calculation of vapor-liquid equilibrium data of binary mixtures using vapor pressure.pdf

Research Article

Calculation of Vapor-Liquid Equilibrium Dataof Binary Mixtures Using Vapor PressureData*

The design of plants or unit operations requires the knowledge of data for multi-component systems. An approach is presented for the regression of binary inter-action parameters required for the calculation of activity coefficients. Binaryvapor pressure data are used to correlate these parameters for calculation ofvapour/liquid equilibrium data. The advantage of the approach is the fact thatonly easily measured vapor pressure data are required. The procedure is demon-strated by means of two binary mixtures. Parameter regression, prediction ofvapour/liquid equilibrium data as well as their experimental verification aredemonstrated.

Keywords: Binary interaction parameter, NRTL equation, Vapor-liquid equilibrium,Vapor pressure

Received: August 17, 2010; accepted: August 17, 2010

DOI: 10.1002/ceat.201000356

1 Introduction

A number of problems in safety engineering and the design ofplants or unit operations in chemical industries require theknowledge of data for multicomponent systems. Withoutknowledge of phase equilibria, the design of separation pro-cesses such as distillation, rectification, or extraction wouldnot be possible with scale-up factors as they are achievednowadays. Examples are vapor pressures of mixtures, the phasebehavior of liquid/liquid biphasic systems, and others. In thepetrochemical industry and automobile sector biofuels like,e.g., ethanol or FAME (fatty acid methyl ester = biodiesel) aswell as blend components like ethers are getting more andmore in the focus. The strongly nonideal behavior of thesepolar substances in a fuel consisting of unpolar alkanes, ole-fins, and aromatics requires several experimental data to fitparameters for calculation of phase equilibria. For multicom-ponent systems in general only few experimental data areavailable which leads to the necessity of mathematical predic-tion of these data. Several equations have been developed todescribe the phase behavior of multicomponent systems [1].

Basically these complex systems are split into a number ofbinary systems. In general, as fundamental equation for calcu-lation of vapor-liquid equilibria (VLEs) Eq. (1) is used:

yipui � xiciu�i p�i exp

v ′i p � p�i� �

RT

� �(1)

The exponential term in Eq. (1) is the Poynting factor whichcould be neglected for low total pressure p. With this simplifi-cation Eq. (1) becomes:

yipui � xicip�i (2)

The variables left in Eq. (2) are the vapor pressure pi* of thepure substance, the activity coefficient ci, and the fugacitycoefficient �i for the correction of real gas behavior. The vaporpressure can be calculated with, e.g., the Antoine equation.The fugacity coefficient is accessible from an equation of statesuch as the Soave, Redlich, and Kwong (SRK) equation [1].For the activity coefficient generally expressions for the freeexcess enthalpy (gE) are used such as the Wilson equation,NRTL (Non-Random-Two-Liquid Model) equation, or theUNIQUAC (Universal Quasiemical) equation. Among theseequations the NRTL equation (and its modifications) is mostwidely used [2]. All these approaches require two or threebinary interaction parameters (BIPs) to describe a binary mix-ture. The interaction parameters are generated with experi-mental data or can be predicted by the group contributionmethod UNIFAC (Universal Quasichemical Functional GroupActivity Coefficients).

Chem. Eng. Technol. 2010, 33, No. 12, 2089–2094 © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.cet-journal.com

Tilman Knorr1

Eberhard Aust2

Karl-Heinz Jacob2

1 University of Erlangen-Nürnberg, Institute ofChemical ReactionEngineering, Erlangen,Germany.

2 Georg-Simon-OhmHochschule, Department ofApplied Chemistry, Nürnberg,Germany.

–Correspondence: Prof. Dr.-Ing. E. Aust ([email protected]), Georg-Simon-Ohm Hochschule, Department of Ap-plied Chemistry, Prinzregentenufer 47, 90489 Nürnberg, Germany.

–* English version of: Chem. Ing. Tech. 2009, 81 (12), 1999.

DOI: 10.1002/cite.200900058

Vapor-liquid equilibrium 2089

2 Calculation of Interaction Parameters

State-of-the-art and established without doubt is the adjust-ment to VLE data (Tpxy data) [3, 4]. Furthermore, the use ofenthalpies of mixtures (hE), activity coefficients for infinitedilution (c∞), or liquid/liquid equilibria data (LLE data) isreported [3]. Recently, a method for determination of VLEdata using differential scanning calorimetry (DSC) coupledwith a flame ionization detector (FID) for analysis of the gasphase was published [5]. This method leads to traditional VLEdata which could be used to adjust binary interaction para-meters.

Barker demonstrated the regression of interaction para-meters from vapor pressure data [6]. However, this methodrequires vapor pressure data of binary mixtures with differentcompositions but for the same temperature and no vapor pres-sure curves over a certain range of temperature. The Redlich-Kister equation was used in combination with a virial equationof state for the gas phase correction. The adjustment of theinteraction parameters is done at a fixed temperature. Unfor-tunately, the experimental effort for this method is consider-able, and the data obtained are valid only for a certain temper-ature. A method requiring only few data leading to parametersvalid for a wide range of temperature would be preferable. Inanother work, published in 1962, the method of Barker wasused for some systems [7]. Furthermore, the use of vapor pres-sure data is mentioned in connection with the calculation ofVLE data but no system for the regression of interactionparameters out of these data is presented.

The determination of binary interaction parameters fromVLE data requires access to these equilibrium data. For mea-surement of phase equilibria several strategies have been devel-oped but most of these methods require a lot of experimentalwork. On the other hand, this work is required to obtain rea-sonable data for a set of interaction parameters one can trust.The classical methods use a variety of different equilibriumstills to take samples from the liquid and the condensed vaporphase at a certain temperature and pressure [5, 9, 10]. Thesesamples have to be analyzed by gas chromatography or othermethods to determine the compositions.

3 New Method for Determination of BinaryInteraction Parameters

3.1 Idea

Since a lot of effort has to be put into the generation of VLEdata, it would be desirable to use more accessible experimentaldata for adjustment of the parameters. For this purpose anapproach using binary vapor pressure data is characterized.This method allows calculation of the parameters for a certainrange of temperature. The NRTL equation was used to calcu-late the liquid phase activity coefficient.

gE

RT� xixj

sijGji

xi � xjGji� sijGji

xj � xiGij

� �(3)

Gij � exp �aijsji

� �Gji � exp �ajisij

� �aij � aji

(4)

sij �gij � gji

RTsij �

gji � gij

RT(5)

Aij � gij � gji Aji � gji � gij (6)

The nomenclature Aij and Aji for the binary interactionparameters was chosen according to the Vapor-Liquid Equilib-rium Data Collection published by the DECHEMA, Germany[11]. In the NRTL equation, the parameters gij and gji are inter-preted as a number for the interaction between the two speciesi and j, and aij represents the nonideal behavior of the mixture.However, this study is based on the parameters A which can becalculated from the g-values according to Eq. (6).

To calculate a binary VLE, the solution of a system of twoequations is required. For both components the distributionbetween gas and liquid phase is given by Eq. (1). In this study,the simplified version of Eq. (1), namely Eq. (2), was used.

yipui � xicip�i (7)

yjpuj � xjcjp�j (8)

The solution of the system of equations given above is possi-ble if values for the coefficients c and � as well as for the vaporpressure p* are known. In the following study the fugacitycoefficient � was calculated according to the SRK equation ofstate [12]. The vapor pressure was calculated using the Antoineequation, and the activity coefficient was obtained from theNRTL equation as already mentioned. From the solution ofEqs. (7) and (8) the total pressure of the system can be calcu-lated as the sum of all partial pressures using Dalton’s law.

pi � yip p ��

i

pi (9)

This total pressure over the mixture of two species is equalto the vapor pressure of the liquid phase. With given values Aij

and Aji for the binary interaction parameters, the vapor pres-sure can be calculated over a liquid phase with known compo-sition and at a certain temperature. If the procedure is re-peated, the vapor pressure for a defined range of temperaturecan be calculated as a function of the given values for the inter-action parameters Aij and Aji. These calculated vapor pressurecurve can now be compared to the measured data for the sameinterval of temperature and the same composition of the liquidphase. The differences between measured and calculated pres-sures lead to the sum of squares, again as a function of the giv-en values for the interaction parameters according to Eq. (10).

Sum of squares ��

i

p�measured � p�calculated

� �2

p�measured

� �(10)

For a simple visualization of the method, the third param-eter in the NRTL equation, aij, was kept constant for a certain

www.cet-journal.com © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Chem. Eng. Technol. 2010, 33, No. 12, 2089–2094

2090 T. Knorr et al.

binary system. Hence, the number of variables is set to twoand the sum of squares could be displayed easily in a Cartesiancoordinate system on the Z-axis while Aij and Aji are on theX-axis and the Y-axis, respectively, as shown in Fig. 1.

As a first indication for the value of aij, the commendationsof Renon and Prausnitz are used [2]. However, the group con-tribution method UNIFAC will provide useful values as well.Besides the NRTL equation, other approaches for activity coef-ficients can be treated in a similar way.

3.2 Validation

For verification of the method two binary systems have beenchosen. The first system is a rather ideal one, while the secondsystem shows an azeotrope and thus exhibits strong nonidealcharacteristics. The system n-pentane/o-xylene is known as analmost ideal system. The mixture n-hexane/ethanol shows anazeotrope and is well-known from literature [13–15]. For theregression of the interaction parameters of both systems vaporpressure curves for different mixtures have been measured (seeTabs. 1 and 2). For the vapor pressures an equipment devel-oped for precise measurement was available and used in thepresent work [16]. The accessible temperature range isapproximately from –30 °C to 200 °C. For the measurementsof vapor pressures the above-mentioned substances (fromE. Merck, Germany) were used. The purities of all chemicalswere either for synthesis or for spectroscopy: n-pentane> 99 %, o-xylene > 98 %, n-hexane > 99 %, ethanol > 99.9 %.The purities have been controlled by gas chromatography. Forthe pure substances vapor pressure curves have been deter-mined, being in good agreement with the ones calculated bythe Antoine equation (using the process simulator Chem-CAD) [17].

3.3 Computation

Based on the process simulator Chem-CAD (version 6.0.2)[17] in combination with Excel/VBA (Visual Basic for Applica-tion, version 2003), regression of the interaction parameterswas carried out using measured vapor pressure data as de-scribed in Section 3.1. For a reasonable number of supportingpoints the sum of squares for the measured and calculatedvapor pressures was calculated. The scheme of the regressionprocedure using Excel/VBA and Chem-CAD is illustrated inFig. 2.

When plotted for all studied parameters Aij and Aji asdescribed in Section 3.1, a clear connection between bothinteraction parameters and the sum of squares is displayedin Fig. 3. The results for the first mixture (see Tab. 1) of

the system n-pentane/o-xylene are demon-strated.

The parameters have been varied in therange of 0 to 1000 and –500 to 400, respec-tively. Fig. 3 shows that many pairs of Aij

and Aji can be given describing the chosenvapor pressure data quite similarly. For allmixtures of all systems similar results areobtained. The sets of possible parametersrepresent a line in the plot of the sum ofsquares, a line of minima. If one comparesthe trends for the minimum sum ofsquares for both mixtures in the systemn-pentane/o-xylene, a clear intersectionpoint is found as shown in Fig. 4.

It is obvious to choose the parameters atthe intersection point as solution for theinteraction parameters since this is theonly set of values describing both vaporpressure curves with good accuracy. Alsofor the system n-hexane/ethanol a similarcorrelation was found. Due to the stronglynonideal behavior of the system a thirdcomposition was added (see Tab. 2). Even


Figure 1. Scheme of calculation of the sum of squares.

Table 1. Compositions of n-pentane/o-xylene mixtures.

CompositionMolar fractions

xn-pentane xo-xylene

1 0.59 0.41

2 0.19 0.81

Table 2. Compositions of n-hexane/ethanol mixtures.

CompositionMolar fractions

xn-hexane xethanol

1 0.91 0.09

2 0.50 0.50

3 0.09 0.91


for the nonideal system with azeotrope a similar trend for theminimum sum of squares was found, as can be seen in Fig. 5.All three curves intersect in a small area.

4 Validation of Results and Discussion

For validation of the interaction parameters obtained from theprocedures described above for both systems, VLE data havebeen calculated using the found parameters. The calculateddata were compared to measured data. For this purpose, anisobaric VLE was measured for the system n-pentane/o-xyleneusing the equilibrium still of Röck and Sieg manufactured bythe NORMAG AG Company, Germany [18]. Comparing bothexperimental and calculated data, one can see a good agree-ment in Fig. 6. Again, as done for regression, the gas phasebehavior was corrected by the SRK equation.

For the system n-hexane/ethanol two different VLEs, an iso-thermal and an isobaric one, were calculated and compared toliterature data [13, 14] (see Figs. 7 and 8). These plots pointout that the method presented in this work is able to lead to


Figure 2. Scheme of the regression of interaction parametersusing Excel and Chem-CAD.

Figure 3. 3D plot of the sum of squares for a mixture of n-pen-tane and o-xylene as a function of the two binary interactionparameters of the NRTL equation.

Figure 4. Trends for the minimal sum of squares for the twocompositions given in Tab. 1 for the system n-pentane/o-xylene.

Figure 5. Trends for the minimal sum of squares for the threecompositions given in Tab. 2 for the system n-hexane/ethanol.

Figure 6. Isobaric vapor-liquid equilibrium for n-heptane/o-xy-lene at 940 mbar. The calculation was done using the NRTLequation in combination with the SRK equation of state for gasphase correction. The data have been measured at theHochschule Nürnberg.


parameters even for strongly nonideal mixtures like azeo-tropes. Only minor deviations between measurement and cal-culation could be observed.

However, Fig. 5 illustrates that several sets of parameters areable to describe the measured vapor pressure curves with suffi-cient precision. In the following, a sensitivity study proves thatonly the set of parameters at the intersection point of thecurves will also describe VLE data properly. The sets of param-eters, indicated in Fig. 5 by the numbers I to IV, reproduce thevapor pressure curve used for their calculation with good accu-racy. Obviously the results are different when these sets ofparameters are used to calculate VLE data. In Fig. 9 the calcu-lated isothermal VLE data were compared to those of the lit-erature [12]. The azeotropic point, the dew point line, and thebubble point line vary with the set of parameters.

These sets of parameters were generated from composition 1(see Tab. 2) with 9 mol.-% of ethanol. This explains why all thecurves almost look the same for low ethanol concentrationand vary strongly for higher ethanol concentrations. Since

interaction parameters should be valid for the total range ofconcentration, the regression using vapor pressure data has tobe done with at least two curves for different liquid phasecompositions.

5 Conclusions

Calculation of phase equilibria requires accurate interactionparameters of the equation used to calculate activity coeffi-cients. As an alternative to the regression from VLE data it isalso possible to use vapor pressure curves of binary mixturesfor the fit of the parameters. Here, the parameters are fitteduntil the calculated data are in good agreement with the ex-perimental curves using the least square method. To achieveparameters for the full range of concentration, at least twovapor pressure curves of binary mixtures with different con-centrations are required. The method was demonstrated fortwo different binary mixtures. Even for a strongly nonidealsystem, namely n-hexane/ethanol with an azeotropic point,good agreement between calculated and experimental data wasfound. Since conventional VLEs require a lot of samples to beanalyzed (e.g., GC or refraction index), this method demandssimply the measurement of the pressure over the liquid phaseas well as the composition of the liquid phase.

The authors have declared no conflict of interest.

Symbols used

Aij [cal mol–1] interaction parameter of NRTLequation

gE [J mol–1] free excess enthalpyhE [J mol–1] enthalpy of mixingp [Pa] pressurepi [Pa] partial pressurepi* [Pa] vapor pressureR [J mol–1K–1] universal gas constant


Figure 7. Isobaric vapor-liquid equilibrium for n-hexane/ethanolat 1013 mbar. The calculation was done using the NRTL equa-tion in combination with the SRK equation of state for gas phasecorrection. Measured data from [10].

Figure 8. Isothermal vapor-liquid equilibrium for n-hexane/etha-nol at 318.15 K. The calculation was done using the NRTL equa-tion in combination with the SRK equation of state for gas phasecorrection. Measured data from [11].

Figure 9. Isothermal vapor-liquid equilibrium for n-hexane/etha-nol at 318.15 K. The curves Parameter I to Parameter IV representthe sets of BIPs indicated in Fig. 5. Calculation was done usingthe NRTL and SRK equations. Measured data from [11].


T [K] temperaturevi′ [m3mol] molar volume of liquid phasexi [moli mol–1] molar fraction in liquid phaseyi [moli mol–1] molar fraction in gas phaseui [–] fugacity coefficientaij [–] interaction parameter of NRTL

equationci [–] activity coefficient

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calculation of vapor-liquid equilibrium data of binary mixtures using vapor pressure.pdf

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