computing all the azeotropes in refrigerant mixtures through equations of state

Fluid Phase Equilibria 224 (2004) 97–109

Computing all the azeotropes in refrigerant mixturesthrough equations of state

Naveed Aslam, Aydin K. Sunol∗

Department of Chemical Engineering, University of South Florida, 4202 E. Fowler Avenue, Tampa, FL 33620, USA

Received 14 May 2003; accepted 16 March 2004

Available online 6 August 2004

Abstract

Azeotropic mixtures of fluorocarbon (FC) and hydro fluorocarbon (HFC) with hydrocarbons are gaining popularity as drop-in substitutesfor CFCs and HCFCs. A method to compute all the azeotropes in a refrigerant mixture through the equation of state approach is described.The method allows prediction of all the azeotropes in a refrigerant mixture and is in close agreement with the experimental data. Boththe vapor and the liquid phase non-idealities are incorporated through fugacity coefficients modeled using Peng–Robinson–Stryjek–Veraequation of state with Wong-Sandler and van der Waals mixing rules. Homotopy continuation based methodology guarantees computationof all the solutions of necessary and sufficient condition of azeotropy in multicomponent refrigerant mixtures. The method establishes thepressure dependency of azeotropic composition allowing prediction of bifurcation pressure where refrigerant azeotropes may appear ordisappear and predicts azeotropes at elevated pressures. The approach is independent of equation of state and mixing rules but rely on theirability to represent the phase behavior. The approach is tested with R23–R13, propane–R227ea binary mixtures and a ternary mixture ofR32–R125–R143a.© 2004 Elsevier B.V. All rights reserved.

Keywords:Azeotropes; Vapor–liquid equilibria; Cubic equations of state; Mixing rules; Refrigerants; Mixtures

1. Introduction

The scheduled phasing out of ozone depleting refriger-ants like chlorofluoro carbon (CFCs) and hydrochlorofluorocarbon (HCFCs) necessitated the determination and the de-velopment of environmentally safe refrigerants. The searchfor alternative is quite challenging since not many singlecomponent fluids have saturation behavior similar to popu-lar refrigerants such as HCFC-22. There is no single fluidcomponent that can offer the same refrigeration character-istics. This limitation shifted the focus on refrigerant alter-natives, which are obtained as a mixture of two or morecomponents. Both zeotropic and azeotropic mixtures havebeen investigated. Zeotropic mixtures are one, which doesnot form a constant boiling mixture i.e., at a given pres-sure if such a mixture is distilled the vapor and liquid phasewill have different compositions. Unlike the azeotropic mix-

∗ Corresponding author. Tel.:+1-813-974-3566;fax: +1-813-974-3651.

E-mail address:[email protected] (A.K. Sunol).

tures, zeotropic mixtures have vapor pressure between itscomponents. Several zeotropic mixtures are in commercialuse as possible alternatives for R-22. However, the diffi-culty in maintaining the design composition during recharg-ing, and change in composition due to leaking is hinderingtheir popularity in the refrigeration industry (Didion et al.[1,2]).

The azeotropic mixtures have the same composition bothfor vapor and liquid phases and are always preferred amongdifferent types of refrigerant mixtures. Therefore, from arefrigeration perspective, the azeotropy is a desirable con-dition as all azeotropes behave like pure components whenboiling. The azeotropic mixtures of hydrocarbons with chlo-rofluoro carbons are quite promising alternatives for R-22and other categories of HCFCs. Several binary mixtures andsome ternary mixtures have been developed as potential al-ternative refrigerants (Didion et al.[2]; Doring et al.[3]). Inorder to replace CFCs and HCFCs with azeotropic mixtures,the accurate prediction of azeotropic composition in binaryand multi-component mixtures is required throughout theentire pressure range. A comprehensive experimental pro-

0378-3812/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.fluid.2004.03.014

98 N. Aslam, A.K. Sunol / Fluid Phase Equilibria 224 (2004) 97–109

gram to obtain all the data covering azeotropic compositionsover entire pressure range for binary and multi-componentmixtures of all the possible refrigerant and hydrocarbon mix-tures would be unnecessarily expensive and time consum-ing. The alternative approach is to perform the phase equi-librium calculations for complex mixtures of refrigerantsand hydrocarbons through cubic equations of state with dif-ferent mixing rules (Orbey and Sandler[4]; Ioannidis andKnox [5]). However, even these phase equilibrium calcula-tions will be expensive and may not provide accurate in-formation regarding the azeotropes over the entire range ofpressure and temperature, particularly with implicit or adhoc procedures to isolate an azeotrope. Therefore, a reliableand efficient method is needed to compute all the azeotropesin a refrigerant mixture over the wide range of pressures.In this paper, we present a method that can reliably be usedto calculate all the azeotropes for a given refrigerant mix-ture. The method is also able to predict the conditions underwhich an azeotrope may exist or disappear. It can also estab-lish the pressure dependency of hydrocarbon and refrigerantazeotropes.

Fidkowski et al.[6] described an approach to compute allthe homogeneous azeotropes in a multi-component mixture.He used the homotopy continuation method to solve the nec-essary conditions for azeotropy. While the homotopy param-eter in the model is varied from zero to one, the equilibriumvaries from ideal equilibrium (γi = 1.0) to non-ideal equi-librium (γi �= 1.0) and this constitutes the necessary condi-tion of azeotropy. They used an activity coefficient modelto model the liquid phase non-idealities and neglected thevapor phase non-idealities.

Tolsma and Barton[7] extended Fidkowski’s approach tocompute heterogeneous azeotropes and presented the nec-essary proofs regarding the computation of all the homo-geneous and heterogeneous azeotropes for multi-componentmixtures. They further analyzed the variation of phase equi-librium structure with pressure. They used activity coeffi-cients to model the liquid phase and assumed that vaporphase is ideal.

Our approach is based on Tolsma and Barton’s work andextends it to refrigerant mixtures through an EOS incorpo-rating vapor phase non-idealities as well. We obtained fugac-ity coefficients from Peng–Robinson–Stryjek–Vera (PRSV)equation of state with van der Waals and Wong-Sandlermixing rules and mapped the vapor and liquid phasenon-idealities. Coupled with the homotopy continuationbased methodology, our approach is capable of reliablypredicting the azeotropy in refrigerant mixtures both atlower and elevated pressures. We analyzed three binarymixtures R23–R13, propane–R227ea and a ternary mixtureof R32–R125–R143a. Our method is also capable of pre-dicting the bifurcation pressures i.e., the pressures whererefrigerant azeotropes may appear or disappear. At ele-vated pressures, the bifurcation points of pure componentrefrigerant branches are predicted by using infinite dilutionK-values obtained from EOS.

2. Azeotropy in refrigerant mixtures

Fidkowski et al.[6] solved the necessary conditions ofazeotropy (xi = yi) using homotopy continuation. The ho-motopy map is given as follows:

hi(x, λ) ≡ λ× (xi − yReali )+ (1 − λ)× (xi − yIdeal

i ) (2.1)

where� is a homotopy parameter, which varies from 0 to1 andxi is the liquid composition. TheyIdeal

i is the vaporcomposition for an ideal gas and ideal solution andyReal

i isfor a real gas and non-ideal solution. Forλ = 0, thexi =yIdeali and forλ = 1, thexi = yReal

i .

yIdeali = KIdeal

i × xi (2.2)

yReali = KReal

i × xi (2.3)

TheKIdeali is the ideal solutionK-value and is derived from

Raoult’s law whileKReali is a non-idealK-value obtained

from EOS to account for both the liquid and the vapor phasenon-idealities in a refrigerant mixture.

KIdeali = Psat

i

P(2.4)

and

KReali = φ̂Li (xi, xj, T, P)

φ̂Vi (yi, yj, T, P)(2.5)

Tolsma and Barton[7] further enhanced Fidkowski’s ap-proach for isobaric conditions. They developed the followingformulation shown inEqs. (2.6)–(2.9), using activity coeffi-cients to map the liquid phase non-idealities and assumingideal vapor phase.

F(x, y, T, l) =

x1 − y1

...

xn − yn

y1 −K01 × x1

...

yn −K0n × xn

n∑i=1

xi − 1

(2.6)

K0i is the pseudo equilibrium constant.

K0i = [λ×KReal

i + (1 − λ)×KIdeali ] (2.7)

They removed the perturbed vapor composition fromEq. (2.6), obtaining the following relation:

F(x, T, λ) =

x1 −K01 × x1

...

xn −Kon × xnn∑i=1

xi − 1

(2.8)

N. Aslam, A.K. Sunol / Fluid Phase Equilibria 224 (2004) 97–109 99

when we set F(x,T,λ) = 0, we have defined a solutionspace inx, T, λ for a fixed pressure. The mapped solutionof above function is called the homogeneous homotopypath.

The Jacobian matrix ofEq. (2.8)is given as:

∇F(x, T, λ) =

α1 − λ× x1 ×K1,1 −λ× x1 ×K1,2 · · · −λ× x1 ×K1,n −x1 × β1 −x1 × ϕ1

−λ× x2 ×K2,1 α2 − λ× x2 ×K2,2 · · · −λ× x2 ×K2,n −x2 × β2 −x2 × ϕ2

......

. . ....

......

−λ× xn ×Kn,1 −λ× xn ×Kn,2 · · · αn − λ× xn ×Kn,n −xn × βn −xn × ϕn

1 1 1 0 0

(2.9)

where, one can start incorporating the equation of state basedreformulation for refrigerant mixtures.

αj = 1 −[(λ×

(φ̂Lj (xi, xj, T, P)

φ̂Vj (yi, yj, T, P)

)

+ (1 − λ)×(Psatj

P

))](2.10)

∇F(c(s)) =

−λ× x1 ×K1,1 −λ× x1 ×K1,2 · · · −λ× x1 ×K1,k+1 · · · −λ× x1 ×K1,n −x1 × β1 −x1 × ϕ1

......

. . ....

. . ....

......

−λ× xk ×Kk,1 −λ× xk ×Kk,2 · · · −λ× xk ×Kk,k+1 · · · −λ× xk ×Kk,n −xk × βk −xk × ϕk

0 0 · · · αk+1 · · · 0 0 0

......

. . ....

. . ....

......

0 0 · · · 0 . . . αn 0 0

1 1 · · · 1 . . . 1 0 0

(2.18)

βj = λ×(∂ × (φ̂Lj (xi, xj, T, P))/(φ̂

Vj (yi, yj, T, P))

∂T

)x,y,P

+(1 − λ)× dPsat

j /dt

P(2.11)

ϕj =φ̂Lj (xi, xj, T, P)

φ̂Vj (yi, yj, T, P)−Psatj

P(2.12)

Ki,j =(∂Ki

∂xj

)xi[j] ,y,T,P

(2.13)

Without loss of generality, ak-ary branch of ann-component mixture satisfies the following.

xj �= 0∀j ≡ 1, . . . , k (2.14)

αj = 0∀j ≡ 1, . . . , k (2.15)

xj = 0∀j ≡ k + 1, . . . , n (2.16)

αj �= 0∀j ≡ k + 1, ..., n (2.17)

Constraints (2.14) and (2.17) may be simultaneously vio-lated at isolated points along the homotopy branch. If we let

c(s) ∈ F−1(0) that represents a continuation branch where“s” is arc length and we are following ak-ary branch, thenon c(s) the following relation will hold.

αj = 0

for all j = 1, . . . , k.Thus,

Lemma 1. A necessary condition for a trans-critical bifur-cation from a k-ary branch on to a(k + 1)-ary branch at apoint c(s) is

αj(s) = 0 for somej ∈ {k + 1, . . . , n} (2.19)

A necessary condition for trans-critical bifurcation froma (k + 1)-ary branch to a k-ary branch at a point c(s) is

xj(s) = 0 for somej ∈ {1, . . . , k + 1} (2.20)

The condition(2.20) becomes necessary and sufficient forthe bifurcation from a(k + 1)-ary branch on to a k-arybranch by adding the following.

(1) dxj/ds|s=s− �= 0(2) rank ∇F(s) = N − 1 whereN = n+ 1, and(3) ∂F/∂xj|s=s− ∈ R(∇[j]F(s)) where ∇[j] denotes par-

tial derivatives with respect to all variables exceptxj.


Fig. 1. Hypothetical temperature and mole fraction bifurcation diagram for a ternary system (A, B, C).

2.1. Prediction of bifurcation points on pure componentbranches

The system of equations for a binary azeotropic mixturecan be extracted fromEq. (2.8).

F(x, T, λ) =

x1 −K0

1 × x1

x2 −K02 × x2

n=2∑i=1

xi − 1

(2.21)

As λ → 0, the pure components will be the solution ofabove system of equations provided that the pure compo-nents have distinct boiling points. In order to start the con-tinuation, we have all the pure component branches extend-ing from λ = 0 and moving towardsλ = 1.0. Fig. 1 con-tains a hypothetical diagram for a ternary system (A, B andC) illustrating that atλ → 0 all the pure components (A,B and C) are solutions ofEq. (2.8)extending fromλ = 0and moving towardsλ = 1.0. On these pure componentbranches, there will be a point where the binary branch is

going to intersect. This point of intersection between a purecomponent and binary branch needs to be properly iden-tified for ensuring the successful branch tracking. Fortu-nately, these points can be calculated without any branchtracking.

FromEq. (2.21), we get the following equality.

x1 × (1 −K01) = 0

which reduces to

(1 −K01) = 0 (2.22)

Using the value of pseudo equilibrium constantK01 from

Eq. (2.7)in Eq. (2.22), we get

K0i = [λ×KReal

i + (1 − λ)×KIdeali ] = 1.0 (2.23)

and solving (2.23) forλ one obtains:

λ =(

1 −KIdeali

KReali −KIdeal

i

)(2.24)


For a binary mixture of componenti andj, theEq. (2.24)becomes

λij =(

1 − Psatj (T

bi )/P

Kj − Psatj (T

bi )/P

)(2.25)

For low to moderate pressures,Kj is usually expressed asa function of infinite dilution activity coefficient,γ∞

j . Sincewe are exploring elevated pressure systems, infinite dilutionK values computed from EOS will be more appropriate asshown below:

Kj = K∞ij =

φ̂L∞j (xi, xj, T, P)

φ̂V∞j (yi, yj, T, P)

(2.26)

where φ̂L∞j and φ̂V∞

j are the infinite dilution liquid andvapor fugacity coefficients for component j whenxi = 1.0andxj = 0.0.

Eq. (2.25)gives the bifurcation point on a pure compo-nent refrigerant branchi. The bifurcation point is actuallythe point of intersection of pure component branch i anda binary branchij . These bifurcation points act as start-ing points of binary branches. Only the binary brancheswhich are intersecting with pure component branches be-tween the values of 0≤ λi,j ≤ 1 will lead to physicallymeaningful solutions atλ = 1.0. For a binary mixture,a minimum boiling azeotrope bifurcates from lower boil-ing refrigerant and a maximum boiling azeotrope bifurcatesfrom higher boiling refrigerant. The hypothetical diagram ofFig. 1also depicts that a bifurcation point appearing at purecomponent branch C acts as a starting point for a binary

Fig. 2. Variation of bifurcation point with pressure for R23–R13 system (PRSV with Wong Sandler mixing rules is used).

branch which leads to a minimum boiling binary azeotropeat λ = 1.0.

2.2. Criteria 2.1

The following criteria will hold because azeotropy occurswhen the relative volatility (α) passes through unity. Hence,the general criteria for a high-boiling azeotrope to exist in asystem in which gas phase is ideal is given as follows

γ∞1 <

Ps2

Ps1<

1

γ∞2

(2.27a)

while for a low-boiling azeotrope with ideal gas phase isgiven as,

γ∞1 >

Ps2

Ps1>

1

γ∞2

(2.27b)

The above criteria will also hold with non-ideal vaporphase. Thus, for a minimum boiling azeotrope in a binarymixture the following should hold

K∞21 > 1 and K∞

12 > 1 (2.27c)

and for a maximum boiling azeotrope,

K∞21 < 1 and K∞

12 < 1 (2.27d)

where,K∞12 represents the infinite dilutionK value when

component 2 is at infinite dilution whileK∞21represents the

infinite dilution K value when component 1 is at infinitedilution.


Fig. 3. Homotopy branches for R23–R13 system for different pressures.

2.3. Prediction of bifurcation points on binary branches

The branch leading to a ternary azeotrope will alwaysbifurcate from one of the binary branches. The system ofequations for a ternary system is extracted fromEq. (2.8).

F(x, T, λ) =

x1 −K01 × x1

x2 −K02 × x2

x3 −K03 × x3

n=3∑i=1

xi − 1

(2.28)

As pointed out in Lemma (1) andEq. (2.19)that whilemoving along a binary azeotropic branch a necessary con-dition for a bifurcation point to occur,αj(s) = 0 for some

Table 1Pure and binary interaction parameters

Component Tc (K) Pc (bar) ω Kappa

R23 299.06 48.41 0.264 −0.1513R13 301.90 38.77 0.174 −0.0758Propane 369.82 42.49 0.154 0.0313R227ea 375.95 29.80 0.363 0.1191R32 351.26 57.82 0.276 −0.0424R143a 346.04 37.76 0.261 −0.0768R125 339.33 36.29 0.303 −0.0461

System T12 T21 α k12

Wong Sandler parametersR23(1)–R13(2) 0.988 0.988 0.5 −0.109Propane(1)–R227(2) 0.755 0.218 0.3 0.289

System kij

van der Waals parametersR32 (i)–R143a (j) 0.0116R32 (i)–R125 (j)

−0.0466+0.0019× T

R143a (i)–R125 (j) −0.0126

j ∈ {k + 1, . . . , n}. It means that a necessary condition forbifurcation point on binary azeotropic branch corresponds toan intersection of a ternary azeotropic branch with a binarybranch. The bifurcation point on a ternary branch is locatedby constructing an interpolating polynomial, which locatesthe point whereα3 = 0 in Eq. (2.28).

The bifurcation point once located acts as a starting pointof ternary branch. The ternary azeotropic branch is trackedby solving following system of equations

F(x, T, λ) =

x1 −K01 × x1

x2 −K02 × x2

n=3∑i=1

xi − 1

α3(x, T, λ)

x3 − ε

= 0 (2.29)

whereε is a small constant. The ternary branch is followedby solving system ofEq. (2.28). The solution atλ = 1.0corresponds to a ternary azeotrope.

3. Results and discussion

Predicting the exact composition for an azeotropic re-frigerant mixture is very important before it can be testedfor its refrigeration characteristics. There is limited amountof experimental information available for some binary andternary azeotropic refrigerant mixtures[1,2,10,13,14]. Ourwork will provide a robust and efficient method for predict-ing the azeotropic composition in a refrigerant mixture overentire range of pressures. The prediction of azeotropes inbinary or multi-component mixtures will minimize the needfor phase equilibrium experiments and calculations. It willalso help in developing novel multi-component azeotropicrefrigerant mixtures by locating their azeotropic points over


Fig. 4. Variation of R23–R13 azeotropic composition with pressure.

a wide range of pressures. It may also provide guidelines re-garding the operating conditions of a refrigeration plant as atsome pressures azeotropes may disappear or new azeotropesmay appear in a multi-component mixture. The followingthree systems are used to evaluate our approach.

• Trifluoromethane–trifluorochloromethane (R23–R13).• Propane-1,1,1,2,3,3,3–heptafluoropropane (R290–R227ea).

Fig. 5. Variation of bifurcation point with pressure for propane–R227ea system (PRSV with Wong Sandler mixing rules is used).

• Difluoromethane–pentafluoroethane-1, 1,1-trifluoroethane(R32–R125–R143a).

As λ → 0 in Eq. (2.8), the solution becomes (x, T,�)= (xk, T

sk ,0), wherek = 1, . . . , n, and, xk denotes pure

componentk with boiling point Tk at a specified pressure.Thus, we start with the solution ofEq. (2.8)for pure compo-nent branches. For some of these pure component branches,


Fig. 6. Homotopy branches for propane–R227ea system for different pressures.

a bifurcation point (λij) will exist depending on the boilingnature of azeotrope. If a componenti forms an azeotropewith a componentj, a bifurcation point will exist on thepure component branchi. Tolsma and Barton[7,8] pointedout that for a maximum boiling azeotrope componentiwill be the higher boiling specie and for a minimum boil-ing azeotrope, componenti will be a lower boiling specie.These bifurcation points are points of intersection of a pure

Fig. 7. Variation of bifurcation point with pressure for R125–R143a system (PRSV with van der Waals mixing rules is used).

component branchi with a binary branchij . Fig. 1 containsa hypothetical temperature and composition bifurcation di-agram for ternary system (A, B and C), which shows that atλ → 0 all three pure components (A, B and C) are solutionsof Eq. (2.8). A bifurcation point appears on pure componentbranch C and binary branch starting from this bifurcationpoint will lead to a minimum boiling binary azeotrope(CB). On this binary branch another bifurcation point


Fig. 8. Variation of bifurcation point with pressure for R32–R125 system (PRSV with van der Waals mixing rules is used).

appears which acts as a starting point for a ternary branchleading to a ternary azeotrope (CBA) atλ → 1.0 as shownin hypothetical diagram ofFig. 1. As indicated inSection 2,and can be noted fromFig. 1, the branches starting from thebifurcation points between zero and one can lead to solutionat λ = 1 and satisfy the necessary condition for azeotropy

Fig. 9. Variation of bifurcation point with pressure for R32–R143a system (PRSV with van der Waals mixing rules is used).

at λ = 1. These bifurcation points can be predicted throughEq. (2.25)prior to any branch tracking. We are using in-finite dilution K values,K∞

i,j, in Eq. (2.25), as opposed toinfinite dilution activity coefficient,γ∞

ij . The infinite dilu-tion K values will account for both vapor and liquid phasenon-idealities at higher pressure.


Fig. 10. Homotopy branch for R125–R143A system at atmospheric pressure (PRSV with van der Waals mixing rules is used).

3.1. Trifluoromethane–trifluorochloromethane (R23–R13)system

Fig. 2 shows the variation of bifurcation point with pres-sure, based on EOS for R23–R13 system. As can be seen,the bifurcation point based on infinite dilutionK-values fromEOS increases as we increase the pressure. This might bedue to the incorporation of vapor phase non-idealities. Fur-thermore, the infinite dilutionK value provides a better scal-ing in Eq. (2.25). At higher pressures, neglecting the vaporphase non-idealities may not be a realistic. Another impor-

Fig. 11. Homotopy branches for R125–R143a system for different pressures (PRSV with van der Waals mixing rules is used).

tant information, which can be extracted fromFig. 2, is thebifurcation pressure i.e., pressure where azeotrope may ap-pear or disappear. Pressures at which bifurcation point eitherbecomes equal to zero or one indicates the disappearanceof azeotrope while the pressures at which bifurcation pointis in the range of zero to one i.e., 0≤ λi,j ≤ 1 indicatesthe persistence of azeotropes.Fig. 2 shows that R23–R13azeotrope persists from atmospheric pressure to 45.0 atmwhere it finally disappears into critical line of R23.

Once we locate the bifurcation point for R23–R13 sys-tem, we can utilize it for tracking the homogeneous binary


Fig. 12. Homotopy branches for R32–R125 system for different pressures (PRSV with van der Waals mixing rules is used).

branch. As pointed out inSection 2, the binary branchesstarting from 0≤ λi,j ≤ 1 will satisfy the necessary condi-tion of azeotropy at,λij = 1.0. TheFig. 3 shows the binarybranches for R23–R13 system bifurcating from the lowerboiling specie i.e., R23 at various pressures. We used PRSVwith Wong Sandler mixing rules for tracking these branches.The PRSV and Wong Sandler parameters are obtained fromIoannidis and Knox[5] and are reported inTable 1. Fig. 4shows the comparison of calculated azeotropic compositionswith experimental data of Proust and Stein[15,16]. As it canbe noted the calculated azeotropic composition is in closeagreement to experimental data through out the entire pres-sure range.

Fig. 13. Homotopy branches for R32–R143A system for different pressures (PRSV with van der Waals mixing rules is used).

3.2. Propane-1, 1,1,2,3,3,3-heptafluoropropane(R290–R227ea) system

Fig. 5shows the variation of Bifurcation Point with pres-sure for propane–R227ea system. Again the azeotrope per-sists till 35 atm before disappearing into critical locus ofpropane. We also modeled the Propane–R227ea system byour approach.Fig. 6 shows the binary homotopy branchesfor this system at two different pressures. The computed re-sults of binary azeotropie in terms of propane compositionfrom our approach at 22.54 and 27.85 atm are 0.799 and0.7767, respectively, which is closely matching with exper-imental values of 0.7813 and 0.7754 reported by Valtz et al.


Fig. 14. Homotopy branches for R32–R125–R143A system at atmospheric pressure.

[11]. PRSV and Wong Sandler mixing rules parameters forthis system are reported inTable 1.

3.3. Difluoromethane–pentafluoroethane-1,1,1-trifluoroethane (R32–R125–R143a)

Fig. 7 depicts the pressure dependency of bifurcationpoint for R125–R143a system. It can be noted that at pres-sure around 10 atm the bifurcation point attains a value ofunity indicating the disappearance of azeotrope.Fig. 8is forR32–R125 system. The bifurcation point increases with thepressure and is in the range of 0≤ λi,j ≤ 1 till 58 atm. Atthis pressure two phase region ceases to exist as only onereal root is obtained from compressibility equation. There-fore, we conclude that at this pressure azeotrope has dis-appeared into critical line of mixture.Fig. 9 represents thevariation of bifurcation point with pressure for R32–R143asystem. As can be noted fromFig. 9the bifurcation pressurefor this system is 14 atm. With increase in pressure the bi-furcation point starts turning back but at 59.0 bar bifurcationpoint has value of 1.008 and at this pressure two-phase re-qoin for this system disappears. The exact location of thesebifurcation points at given pressure is very important forsuccessful tracking of binary branches which will lead tofinal azeotropic compositions atλij = 1.0. If these pointsare not correctly calculated, the binary branches may leadto different solutions or may not converge at all.

The R32–R125–R143a system has three binary and aternary azeotrope[12]. We used PRSV with van der Waalsmixing rules to model this mixture.Fig. 10 shows homo-topy branches for R125–R143a binary system at atmosphericpressure. The maximum boiling binary azeotrope bifurcatesfrom maximum boiling component i.e., R143a. The com-puted azeotrope composition is R125–R143a 0.3675/0.6325.

Fig. 11represents the bifurcation diagram for R125–R143asystem at different pressures. As indicated above around10 atm this binary azeotrope disappears as bifurcation pointattains the value of unity.

Fig. 12depicts the homotopy branches for R32–R125 sys-tem. As it can be seen that minimum boiling binary azeotropebifurcates from minimum boiling component that is R32 inthis case.

Fig. 13shows the homotopy branches for R32–R143a bi-nary system at two different pressures. The computed com-position of R32–R143a azeotrope at atmospheric pressure is0.768/0.232, which is in close agreement to experimentallyreported value of 0.782/0.218 by Barley et al.[13].

3.4. Ternary azeotrope

As mentioned inSection 2, the ternary azeotropes areemerging from binary branches. All three binary branchesin the ternary mixture are tested for the presence of bi-furcation point. Only the binary branch of R32–R143a hasa bifurcation point atλ = 0.778 which corresponds toa ternary branch from which a ternary azeotrope is ob-tained.Fig. 13represents the ternary branch bifurcating froma binary branch. The ternary azeotrope has compositionR32–R125–R143a as 0.7777/0.1706/0.0517 with a boilingtemperature of 219.945 K. The PRSV and van der Waals pa-rameters for this ternary system are reported inTable 1andFig. 14.

4. Conclusion

This paper presents a methodology for reliably computingall the azeotropes for a multi-component refrigerant mix-


ture. The approach extends Tolsma and Barton’s[7,9] workto refrigerant systems and reformulates it with equations ofstate. Both the liquid and vapor phase non-idealities are in-corporated through liquid and vapor phase fugacity coeffi-cients based on equation of state (PRSV with Wong Sandlerand van der Waals mixing rules). At higher pressures, thephase equilibrium model is deformed from an ideal equilib-rium (based on Raoults law ) to highly non-ideal equilibrium(based on equation of state) with homotopy parameter vary-ing from 0 to 1. The approach is also capable of predictingthe bifurcation pressure. The reliable and efficient methodof predicting binary and ternary refrigerant azeotropes overentire pressure range will reduce the requirements of exper-imental data for refrigerant mixtures. Azeotropy in refriger-ant mixture is a desirable condition as azeotropes behavesas pure components. The technique can be used in reliablycomputing the multi-component azeotropes, which may befurther investigated to develop novel refrigerants as drop-insubstitutes for CFCs and HCFCs.

List of symbols[F] matrix for the necessary condition of azeotropy[∇F] Jacobian matrix ofFKIdeali equilibrium constant for ideal solution

KReali equilibrium constant for non-ideal solution

K0i pseudo equilibrium constant

K∞ij infinite dilution equilibrium constant

Kij derivative of equilibrium constant with respectto componentj

P total pressure of the systemPsati vapor pressure of componenti

s arc lengthT temperaturexi mole fraction of the componenti in the

liquid phaseyIdeali mole fraction of the componenti in the

vapor phase for an ideal gas and ideal solution

yReali mole fraction of the componenti in the vapor

phase for a real gas and non-ideal solutionGreek lettersφ̂Lj fugacity coefficient of componentj in liquid phase

φ̂Vj fugacity coefficient of componentj in vapor phase

φ̂L∞j fugacity coefficient of componentj in liquid phase

at infinite dilutionφV∞j fugacity coefficient of componentj in vapor phase

at infinite dilutionλ homotopy parameterλij bifurcation point

References

[1] D.A. Didion, D.B. Bivens, Int. J. Refrig. 13 (1990) 163–175.[2] D.A. Didion, W.J. Mulroy, G. Morrison, M.S. Kim, A Study to

Determine the Existence of an Azeotropic R-22 Drop-In Substitute,Building and Fire Research Laboratory, 1996.

[3] R. Doring, H. Buchwald, J. Hellmann, Int. J. Refrig. 20 (1997) 78–84.

[4] H. Orbey, S.I. Sandler, Ind. Eng. Chem. Res. 34 (1995) 2520–2525.

[5] S. Ioannidis, D.E. Knox, Fluid Phase Equilib. 140 (1997) 17–35.[6] Z.T. Fidkowski, M.F. Malone, M.F. Doherty, Compt. Chem. Eng. 17

(1993) 1141–1155.[7] J.E. Tolsma, P.I. Barton, Chem. Eng. Sci. 55 (2000) 3817–3834.[8] J.E. Tolsma, P.I. Barton, Chem. Eng. Sci. 55 (2000) 3835–3853.[9] J.E. Tolsma, Simulation and analysis of heteroazotropic systems,

Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge,MA, 1999.

[10] M. Nagel, K. Bier, Int. J. Refrig. 18 (1995) 534–543.[11] A. Valtz, C. Coquelet, A. Baba-Ahmed, D. Richon, Fluid Phase

Equilib. 202 (2002) 29–47.[12] M.H. Barley, J.D. Morrison, O. Donnel, I.B. Parker, S. Petherbridge,

R.W. Wheelhouse, Fluid Phase Equilib. 140 (1997) 183–206.[13] I. Kul, D.D. DesMarteau, A.L. Beyerlein, Fluid Phase Equilib. 185

(2001) 241–253.[14] E. Granryd, Int. J. Refrig. 24 (2001) 15–24.[15] P.C. Proust, F.P. Stein, Fluid Phase Equilib. 3 (1979) 313–322.

computing all the azeotropes in refrigerant mixtures through equations of state

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