Fluid Phase Equilibria 236 (2005) 261–266

Correspondence

Comments on “Computing all the azeotropes in refrig-erant mixtures through equations of state” by NaveedAslam and Aydin K. Sunol [Fluid Phase Equilib. 224(2004), 97–109]

In a very recent paper in Fluid Phase Equilibria, Aslamand Sunol[1] proposed a method for computing azeotropesthrough Equations of State, as an application of the workpresented in previous references[2,3] by the same authors.Briefly, the method is based on a homotopy-continuationscheme with the homotopy maphi given by

hi(x, λ) = λ(xi − yi) + (1 − λ)(xi − yideali ) (1)

whereλ is the homotopy parameter (1≥ λ ≥ 0), xi , yi arethe liquid and vapor phase concentrations andyideal is thevCh

λ

tdf

According to Aslam and Sunol, an azeotrope diluted in com-ponenti occurs at the pure componentj whenλij = 1 (thiscondition is rigorous and follows directly from Eq.(1)). Inaddition, an stable or unstable azeotrope exists (or persists)in a binary system if

0 < λ12 < 1 and/or 0 < λ21 < 1. (6)

Although this last claim is true in many cases, in this work wepresent some counterexamples that question their validity forgeneral prediction purposes. First, let us consider Eq.(5) atthe light of the modified Raoult’s law[5], where vapor phasenon idealities are neglected. In this case

φLi = f L

i

x P= xiγif

0i

x P= γif

0i

P≈ γiP

0i

P

φL

i

apor phase concentration as calculated from Raoult’s law.onsidering that an stable physical azeotrope is met when

i = 0 andλ = 1, Aslam and Sunol explored the function

= 1 − yideali /xi

yi/xi − yideali /xi

(2)

hat can be directly derived from Eq.(1), as a potential can-idate for startingλ in homotopy computations. It follows

i i

φVi = fV

i

yiP≈ 1

⇒ i

φVi

= γiP0i

P⇒ φ

L,∞i

φV,∞i

= γ∞i P0

i

P0j

(7)

so that, replacing Eq.(7) in Eq.(5) yields

P0 − P0 P0 − P0

rom the real and ideal vapor–liquid equilibrium approaches

re

s

λij = j i

P0i (γ∞

i − 1); λji = i j

P0j (γ∞

j − 1). (8)

F ropicb ar-g

w ed

l

a er

C

F top ropic

.

ig. 1 presents the global phase diagram for the azeotehavior of Margules binary mixtures. According to the Mules model[6], the excess Gibbs energyGE is

GE

RT= x1x2(Ax2 + Bx1) (9)

here the parametersA andB are directly related to infinitilution activity coefficientsγ∞

i by

n γ∞1 = A; ln γ∞

2 = B (10)

nd the parameterC that appears inFig. 1, in the low pressurange, is defined as

= lnP0

2

P01

(11)

rom Fig. 1 it follows that the Margules model is ableredict zeotropic, single azeotropic and double azeot

that[4]

yideali

xi

= P0i

P;

yi

xi

= φLi

φVi

(3)

whereP is pressure,P0 is the pure component vapor pressuandφ�

i is the effective fugacity coefficient of componenti inthe�-phase (liquid L or vapor V). Replacing Eq.(3) in Eq.(2) yields

λ = 1 − P0i /P

φLi /φV

i − P0i /P

(4)

from which it is possible to deduce the following functionat infinite dilution for a binary mixture

λ12 = 1 − P01/P0

2

φL,∞1 /φ

V,∞1 − P0

1/P02

λ21 = 1 − P02/P0

1

φL,∞2 /φ

V,∞2 − P0

2/P01

.

(5)

0378-3812/$ – see front matter © 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.fluid.2005.07.001

262 Correspondence / Fluid Phase Equilibria 236 (2005) 261–266

Fig. 1. Global phase diagram for azeotropic behavior in Margules binaryfluids.

behavior in binary mixtures. In addition, considering Eqs.(8), (10)and(11)we have

λ12 = eC − 1

eA − 1; λ21 = 1 − eC

eC(eB − 1)(12)

The ranges drawn inFig. 1, together with Eq.(12), could beused for testing the prediction capability of the criteria givenby Eq.(6) and results are shown inTable 1. As follows fromthese results, it is clear that the method proposed by Aslamand Sunol succeeds in predicting double azeotropy for range1, single azeotropy for range 2 and no azeotropy for range3. However, the method under consideration does not predictazeotropy in region 4 where, according toFig. 1, at least twoazeotropes (stable or unstable) should exist (Table 1).

Consequently, in our opinion, the method proposedby Aslam and Sunol fails in predicting some cases ofpolyazeotropy. One of these cases is directly related to refrig-erant mixtures, as the case of the mixture 1,1,1,2,3,4,4,5,5,5-decafluoropentane (HFC-4310 mee) + oxolane (THF). Fromthe experimental determination made by Kao et al.[7] andLoras et al.[8], it follows that the binary mixture HFC-4310mee (1) + THF (2) has two azeotropes that evolve to tangent

Table 2Parameters for the excess Gibbs energy model and Antoine’s coefficients ofthe binary system HFC-4310 mee (1) + THF (2)a

(A) Parameters (Cj

i ) for Eq.(13)

k →j ↓ 1 2 3 4

0 2.826 0.625 −0.879 0.1211 −1063.080 −182.011 352.862 0.000

(B) Antoine coefficients, Eq.(15).

Compound Ai Bi Ci

HFC- 4310 mee 6.43876 1242.510 46.568THF 6.14852 1211.079 46.627

a Data taken from Loras et al.[8].

azeotropy[9] atT ≈ 314 K,P ≈ 42 kPa andx1 ≈ 0.17. In addi-tion, Loras et al. proposed the following Redlich-Kister[10]type model

GE

RT= x1x2

[C1 + C2(x2 − x1) + C3(x2 − x1)2

+C4(x2 − x1)3]

(13)

where the parametersCk have been considered temperaturedependent according to

Ck = C0k + C1

k

T/K, k = 1, 4 (14)

and the parametersC0k andC1

k , together with the parametersof vapor pressure Antoine’s equation

log(P0

i /kPa)

= Ai − Bi

(T/K) − Ci

(15)

are presented inTable 2. Following the method of Aslamand Sunol, and considering the low pressure approximationfor λij (see Eq.(8)), we analyze a range of isobars of thebinary mixture HFC-4310 mee (1) + THF (2) as predictedfrom Eqs.(13)–(15). As shown inFig. 2, and according tothe approach of Aslam and Sunol, it is possible to observethat azeotropy is predicted below the pressure of point A( ich-K ween

Table 1T ing to

C gle aze zeotropy)

CACACACA

esting of Eq.(5) for predicting azeotropic in Margules mixtures accord

ase Region 1 (double azeotropy) Region 2 (sin

> 0 0 <λ12 < 1 0 <λ12 < 1+ B > 0 0 <λ21 < 1 λ21 < 0> 0 0 <λ12 < 1 λ12 < 0+ B < 0 0 <λ21 < 1 0 <λ21 < 1< 0 0 <λ12 < 1 0 <λ12 < 1+ B < 0 0 <λ21 < 1 λ21 < 0< 0 0 <λ12 < 1 λ12 < 0+ B > 0 0 <λ21 < 1 0 <λ21 < 1

PA ≈ 14.55 kPa). However, the prediction of the Redlister model indicates that double azeotropy persists bet

the global phase diagram shown inFig. 1

otropy) Region 3 (no azeotropy) Region 4 (double a

λ12 < 0 λ12 > 1λ21 < 0 λ21 < 0λ12 < 0 λ12 < 0λ21 < 0 λ21 > 1λ12 < 0 λ12 > 1λ21 < 0 λ21 < 0λ12 < 0 λ12 < 0λ21 < 0 λ21 > 1

Correspondence / Fluid Phase Equilibria 236 (2005) 261–266 263

Fig. 2. Application of the method of Aslam and Sunol to the predictionof the azeotropic behavior of the system HFC-4310 mee (1) + THF (2).λij

calculated from Eqs.(13)–(15).

the pressures of points A and B (PB ≈ 42 kPa), as shown inthe predicted VLE diagram atP = 35 kPa inFig. 3.

Let us consider additional examples concerning the pre-diction of azeotropy from equations of state (EOS). Kolafa[11] has presented an authoritative analysis concerning thecapability of prediction of azeotropic phenomena in cubics,and we borrow some points from his theory in what follows.Fig. 4depicts the global phase diagram of the traditional vander Waals cubic for molecules of different sizes (b2/b1 = 1/2).Following the notation of van Konynenburg and Scott[12],

F mee( ld

Fig. 4. Global phase diagram for van der Waals fluids (ξ = 1/3). ( ): tricrit-ical point, (-©-): zero temperature end point, (· · ·): pure critical azeotropicpoint, (·+·): zero temperature limited azeotropy, (- - -): critical quadruple endpoint. ( ): critical azeotropic end point.

the coordinates of the GPD are given by

ξ = b2 − b1

b2 + b1

ζ = a2/b22 − a1/b

21

a2/b22 + a1/b

21

λ = a2/b22 − 2a12/b

212 + a1/b

21

a2/b22 + a1/b

21

(16)

whereai, bi are the parameters of the van der Waals EOS forpure fluids while the cross parametersaij, bij are defined as

a12 = √a1a2(1 − k12)

b12 = b1 + b2

2

(17)

where k12 is the binary interaction parameter. InFig. 4,the shaded area represents the range of parameters ofthe vdW model where stable azeotropy (single or double)could be predicted for binary mixtures. FromFig. 4 it ispossible to deduce that azeotropy affects systems Type Ito III and persists up to the shield region. In addition,azeotropic phenomena could also affect—although withlower probability—systems Type IV, IV* and V (whichhave been not shown inFig. 4). Of particular interest isthe capability of the original vdW cubic for predictingd fromt fg out-s glea

u ure

ig. 3. Vapor–liquid equilibrium diagram for the system HFC-43101) + THF (2) at 35 kPa. (—): calculated from Eq.(13), (©): experimentaata of Loras et al.[8].

ouble-azeotropy when the global coordinates are takenhe so called horn-region[11]. Every other combination olobal parameters that lie inside the shaded area andide the horn region ofFig. 4corresponds to a case of sinzeotropy.

As in the case of the Margules model,Fig. 4 may besed for testing the prediction capability of the proced

264 Correspondence / Fluid Phase Equilibria 236 (2005) 261–266

proposed by Aslam and Sunol. In this case we considerdirectly Eq.(5) for calculatingλij, together with the tradi-tional vdW-EOS and quadratic mixing rules. General resultsindicate that the method of Aslam and Sunol is effective forpredicting single azeotropy from the lowest temperature atwhich vapor pressures can be calculated to the critical tem-perature of the most volatile component. This is so because,as it can be seen in Eq.(5), the vapor pressure of all thecomponents of the mixture should exist at the condition inwhich azeotropy is being tested. As known, vapor pressurescan be calculated below the critical point and, in the case ofmixtures, the most volatile component establishes the tem-perature range where the complete set ofP0s can be predictedfrom the EOS. In addition, at very low temperatures (althoughstill above the triple point), the calculation of vapor pres-sures is a ill-conditioned numerical problem and alternativeapproaches are required for estimation purposes[13]. Con-sequently, for the case of single azeotropy, the method ofAslam and Sunol does not work for azeotropes that appearfundamentally in supercritical ranges, as the case shown inFig. 5.

Calculations based on binary systems taken from the horn-region (where the vdW model predicts double azeotropy)indicate that the procedure of Aslam and Sunol may fail again.For example,Fig. 6shows a vdW mixture with global coor-dinatesξ = 1/3, ζ =−0.405,λ = 0.135. The left hand side oftλ theTt of am tartsi pure

Fig. 5. T–x projection for a Type I mixture with global coordinatesξ = 1/3,ζ =−1/2, λ = 2/25. ( ): critical line, (· ·): azeotropic line, (�): purecomponent critical point, (♦): critical azeotropic end point, (�): pure com-ponent azeotropic end point.

component azeotropy in points A and B. The comparisonof Fig. 6 reveals that the method of Aslam and Sunol pre-dicts azeotropy between the temperatures of points C to A,although it fails in predicting azeotropy (or polyazeotropy)below the temperature of point C (where a Bancroft pointoccurs). FromFig. 6a it follows also that the occurrenceof polyazeotropy is reflected by multiple branches where0 <λij < 1. As pointed out by van Konynenburg and Scott,double azeotropy in van der Waals mixtures occurs in the

F with gc opic po

he figure shows the trend of the homotopy parametersλ12,21 in temperature, while the right hand side plot depicts–x projection of the example mixture. In theT–x projec-

ion, the line A–E–B corresponds to the azeotropic lineixture that exhibits double azeotropy, a behavior that s

n tangent azeotropy in point E and, then, diverges to

ig. 6. Homotopy parameter plot andT–x projection for a vdW mixtureomponent azeotropic end point, (©): Bancroft point, (�): tangent azeotr

lobal coordinatesξ = 1/3, ζ =−0.405,λ = 0.135. ( ): λij curve, (�): pureint, (· ·): azeotropic line.

Correspondence / Fluid Phase Equilibria 236 (2005) 261–266 265

Fig. 7. Homotopy parameter plot andT–x projection for a RKS mixture with properties inTable 3. ( ): λij curve, ( ): critical line, (· ·): azeotropicline, (�): pure component critical point, (♦): critical azeotropic end point, (�): pure component azeotropic end point.

vicinity of Bancroft points,1 where the vapor pressures of thecomponents become identical. It is convenient to note that,according to Eq.(5), λij vanishes at the Bancroft point. Inaddition, in the example system, it is observed that the twoazeotropes collapse in tangent azeotropy, below the temper-ature of the Bancroft point. The evidence shown here pointsto the fact that the prediction capability of the method ofAslam and Sunol depends strongly on the relative position ofBancroft points and tangent azeotropes.

As it can be seen inFig. 6, and contrary to Brandani’s[14,15] method, the approach of Aslam and Sunol is capa-ble to predict polyazeotropy when, at the same tempera-ture or pressure, variousλij branches verify the condition0 <λij < 1. However, this is not a general rule for the differentpolyazetropes that EOS models could predict. For example,in Fig. 7 we present the azeotropic behavior predicted bythe RKS-EOS with quadratic mixing rules, where the crossparameters are given by

a12 = √a1a2(1 − k12)

b12 = b1 + b2

2(1 − l12)

(18)

and the critical properties of the mixture, together with theinteraction parametersk12, l12, are shown inTable 3. InFig. 7.a it is seen that the trend ofλij indicates that singlea ow-e isa ngea em-

ropicc

Table 3Parameters for the mixture shown inFig. 7

Tc2/Tc1 Pc2/Pc1 m1a m2 k12 l12

1.181851 5.132075 1.345005 1.000172 0.51 0.75a mi corresponds to the Soave’s factor in the thermal cohesion function

α =(

1 + mi

[1 − √

T/Tci

])2.

perature of point A. It is interesting to note thatFig. 7a doesnot suggest the possibility of polyazeotropy, as follows fromthe fact that theλ12 branch never crosses the range 0 <λij < 1.

In conclusion, the method proposed by Aslam and Sunol isvaluable because of its simplicty and capability of predictingazeotropy and—with some constraints—polyazetropy in thetemperature ranges where the approach is applicable. How-ever, as follows from the examples presented in this work,the prediction reliability is still limited for completely gen-eral purposes.

Acknowledgement

This work has been financed by Fondecyt Project No.1020340.

References

[1] N. Aslam, A.K. Sunol, Fluid Phase Equilib. 224 (2004) 97–109.[2] N. Aslam, A.K. Sunol, Phys. Chem. Chem. Phys. 6 (2004)

2320–2326.[3] N. Aslam, A.K. Sunol, Chem. Eng. Sci. 59 (2004) 599–609.[4] H.C. Van Ness, M.M. Abbott, Classical Thermodynamics of Non

Electrolyte Solutions, McGraw-Hill, New York, 1982.

zeotropy occurs below the temperature of point A. Hver, fromFig. 7b it is clear that this example mixturezeotropic from low temperature to the supercritical rand, in addition, it exhibits double azeotropy from the t

1 This is not necessarily so in experimentally observed polyazeotases, as the mixture HFC-4310 mee + THF.

266 Correspondence / Fluid Phase Equilibria 236 (2005) 261–266

[5] J.M. Smith, H.C. Van Ness, Introduction to Chemical EngineeringThermodynamics, McGraw-Hill, New York, 1982.

[6] S.M. Walas, Phase Equilibria in Chemical Engineering, ButterworthPublishers, Boston, 1985.

[7] C.-P.C. Kao, R.N. Miller, J.F. Sturgis, J. Chem. Eng. Data 46 (2001)229–233.

[8] S. Loras, A. Aucejo, J.B. Monton, J. Wisniak, H. Segura, J. Chem.Eng. Data 47 (2002) 1256–1262.

[9] K.Z. Guminski, Rocz. Chem. 32 (1958) 569–582.[10] O. Redlich, A.T. Kister, Ind. Eng. Chem. 40 (1948) 345–348.[11] J. Kolafa, Phys. Chem. Chem. Phys. 1 (1999) 5665–5670.[12] P. van Konynenburg, R.L. Scott, Philos. Trans. R. Soc. London 298

(Ser. A) (1980) 495–540.[13] H. Segura, J. Wisniak, Comput. Chem. Eng. 21 (1997) 1339–1347.[14] V. Brandani, Ind. Eng. Chem. Fundam. 13 (1974) 154–156.[15] J. Wisniak, H. Segura, R. Reich, Ind. Eng. Chem. Res. 35 (1996)

3742–3758.

Hugo Segura∗Rafael A. Gonzalez

Department of Chemical EngineeringUniversidad de Concepcion

POB 160-C, Concepcion, Chile

Jaime WisniakDepartment of Chemical Engineering

Ben-Gurion University of the NegevBeer-Sheva 84105, Israel

∗ Corresponding author. Tel.: +56 41 203562fax: +56 41 247491

E-mail address: hsegura@diq.udec.cl (H. Segura)

6 September 2004

Available online 9 August 2005