. Two isotherms for each of the binary and ternaryixtures were considered (CO2 + 2,3DMN, CO2 + 2,6DMN

nd CO2 + 2,3DMN + 2,6DMN at 308 and 318 K; andO2 + capsaicin, CO2 + �-carotene and CO2 + capsaicin + �-arotene at 298 and 313 K). Table 3 presents details on thexperimental solubility data used for binary mixtures whileable 4 presents similar information for the ternary mix-

ures. The literature sources of the data are also shown in theables.

J.O. Valderrama, P.A. Robles / Fluid Phase Equilibria 242 (2006) 93–102 99

Table 3Details on the phase equilibrium data for the binary mixtures considered in this study

System: CO2+ N T (K) Range of data Reference

P (MPa) y2 (×104)

2,3-DMN 5 308 10–28 22.0–64.3 Kurnik et al. [6]5 318 10–28 12.8–71.9

2,6-DMN 5 308 10–28 19.0–44.7 Kurnik et al. [6]5 318 10–28 7.57–67.7

Capsaicin 10 298 15–29 0.8–1.1 de la Fuente et al. [10]14 313 15–35 1.3–2.4

�-Carotene 10 298 13–27 0.0011–0.0017 Skerget et al. [14]10 313 13–27 0.0015–0.0039 Mendes et al. [16]

The temperature and pressure values have been rounded to the closest integer

Table 4Details on the phase equilibrium data for the ternary systems considered in this study

CO2 (1)+ N T (K) Range of data Reference

P (MPa) y2 (×104) y3 (×104)

2,3-DMN(2) + 2,6-DMN(3) 9 308 12–28 39.2–64.0 30.4–47.4 Kurnik and Reid [7]9 318 12–28 36.7–101.0 34.0–81.3

Capsaicin(2) + �-carotene(3) 10 298 10–28 0.28–0.33 0.001–0.003 Skerget and Knez [19]11 313 10–30 0.24–0.45 0.002–0.006

The temperature and pressure values have been rounded to the closest integer.

6. Results and discussion

Table 5 presents the relative and absolute deviations for thecalculated solubilities and the calculated sublimation pressuresfor the solids, determined using binary solubility data and thePR/WS/VL model. As explained above, the modeling of binarydata using the PR/WS/VL model has the only objective of deter-mining the sublimation pressure of the solids. Binary systemscannot be accurately modeled using the simple PR/VDW modelwith a single interaction parameter, so sublimation pressurescannot be estimated with this model.

As seen in Table 5, deviations are higher for the more complexbinary mixtures capsaicin + CO2 and �-carotene + CO2. Thereare a couple of reasons for this: (i) the sublimation pressures forcapsaicin and �-carotene are much lower than those for 2,3DMNand 2,6DMN; and (ii) the solubilities are much lower than those

of the simple DMN’s mixtures. These two situations impose anadditional problem for the numerical technique because of thesmall numbers that must be treated.

Table 6 shows the modeling results for the ternary mixtures.This table shows the deviations between experimental and cal-culated values for the solute solubilities, %�y2, |%�y2|, %�y3,and |%�y3|, and the interaction parameters k12, k13, and k23 cal-culated using ternary solubility data and the PR/vdW model. Inall cases the deviations for the gas solvent is lower than 1%,so numbers are not shown. The sublimation pressures requiredfor the calculations were those values determined from binarymixture data.

The solvent–solute parameters k13 and k23 are of the order of0.1 and below, similar to values found for several other mixtures[23]. The solute–solute interaction parameters are higher (from0.05 up to 0.6), indicating the weaker interaction between the

Table 5Sublimation pressures calculated from solubility data of the solid solutes in binary CO2 + solute mixtures using the PR/WS/VL model

CO2 (1)+ T (K) N kij Lij Lij P s2 (bar) %�y2 |%�y2|

2,3-DMN(2) 308 5 0.11227 3.430 2.647 1.26 × 10−5 −0.03 2.11318 5 0.06411 4.220 2.996 3.53 × 10−5 −3.60 5.11

2,6-DMN(2) 308 5 0.10009 4.427 2.928 1.25 × 10−5 −1.67 5.04318 5 0.08026 4.315 2.047 3.44 × 10−5 0.31 2.41

C .344.301

� 3.743.59

R

apsaicin(2) 298 10 0.00102 6313 14 0.00097 6

-Carotene(2) 298 10 0.10173 1313 10 0.05184 1

elative and absolute deviations are those defined by Eqs. (6) and (7).

8.950 8.17 × 10−18 −2.84 3.159.229 7.22 × 10−16 −8.43 10.5

7.082 9.03 × 10−22 −3.55 7.006.822 2.78 × 10−20 −6.86 9.36

100 J.O. Valderrama, P.A. Robles / Fluid Phase Equilibria 242 (2006) 93–102

Table 6Calculated deviations and optimum interaction parameters for the ternary mixtures for the solutes at all temperatures studied using the PR/VdW model

CO2(1)+ T (K) N P (MPa) k12 k13 k23 %�y2 |%�y2| %�y3 |%�y3|2,3-DMN(2) + 2,6-DMN(3) 308 9 12–28 0.0918 0.1019 0.3139 0.40 2.72 1.16 3.97

318 9 12–28 0.0792 0.0843 0.5849 −0.87 6.51 0.16 1.50

Capsaicin(2) + �-carotene(3) 298 10 10–28 0.0569 0.0693 0.0010 −6.63 16.90 −3.03 6.23313 10 10–30 0.0715 0.0578 0.2957 −3.49 10.97 −1.64 9.33

Table 7Details of the consistency test for the ternary systems studied using the PR/vdW model

CO2(1)+ T (K) N Range P (MPa) Max. [%�A2]i Max. [%�A3]i Aver. [%�A2]i Aver. [%�A3]i Consistency result

2,3-DMN(2) + 2,6-DMN(3) 308 9 12–28 −5.3 −7.7 −1.0 −1.3 TC318 9 12–28 15.1 2.1 0.6 −0.9 TC

Capsaicin(2) + �-carotene(3) 298 10 14–26 59.0 19.1 – – NFC298 7 18–26 −14.7 19.1 −3.6 5.7 TC313 10 21–30 29.0 24.6 – – NFC313 8 23–30 13.1 −12.0 −0.7 −1.5 TC

TC, thermodynamically consistent; TI, thermodynamically inconsistent; and NFC, not fully consistent.

highly diluted solutes in the compressed gas phase. This can beeasily seen by observing the solute–solute contribution term a23that appears in the mixing rule for am: a23 = 2y2y3

√a2a3(1 −

k23). The higher the value of k23 the lower is the contribution ofa23 to the force term am.

Table 7 presents the results of the consistency test for theternary mixtures. Of the two mixtures, the nine data points setof the simple ternary system CO2 + 2,3DMN + 2,6DMN showedto be consistent at both temperatures (308 and 318 K) while the10 data points set at both temperatures (298 and 313 K) for themore complex mixture CO2 + capsaicin + �-carotene showed tobe not fully consistent. This is so because the maximum devi-ations in the individual areas, designated as Max.[%�A2]i andMax.[%�A3]I, are outside the acceptable interval of −20% to+20%. Once the data with high area deviations are removed, theremaining set of seven data points at 298 K and eight data pointsat 313 K showed to be consistent.

Table 7 also shows the average deviations for all the isothermsthat resulted to be consistent. These are designated in the table asAver.[%�A2]i and Aver.[%�A3]i. The subindexes 2 and 3 indi-cate the average deviation with respect to solutes 2 and 3, respec-tively. As observed, the average values are lower for the simpleternary mixture CO2 + 2,3-DMN + 2,6-DMN (between −1.3%and +0.6%) and a little higher for the more complex ternary mix-ture CO2 + capsaicin + �-carotene (between −3.6% and 5.7%).This is an expected result considering that the modeling of thiscftdsb

wiC

CO2 + capsaicin + �-carotene. The behavior of these deviationsshows that the values are reasonably distributed within the inter-val established as acceptable for consistency purposes in theproposed method.

Fig. 4. Individual percent area deviation [%�Aj]i with respect at solutes 2 and3, [%�Aj]2 and [%�Aj]3, for the ternary system CO2(1) + 2,3-DMN(2) + 2,6-DMN (3): (�) 308 K and (�) 318 K.

omplex mixture also gave deviations higher than those foundor the simpler mixture CO2 + 2,3-DMN + 2,6-DMN. Althoughhe average percent deviation is not explicitly considered as aetermining parameter for the decision of consistency or incon-istency, the results presented in Table 7 show the expectedehavior.

A graphical picture of the deviations in the individual areasith respect to the solutes 2 and 3 [%�Aj]2 and [%�Aj]3,

s presented in Figs. 4 and 5; in Fig. 4 for the ternaryO2 + 2,3-DMN + 2,6-DMN and in Fig. 5 for the ternary

J.O. Valderrama, P.A. Robles / Fluid Phase Equilibria 242 (2006) 93–102 101

Fig. 5. Deviations in the individual areas with respect to the solutes 2 and3 [%�Aj]2 and [%�Aj]3 for the ternary system CO2(1) + capsaicin(2) + �-carotene(3): (�) 298 K and (�) 313 K.

7. Conclusions

According to the results, the following conclusions can bedrawn: (i) the Peng–Robinson equation with the Wong–Sandlermixing rule can be used to accurately correlate experimentalbinary high pressure solubility data; (ii) the Peng–Robinsonequation with the classical van der Waals mixing rules can beused to accurately correlate experimental ternary high pressuresolubility data; (iii) the concentration of the solvent carbon diox-ide in the gas phase can be correlated with deviations lower than0.1% for any temperature; (iv) the numerical method based ongenetic algorithms shows to be efficient for searching a globaloptimum; (v) the consistency method used for ternary mixturesrepresents a novel contribution to the analysis of high pressurephase equilibrium data.

List of symbolsa force constant in the PR equation of stateac force constant in the PR equation of state at the critical

pointaij cross force constant in the EOS mixing ruleam force constant for a mixtureAPj integral for two consecutive points using P–y experi-

mental dataAϕj integral for two consecutive points using a thermody-

[%�Aj]i individual percent area deviationb volume constant in the PR equation of statebm volume constant for a mixturebij cross volume constant in the EOS mixing ruled derivative operatore base of the natural logarithm function (2.71828)EA error in the calculated areas%EA percent error in the calculated areasF parameter in the �(Tr) function of the PR equation of

statef sol

i fugacity of component “i” in the solid phasef

gasi fugacity of component “i” in the gas phase

fgas2 , f

gas3 solute fugacity in the gas phase

f sol.2 , f sol

3 solute fugacity in the solid phasegE gibbs free energy at low pressurekij binary interaction parameters for the force constant in

an EoSln natural logarithmLij parameters in the van Laar modelM molecular weightn number of molesN number of data points in a data setP pressurePi pressure for a point “i” in the data setPc critical pressure of componentP s

RTTTVVV

WWyy

y

%

|

Z

Gα

γ

∆

ϕ

ω

∂

Sce

i sublimation pressure of the pure solid “i”ideal gas constanttemperature

c critical temperaturer reduce temperature (Tr = T/Tc)

volumeE excess volumesoli molar volume of solid “i”y objective function for binary systems modelingyy objective function for ternary mixture modeling

i, yj mole fraction of components i and j in the gas phasecal2 , ycal

3 calculated mole fraction of the solid solutes in the gasphase

exp2 , y

exp3 experimental mole fraction of the solid solutes in the

gas phase�yi, %�y2, %�y3 average relative deviation in the gas phase

solute concentration%�yi|, |%�y2|, |%�y3| average absolute deviation in the gas

phase solute concentrationcompressibility factor (Z = PV/RT)

reek letterstemperature function for the PR equation of stateactivity coefficientdeviationfugacity coefficientacentric factorpartial derivative

uper/subscriptsal calculatedxp experimental

102 J.O. Valderrama, P.A. Robles / Fluid Phase Equilibria 242 (2006) 93–102

gas gasi, j component i or js sublimationsol solid

Acknowledgements

The authors thank the Direction of Research of the Universityof La Serena-Chile for permanent support, the Center for Tech-nological Information of La Serena-Chile for using its computerand library facilities, and of the National Council for Scientificand Technological Research of Chile (CONICYT) for its supportthrough the research grant FONDECYT 1040285.

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