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Fluid Phase Equilibria 316 (2012) 55– 65

Contents lists available at SciVerse ScienceDirect

Fluid Phase Equilibria

j our na l ho me page: www.elsev ier .com/ locate / f lu id

xperimental determination and theoretical modeling of the vapor–liquidquilibrium and surface tensions of hexane + tetrahydro-2H-pyran

ndrés Mejíaa,∗, Hugo Seguraa,∗, Marcela Cartesa, J. Ricardo Pérez-Correab

Departamento de Ingeniería Química, Universidad de Concepción POB 160 – C, Correo 3, Concepción, ChilePontificia Universidad Catoılica de Chile, Department of Chemical and Bioprocesses Engineering, Avenida Vicuna Mackenna 4860, Santiago, Chile

r t i c l e i n f o

rticle history:eceived 10 November 2011eceived in revised form0 November 2011ccepted 5 December 2011vailable online 19 December 2011

eywords:apor–liquid equilibriumurface tension

a b s t r a c t

Isobaric vapor–liquid equilibrium (VLE) data have been measured for the binary system hex-ane + tetrahydro-2H-pyran at 50, 75, and 94 kPa and over the temperature range 321–358 K using avapor–liquid equilibrium still with circulation of both phases. Atmospheric surface tension data havebeen also determined at 303.15 K using a maximum bubble pressure tensiometer. Experimental resultsshow that the mixture is zeotropic and exhibits slight positive deviation from ideal behavior over theexperimental range. Surface tensions, in turn, exhibit negative deviation from the linear behavior.

The VLE data of the binary mixture satisfy the Fredenlund’s consistency test and were well-correlatedby the Wohl, nonrandom two-liquid (NRTL), Wilson, and universal quasichemical (UNIQUAC) equations.The dependence of surface tensions on mole fraction was satisfactorily smoothed using the Redlich–Kister

quare gradient theoryHV

eng–Robison EoSHPexane

equation.The experimental VLE and surface tension data were accurately predicted by applying the square

gradient theory to the Peng–Robinson Stryjek–Vera equation of state (EoS), appropriately extended tomixtures with a modified Huron–Vidal mixing rule. This theoretical model was also applied to describethe surface activity of species along the interfacial region, from which it was concluded that hexanepresents interfacial accumulation and, therefore, a positive relative Gibbs adsorption isotherm on THP.

. Introduction

Cyclic ethers (such as tetrahydrofuran or THF, 1,4-dioxane andetrahydropyran or THP) are frequent chemicals in several indus-rial processes [1,2], where they found applicability as solvents (foroating, adhesives and varnishes), extracting agents, supportingedia for organometallic syntheses and potential oxygenates for

ow pollutant gasoline blends. Concerning this latter application,yclic ethers can be considered as potentially attractive substitutesor branched ethers since – besides exhibiting similar normal boil-ng temperatures – they are characterized by additional desirablehermo-physical properties such as higher densities and surfaceensions. In addition, cyclic ethers constitute environmentallyriendly solvents because they are easily recovered by conventionaldsorption and/or recycled by distillation [2,3]. Finally, becausef their donor ability as well as their characteristic molecularnteractions in mixtures (see Ref. [4] and references therein), the

heoretical treatment of quoted ethers constitutes a challengingask for developing predictive models able to optimize their presentse or to rationalize new potential applications.

∗ Corresponding authors. Tel.: +56 41 2203897; fax: +56 41 2247491.E-mail addresses: amejia@udec.cl (A. Mejía), hsegura@udec.cl (H. Segura).

378-3812/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2011.12.007

© 2011 Elsevier B.V. All rights reserved.

In spite of their importance, experimental and theoretical inves-tigations concerning key properties of cyclic ethers, such as vaporliquid equilibrium (VLE) and surface tension (ST) are scarce andlimited to narrow experimental conditions, especially for the caseof THP and its mixtures (see Ref. [5] and references therein). Exper-imental mixing enthalpies [6,7] and volumes [7,8] for hexane + THPmixtures have been reported at 298.15 and 303.15 K, from which itcan be concluded that mixing properties exhibit positive deviationover the whole mole fraction range.

To the best of our knowledge, neither VLE nor IFT data havebeen reported previously for THP + hexane and, additionally, notheoretical model has been proposed for explaining its behavior.Consequently, and as part of our ongoing research program devotedto the characterization of the thermo-physical properties of THPmixtures (THP + n-alkanes and + alcohols [5,9]), this work is under-taken to determine VLE and interfacial properties of hexane + THPand to analyze its phase an interface behavior at the light of pre-dictive theories. Specifically, a primary goal of this contribution isto report isobaric VLE data at 50, 75 and 94 kPa for hexane + THP,together with their IFTs at 303.15 K and 101.3 kPa. An additional

goal is to simultaneously predict both bulk phase (VLE) and interfa-cial properties (ST, surface activity, and Gibbs adsorption isotherm)of the mixture. For that purpose, bulk phases are described by usinga Peng–Robinson equation [10] with modified Huron–Vidal mixing

5 se Equ

rpEsfs

2

2

at(niir

2

2

mwm2apcvamtScdsdmbTida

(tct

TG

6 A. Mejía et al. / Fluid Pha

ules (MHV) [11], while the corresponding interfacial properties areredicted by applying the square gradient theory (SGT) [12] to thatoS model. As we have demonstrated in previous works [13,14],uch a modeling approach provides a full predictive scheme bothor bulk and interfacial properties from experimental � values andurface tensions of the pure components.

. Experimental

.1. Purity of materials

Hexane and tetrahydro-2H-pyran were purchased from Mercknd Aldrich, respectively. Both chemicals were used without fur-her purification. Table 1 reports the purity of the componentsas determined by gas chromatography, GC), together with theormal boiling points (Tb), the mass densities ( �), the refractive

ndexes (nD) at 298.15 K and the surface tensions (�) of pure flu-ds at 303.15 K. The reported values are also compared with thoseeported in the literature [15–19].

.2. Apparatus and procedure

.2.1. Vapor–liquid-equilibrium cellAn all-glass vapor–liquid-equilibrium apparatus model 601,

anufactured by Fischer Labor and Verfahrenstechnik (Germany),as used in the equilibrium determinations. In this circulation-ethod apparatus, the mixture is heated to its boiling point by a

50 W immersion heater. The vapor–liquid mixture flows throughn extended contact line (Cottrell pump) that guarantees an intensehase exchange and then enters to a separation chamber whoseonstruction prevents an entrainment of liquid particles into theapor phase. The separated gas and liquid phases are condensednd returned to a mixing chamber, where they are stirred by aagnetic stirrer, and returned again to the immersion heater. The

emperature in the VLE still was determined with a Systemteknik1224 digital temperature meter and a Pt 100 probe, which wasalibrated against the experimental fusion and boiling points ofistilled water. The reliability of such a calibration procedure wasuccessfully checked using the experimental boiling temperatureata of the pure fluids used in this work. The accuracy is esti-ated as ±0.02 K. The total pressure of the system is controlled

y a vacuum pump capable of work under vacuum up to 0.25 kPa.he pressure is measured with a Fischer pressure transducer cal-brated against an absolute mercury-in-glass manometer (22 mmiameter precision tubing with cathetometer reading); the overallccuracy is estimated as ±0.03 kPa.

On average the system reaches equilibrium conditions after

2–3) h operation. The 1.0 �L samples taken by syringe afterhe system had achieved equilibrium and were analyzed by gashromatography on a Varian 3400 apparatus provided with ahermal conductivity detector and a Thermo Separation Products

able 1as Chromatography (GC) purities (mass fraction), refractive index (nD) at Na D line, mass d

Component (purity/mass fraction) nD �/g cm−

T/K = 298.15 T/K = 298

Exp. Lit. Exp.

Hexane (0.999) 1.37374 1.37230b 0.6551THP (0.998) 1.42000 1.41950c 0.8790

a The measurement uncertainties are: nD ± 10−5; � ± 5 × 10−6 g cm−3; � ± 0.1 mN m−1

b Daubert and Danner [15].c Riddick et al. [16].d Giner et al. [17].e Gill et al. [18].f Interpolated data from Villares et al. [19].

ilibria 316 (2012) 55– 65

model SP4400 electronic integrator. The column was 3 m longand 0.3 cm in diameter, packed with SE-30. Column, injector, anddetector temperatures were 373.15, 393.15, and 493.15 K, respec-tively. Good separation was achieved under these conditions, andcalibration analyses were carried out to convert the peak area ratioto the mass composition of the sample. The pertinent polynomialfit of the calibration data had a correlation coefficient R2 betterthan 0.99. At least three analyses were made of each sample. Themaximum standard deviation of these analyses was 0.002 in areapercentage. Concentration measurements were accurate to betterthan ±0.001 in mole fraction.

2.2.2. Density and refractive indexes measurementsThe mass density ( �) of the pure fluids was measured at 298.15 K

using a DMA 5000 densimeter (Anton Paar, Austria) with an accu-racy of 5 × 10−6 g cm−3. The density determination is based onmeasuring the period of oscillation of a vibrating U-shaped tubefilled with the liquid sample. During the operation, the temperatureof the apparatus was maintained constant to within ±0.01 K. Therefractive indexes (nD) of pure liquids were measured at 298.15 Kusing a Multiscale Automatic Refractometer RFM 81 (Belling-ham + Stanley, England). During the operation, temperature wascontrolled to within ±0.01 K by means of a thermostatic bath(Haake DC3, Germany). The uncertainties in refractive index mea-surements are ±10−5.

2.2.3. Surface tension measurementsA maximum differential bubble pressure tensiometer model

PC500-LV manufactured by Sensadyne Inc. (USA), was used in sur-face tension measurements. In this equipment, two glass probes ofdifferent orifice radii (r1 = 0.125 ± 0.01 mm and r2 = 2.0 ± 0.01 mm)are immersed in a vessel that contains the sample to be measured.In order to guarantee high accuracy and reduce the measurementerrors, the glass probe with the small orifice was located 2.5 mmbelow the probe with the larger orifice. Ultra high purity nitrogen(UHP = 99.995%) is then blown through the probes and the differ-ential pressure (�p) between them is recorded.

According to the Laplace’s equation, �p and r1, r2 are related tothe surface tension, �, as:

�p = p1 − p2 = 2�(

r−11 − r−1

2

)(1)

where pi is the pressure exerted by the gas flow in the probe ofradius ri. The gas flow is controlled by a sensor unit connectedto a personal computer through an interface board (PCI-DAS08,Measurement Computing, USA). Besides a constant volume flowcontroller, this sensor unit contains a differential pressure trans-

ducer, a temperature transducer and a pressure regulator. Thetemperature of the sample in the vessel is measured by means ofa Pt 100 probe, and maintained constant to within ±0.01 K using athermostatic bath (Julabo, Germany).

ensities ( �), normal boiling points (Tb) and surface tensions (�) of pure components.a

3 Tb/K �/mN m−1

.15 p/kPa = 101.33 T/K = 303.15

Lit. Exp. Lit. Exp. Lit.

9 0.65603b 341.94 341.88b 17.40 17.37b

3 0.87880d 361.36 361.31e 27.10 26.68f

; p ± 0.03 kPa; T ± 0.01 K.

se Equilibria 316 (2012) 55– 65 57

caiacosaiwmsstTts

cciCtA

3

3

ic

l

wppvvom

ı

ctwiapcftTmvocpo�

v

Table 2Physical properties of the pure components.a, .b

Fluid Tc/K pc/MPa k1 k2 1020 × cii/J m5 mol−2

Hexane 507.6 3.025 0.81075 0.02828 41.72503THP 572.2 4.770 0.69529 0.04434 25.26504

a Tc and Pc are the critical temperature and pressure, respectively. k1 and k2 arethe Stryjek–Vera’s constants. cii is the influence parameter.

b Critical properties were taken from Riddick et al. [16]. k1 and k2 were fitted from

of the liquid bulk phase is approached by using activity coefficient(�) models (e.g. Wohl, NRTL, Wilson, UNIQUAC, etc.), whereas thevapor bulk phase is described by the virial equation of state (seeEq. (2)). In the �–� approach, bulk phases are treated by using a

Table 3Antoine coefficients (Ai , Bi and Ci) in Eq. (4).a

A. Mejía et al. / Fluid Pha

The experimental method for determining surface tensions pro-eeds as follows: the mixture to be analyzed is prepared by addingppropriate volumes of each pure fluid. The sample is then placednto the vessel, where it is heated to the experimental temperaturend stirred during 5 min by a magnetic stirrer. At that moment, theoncentration of the sample is determined in triplicate by meansf GC. Thereafter, UHP nitrogen flows through the probes, and theensor unit translates the measurement of voltage signal (�v) to

�p signal. The relation between �v and �p is obtained by cal-brating the sensor unit software using two reference fluids with

ell-characterized surface tensions over the range of expectedeasurements. In this work, we used de-ionized water as the high

urface tension reference fluid, while ethanol as low surface ten-ion reference fluid. Finally, the surface tension is calculated fromhe Sensadyne software according to Eq. (1) within ±0.01 mN m−1.his calculation is carried out for a mixture of constant concentra-ion, and over a period of time where the surface tension reaches atatic or constant value.

As part as experimental procedure, the probes are periodicallyleaned before each experimental measurement. The cleaning pro-edure consists in washing the probes with acetone and then,n drying them with additive-free wipes (Kimwipes, Kimberly –lark co.). Additional details concerning to the maximum differen-ial bubble pressure technique have been extensively described bydamson and Gast [20] and Rusanov and Prokhorov [21].

. Theoretical

.1. Experimental data treatment and consistency

Isobaric VLE measurements have been used to predict the activ-ty coefficients (� i) and then evaluate their consistency. � i arealculated from the following equation [22]:

n �i = lnyip

xip0i

+ (Bii − VLi

)(p − p0i)

RT+ y2

j

ıijp

RT(2)

here p is the total pressure and p0i

is the pure component vaporressure. R is the universal gas constant. T is the equilibrium tem-erature. xi and yi are the mole fraction of the liquid and theapor-phase of component i, respectively. VL

iis the liquid molar

olume of component i, Bii and Bjj are the second virial coefficientsf the pure gases, Bij is the cross second virial coefficient, and theixing rule of second virial coefficients (ıij) is given by

ij = 2Bij − Bjj − Bii (3)

According to Eq. (2), the standard state for calculating activityoefficients is the pure component at the pressure and tempera-ure of the solution. Eq. (2) is valid from low to moderate pressures,here the virial equation of state truncated after the second term

s adequate for describing the vapor phase of the pure componentsnd their mixtures and, additionally, the liquid molar volumes ofure components are incompressible over the pressure range underonsideration. In this work, liquid molar volumes are estimatedrom the correlation proposed by Rackett [23]. Critical proper-ies were taken from Riddick et al. [16] and presented in Table 2.he molar virial coefficients Bii, Bjj and Bij were estimated by theethod of Hayden and O’Connell [24] using the molecular and sol-

ation parameters � suggested by Prausnitz et al. [25] for the casef hexane. For the case of THP, molecular parameters and physi-al properties were also taken from Ref. [16] while the solvationarameter was estimated by smoothing experimental data of sec-

nd virial coefficients reported in Ref. [26], thus yielding the value

= 0.05.The vapor pressures of the pure components have been pre-

iously reported as a function of temperature [5,14]. These

experimental vapor pressure data reported in previous works [5,14]. cii were fittedfrom experimental surface tension data reported in this work for the case of THP,and taken from Ref. [14] for hexane.

experimental data have been measured using the same equipmentas that for obtaining the present VLE data. The temperature depen-dence of the vapor pressure p0

iwas correlated using the Antoine

equation:

log(p0i /kPa) = Ai − Bi

(T/K) + Ci(4)

where Ai, Bi, and Ci are the Antoine constants. Table 3 summarizedtheir values, which have been taken from Ref. [5] for THP, and fromRef. [14] for hexane.

In order to test the thermodynamic consistency of the presentVLE data, we applied the point-to-point method of Van Ness et al.[27] as modified by Fredenslund et al. [28]. In the latter approach,a isobaric VLE dataset is consistent when an average deviation of�y < 0.01 is met by fitting the equilibrium vapor pressure accordingto the Barker’s [29] reduction method.

The VLE data obtained in this work were also correlated with theWohl, nonrandom two-liquid (NRTL), Wilson and universal quasi-chemical (UNIQUAC) equations [30], whose adjustable parameterswere determined by minimizing the following objective function(OF):

OF =N∑

i=1

(∣∣pexpi

− pcali

∣∣pexp

i

+∣∣yexp

i− ycal

i

∣∣)2

(5)

In Eq. (5), the superscript exp stands for experimental data whilecal means calculated quantity. N is the number of data points.

Experimental data of the surface tension (�) have been corre-lated by using the Redlich–Kister expansion [31]:

� = x1x2

m∑k=0

ck(x2 − x1)k + x1�1 + x2�2 (6)

where, m denotes the number of ck parameters, which can be foundby a Simplex optimization technique. �i denotes the surface tensionof the pure components, which are presented in Table 1.

3.2. Theoretical modeling of phase equilibrium and interfacialbehavior

Theoretical modeling of VLE can be accomplished either by �–�or �–� approaches [25,30,32]. In the �–� approach, the behavior

Compound Ai Bi Ci Temperature range/K

Hexane 6.02073 1182.8673 −47.3254 307.66–342.08THP 5.60624 1010.3546 −80.7458 312.59–361.55

a Parameters have been taken from Ref. [14] for hexane and from Ref. [5] for THP.

5 se Equ

cEwatisp

dteicmmnstworbfi

mo[r

3

p

wtaa

a

b

pm

˛

wvdTd

rc

b

wm

8 A. Mejía et al. / Fluid Pha

ommon EoS model, such as a cubic EoS [33] or molecular basedoS [34]. The �–� approach is customarily used to model VLE dataithout considering an explicit interface, and provides a simple

nd straightforward route for testing VLE consistency and for fit-ing the activity coefficient model parameters. The �–� approachs adequate when the intrinsic interface is considered, or when theelected EoS model is able to accurately describe vapor and liquidhases.

In this work, we apply the �–� approach to assess the thermo-ynamic consistency of the reported experimental data and to fithe parameters involved in conventional activity coefficient mod-ls. The �–� approach is then applied to predict the phase andnterface behavior by considering the following structure: First, weonsider the Peng–Robinson EoS [10] with the Stryjek–Vera’s ther-al cohesion function [35] and extended it to mixtures using theodified Huron–Vidal mixing rule (MHV) [11]. The latter combi-

ation will be used to describe bulk phase equilibrium. Second, thequare gradient theory (SGT) [12,13] will be applied to EoS modelo predict the interfacial behavior at the equilibrium conditions. Ase demonstrated in previous works [13,14], the main advantage

f this second methodological framework is that all parametersequired to model both bulk phases and interfacial properties cane obtained from pure component properties, while activity coef-cients can be obtained from the �–� approach.

In this section we briefly discuss the basic relationships forodeling phase and interfacial behavior. A complete description

f these equations can be found elsewhere in the original works10–12] as well as in some selected works (see Refs. [36–42] andeferences therein).

.2.1. The EoS model and MHV mixing ruleThe Peng–Robinson EoS is given by [10]

= RT�

1 − �b− a�2

1 − 2�b + �2b2(7)

here p is the total pressure, R is the universal gas constant, T ishe temperature. � is the molar density. a is the cohesion parameternd b is the covolume that, for the case of pure fluids, are defineds

i = 0.457235(RTc,i)

2

Pc,i˛(T, Tc,i) (8a)

i = 0.077796RTc,i

Pc,i(8b)

In Eq. (8), Tc,i, Pc,i are the critical temperature and pressure ofure fluids, respectively, and is given by the Stryjek–Vera’s ther-al cohesion function [35],

i =(

1 + k1

[1 −√

T

Tc,i

]+ k2

(1 − T

Tc,i

) (0.70 − T

Tc,i

))2

(9)

here k1 and k2 are parameters, which were directly fitted fromapor pressure data of pure components (see Wisniak et al. [43] foretails). Table 2 reports the k1 and k2 parameters for hexane andHP which were obtained from the experimental vapor pressureata reported in previous works [5,14].

Eq. (8) can be extended to mixtures by using adequate mixingules. In this work, we have considered the MHV mixing rule [11],ase in which the parameters a and b are defined as follows:

a

b= � RT (10a)

= 1�

(�1b1 + �2b2) (10b)

here �i is the molar density of specie i. �i and � are related by theole fraction xi according to �i = xi �. �, in turn, corresponds to an

ilibria 316 (2012) 55– 65

implicit function which is given by the numerical solution of thefollowing non-linear system

GE

RT+ ln(u − 1) − �

2√

2ln

(u + 1 −

√2

u + 1 +√

2

)

−2∑

i=1

�i

�

[ln(ui − 1) − �i

2√

2ln

(ui + 1 −

√2

ui + 1 +√

2

)]= 0 (11a)

u − � + 2 + 12

√(� − 2)2 − 4(� − 1) = 0 (11b)

and ui, �i are given by

ui = �i − 2 − 12

√(�i − 2)2 − 4(�i − 1) (12a)

�i = ai

biRT(12b)

The non-linear system defined in Eq. (11) yields an implicit mix-ing rule by means of which the EoS exactly predicts the value of theGE/RT function (in Eq. (11a)) at zero pressure. In this work, the GE/RTfunction is parameterized from an activity coefficient model suchas Wohl, NRTL, Wilson, UNIQUAC, etc, considering the parametersobtained from the reported VLE data. In this way, the EoS model,appropriately parameterized from the �–� approach, will be usedto predict the VLE and the IFT by using a �–� methodology.

3.2.2. The square gradient theoryAccording to the square gradient theory (SGT), the population

of species (�i) along the interfacial region is governed by the condi-tion of minimum Helmholtz energy density of the inhomogeneoussystem at fixed temperature (T), volume (V) and moles (N).

For the case of a flat interface connecting a liquid (L) binary mix-ture in equilibrium with its vapor (V), the constraint of minimumHelmholtz energy density can be described using the followingsystem of ordinary differential equations [13,36–41,44]:

c11d2�1

dz2+ c12

d2�2

dz2= 0

1 − 1

c21d2�1

dz2+ c22

d2�2

dz2= 0

2 − 2

(13)

The boundary conditions of Eq. (13) are given by the limits ofthe bulk fluid phase equilibrium

�i(z → −∞) = �Vi and �i(z → +∞) = �L

i (14)

In Eqs. (13) and (14), �i is the molar concentration of speciesi. �i is related with the molar concentration of the mixture (�) bythe molar fraction (xi): �i = xi�. �V

iand �L

icorresponds to the molar

concentration of component i in the V and L bulk phases, respec-tively. cij is the cross influence parameter (cij = cji), z is a coordinatenormal to the interface, and i is the chemical potential of speciesi. The superscript 0 in i denotes that it is evaluated at the phaseequilibrium condition of the bulk phases (V, L). i is related with�i through the use of an equation of state (EoS).

Integration of Eq. (13) allows quantification of the populationof species at the interface (�i(z)) from which the surface activity(or absolute adsorption/desorption of species along the interfaceregion), relative Gibbs adsorption isotherm and the interfacial orsurface tension can be calculated.

Fig. 1 shows, schematically, the most frequently �i(z) distribu-tion obtained from the integration of Eq. (13). In this projection,it is possible to observe that the concentration profile may be a

monotonic (Type a) or non-monotonic (Type b) function. Accordingto our experience, Type a profiles (or hyperbolic tanh profiles) arefound in pure fluids and in mixtures of light n-alkanes or consecu-tive n-alkanes. Type b profiles, in turn, are typical found, for at least

A. Mejía et al. / Fluid Phase Equilibria 316 (2012) 55– 65 59

Fr

ownpva(

cssdmaanivaIcb

ai

adfi

za

sdA

�RT=

0 RT�2−

�d� +

�(18)

ig. 1. Schematic representation of concentration profiles along the interfacialegion. (�) bulk phases: vapor (�v) and liquid (�l); (©) SP: stationary point.

ne component, in asymmetric mixtures such as water + n-alkanes,ater + alcohols, mixtures of heavy n-alkanes, CO2 + n-alkanes, and-alkanes + branched ethers. The reliability of the concentrationrofiles obtained from the integration of Eq. (13) has been alsoerified by using molecular dynamics simulations for pure n-lkanes (from nC10 to nC100) [45], as well as for binary mixturesLennard–Jones, n-alkanes + n-alkanes, CO2 + n-decane) [46,47].

It should be pointed out that the non-monotonic behavior of theoncentration profiles (Type b) is interesting because it reflects theurface activity at the interface. Specifically, the accumulation of apecies i at the interface region is characterized by the condition,�i/dz = 0 (see SP point in Fig. 1), and it may be positive (most com-on case) or negative. The positive surface activity reflects absolute

dsorption of species along the interface region and is reflected in negative second derivative, d2�i/dz2 < 0. In this case, the compo-ent i is found in the surface zone in higher concentration than

s to be expected from its concentration in the bulk phases. Con-ersely, the negative surface activity denotes desorption of specieslong the interface region and its condition is given by d2�i/dz2 > 0.n this case, the component i is found in the surface zone in loweroncentration than is to be expected from its concentration in theulk phases.

The relative Gibbs adsorption isotherm of a species i relative to species j ( ij) can be expressed in terms to �i (z) by the followingntegral equation [44,48]:

ij =∫ zj

0

−∞

[�i(z) − �˛

i

]dz +

∫ +∞

zj0

[�i(z) − �ˇ

i

]dz (15)

In Eq. (15), zj0 is the localization of the divide position relative to

species j. zj0 is calculated from Eq. (15) considering that species j

oes not have adsorption along the interfacial region (Type a pro-le). In other words, Eq. (15) is solved for the case that jj = 0. Onesj0 is fixed, the relative Gibbs adsorption isotherm of a species i rel-

tive to a species j ( ij) is calculated from Eq. (15). Fig. 2 shows,

chematically, the localization and application to zj0. In Fig. 2a, zj

0 isefined as the geometry localization where Area 1 (A1) is equal torea 2 (A2). According to Eq. (15), jj = (A2) + (−A1) = 0 implies that

Fig. 2. Schematic representation of the localization of the divide position (z0) in therelative Gibbs adsorption isotherm. (a) Interfacial concentration for specie j witharea 1 (A1) equals to area (A2). (b) Interfacial concentration for specie i with A2 > A1.

jj = 0. In Fig. 2b, the localization of z0j conducts to A2 > A1 therefore

ij > 0.In the framework of SGT, IFT (�) can be calculated from the

following integral expression:

� =∫ ∞

−∞

[c11

(d�1

dz

)2

+ 2c12d�1

dz

d�2

dz+ c22

(d�2

dz

)2]

dz (16)

Inspection of Eqs. (13), (15) and (16) reveals that the calculationof �i (z), ij and � depend on the EoS model and on the cij values.

The role of a specific EoS is to provide analytical relations forchemical potential and to predict the equilibrium state at whichphases coexist.

Chemical potentials (i and 0i) can be calculated from the EoS

model, according to the following relationships [22]

i =(

∂f0∂�i

)T0,V0,�0

(17)

In Eq. (17), f0 represents the homogenous Helmholtz energydensity, which can be directly obtained from the EoS (see Eq. (7))through the following integral expression [22]

f0∫ �(

P 1)

�1 ln �1 + �2 ln �2

Replacing Eq. (7) in Eq. (18), and then substituting the integralresult in Eq. (17), ones obtained the following final expressions forf0 and i:

6 se Equilibria 316 (2012) 55– 65

f

�cP

g(csp

c

cww

t

c

mD0p

fmf

4

4

awv(cicd

dec

Table 4Experimental VLE data for hexane (1) + THP (2) at p = 50.00 kPa.a

T/K x1 y1 �1 �2 −Bij/cm3 mol−1

11 22 12

339.33 0.000 0.000 1.000 1258337.38 0.042 0.108 1.504 0.996 1332 1277 1237335.21 0.086 0.204 1.477 1.004 1355 1300 1259334.32 0.117 0.260 1.429 0.996 1365 1309 1268332.83 0.157 0.330 1.412 0.998 1381 1325 1284331.63 0.203 0.388 1.340 1.005 1395 1339 1296330.60 0.243 0.432 1.284 1.021 1407 1350 1307329.54 0.290 0.483 1.247 1.030 1419 1362 1319328.26 0.335 0.526 1.225 1.057 1434 1377 1333327.38 0.383 0.569 1.196 1.069 1445 1387 1343326.67 0.433 0.608 1.157 1.088 1454 1396 1351325.86 0.490 0.651 1.124 1.112 1464 1405 1360325.10 0.548 0.693 1.097 1.138 1473 1415 1369324.31 0.607 0.733 1.077 1.171 1483 1424 1378323.75 0.663 0.770 1.056 1.204 1490 1431 1385323.17 0.718 0.806 1.042 1.237 1497 1439 1392322.76 0.773 0.842 1.025 1.274 1503 1444 1396322.34 0.821 0.873 1.016 1.317 1508 1449 1402321.95 0.868 0.905 1.009 1.360 1513 1454 1406321.49 0.926 0.946 1.005 1.412 1519 1460 1412321.03 1.000 1.000 1.000 1525

are reported in Table 8, together with the relative deviation for thecase of bubble and dew point pressures.

Table 5Experimental VLE data for hexane (1) + THP (2) at p = 75.00 kPa.a

T/K x1 y1 �1 �2 −Bij/cm3 mol−1

11 22 12

351.53 0.000 0.000 1.000 1145349.63 0.041 0.099 1.442 0.997 1211 1161 1126347.45 0.086 0.196 1.458 0.999 1231 1181 1144346.64 0.113 0.243 1.403 0.995 1239 1188 1152345.04 0.157 0.312 1.360 1.002 1254 1203 1166343.86 0.198 0.371 1.326 1.001 1266 1214 1176342.67 0.244 0.425 1.280 1.009 1277 1225 1187341.57 0.290 0.473 1.237 1.022 1288 1235 1197340.34 0.334 0.515 1.216 1.043 1301 1248 1209339.54 0.381 0.555 1.176 1.059 1309 1255 1216338.70 0.430 0.596 1.148 1.075 1318 1264 1225337.86 0.483 0.639 1.124 1.088 1327 1272 1233336.95 0.546 0.686 1.097 1.114 1336 1282 1242336.15 0.606 0.728 1.075 1.143 1345 1290 1250335.54 0.663 0.765 1.054 1.178 1351 1296 1256334.97 0.716 0.802 1.040 1.204 1358 1302 1262334.46 0.773 0.839 1.025 1.248 1363 1308 1267334.04 0.819 0.871 1.018 1.273 1368 1312 1271

0 A. Mejía et al. / Fluid Pha

0 =2∑

i=1

RT�i ln(

�i

�

)− RT� ln(1 − b�)

− a�

2√

2bln

[1 + b� +

√2b�

1 + b� −√

2b�

]− RT� ln

(Pref

RT�

)(19)

i = −RT ln (1 − b�) −(

∂�2a

∂�i

)1

2√

2b�ln

[1 + b� +

√2b�

1 + b� −√

2b�

]

− RT ln(

Pref

RT�i

)+ RT +

(∂�b

∂�i

)a

2√

2b2ln

[1 + b� +

√2b�

1 + b� −√

2b�

]

−(

∂�b

∂�i

)a�

[1 − (b� − 2)b�] b+(

∂�b

∂�i

)RT�

(1 − b�)(20)

In Eqs. (19) and (20), �i is the molar concentration of species i. is the molar concentration of the mixture, R is the universal gasonstant, T is the temperature, a and b are given by Eq. (10), andref is some freely chosen reference pressure (e.g. Pref = 1).

In this work, cij is calculated using the procedure originally sug-ested by Carey et al. [38,49] and used extensively by other authorssee Refs. [36,37,39–41,45–47]) Briefly, for the case of pure fluids,ii (i = j) is calculated at the boiling temperature from experimentalurface tension data values (�exp) and using Eq. (16) for the case ofure fluids, which simplify to [38,49]:

ii(T0) = �2

exp(T0)

(∫ �Li

�Vi

√2(f0 − �i

0i

+ P0) d�i

)−2

(21)

Table 2 includes the fitted cii values for hexane and THP. For thease of hexane, c11 has been taken from our previous work [14]hereas for THP, c22 is fitted by using new IFT data reported in thisork and Eq. (21), as we described in Results Sec.

For the case cross influence parameter (cij), it is calculated fromhe following geometric average of pure fluids:

ij = (1 − �ij)√

ciicjj (22)

In Eq. (22), �ij is a symmetric adjustable parameter that, in turn,ay be obtained from the fit of experimental � data of mixtures.ue to stability requirements, �ij should be bounded to the range

≤ �ij < 1 [49,50]. This work, �ij is set to 0 in order to preserve theredictive scheme of SGT.

As we can observe, all the required SGT data input is obtainedrom pure fluid properties, and the GE model parameters deter-

ined from � to � approach. Henceforward, the SGT operates as aull predictive scheme.

. Results and discussions

.1. Vapor–liquid equilibrium

The experimental VLE data for hexane (1) + THP (2) at p = 50, 75,nd 94 kPa are reported in Tables 4–6 and in Fig. 3. In these tablese have included the equilibrium temperature T, the liquid xi and

apor yi phase mole fractions of component i, the pure componentBii and Bjj) and cross (Bij) second virial coefficients and the activityoefficients (� i) which were calculated from Eq. (2) and depictedn Figs. 4–6. The experimental data reported in Tables 4–6 allowoncluding that the hexane + THP mixture exhibits slight positiveeviation from ideal behavior and that no azeotrope is present.

The VLE data reported in Tables 4–6 were found to be thermo-ynamically consistent by the point-to-point method of Van Nesst al. [27] as modified by Fredenslund et al. [28] For each isobaricondition, consistency criterion (�y < 0.01) was met by fitting the

a The measurement uncertainties are: p ± 0.03 kPa; T ± 0.01 K; x1, y1 ± 0.001.

equilibrium vapor pressure according to the Barker’s [29] reduc-tion method. Statistical analysis [51] revealed that a two parameterLegendre polynomial is adequate for fitting the equilibrium vaporpressure in each case. Pertinent consistency statistics and Legendrepolynomial parameters are presented in Table 7. The activity coef-ficients presented in Tables 4–6 are estimated accurate to within1.8%.

The VLE data reported in Tables 4–6 were correlated with theWohl, nonrandom two-liquid (NRTL), Wilson and universal quasi-chemical (UNIQUAC) equations [30], whose adjustable parameterswere obtained by minimizing the objective function presented inEq. (5). For the case of UNIQUAC model, the area and volume param-eters of pure fluids were taken from Ref. [52]. Pertinent parameters

333.63 0.867 0.904 1.010 1.313 1373 1317 1275333.13 0.926 0.945 1.005 1.370 1378 1322 1280332.65 1.000 1.000 1.000 1383

a The measurement uncertainties are: p ± 0.03 kPa; T ± 0.01 K; x1, y1 ± 0.001.

A. Mejía et al. / Fluid Phase Equilibria 316 (2012) 55– 65 61

Table 6Experimental VLE data for hexane (1) + THP (2) at p = 94.00 kPa.a

T/K x1 y1 �1 �2 −Bij/cm3 mol−1

11 22 12

358.85 0.000 0.000 1.000 1085356.39 0.043 0.103 1.482 1.008 1152 1105 1071354.78 0.085 0.188 1.431 1.000 1166 1118 1084353.96 0.111 0.231 1.377 0.999 1173 1125 1090352.27 0.156 0.303 1.351 1.004 1188 1139 1104351.11 0.195 0.359 1.315 1.004 1198 1149 1114349.92 0.241 0.413 1.265 1.012 1209 1159 1123348.76 0.291 0.468 1.231 1.017 1219 1169 1133347.61 0.335 0.511 1.206 1.033 1230 1179 1143346.69 0.382 0.551 1.170 1.052 1239 1188 1151345.86 0.428 0.594 1.151 1.057 1246 1195 1158345.01 0.482 0.633 1.119 1.082 1255 1203 1166344.14 0.543 0.679 1.091 1.104 1263 1211 1174343.38 0.599 0.720 1.071 1.129 1270 1218 1181342.61 0.662 0.763 1.052 1.161 1278 1226 1188342.02 0.716 0.799 1.037 1.191 1284 1231 1193341.43 0.773 0.838 1.026 1.224 1290 1237 1199340.97 0.818 0.869 1.018 1.263 1295 1241 1203340.54 0.867 0.902 1.011 1.302 1299 1246 1207340.06 0.924 0.943 1.005 1.349 1304 1250 1212339.57 1.000 1.000 1.000 1309

a The measurement uncertainties are: p ± 0.03 kPa; T ± 0.01 K; x1, y1 ± 0.001.

Table 7Consistency test statistics for the binary system hexane (1) + THP (2).

P/kPa L1a L2

a 100 × �yb �Pc/kPa

50.00 0.4459 0.0083 0.2 0.175.00 0.4101 0.0014 0.4 0.194.00 0.3981 −0.0003 0.4 0.2

a Parameters for the Legendre polynomial [28] used in consistency.b Average absolute deviation in vapor phase mole fractions �y =

(∑N

∣∣ exp cal∣∣

tscpt

Fig. 3. Boiling temperature (T) as a function of the liquid (x1) and vapor (y1) molefractions for the system hexane (1) + THP (2). Experimental data at (�) 50.00 kPa;(�) 75.00 kPa; (�) 94.00 kPa; (—) predicted from the two-parameter Legendre poly-nomial used in consistency analysis; (– –) predicted from PR-EoS with MHV mixing

E

TP

1/N)i=1

yi

− yi

(N: number of data points).

c Average absolute deviation in vapor pressure �P = (1/N)∑N

i=1

∣∣Pexpi

− Pcali

∣∣.From the results presented in Table 8, it is possible to conclude

hat all the fitted models gave a reasonable correlation of the binary

ystem and that the best fit is obtained with the Wohl model. Theapability of simultaneously predicting the bubble- and dew-pointressures and the vapor and liquid phase mole fractions, respec-ively, has been used as the ranking factor.

able 8arameters and prediction statistics for different Gibbs excess (GE) models in hexane (1) +

Model P/kPa A12 A21 ˛12

Wohl 50.00 0.443 0.446 0.818d

75.00 0.396 0.408 0.690d

94.00 0.390 0.394 0.863d

NRTL 50.00 592.49 698.55 0.400e

75.00 556.88 662.19 0.400e

94.00 512.68 678.73 0.400e

Wilsonb 50.00 −350.46 1655.02

75.00 −407.00 1642.97

94.00 −423.20 1627.09

UNIQUACc 50.00 934.61 −472.29

75.00 922.43 −482.52

94.00 916.35 −487.24

a A12 and A21 are the GE model parameters in J mol−1.b Liquid molar volumes have been estimated from the Rackett equation [23].c The molecular parameters r and q are those reported in DECHEMA [52]: r1 = 4.4998, rd “q” parameter for the Wohl’s model.e “˛1” parameter for the NRTL’s model.f �P = (100/N)

∑N

i

∣∣Pexpi

− Pcali

∣∣/Pexpi

.

g �ı = 1/N∑N

i

∣∣ıexpi

− ıcali

∣∣with ı = y or x.

rule and Wohl G model with the parameters indicated in Table 8.

According to the previous results, we can conclude that theGibbs excess (GE) parameters reported in this work can be used toaccurately and consistently predict the VLE of this binary system.

In order to establish the coherency of the reported binary dataand, additionally, to test their predictive capability and transferabil-ity from �–� to �–� approach, we have used the best-ranked model(Wohl’s model) to predict the binary VLE data from the PR-EoS withMHV mixing rules. In Fig. 3, it is possible to observe that the param-eters reported in Table 8 reproduce with an excellent agreement(no visible difference) the VLE data, both in the predicted bubble-

point ( P < 0.3%, �yi < 0.4%) and dew-point pressures (�P < 0.4%,�xi < 0.4%).

THP (2).a

Bubble-point pressures Dew-point pressures

�P (%)f 100 × �yig �P (%)f 100 × �xi

g

0.29 0.2 0.38 0.30.19 0.4 0.34 0.40.21 0.4 0.39 0.50.30 0.3 0.43 0.30.20 0.5 0.38 0.50.22 0.5 0.44 0.60.31 0.3 0.43 0.30.21 0.5 0.40 0.50.21 0.5 0.44 0.60.31 0.3 0.43 0.30.20 0.4 0.39 0.50.21 0.5 0.72 0.6

2 = 3.6159, q1 = 3.8560, q2 = 2.9400.

62 A. Mejía et al. / Fluid Phase Equilibria 316 (2012) 55– 65

Fst

tbhs

4

T

Fst

ig. 4. Activity coefficients (� i) as a function of the liquid mole fraction (x1) for theystem hexane (1) + THP (2) at 50.00 kPa. (�) Experimental data; (—) predicted fromhe two-parameter Legendre polynomial used in consistency analysis.

Based on the previous results, it is possible to conclude thathe parameters presented in Table 8 may be effectively transferredetween �–� and �–� approaches and, consequently, their valuesave been used to predict the excess volumes and interfacial orurface tensions.

.2. Interfacial properties

The temperature dependence of the surface tensions (�) of pureHP were experimentally determined using the same equipment

ig. 5. Activity coefficients (� i) as a function of the liquid mole fraction (x1) for theystem hexane (1) + THP (2) at 75.00 kPa. (�) Experimental data; (—) predicted fromhe two-parameter Legendre polynomial used in consistency analysis.

Fig. 6. Activity coefficients (� i) as a function of the liquid mole fraction (x1) for thesystem hexane (1) + THP (2) at 94.00 kPa. (�) Experimental data; (—) predicted fromthe two-parameter Legendre polynomial used in consistency analysis.

as that for obtaining the properties of the mixture. Table 9 presentsthe experimental � data, while Fig. 7 shows a comparison betweenthe surface tensions predicted from the SGT using the influenceparameter obtained from Eq. (21) and reported in Table 2 withthe experimental data reported by Villares et al. [19,53] and Gineret al. [54]. In Fig. 7, the good agreement between the predicted andreported values [19,59] with an average of the absolute percentagedeviation (AAPD) of 1.80%, allows concluding about the reliabilityof the adjustable parameter presented in Table 2.

The surface tension measurements for the mixture at 303.15 Kand 101.3 kPa are reported in Table 10 and depicted in Fig 8. Theseexperimental data were also correlated using the Redlich–Kisterexpansion proposed by Myers and Scott [31] (see Eq. (6)). Thecorresponding parameters together with the correlation statisticsare reported in Table 11. From the latter figure, it is possible toobserve that the surface tensions of hexane (1) + THP (2) exhibitnegative deviation from the linear behavior (x1�1 + x2�2). In addi-tion, it is possible to observe that the surface tensions of the mixturedecrease as the hexane concentration increases. Fig. 8 also included

the predicted surface tension, which was obtained from Eqs. (13)and (16) using the pure influence parameters reported in Table 2(where �ij = 0), the PR-EoS with MHV mixing rule and Wohl GE

model (Table 8). From these results it is possible to conclude that, as

Table 9Surface tensions (�) as a function of the temperature (T) for THP at 101.3 kPa.a

T/K �/mN m−1

283.15 29.61288.15 29.15293.15 28.56298.15 28.00303.15 27.14308.15 26.50313.15 25.55318.15 25.25323.15 24.23328.15 23.60333.15 22.75

a The measurement uncertainties are: � ± 0.01 mN m−1; T ± 0.01 K.

A. Mejía et al. / Fluid Phase Equ

F([

ibtatm

top

Fts(i

12increases from x1 > 0 to x1 < 0.63. However, the magnitude of therelative Gibbs adsorption of hexane in THP is modest, as expectedfrom inspection of Fig. 9.

ig. 7. Surface tension (�) as a function of the temperature (T) for THP at 101.3 kPa.�) this work (Table 9); (♦) Villares et al. [19]; (©) Villares et al. [53]; (�) Giner et al.54]; (—) calculated from SGT with the influence parameter reported in Table 2.

n the case of VLE results, the parameters presented in Table 8 maye transferred between �–� and �–� approaches and, therefore,heir values have been used to predict the surface tensions withn absolute percentage deviation in tension of 2.38%. These lat-er results clearly exemplify the capability of a theoretically based

odel for predicting the surface tensions of the present mixture.As it was described before, in addition to predict the surface

ension of the mixture, the SGT provides a rigorous route to analyzether interfacial properties such as the interfacial concentrationopulation and the Gibbs adsorption isotherm. Fig. 9 shows the

ig. 8. Surface tension (�) as a function of the liquid mole fraction (x1) for the sys-em hexane (1) + THP (2) at 303.15 K and 101.3 kPa. (�) this work (Table 10); (—)moothed by a Redlich–Kister expansion with the parameters shown in Table 11;. . .) lineal behavior (x1�1 + x2�2); (– –) predicted from SGT-PR-EoS with MHV mix-ng rule and the Wohl GE model using the parameters indicated in Table 8.

ilibria 316 (2012) 55– 65 63

z − �i projection as a function of the liquid mole fraction. In thatfigure, we observe that hexane exhibits positive surface activity(i.e. d�1/dz = 0; d2�1/dz2 < 0 inside the interfacial region), whereasTHP does not show surface activity. As it is shown in Fig. 9a, thesurface activity of hexane increases as its mole fraction increases.According to our calculations, the surface activity or absoluteadsorption of hexane is present over the range 0 < x1 < 0.63, whereit is found its maximum value. Finally, Fig. 10 shows the relativeGibbs adsorption isotherm of hexane (1) with respect to THP (2).It is seen how increases as the liquid mole fraction of hexane

Fig. 9. Concentration profiles (�i) along the interfacial region, z, for hexane (1) + THP(2) mixture at 303.15 K as a function of liquid mole fraction (x1). (a) z − �1; (b) z − �2;predicted from SGT-PR-EoS with MHV mixing rule and the Wohl GE model using theparameters indicated in Table 8 at (—) x1 = 0.25; (. . .) x1 = 0.50; (– –) x1 = 0.75.

64 A. Mejía et al. / Fluid Phase Equ

Table 10Surface tensions (�) as a function of the liquid mole fraction (x1) for the binarysystem hexane (1) + THP (2) at 303.15 K and 101.3 kPa.a

x1 �/mN m−1

0.000 27.100.045 26.470.090 25.640.181 24.390.282 23.170.383 22.030.487 21.130.589 20.390.674 19.470.786 18.770.899 18.120.964 17.461.000 17.40

a The measurement uncertainties are: � ± 0.01 mN m−1; T ± 0.01 K; x1 ± 0.001.

Table 11Coefficients (c0, c1, and c2) and deviations (maximum (max dev), average (avg dev)and standard (st dev)) obtained in correlation of surface tension, Eq. (6), for thehexane (1) + THP (2) at 303.15 K and 101.3 kPa.

c0 (mN m−1) c1 c2 Max dev Avg dev (103 mN m−1) st dev

−4.9944 −1.8078 −0.6226 24.99 6.46 8.49

Fig. 10. Relative Gibbs adsorption isotherm ( 12) for hexane (1) + THP (2) mixtureaEi

5

paborartt

� molar density

t 303.15 K as a function of the liquid mole fraction (x1). (—) predicted from SGT-PR-oS with MHV mixing rule and the Wohl GE model using the parameters indicatedn Table 8.

. Conclusions

Phase equilibrium and interfacial properties (concentrationrofile along the interfacial region, surface activity or absolutedsorption, relative Gibbs adsorption, and surface tension) for theinary system hexane + tetrahydro-2H-pyran have been describedver the whole mole fraction range. According to experimental VLEesults, the mixture is zeotropic and exhibits slight positive devi-tion from the ideal behavior over the considered experimental

ange. The phase equilibrium data of the binary mixture satisfieshe Fredenlund’s consistency test, and were well correlated usinghe �–� approach with the Wohl, NRTL, Wilson and UNIQUAC

ilibria 316 (2012) 55– 65

equations for all the measured isobars. Based on the capability ofpredicting simultaneously the bubble- and dew-point pressuresand the vapor and liquid phase mole fractions, the best-rankedactivity coefficient model was the Wohl model. Surface tensionsexhibit negative deviation from the linear behavior, and the exper-imental data were satisfactorily correlated using the Redlich–Kisterequation.

Besides the new experimental data reported for the VLE and sur-face tensions of the mixture, a full predictive theoretical schemewas used to predict both phase equilibrium and interfacial prop-erties. The theoretical approach is based on SGT and an improvedPeng–Robinson equation of state, appropriately extended to mix-tures within the framework of a predictive MHV mixing rule.According to the results, accurate predictions of the experimentalVLE and IFT data were obtained. In addition, the present approachapplied to a characterization of the interfacial behavior allows con-cluding that:

• hexane is adsorbed at the interface region• the surface activity of the mixtures decreases as the concentration

of hexane increases, the relative Gibbs adsorption of hexane onTHP increases with hexane concentration, within ordinary rangesand without reaching a saturation limit.

List of symbolsa cohesion parameter in the EoSAi, Bi, Ci antoine constants in Eq. (4)Aij Wohl parameterb covolume parameter in the EoSB second virial coefficients of the pure gasesc influence parameterck Redlich–Kister parametersf0 Helmholtz energy density of the homogeneous systemG Gibbs energyk1, k2 Stryjek–Vera’s parametersm numer of parameter in Redlich–Kister expansionN molesnD refractive indexp absolute pressureri radii of probe orifice iR universal gas constantT absolute temperatureTb normal boiling temperatureV volumev voltageu MHV functionx, y mole fraction of the liquid and vapor phasesz spatial coordinate normal to the surface area

Greek˛i thermal cohesive function of the species i

NRTL parameter� adjustable parameter for influence parameter of mixtures� mixing rule of second virial coefficients� differential� fugacity coefficient relative Gibbs adsorption isotherm� activity coefficient� solvation parameter� Wilson parameter chemical potential� mass density

� surface tension� NRTL parameter� MHV function

se Equ

S1ci

ScelLV0

F

1

R

[[[

[

[[

[

[

[[

[

[

[

[[[

[[[

[[

[[

[[[[

[[

[[

[

[

[[

[[

[[[

[[[

A. Mejía et al. / Fluid Pha

ubscripts,2 number of orifice

critical state, j component i, j, respectively

uperscriptsal calculatedxp experimentalit literature

liquid vapor

equilibrium

unding sources

This work was financed by FONDECYT, Santiago, Chile (Project100357).

eferences

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[

[

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