Prediction of Phase Equilibria for Refrigerant Mixtures Of

7632 J. Phys. Chem. B 1998, 102, 7632-7639

Prediction of Phase Equilibria for Refrigerant Mixtures of Difluoromethane (HFC-32), 1,1,1,2-Tetrafluoroethane (HFC-134a), and Pentafluoroethane (HFC-125a) Using SAFT-VR

Amparo Galindo,† Alejandro Gil-Villegas,‡ Paul J. Whitehead, and George Jackson*,† Department of Chemistry, UniVersity of Sheffield, Sheffield S3 7HF, U.K.

Andrew N. Burgess Research and Technology, ICI Chemicals and Polymers, PO Box 8, The Heath, Runcorn, Cheshire WA7 4QD, U.K. ReceiVed: January 29, 1998; In Final Form: July 1, 1998

The statistical associating fluid theory for chain molecules with attractive potentials of variable range (SAFT- VR) is used to model the phase equilibria for three binary mixtures formed by difluoromethane (HFC-32), 1,1,1,2-tetrafluoroethane (HFC-134a), and pentafluoroethane (HFC-125a). Molecules are represented as chains of spherical segments with short-ranged attractive sites. The intermolecular van der Waals forces are modeled with variable-range square-wells. The optimized values of the parameters of the model are obtained by fitting to experimental data for the vapor pressures and saturated liquid densities of each of the pure components. These parameters are the number and diameters of the spherical segments and the strengths and ranges of the potentials describing the site-site and segment-segment interactions. Using the values of the pure-component parameters and standard combining rules, the phase equilibrium of the mixtures is described very accurately. SAFT-VR improves the predictive power of mean-field versions of SAFT.

Introduction The HFCs do not contain chlorine atoms, have a zero ozone depletion potential, and have lifetimes of the order of 10 times Chlorofluorocarbons (CFCs) are stable, nontoxic compounds, shorter than those of the CFCs. The thermodynamical properties 1 first used as refrigerants in 1930, when they replaced the toxic of pure HFCs have been extensively measured over the past 10 fluids that were used as coolants in refrigeration systems years and are well-known; it now seems likely that blends of (CH3Cl, SO2,NH3,CH2CH2, and hydrocarbons). They have a new refrigerants rather than pure substances will replace the very good refrigerating performance, and their nontoxic nature common CFCs; e.g., mixtures of HFC-32 (difluoromethane), encouraged widespread industrial applications, being used as HFC134a (1,1,1,2-tetrafluoroethane), HFC-125 (pentafluoroet- foam-blowing agents for polystyrene and polyurethane, solvents, hane), and HFC-152a (1,1-difluoroethane) are used in air and cleaning agents over the following 50 years. Their stability, conditioning systems. The number of mixtures of HFCs however, causes them to reach the stratosphere where they potentially viable as replacements for the old CFCs is, however, eventually decompose under solar ultraviolet radiation releasing very extensive. Equilibrium experimental data is available for chlorine radicals that react with ozone molecules (O3) to yield a small number of mixtures and only over limited ranges of 2 molecular oxygen O2. Molina and Rowland were the first to temperature and pressure. Further, the synthesis of the replace- suggest a connection between the CFCs in the stratosphere and ment refrigerants requires separation processes involving mul- the depletion of the ozone layer, a hypothesis that was confirmed ticomponent mixtures of these compounds and various other in the late 70s and early 80s. A detailed account on the role of components. It becomes then of crucial importance to develop chlorine in stratospheric chemistry can be found in a recent paper reliable theoretical models to predict the phase equilibria and by Molina.3 other thermodynamical properties of such mixtures. The Montreal Protocol of 1987 provided the first international Several equations of state (EOSs) have already been used in agreement limiting the production of CFCs worldwide. It the prediction and correlation of experimental data of refrigerant initially called only for a 50% reduction in the manufacture of systems. In a recent paper Gow4 reviews the achievements of CFCs by the year 2000. The London and Copenhagen amend- two- and three-parameter cubic EOSs in the description of such ments of 1990 and 1992 strengthened the ban to ask for a mixtures. He uses a three-parameter cubic equation with a phaseout of production by the end of 1995. CFCs are now being temperature-dependent attractive term to correlate binary vapor- replaced by hydrochlorofluorocarbons (HCFCs) and hydrofluo- liquid equilibrium data and extends it to predict the phase rocarbons (HFCs); the presence of the hydrogen atoms allows equilibria of a ternary refrigerant mixture. Good agreement with water-soluble compounds to be formed, which are removed from experimental data is found, although only one isotherm of the the atmosphere by rainfall. ternary mixture is studied. Obey and Sandler5 have examined several refrigerant mixtures with a cubic EOS and a number of † Current address: Department of Chemical Engineering and Chemical mixing rules taking special interest in the predicting capabilities Technology, Imperial College of Science Technology and Medicine, of their approach for extended ranges of temperature. University of London, Prince Consort Road, London SW7 2BY. ‡ Current address: Instituto de Fı´sica, Universidad de Guanajuato, Leo´n Cubic EOSs coupled with group contribution methods have 37150, Mexico. also been extensively used to study the phase equilibria of binary S1089-5647(98)00943-2 CCC: $15.00 © 1998 American Chemical Society Published on Web 09/09/1998 Phase Equilibria for Refrigerant Mixtures J. Phys. Chem. B, Vol. 102, No. 39, 1998 7633 mixtures. The fundamental assumption of the group contribution methods is that the properties of a molecular group are the same in every molecule and have no influence on other segments. On the basis of this idea the thermodynamic properties of new substances are predicted with a database of the interaction parameters of the “groups” involved. The interaction parameters for the different “groups” are obtained by comparisons with experimental data. It should be noted, however, that the assumption of a new molecular group with no influence on other segments of a molecule is rarely true. Barolo et al.6 used the Soave-Redlich-Kwong EOS with a group contribution method derived from UNIFAC to study mixtures of refrigerants and proposed a list of groups and subgroups for CFCs, HCFCs, HFCs, and FCs. Similarly, Kleiber7 has extended the UNIFAC group assignment with the UNIQUAC equation to deal with replacement refrigerants; 10 Figure 1. Models for (a) HFC-32, (b) HFC-134a and HFC-R125. A new groups were described. number of off-center square-well sites are placed on a sphere of diameter 8 σi. The sites are placed at a distance rd from the centre of the sphere Blindenbach et al. used the perturbed anisotropic chain theory and have a cut-off range r . The two different types of sites are 9,10 c (PACT) developed by Vimalchand and Donohue, in order colored white and grey; only white-grey bonding is allowed. The white to model the thermodynamic properties of pure CFCs and and grey sites interact with a hydrogen-bonding energy hb when the - - HCFCs, and the vapor liquid equilibria of their mixtures, site site distance is less than rc. HFC-32 is modeled with a single including mixtures with hydrocarbons. PACT is an EOS that hard sphere (m ) 1 ) and HFC-134a and HFC-125 are modeled with ) ) takes into account dispersion, polar, and induced polar interac- two overlapping hard spheres (m 1.4 and m 1.35, respectively). Long-range dispersion interactions are modelled via a high-temperature tions. On the basis of pure component parameters, very good perturbation expansion for a site-site square-well potential of depth agreement with experimental data was found and better predic- and range λ. tive results than the ones obtained with cubic equations of state. 11 Economou et al. have measured the properties of the mixture that is in very good agreement with experimental data for + HFC-22 (chlorodifluoro ethane) HFC-134a up to the critical strongly associated systems such as the mixtures containing region and have compared their data with PACT, obtaining very hydrogen fluoride. SAFT-VR introduces a more accurate good agreement. The statistical associating fluid theory (SAFT) description of the thermodynamics due to attractive van der also takes into account the anisotropies of the fluid, both due Waals forces by using a second-order perturbation theory. With to molecular shape and to directional forces. In the SAFT this approach is possible to describe mixtures of refrigerants in approach the free energy can be written as separate contribu- a more accurate way than the simplified SAFT-HS approach, tions, describing the effects of molecular shape, dispersion as we will show in this paper. forces, and molecular association. In the original SAFT approach,12,13 the molecules are modeled as chains of Lennard- Jones segments, with embedded short-range sites to describe 2. Models and Theory association. In its many versions, the SAFT approach has been used to examine a wide range of fluids, from linear small alkanes It is useful to illustrate the models used in the description of - to polymeric fluids (see refs 14 16 and references therein). An these molecules with the SAFT-VR approach and then give a extended version of the SAFT equation, which accounts for short reminder of the specific equations involved in each of explicit contributions to the free energy due to dipolar interac- the mixtures. The comparison with experimental data is 17 tions (based on previous work of Mu¨ller and Gubbins and presented in the following section. 18 the well-known Pade´ approximation due to Stell et al. ), was 2.1. Models. We model our molecules as hard-spherical 19,20 recently used by Kraska and Gubbins to study the phase cores formed from tangentially bonded segments of diameter σ equilibria of mixtures of alcohols. This extended version of with a number of embedded off-center square-well bonding sites. the SAFT approach gives very good agreement with experi- The sites are placed at a distance rd from the centre of the sphere mental data. and have a cut-off range rc. These two parameters define the We have recently described a new version of the SAFT available volume for bonding.24 When two sites come closer approach for attractive potentials of variable range (SAFT- to each other than the cut-off distance rc there is an attractive VR).21,22 In this approach the fluids are modeled as chains of interaction hb. These features are equivalent to those of a tangentially bonded hard spheres interacting with attractive SAFT-HS model molecule. However, the thermodynamics of potentials of variable range. These variable-range potentials the long-range attractive forces are described here by a second- model van der Waals long-range forces including dispersion as order high-temperature expansion of the Helmholtz free energy well as permanent and induced polar forces (in the last two cases for a square-well potential. The attractive square well is at the level of the Keesom and Debye angle-averaged ap- characterized by a depth- and a range λ. In HFC-32 four sites proximations, respectively). Short-ranged attractive sites medi- are included in a central hard sphere (see Figure 1a). Previously, ate molecular association, and they also model, in a geometric HFC-32 was modeled by including only two attractive sites on sense, the short-ranged polar anisotropic effects. In earlier the central core. It was observed that SAFT-VR gives a much work23 we used a simplified SAFT approach (SAFT-HS) to better description of the properties of HFC-32 when four sites study mixtures of hydrogen fluoride and replacement refriger- are used. Two of the sites are of type a (say, white), two are ants. At this level of approximation, the long-range attractive of type b (black), and only a-b bonding is allowed. This gives forces are treated at the mean-field (van der Waals) level. This the possibility of treelike structures being formed in the fluid very simple model provides a description of the phase equilibria rather than just chains of associated aggregates. HFC-134a and 7634 J. Phys. Chem. B, Vol. 102, No. 39, 1998 Galindo et al.

HFC-125 (see Figure 1b) are modeled with two overlapping factor Z as hard spheres to describe the nonspherical shape of these pV molecules. Two different attractive sites of type a and b account Z ) for the anisotropy of the dipolar interactions. The exact position NkT of the sites is not important at this level of approximation, but n they describe the fact that chains of associated monomers may µi A ) - be formed, but not treelike aggregates. Two sites allow the ∑xi (4) kT NkT molecules to form ring structures as well as chainlike aggregates, i but ring aggregates are not accounted for at this level of where n is the total number of components in the mixture and approximation. In HFC-134a m ) 1.4 spheres form the x ) N /N is the mole fraction of component i. backbone of the model, and in HFC-125 m ) 1.35; these i i 2.2.1. The Ideal Mixture. The free energy of the ideal noninteger values of m indicate a nonsphericity that is inter- mixture is given by25 mediate between the monomer and the dimer. This parameter is, to some extent, fitted to give the best agreement with AIDEAL n experimental data, although it is important to retain a physically ) F 3 - ∑xi ln iΛi 1 reasonable molecular shape. NkT i Before the properties of the mixtures are examined, the calculated vapor pressures and saturated liquid densities of the ) F 3 + F 3 - x1 ln 1Λ1 x2 ln 2Λ2 1 (5) pure components are fitted to experimental data, from the triple point to the critical point. In this way we obtain values for the where Fi ) Ni/V is the number density and Λi is the thermal de size of the hard spheres σ, the strength of the square-well Broglie wavelength of species i. attractive interaction , its range λ, and for the strength of the 2.2.2. Monomer Contribution. The monomer free energy - hb site site hydrogen bonding interaction and its cut-off range is rc. The sites are always placed at a distance rd/σ ) 0.25 from the center of the sphere so that once rc is determined the bonding AMONO. AM volume K is known.24 ) (∑ximi) 2.2. SAFT-VR Equation of State. We briefly recall the NkT i NskT specific expressions of SAFT-VR for the contributions to the ) M free energy. The equation of state for a mixture of associating (∑ximi) a (6) chain molecules is written in terms of four separate contributions i to the Helmholtz free energy, A21,22 where mi is the number of spherical segments in each chain i A AIDEAL AMONO. ACHAIN AASSOC. and Ns is the total number of spherical segments. The monomer ) + + + (1) free energy per segment of the mixture aM ) AM/(N kT)is NkT NkT NkT NkT NkT s obtained from the Barker and Henderson high-temperature expansion26-28 where N is the number of chain molecules in the mixture, k is the Boltzmann constant, and T is the temperature. In this M ) HS + + 2 + equation AIDEAL is the ideal free energy, AMONO. is the residual a a âa1 â a2 ... (7) free energy due to the monomer segments, ACHAIN is the HS ) contribution due to the formation of chains of monomers, and where a is the free energy for a mixture of hard spheres, â AASSOC. is the term that describes the contribution to the free 1/kT; a1 and a2 are the first two perturbation terms associated - energy due to intermolecular association. In this paper we will with the attractive energy ij. The three mixtures studied here + be concerned only with mixtures that interact with the square- are HFC-32 (component 1) HFC-125 (component 2), + + well potential HFC-32 (1) HFC-134a (2), and HFC-125 (1) HFC-134a (2). The numbers of segments m take values of 1, 1.35, and 1.40 for HFC-32, HFC-125, and HFC-134a respectively. +∞ if r < σ ij The free energy of the reference hard-sphere mixture is ) - if σ e r < λ σ uij(r) { ij ij ij ij (2) obtained from the expression of Boublı´k29 and Mansoori et al.30 g 0 if r λijσij 3 3 6 ú2 3ú1ú2 ú2 where r is the distance between the two segments, defines aHS ) - ln (1 - ) + + σij F [( 2 ú0) ú3 - 2] π s 1 ú3 - the contact distance between spheres, and λij and ij are the range ú3 ú3(1 ú3) - and depth of the potential well for the i j interaction respec- (8) tively. For two or more phases to be in equilibrium with one another the pressures, temperatures, and chemical potentials of In this expression Fs ) Ns/V is the number density of the mixture each component must be equal in the coexisting phases. The in terms of the number of spherical segments. Note that Fs ) chemical potential, , of species i can be written in terms of µi F(∑iximi), where F is the total number density of the mixture. the free energy The reduced densities úl are defined as

n µi ∂A/kT π ) ( ) (3) ) F l úl s ∑xs,i(σi) (9) kT ∂Ni T,V,N j*i 6 [i)1 ] where Ni is the number of chain molecules of species i. The where σi is the diameter of spherical segments of chain i and overall pressure p may be calculated through the compressibility xs,i is the mole fraction of segments of type i in the mixture, Phase Equilibria for Refrigerant Mixtures J. Phys. Chem. B, Vol. 102, No. 39, 1998 7635 given by in a binary mixture is written as21,22

m x ACHAIN n x ) i i (10) )- - SW s,i n ∑xi(mi 1) ln yii (σii) ∑j)1mjxj NkT i)1 )- - SW - - SW The overall packing fraction of the mixture is thus given by ú3, x1(m1 1) ln y11 (σ11) x2(m2 1) ln y22 (σ22) which is equivalent to η in the pure component case. (20) The mean-attractive term a1 is given by SW ) SW - SW n n where yij (σij) gij (σij) exp( âij). We obtain yij (σij) from ij SW ) the high-temperature expansion of gij (σij) a1 ∑∑xs,ixs,ja1 (11) ) ) i 1 j 1 SW ) HS + gij (σij) gij (σij) âijg1(σij) (21) where HS The hard-sphere term gij is obtained from the expression of ij )-FRVDW HS eff Boublı´k29 a1 s ij gij [σij; ú3 ] (12) D (D )2 RVDW - HS 1 ijú3 ijú3 Here ij is the van der Waals attractive constant for the i j g (σ ; ú ) ) + 3 + 2 (22) ij ij 3 1 - ú - 2 - 3 interaction 3 (1 ú3) (1 ú3) RVDW ) 3 3 - with the parameter D defined as ij 2πijσij(λij 1)/3 (13) ij n 2 HS σiiσjj ∑i)1xs,iσii gij is the radial distribution function for a mixture of hard D ) (23) eff ij σ + σ n 3 spheres, and ú3 is an effective packing fraction given by ii jj ∑i)1xs,iσii eff ) + 2 + 3 The term g ( ) is obtained from a self-consistent method for ú3 (ú3, λij) c1(λij)ú3 c2(λij)ú3 c3(λij)ú3 (14) 1 σii the pressure p from the Clausius virial theorem and from the 21 21,22 The coefficients c1, c2, and c3 are given by density derivative of the Helmholtz free energy g (σ ; ú ) ) g HS[σ ; úeff] + c1 2.25855 -1.50349 0.249434 1 1 ij 3 φ ij 3 HS eff eff eff c ) -0.669270 1.40049 -0.827739 λij (15) ∂g [σ ; ú ] λ ∂ú ú ( 2 ) ( )( ) 3 φ ij 3 ii 3 3 2 (λ - 1) ( - ú ) (24) c - ii 3 3 10.1576 15.0427 5.30827 λij eff 3 ∂λ ∂ú ∂ú3 ij 3 This corresponds to the MX3b mixing rule of ref 22. 2.2.4. Association Contribution. The contribution to the free The fluctuation term of the free energy is given by energy due to association of si sites on chain molecules of species i is obtained from the theory of Wertheim32 as n n ij a ) ∑∑x x a (6) ASSOC. n si 2 s,i s,j 2 A Xa,i si i j ) ∑x ∑(ln X - ) + (25) i[ a,i ] NkT i)1 a)1 2 2 ij where each of the terms a2 are obtained with the local compressibility approximation (LCA)26,27 where the first sum is over the species i and the second over all si sites a on a molecule of species i. Xa,i is the fraction of ij molecules of type i not bonded at site a, and is given by the ∂a1 ij ) 1 HS F 13,32 a2 K ij s F (17) mass action equation as 2 ∂ s 1 HS X ) (26) and K is the isothermal compressibility for a mixture of hard a,i n sj 1 + ∑ ) ∑ ) Fx X ∆ spheres, given by the Percus-Yevick expression31 j 1 b 1 j b,j a,b,i,j where, in the case of square-well segments, - 4 HS ú0(1 ú3) K ) (18) ) SW - 2 + + 3 ∆a,b,i,j Ka,b,i,jfa,b,i,jgij (σij) (27) ú0(1 ú3) 6ú1ú2(1 - ú3) 9ú2 13,32 Ka,b,i,j is the volume available for bonding, fa,b,i,j is the Substituting for aij in eq 17 gives 1 Mayer f -function of the a-b site-site interaction φa,b,i,j, fa,b,i,j SW ) exp(-φa,b,i,j/kT) - 1, and gij (σij) is obtained with eqs 21, 1 a )- F KHS∑∑x x RVDW × 22, and 24. 2 s s,i s,j ij ij + 2 i j In the mixtures HFC-32 (1) HFC-125 (2), and HFC-32 HS eff (1) + HFC-134a (2), eq 25 is ∂gij [σij; ú3 ] HS eff + (gij [σij; ú3 ] ú3 ) (19) ASSOC. A ∂ú ) 3 NkT X X 2.2.3. Chain Contribution. The contribution to the free x [4(ln X - 1) + 2] + x [2(ln X - 2) +1] (28) energy due to the formation of chains of square-well segments 1 1 2 2 2 2 7636 J. Phys. Chem. B, Vol. 102, No. 39, 1998 Galindo et al. since HFC-32 has four attractive sites and HFC-125 has two. In a mixture all the parameters are reduced relative to one of / ) All the sites in a molecule are assumed to be equivalent in what the components, component 1 in our case, so that σij σij/σ11, concerns the fraction of molecules not bonded at a given site; R/ )R R ij ij/ 11, etc. The range and position of the bonding sites X1 is the fraction of HFC-32 molecules not bonded at a given are reduced with respect to the diameter of the particular sphere / / site, and X2 the fraction of HFC-125 or HFC-134a molecules ) ) in which they are included, i.e., rc rc1/σ11 and rd rd1/σ11 not bonded at a given site. For the HFC-125 (1) + HFC-134a / / 1 1 while r ) rc /σ22 and r ) rd /σ22, and the reduced bonding c2 2 d2 2 (2) mixture, the contribution to the free energy due to association volume for each of the pure components is calculated as32 is given by K ASSOC. X X / ii / / /3 /2 / /3 A 1 2 K ) ) 4π[ln((r + 2r ))(6r + 18r r - 24r ) + ) x [2(ln X - ) + 1] + x [2(ln X - ) +1] ii 3 ci di ci ci di di 1 1 2 2 NkT 2 2 σii / / / / / / / (29) (r + 2r - 1)(22r 2 - 5r r - 7r - 8r 2 + r + 1)]/ ci di di ci di di ci ci / (72r 2) (35) since both components of the mixtures have two bonding sites. di The fractions of molecules not bonded at a given site Xi can be obtained by a self-consistent iterative solution of the mass action and can then be written in terms of one of the two components. 32 + equations. In the mixtures of HFC-32 (1) R125 (2), and The reduced energy for the site-site interaction is defined in HFC-32 (1) + HFC-134a (2) hb/ ) terms of the mean-field interaction of component 1, ii hb 1 ii /11. The reduced dimensionless temperatures and pres- X ) (30) / VDW / 2 VDW 1 + F +F sures are defined as T ) kTbii/R and p ) pb /R , 1 2 x1X1∆11 x2X2∆12 ii ii ii ii where the subscript c is sometimes used to denote the critical and point.

) 1 X2 + F +F (31) 3. Prediction of Experimental Data 1 2 x1X1∆21 x2X2∆22

Similarly, the fraction of molecules not bonded at a given site The parameters of the potential models presented in the in the HFC-125 (1) + HFC-134a (2) mixture are given by previous section are determined for the pure components before studying the mixtures. The calculated vapour pressures and ) 1 saturated liquid densities of each of the pure components are X1 +F +F (32) 35 1 x1X1∆11 x2X2∆12 fitted to experimental data from the triple point to the critical point using a simplex method.33 The optimized values for the and number of spherical segments in the molecule m (note that in the case of HFC-32 it is fixed to m ) 1 during the minimization), ) 1 the range of the attractive square-well λ, the diameters of the X2 +F +F (33) 1 x1X1∆21 x2X2∆22 spherical segments σ/Å, the strength of the square-well interaction /K, the strength of the association site-site energy hb/K, 3 The function ∆ij characterizes the association between a and the bonding volume K/Å (with the corresponding rc) are molecule of species i and a molecule of species j since all sites presented in Table 1. in a molecule are equivalent. ∆ij can be written in terms of the In Figure 2 the vapour-pressure curve (Figure 2a) and SW contact value g (σij) of the radial distribution function of the coexistence densities (Figure 2b) obtained for HFC-32 with the reference unbonded mixture (a mixture of square-well seg- SAFT-VR approach are compared with experimental data.35 We ) - hb - - ments), the Mayer f function Fij exp( ij /kT) 1ofthe show the vapor-pressure curve as a Clausius Clapeyron site-site bonding interaction, and the volume Kij available for representation since, in this way, the low-temperature region is bonding better observed. The experimental properties of HFC-32 are reproduced very accurately up to the critical point. The ) SW ∆ij KijFijg (σij) (34) calculated vapor pressures and coexisting densities for HFC- 125 and HFC-134a are compared with experimental data35 in As in all the SAFT approaches based on the first order Figures 3 and 4, respectively. The agreement with experimental thermodynamic perturbation theory of Wertheim, the extent of data is again found to be excellent. association depends on the strength of the interaction via F and We have examined the behavior of three binary mixtures on the position and range of the site-site interaction via K, but formed by a combination of HFC-32, HFC-125, and HFC134a. the precise details of the geometry of the sites are not important The theoretical study of mixtures from pure component proper- at this level of approximation so long as they correspond to the ties requires the determination of a number of unlike interaction same integrated bonding volume. parameters, the unlike parameters σ12, 12, and λ12 and the unlike The other thermodynamical properties of the mixture are hb association parameter 12, and its range. The unlike size and obtained from the Helmholtz free energy using the standard energy parameters are obtained using the Lorentz-Berthelot relationships; the conditions for equilibria in the mixture and combining rules25 the conditions for the critical points are then obtained using a simplex method.33 In refs 25 and 34 these procedures are + σ11 σ22 discussed in detail. We define now a number of reduced σ ) (36) 12 2 parameters that are useful in the calculations. The volume of ) 3 a spherical segment is denoted by bii πσii/6, and the )RVDW ) x (37) integrated energy of the square-well interaction ii ii /bii. 12 11 22 Phase Equilibria for Refrigerant Mixtures J. Phys. Chem. B, Vol. 102, No. 39, 1998 7637

TABLE 1: Optimized Parameters with the SAFT-VR Approach for Difluoromethane (HFC-32), Pentafluoroethane (HFC-125), and 1,1,1,2-Tetrafluoroethane (HFC-134a)a

hb 3 / / / 2 m λσ/Å (/k)/K ( /k)/K K/Å rc Tc pc10 HFC-32 1 1.239 4.116 259.3 619.3 11.55 0.808 0.381710 1.88009 HFC-125 1.35 1.360 4.535 303.8 724.8 2.273 0.669 0.195113 1.02149 HFC-134a 1.4 1.850 3.947 122.7 1301 3.198 0.715 0.151546 0.50311 a m is the number of spherical segments in the model, λ the range of the square-well interaction, σ the diameter of each hard-sphere segment, the attractive square-well energy per spherical segment, hb the energy of the site-site hydrogen bond, K the bonding volume for the site-site / / / interaction, and rc the cut-off range corresponding to each K. Tc and pc are the theoretical critical points in reduced units.

Figure 2. (a) Vapor pressures as a Clausius-Clapeyron representation Figure 3. (a) Vapor pressures as a Clausius-Clapeyron representation and (b) vapor-liquid coexistence densities for HFC-32 compared with and (b) vapor-liquid coexistence densities for HFC-125 compared with the SAFT-VR predictions. The circles represent the experimental data.35 the SAFT-VR predictions. The circles represent the experimental data.35 The results obtained with the parameters (see Table 1) fitted to both The results obtained with the parameters (see Table 1) fitted to both experimental vapor pressures and saturated liquid densities are shown experimental vapor pressures and saturated liquid densities are shown as continuous curves. as continuous curves. while adjusted to give the best representation of the mixture at a constant temperature of 40 °C (313 K). We decided to optimize + λ11σ11 λ22σ22 the unlike parameters at this temperature since the azeotrope is λ ) (38) quite well defined here. It also gives us the opportunity to see 12 σ + σ 11 22 the reliability of our parameters when the temperatures are much and lower; this is the region of main interest for the manufacture of replacement refrigerants. The optimized parameters for this ) hb ) hb ) hb hb mixture are 12/k 282.1 K and 12/k 769.0 K (with 12 x11 22 (39) R/ ) R/ ) corresponding reduced parameters 12 1.670, 22 2.627, hb/ ) hb ) 3 1/3 + 3 1/3 3 and 12 0.820). The unlike bonding volume K12 and / K12 (K1/σ11) (K2/σ11) K ) ) [ ] (40) range of the attractive square-well interaction λ12 are calculated 12 3 2 σ11 from expressions 40 and 38, respectively, and are not read- 3 justed: K12/Å ) 5.715 and λ12 ) 1.302. The agreement with The mixture HFC-32 (1) + HFC-125 (2) exhibits positive experimental data is found to be very good over the temperature azeotropy. In a pressure-composition px slice at constant range examined, although is may be noted that the composition temperature (see Figure 5), a maximum in pressure is observed of the azeotrope predicted with the SAFT-VR approach moves (note that the maximum is not very dramatic in this mixture). to the R125-rich phases as the temperature increases. The real - hb We have readjusted the unlike site site interaction energy 12 system does not appear to exhibit this behavior. However, it and the square-well energy parameter 12 to reproduce the is important to to recall the fact that our unlike parameters are presence of the azeotrope in the mixture. In Figure 5 the not temperature/dependent; once they are adjusted at a particular calculated bubble point pressures at constant temperatures from state point it is possible to predict with good accuracy the -30 °C (243 K) to 50 °C (323 K) are compared with behavior of a system over a wide range of temperatures and experimental data.36 The unlike parameters in the mixture were pressures without further adjustment. 7638 J. Phys. Chem. B, Vol. 102, No. 39, 1998 Galindo et al.

Figure 6. Bubble pressures at constant temperatures for the HFC-32 (1) + HFC-134a (2) mixture compared with the theoretical predictions. The experimental data are shown as circles.36 The continuous curves represent the calculated bubble pressures. The pressures correspond to constant temperatures from -30 °C (343 K), for every 10°,upto50 °C (323 K) with the exception of the bubble pressures at T ) 30 °C. ) hb ) The unlike intermolecular parameters are 12/k 178.3 K, 12/k 3 ) ) - 897.5 K, K12/Å 6.509, and λ12 1.538. The pure component Figure 4. (a) Vapor pressures as a Clausius Clapeyron representation parameters are given in Table 1. and (b) vapor-liquid coexistence densities for HFC-134a compared with the SAFT-VR predictions. The circles represent the experimental data.35 The results obtained with the parameters (see Table 1) fitted to both experimental vapor pressures and saturated liquid densities are shown as continuous curves.

Figure 7. Bubble pressures at constant temperatures for the HFC-125 (1) + HFC-134a (2) mixture compared with the theoretical predictions. The experimental data are shown as circles.36 The continuous curves represent the calculated bubble pressures. The pressures correspond to constant temperatures from -40 °C (233 K), for every 10°,upto60 Figure 5. Bubble pressures at constant temperatures for the HFC-32 °C (333 K) with the exception of the bubble pressures at T ) 30 °C. + (1) HFC-125 (2) mixture compared with the theoretical predictions. ) hb ) The unlike intermolecular parameters are 12/k 193.1 K, 12/k The pressures correspond to constant temperatures from -30 °C (243 3 970.5 K, K12/Å ) 2.709, and λ12 ) 1.588. The pure component K), for every 10°,upto50°C (323 K) with the exception of the bubble parameters are given in Table 1. pressures at T ) 30 °C. The experimental data are shown as circles.36 The continuous curves represent the calculated bubble pressures. The that the experimental data is accurately predicted by the SAFT- ) hb ) unlike intermolecular parameters are 12/k 282.1 K, 12/k 769.0 VR approach without readjusting any of the unlike intermo- 3 K, K12/Å ) 5.715, and λ12 ) 1.302. The pure component parameters lecular parameters; i.e., using the arithmetic and the geometric are given in Table 1. mean. We have studied the HFC-32 (1) + HFC-134a (2) In Figures 6 and 7 the predicted bubble pressures obtained mixture over a range of temperatures from -30 °C (243 K) to with the SAFT-VR approach are compared with experimental 50 °C (323 K) (see Figure 6). The unlike energy-related ) hb ) data for the HFC-32 (1) + HFC-134a (2), and HFC-125 (1) + intermolecular parameters are 12/k 178.3 K and 12/k R/ ) HFC-134a (2) mixtures, respectively. The phase behaviour of 897.5 (with corresponding reduced parameters 12 1.920, R/ ) hb/ ) these mixtures is very close to ideal, and in fact it was found 22 2.461, and 12 0.957). The range for the square-well Phase Equilibria for Refrigerant Mixtures J. Phys. Chem. B, Vol. 102, No. 39, 1998 7639 segment-segment unlike interaction is λ12 ) 1.538, and the References and Notes 3 ) unlike bonding volume K12/Å 6.509. These are obtained as (1) Midgley, T.; Henne, A. C. Ind. Eng. Chem. 1930, 22, 542. discussed earlier. In this mixture none of the parameters are (2) Molina, M. J.; Rowland, F. S. Nature 1974, 249, 810. readjusted. Similarly, the HFC-125 (1) + HFC-134a (2) mixture (3) Molina, M. J. Pure Appl. Chem. 1996, 68, 1749. is studied over a temperature range from -40 °C (233 K) to 60 (4) Gow, A. S. Fluid Phase Equilib. 1993, 90, 219. ° (5) Obey, H.; Sandler, S. I. Ind. Eng. Chem. Res. 1995, 34, 2520. C (333 K) (see Figure 7). The values of the unlike intermo- (6) Barolo, M.; Bertucco, A.; Scalabrin, G. Int. J. Refrig. 1995, 18, ) hb ) lecular parameters in this mixture are 12/k 193.1 K, 12/k 550. R/ ) (7) Kleiber, M. Fluid Phase Equilib. 1995, 107, 161. 970.5 (with corresponding reduced parameters 12 1.075, / / (8) Blindenbach, W. L.; Economou, I. G.; Smits, P. J.; Peters, C. J.; R ) hb ) ) 3 ) 22 0.937, and 12 0.527), λ12 1.588, and K12/Å de Swaan Arons, J. Fluid Phase Equilib. 1994, 97, 13. 2.709. It can be seen from the figure that the agreement with (9) Vimalchand, P.; Donohue, M. D. Ind. Eng. Chem. Fundam. 1985, the experimental data is very good for the entire temperature 24, 246. (10) Vimalchand, P.; Donohue, M. D.; Celmins, I. Am. Chem. Soc. Symp. range. Ser. 1986, 300, 297. (11) Economou, I. G.; Peters, C. J.; Florusse, L. J.; de Swaan Arons, J. 4. Conclusions Fluid Phase Equilib. 1995, 111, 239. (12) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Fluid The SAFT-VR approach has been used used to predict the Phase Equilib. 1989, 52, 31. phase equilibria of the binary mixtures HFC-32 + HFC-125, (13) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Ind. HFC-32 + HFC-134a and HFC-125 + HFC-134a. Very good Eng. Chem. Res. 1990, 29, 1709. (14) Huang, S. H.; Radosz, M. Ind. Eng. Chem. Res. 1990, 29, 2284. agreement with experimental data has been obtained for the (15) Huang, S. H.; Radosz, M. Ind. Eng. Chem. Res. 1991, 30, 1994. mixtures. It is encouraging to see the abilities of the SAFT- (16) Galindo, A.; Whitehead, P. J.; Jackson, G.; Burgess, A. N. J. Phys. VR approach in the description of the phase equilibria of Chem. 1996, 100, 6781. (17) Mu¨ller, E. A.; Gubbins, K. E. Ind. Eng. Chem. Res. 1995, 34, 3662. complex molecules. In the SAFT-VR approach there is no (18) Stell, G.; Rasaiah, J. C.; Narang, H. Mol. Phys. 1974, 27, 1393. direct evaluation of the contribution to the free energy due to (19) Kraska, T.; Gubbins, K. E. Ind. Eng. Chem. Res. 1996, 35, 4727. permanent or induced multipolar interactions. The square-well (20) Kraska, T.; Gubbins, K. E. Ind. Eng. Chem. Res. 1996, 35, 4738. monomer-monomer interaction is an effective potential that (21) Gil-Villegas, A.; Galindo, A.; Whitehead, P. J.; Mills, S. J.; Jackson, G.; Burgess, A. N. J. Chem. Phys. 1997, 106, 4168. describes the van der Waals forces, and the level of description (22) Galindo, A.; Davies, L. A.; Gil-Villegas, A.; Jackson, G. Mol. Phys. of polar effects obtained with this approach is restricted to the 1998, 93, 241. Boltzmann-averaged interaction between permanent multipoles (23) Galindo, A.; Whitehead, P. J.; Jackson, G.; Burgess, A. N. J. Phys. (Keesom interactions) or between permanent and induced Chem. 1997, 101, 2082. (24) Chapman, W. G.; Jackson, G.; Gubbins, K. E. Mol. Phys, 1988, multipoles (Debye interactions). The detailed description of 65,1. these long-range forces with a second-order perturbation theory (25) Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd for an effective variable-range potential improves the agreement ed.; Butterworth Scientific: London 1982. (26) Barker, J. A.; Henderson, D. J. Chem. Phys. 1967, 47, 2856. obtained with the SAFT-HS approach, especially for the (27) Barker, J. A.; Henderson, D. J. Chem. Phys. 1967, 47, 4714. saturated liquid densities. Together with the use of sites for (28) Barker, J. A.; Henderson, D. ReV. Mod. Phys. 1975, 48, 587. the modeling of short-ranged directional forces in the fluids, (29) Boublı´k, T. J. Chem. Phys. 1970, 53, 471. SAFT-VR is accurate enough to reproduce the main features (30) Mansoori, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W. J. Chem. Phys. 1971, 54, 1523. of associating polar and nonpolar fluids. In many cases the (31) Reed, T. M.; Gubbins, K. E. Applied Statistical Mechanics; refrigerant mixtures include long-chain hydrocarbons that act MacGraw-Hill: New York, 1973. as lubricants in large-scale manufacture. The study of multi- (32) Chapman, W. G.; Jackson, G.; Gubbins, K. E. Mol. Phys. 1988, component mixtures with this approach may prove very fruitful 65,1. (33) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. to the chemical engineer. Numerical Recipes in Fortran, 1st ed.; Cambridge University Press: New A.G. thanks Sheffield University and ICI Chemicals and York, 1986. Polymers Ltd. for the award of a Roberts-Boucher Scholarship, (34) Scott, R. L.; van Konynenburg, P. H. Discuss. Faraday Soc. 1970, 49, 87. van Konynenburg, P. H.; Scott, R. L. Philos. Trans. R. Soc. A 1980, and A.G.V. and P.J.W. to thank the ICI Strategic Research Fund 298, 495. for funding Senior Research Fellowships. We also acknowledge (35) Physical Property Data, KLEA 32, ICI Chemical and Polymers support from the Royal Society (574005.G501/1992), and from Ltd., 1993. Physical Property Data, KLEA 125, ICI Chemical and Polymers the Computational (GR/H58810-C91) and ROPA (GR/K34740) Ltd., 1993. Physical Property Data, KLEA 134a, ICI Chemical and Polymers Ltd., 1993. Initiatives of the EPSRC for computer hardware on which the (36) Barley, M. H.; Morrison, J. D.; O’Donnel, A.; Parker, I. B.; calculations were performed. Petherbridge, S.; Wheelhouse, R. W. Fluid Phase Equilib. 1997, 140, 183.