Extension of Tao-Mason Equation of State to Mixtures: Results for

Apr 10, 2009 - Tao and Mason (J. Chem. Phys. 1994, 100, 9075-9084) developed a statistical-mechanical-based equation of state (EOS) for pure substance...
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Ind. Eng. Chem. Res. 2009, 48, 5079–5084

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Extension of Tao-Mason Equation of State to Mixtures: Results for PVTx Properties of Refrigerants Fluid Mixtures Fakhri Yousefi,*,† Jalil Moghadasi,† Mohammad Mehdi Papari,‡ and Antonio Campo§ Department of Chemistry, Shiraz UniVersity, Shiraz 71454, Iran, Department of Chemistry, Shiraz UniVersity of Technology, Shiraz 71555-313, Iran, and Department of Mechanical Engineering, The UniVersity of Texas at San Antonio, San Antonio, Texas 78249

Tao and Mason (J. Chem. Phys. 1994, 100, 9075-9084) developed a statistical-mechanical-based equation of state (EOS) for pure substances. In the present study, we have successfully extended this EOS to fluid mixtures, selecting refrigerant fluid mixtures as the test systems. The considered refrigerant mixtures are R32 + R125, R32 + R134a, R134a + R152a, R125 + R143a, R125 + R134a, R32 + R227ea, R134a + R290, and R22 + R152a. The second virial coefficient, B(T), necessary for the mixture version of the Tao-Mason (TM) EOS, was determined using a two-parameter corresponding-states correlation obtained from the analysis of the speed of sound data and two constants: the enthalpy of vaporization ∆Hvap and the molar density Fnb, both at the normal boiling point. Other temperature-dependent quantities, including the correction factor R(T) and van der Waals covolume b(T), were obtained from the Lennard-Jones (12-6) model potential. The cross parameters B12(T), R12(T), and b12(T), required by the EOS for mixtures, were determined with the help of simple combining rules. The constructed mixture version of the TM EOS was extensively tested by comparison with experimental data. The results show that the molar gas and liquid densities of the refrigerant mixtures of interest can be predicted to within 1.3% and 2.69%, respectively, over the temperature range of 253-440 K and the pressure range of 0.33-158 bar. The present EOS was further assessed through comparisons with the Ihm-Song-Mason (ISM) and Peng-Robinson (PR) equations of state. In the gas phase, the TM EOS outperforms the two other equations of state. In the liquid phase, there is no noticeable difference between the TM EOS and the PR EOS, but both work better than the ISM EOS. 1. Introduction 1

In an earlier work, Tao and Mason developed a statistically based equation of state (EOS) by taking into account a perturbation correction term for the effect of attraction forces on the Ihm-Song-Mason (ISM) EOS.2 This correction led to considerable improvements in predicting vapor pressures and orthobaric densities. In fact, the analytic nature of the ISM EOS leads to the prediction of van der Waals loops in subcritical P-V isotherms instead of the correct first-order vapor-liquid transition, and therefore, it cannot be applied to the two-phase region to predict the vapor pressures of fluids. In recent years, the thermodynamic properties of refrigerant mixtures have been investigated by several researchers.3-6 Despite its drawbacks, one of the equations of state employed for predicting the PVTx properties of refrigerant mixtures has been the ISM equation of state.3,5,7 In the present work, the Tao and Mason (TM) equation of state (EOS),1 which is based on statistical mechanics, is extended to predict the densities of both the liquid and gas phases of refrigerant mixtures, and the results are compared with the ISM2 and Peng-Robinson8 (PR) equations of state, as well as with literature data. The reason for selecting refrigerant mixtures is that the thermodynamic properties of both long-known and new refrigerants are the key data needed for the calculation of refrigeration cycles and the design of refrigeration and air-conditioning equipment. Measurement of the thermodynamic properties of * To whom correspondence should be addressed. E-mail: mojyou54@ yahoo.com. Tel.: +98-711-735-4522. Fax: +98-711-726-1288. † Shiraz University. ‡ Shiraz University of Technology. § The University of Texas at San Antonio.

new refrigerants has been carried out within the scope of international research programs. Furthermore, the PVTx and thermodynamic properties of refrigerant mixtures have attracted particular interest not only from the refrigeration industry in relation to heat pump systems, system design, and reliable assessment of cycle performance but also from the power generation industry as refrigerant mixtures are regarded as new prospective working substances. For example, R134a and R152a and their binary mixture are the most promising candidates to substitute for the chlorofluorocarbon difluorodichloromethane (R12), which is assumed to contribute to the depletion of the stratospheric ozone layer. It is important to note that there is a great deal of interest in binary and ternary refrigerant blends, especially those based on hydrofluorocarbon refrigerants. In the present investigation, the considered refrigerant mixtures are R32 + R125, R32 + R134a, R134a + R152a, R125 + R143a, R125 + R134a, R32 + R227ea, R134a + R290, and R22 + R152a. Our results show that the molar gas and liquid densities of the refrigerant mixtures of interest can be predicted to within 1.3% and 2.69%, respectively, over the temperature range of 253-440 K and the pressure range of 0.33-158 bar. 2. Theory 2.1. TM EOS for Pure System. In general, the equations of state in common use can be classified as belonging to the van der Waals family of cubic equations, the extended family of virial equations, or equations based more closely on the results from statistical mechanics and computer simulations.9-11 The TM EOS falls in the latter category. In 1994, Tao and Mason calculated a perturbation correction term for the effect of attractive forces and combined it with the

10.1021/ie8016658 CCC: $40.75  2009 American Chemical Society Published on Web 04/10/2009

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ISM equation of state2 to present an improved equation of state (TM EOS).1 The final form of the TM EOS for single substances is given by (eκTc/T - A2) P RF ) 1 + (B - R)F + + A1(R - B)bF2 FkT 1 - λbF 1 + 1.8(bF)4 (1) where A1 ) 0.143 A2 ) 1.64 + 2.65[e(κ-1.093) - 1]

(2)

κ ) 1.093 + 0.26[(ω + 0.002)1/2 + 4.50(ω + 0.002)] (3) In eqs 1-3, ω is the Pitzer acentric factor, λ is an adjustable parameter, F is the number density, Tc is the critical temperature, kT has its usual meaning, B is the second virial coefficient, R is the scaling parameter, and b is the effective van der Waals covolume. These parameters can be obtained from an intermolecular potential energy.12,13 In the case that an acceptable potential energy is not known, these parameters can be calculated from corresponding-states correlations.14-18 In a previous work, we developed three correlations for B, R, and b using speed-of-sound data.17 These correlations are

( ) [ ( )] {

∆Hvap BFnb ) 0.1 - 0.054 RT RFnb ) a1 exp -d1

RT ∆Hvap

2

( ) [ ( ) ]}

∆Hvap - 0.00028 RT

+ a2 1 - exp -d2

4

(4)

∆Hvap RT

1/4

(5)

[ ( )] [ ( )] { [ ( ) ] [ ( ) ]}

bFnb ) a1 1 - d1

RT ∆Hvap

exp -d1

∆Hvap a2 1 - 1 + 0.25d2 RT

RT ∆Hvap

+

∆Hvap exp -d2 RT

1/4

1/4

(6)

∑ x x (B i j

ij

ij - Rij) + F

∑xxR G i j ij

ij

ij + F

∑ x x (I )

i j 1 ij

ij

(7) where F is the total molecular number density, xi and xj are mole fractions, Gij is the pair distribution function at contact for species i and j, and the double summation runs over all components. The interaction parameters Bij, Rij, and bij for i * j correspond to the values for a hypothetical single substance whose molecules interact according to a pairwise ij potential. For i ) j, the parameters are those for the pure substance i. The problem is now how to estimate the parameters Gij and (I1)ij. The behavior of Gij, the pair distribution function at contact for real mixtures, has been given by Ihm et al. as19

( )

bibj 1 + 1-η bij

F

1/3

∑

(λk - 1/4)

2/3

xb k k k

(1 - η)(1 - F

∑

(8)

xbλ) k k k k

where η is the packing fraction for the mixture, given by η)

F 4

∑x b

(9)

k k

k

For a pure system, we have I1 ) (R - B)ζ(T) Φ(bF). Tao and Mason1 proposed that Φ(bF) and ζ(T) could be approximated as F Φmix ≈

∑

1 + 1.8F4(

xb k k k

∑

(10)

x b )4 k k k

ζmix ≈ 0.143[exp(κmixTcmix/T) - A2mix]

(11)

where κmix )

∑x κ

(12)

k k

k

Table 1. Parameters for Eight Refrigerants refrigerant

Tnb (K)

∆Hvap/R (K)

Fnb (kg m-3)

λ

ωa

R32 R143a R134a R152a R125 R227ea R290 R22

221.572 261.05 246.672 249.119 224.9 236.45 231.15 232.350

2490.11 2677.4 2737.95 2747.33 2371.83 2351.9 2280.9 2516.17

1213.96 1479.00 1380.05 1011.25 1515.15 1603.00 582.00 1408.82

0.619 0.573 0.578 0.595 0.584 0.568 0.539 0.590

0.280 0.288 0.324 0.275 0.305 0.325 0.152 0.221

a

where Fnb is the normal boiling density. The coefficients in the above equations are a1 ) -0.1162, a2 ) 2.22572, d1 ) 6.58566, and d2 ) 0.71472. The remaining problem now is to find λ. We obtained λ from PVT data for single substances at their normal boiling points. When the temperature-dependent parameters together with the values of λ are known, the entire TM EOS can be predicted. 2.2. Extension of TM EOS to Mixtures. In the case of mixtures, it is mixing and combining rules that allow an EOS developed for pure fluids to be used for mixtures. Accuracy in the prediction of properties of mixtures is one of the major concerns in scientific research and engineering calculations.1-3 The TM EOS can be extended to mixtures as follows P )1+F FKT

Gij )

Pitzer acentric factor.

Table 2. Prediction of the Gas Densities of R32 + R125 Mixtures at Variable Pressures, Temperatures, and Compositions AAD (%) density T (K)

xR32

NP

TM

PR

ISM

ref

380 370 360 350 380 360 340 320 373.15 353.15 338.15 373.15 353.15 323.15 373.15 353.15 323.15 373.15 353.15 323.15 330 350 370 400 420 440 340 360 380 400 420 440

0.5001 0.5001 0.5001 0.5001 0.6977 0.6977 0.6977 0.6977 0.5456 0.5456 0.5456 0.432 0.432 0.432 0.5794 0.5794 0.5794 0.8191 0.8191 0.8191 0.36708 0.36708 0.36708 0.36708 0.36708 0.36708 0.60576 0.60576 0.60576 0.60576 0.60576 0.60576

11 7 7 7 10 8 8 7 6 6 5 13 8 8 13 8 8 12 8 8 4 4 4 4 4 4 4 4 4 4 4 4

0.65 0.99 1.44 1.8 1.36 2.33 2.36 2.29 1.11 1.7 2.74 0.19 0.33 0.55 0.32 0.46 0.46 0.55 0.85 0.98 4.09 2.42 1.42 0.49 0.41 0.6 0.56 0.56 0.64 0.87 1.13 1.36

2.15 2.57 2.51 2.39 1.96 1.92 1.51 1.08 2.49 2.42 2.29 2.1 2.39 1.82 1.97 1.76 1.52 1.68 1.63 0.98 2.51 3.3 3.38 3.12 2.88 2.63 2 2.63 2.74 2.67 2.53 2.35

0.81 0.65 0.52 0.61 0.96 1.21 1.74 1.09 0.58 0.67 0.9 0.9 0.53 0.76 0.82 0.7 1.15 0.91 1.02 1.59 4.2 0.24 0.92 0.49 1.4 1.71 2.48 1.47 0.79 0.21 0.28 0.71

22 22 22 22 22 22 22 22 20 20 20 23 23 23 23 23 23 23 23 23 21 21 21 21 21 21 21 21 21 21 21 21

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A2mix ) 1.64 + 2.65[exp(κmix - 1.093) - 1] Tcmix )

∑x T

(13)

Table 6. Prediction of the Gas Densities of R32 + R227ea Mixtures at Variable Pressures, Temperatures, and Compositions

(14)

k ck

k

with Tcmix as the traditional pseudocritical temperature. Now, the problem is restricted to the determination of the interaction parameters Bij, Rij, and bij. In principle, they can all be calculated from a known ij pair potential energy.13 In the event that the potential is not known, Bij can, in principle, be extracted from measurements of the second virial coefficient of i + j mixtures, and then Rij and bij can be calculated by scaling rules. As stated previously, in the present study, a predictive scheme outlined by Sheikh et al.17 (eq 4) was employed to predict pure and cross parameters of the second virial coefficients of refrigerants. The values of the pure and cross parameters R and b can also be calculated from eqs 5 and 6, respectively. 3. Results and Discussion As already stated, in this work, an EOS that was previously constructed for pure substances using statistical-mechanical perturbation theory, has been extended to mixtures. We emTable 3. Prediction of the Liquid Densities of R32 + R125 Mixtures at Variable Pressures, Temperatures, and Compositions AAD (%) density T (K)

xR32

NP

TM

PR

ISM

ref

294.68 313.57 333.55 294.95 313.84 294.93 313.83 333.76

0.5 0.5 0.5 0.76 0.76 0.24 0.24 0.24

3 3 3 3 3 3 3 3

6.6 4.48 2.72 6.71 3.6 3.88 2.55 3.06

0.62 4.45 6.97 5.03 6.33 0.62 3.43 5.85

6.98 6.84 6.85 5.38 6.98 5.11 5.17 6.94

24 24 24 24 24 24 24 24

Table 4. Prediction of the Gas Densities of R32 + R134a Mixtures at Variable Pressures, Temperatures, and Compositions AAD (%) density T (K)

xR32

NP

TM

PR

ISM

ref

373.15 353.15 338.15 360 380 400 420 440 323.15 373.15 423.15 323.15 373.15 423.15 323.15 373.15 423.15

0.5084 0.5084 0.5084 0.39534 0.39534 0.39534 0.39534 0.39534 0.2502 0.2502 0.2502 0.476 0.476 0.476 0.75 0.75 0.75

6 6 5 4 4 4 4 4 10 26 20 14 20 20 15 23 23

0.78 1.15 0.49 2.54 1.25 0.99 0.83 0.93 0.74 1.34 0.72 1.25 1.63 0.58 3.47 3.06 1.71

2.71 2.68 1.67 8.6 2.25 2.53 2.51 2.14 1.01 1.98 2.15 0.74 5.04 1.92 0.85 1.67 1.29

1.17 0.66 0.91 2.43 1.44 1.05 0.81 0.76 1.01 0.91 0.87 0.96 1.04 0.46 3.05 2.98 1.93

20 20 20 25 25 25 25 25 26 26 26 26 26 26 26 26 26

Table 5. Prediction of the Liquid Densities of R32 + R134a Mixtures at Variable Pressures and Temperatures and Fixed Compositions AAD (%) density T (K)

xR32

NP

TM

PR

ISM

ref

280 290 300 310 320

0.3953 0.3953 0.3953 0.3953 0.3953

4 5 4 3 3

0.54 0.35 1.38 2.49 4.18

1.83 1.17 0.4 0.66 0.2

10.2 10.39 10.95 11.61 12.98

27 27 27 27 27

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AAD (%) density T (K)

xR32

NP

TM

PR

ISM

ref

410 360 320 410 360 320 410 360 320

0.2648 0.2648 0.2648 0.5155 0.5155 0.5155 0.7809 0.7809 0.7809

13 10 7 12 9 7 12 9 9

1.58 2.71 1.98 1.21 2.28 2.28 0.64 1.78 2.24

2.77 2.56 1.53 2.7 2.6 1.66 2.18 2.23 1.29

1.6 2.46 1.35 0.96 1.3 1.13 0.86 1.06 2.02

28 28 28 28 28 28 28 28 28

Table 7. Prediction of the Gas Densities of R125 + R134a Mixture at Variable Pressures and Temperatures and Fixed Compositions AAD (%) density T (K)

xR125

NP

TM

PR

ISM

ref

373.15 353.15 338.15 323.15 303.15

0.495 0.495 0.495 0.495 0.495

7 4 3 2 2

0.38 0.44 0.24 0.32 0.31

2.82 2.72 2.19 1.77 1.26

2.06 2.7 1.59 1.31 1.11

20 20 20 20 20

Table 8. Prediction of the Liquid Densities of R125 + R134a Mixtures at Variable Pressures, Temperatures, and Compositions AAD (%) density T (K)

xR125

NP

TM

PR

ISM

ref

279.987 289.987 309.987 279.987 289.987 299.987 279.987 289.987 299.987

0.2674 0.2674 0.2674 0.4603 0.4603 0.4603 0.6653 0.6653 0.6653

6 5 6 6 6 5 6 6 4

9.67 7.18 5.25 1.14 1.35 2.31 2.17 2.7 3.49

0.87 1.4 1.17 1.56 2 2.5 1.99 1.24 0.34

4.48 6.27 4.97 1.14 1.56 2.22 2.89 3.08 3.68

27 27 27 27 27 27 27 27 27

ployed a two-parameter macroscopic corresponding-states correlation obtained in our previous work17 in conjunction with simple combining rules for the heat of vaporization and molar density to predict the interaction second virial coefficients, as well as the second virial coefficients for single substances. (∆Hvap)ij ) [(∆Hvap)i(∆Hvap)j]1/2

(15)

[(Fnb)i-1/3 + (Fnb)j-1/3] (16) 2 The values of λ appearing in eqs 1 and 8 were found empirically from the PVT data of dense pure components and are listed in the Table 1. This method for determining λ makes the whole procedure self-correcting because, if the input values of ∆Hvap and Fnb are not accurate, the effect will be largely compensated in the determination of λ. After constructing the EOS, to provide extensive testing and evaluation, we performed a comprehensive comparison with experimental data over a a vast range of temperatures, pressures, and compositions of refrigerant mixtures of interest. It is well believed that either a binary or a ternary refrigerant mixture composed of R32, R125, and/or R134a would be the optimum candidate to replace R22. This mixture is a suitable replacement for the refrigerant R22, which is currently being used in air-conditioning and heat-pumping equipment but is expected to be phased out by the year 2020. Five data sets on R32 and R125 mixtures were taken from the literature.20-24 The results of calculations of the gas and liquid densities of this mixture together with the temperatures, (Fnb)ij-1/3 )

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Table 9. Prediction of the Gas Densities of R125 + R143a Mixtures at Variable Pressures, Temperatures and Compositions AAD (%) density T (K)

xR125

NP

TM

PR

ISM

ref

373.15 363.15 353.15 343.15 333.15 320 350 380 320 350 380 320 350 380

0.509 0.509 0.509 0.509 0.509 0.2733 0.2733 0.2733 0.5652 0.5652 0.5652 0.737 0.737 0.737

6 6 6 6 3 7 7 11 8 9 8 7 10 10

0.59 0.74 0.91 1.19 0.4 2.57 1.85 1.98 1.52 1.07 0.42 1.42 0.57 0.10

2.23 2.24 2.23 2.08 1.75 1.11 1.79 1.67 1.49 2.08 1.91 1.55 2.22 2.13

0.98 0.93 0.89 0.81 1 0.95 0.73 2.48 0.65 0.72 1.65 0.78 2.09 1.47

20 20 20 20 20 29 29 29 29 29 29 29 29 29

Table 10. Prediction of the Gas Densities of R134a + R152a Mixtures at Variable Pressures, Temperatures, and Compositions AAD (%) density T (K)

xR134a

NP

TM

PR

ISM

ref

373.15 353.15 293.15 313.15 333.15 293.15 313.15 333.15 303.15 323.15 343.15

0.497 0.497 0.24798 0.24798 0.24798 0.49634 0.49634 0.49634 0.75123 0.75123 0.75123

6 5 9 12 15 9 12 15 11 13 16

1.57 1.69 1.23 1.31 1.64 1.1 1.15 1.28 0.8 0.83 0.87

1.3 1.06 0.65 0.79 0.93 0.64 0.8 0.95 0.78 0.92 1.11

0.6 0.73 0.59 0.73 0.98 0.57 0.69 0.8 0.74 0.75 0.81

20 20 30 30 30 30 30 30 30 30 30

Table 11. Prediction of the Liquid Densities of R134a + R152a Mixtures at Variable Pressures, Temperatures, and Compositions AAD (%) density T (K)

xR134a

NP

TM

PR

ISM

ref

253.25 303.15 253.26 273.25 313.17

0.24817 0.24817 0.49997 0.7509 0.7509

15 16 15 16 16

2.01 3.24 3.44 2.19 5.32

4.79 3.42 1.57 0.47 1.73

1.14 2.73 1.85 3.58 5.38

30 30 30 30 30

mixture compositions, and numbers of data points are summarized in Tables 2 and 3. From these tables, it is found that, in the gas phase, the TM EOS agrees with the measured values20-23 within 1.2%; however, the calculated liquid densities deviate from the measured values reported by Weber24 by up to 4.2%. It is a matter of considerable practical importance to compare the present EOS with other equations of state. In this respect, we have compared our results for this mixture with those obtained using two well-known equations of state, namely,

the Peng-Robinson (PR)8 and Ihm-Song-Mason (ISM)2 equations of state. The results of the calculations for R125 + R32 mixtures are gathered in Tables 2 and 3. NP in these tables represents the number of points. It is evident that, in the gas phase, the averages values of density deviations obtained from the TM are almost the same as those calculated from the ISM EOS. However, in the liquid phase, the TM and PR equations of state are very similar. We extended our calculations to other binary mixtures. The binary refrigerant mixture R32 + R134a has zero ozonedepletion potential (ODP). Because of its flammable characteristics, however, R32 is not considered as a suitable working fluid for air-conditioning systems. The mixture of R32 with a nonflammable refrigerant, such as R134a, therefore, might overcome this drawback for practical applications. Tables 4 and 5 list average absolute deviation (AAD) values of the gas and liquid densities, respectively, of this mixture obtained using the TM, ISM, and PR equations of state from the experimental results reported in refs 20 and 25-27. In the gas phase, the same trend as for R32 + R125 mixtures is observed. However, in the liquid phase, the TM EOS surpasses the PR EOS at some temperatures. In general, however, the PR EOS performs slightly better than the TM EOS. Table 6 reports the average absolute deviations of gas densities for the binary mixture R32 + R227ea from the experimental values reported by Feng et al.28 This mixture is a promising alternative refrigerant to replace R22. The experimental density values for this mixture were measured28 with an accuracy of 0.01% using the Burnett isochoric coupling method. The AADs from the literature values20,27 for the gas and liquid densities of the R125 + R134a system are listed in Tables 7 and 8, respectively. In this case, in the gas phase, the TM EOS is superior to the other equations of state. In the liquid state, the PR EOS outperforms the other equations of state. The deviations of the gas densities of R125 + R143a from measurements20,29 are listed in Table 9. In this case, the TM EOS is almost as accurate as the ISM EOS and works better than the PR EOS. Tables 10 and 11 include the deviations of the gas and liquid densities, respectively, of the R134a + R152a system from the literature values.20,30 The mixture R134a + R152a can also be used as a substitute in certain applications because its thermodynamics properties are similar to those of R12. It should be mentioned that, in ref 30, the liquid densities were measured using a vibrating tube densimeter at temperatures between 243.27 and 413.15 K and for pressures up to 16 MPa and a Burnett apparatus was employed for the gas-phase measurements between temperatures of 293.15 and 453.15 K and for pressure up to 16 MPa.

Table 12. Results of Density Predictions for All Considered Mixtures (Liquid and Gaseous Phases) AAD (%) density mixture

phase

∆T (K)

∆p (bar)

NP

TM

PR

ISM

ref(s)

R32 + R125 R32 + R134a R32 + R227ea R125 + R134a R125 + R143a R134a + R152a R134a + R290 R32 + R134a R125 + R134a R134a + R152a R22 + R152a R32 + R125

gas gas gas gas gas gas gas liquid liquid liquid liquid liquid

320–440 323.15–440 320–410 303.15–373.15 320–380 293.15–373.15 314–400 280–320 279.98–309.98 253.25–313.17 279.99–330.00 294.68–333.76

0.9068–52.427 0.33–75.63 1.06–36.58 0.55–40.31 1.15–47.63 0.92–29.23 16.60–29.18 10–30.02 5.12–30.16 7.27–158.34 4.3–18.8 14.53–39.78

216 208 88 18 104 123 29 19 50 78 16 24

1.18 1.38 1.85 0.33 1.09 1.22 2.03 1.49 3.91 3.24 0.93 4.2

2.24 2.45 2.16 2.15 1.89 0.9 8.43 0.71 1.45 2.4 2.09 4.16

1.03 1.32 1.41 1.75 1.15 0.73 6.09 9.35 3.36 2.94 0.98 6.28

20-24 20, 25, 26 28 20 20, 29 20, 30 31 27 27 30 32 24

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Table 12 contains a comparison of the predicted gas densities of the R134a + R290 and R22 + R152a binary mixtures with the experimental values.31,32 Table 12 also summarizes our results for all binary mixtures. Furthermore, this table includes the pressure and temperature ranges of all studied refrigerant mixtures. A close inspection of the deviations given in this table reveals that, in the gas phase, the TM EOS predicts the densities with a mean AAD of 1.3%, which, therefore, means that it outperforms the other equations of state. In the liquid phase, the TM EOS is almost as accurate as the PR EOS with a mean AAD within 2.69%. To assess the capabilities of the TM EOS, the computations were extended to wider ranges of temperature and pressure. The results of the calculations were compared with those obtained by the COSTALD method.33 The AADs for 500 data points for the temperature range of 200-350 K and the pressure range of 10-800 bar were found to be on the order of 4.41%. 4. Conclusions In the present work, we have successfully extended the Tao-Mason equation of state to mixtures. On the basis of the results tabulated herein, the following conclusions can be drawn: (1) When tested against experimental data, the calculated gas and liquid densities in the single- and two-phase regions gave mean average absolute deviations of 1.30 and 2.69, respectively. (2) To assess the TM EOS further, we compared it with the PR and ISM equations of state. In the gas phase, the TM EOS outperforms the two other equations of state. In the liquid phase, there is no noticeable difference between the TM EOS and the PR EOS, but both work better than the ISM EOS. This study demonstrates that the mixture version of the TM EOS is able to provide reliable information on the PVTx properties of both the compressed liquid phase and the gas phase of refrigerant fluid mixtures. Once the second virial coefficient from the macroscopic corresponding-states correlation equation, the values of λ from some high-density information, and the two constants ∆Hvap and Fnb for each refrigerant in conjunction with a simple combining rule for estimating the cross values of these parameters are known, the entire EOS can be employed to predict the volumetric behaviors of fluid mixtures. Application of the mixture version of this EOS to other classes of fluids remains for future work. The results obtained from the TM EOS are remarkable in the low- and high-density regions, especially considering that they are based essentially on the second virial coefficient that characterizes the binary interaction in the low-density gas phase. Acknowledgment The authors thank Shiraz University of Technology and Shiraz University for supporting this project. List of Symbols a1 ) constant coefficient in eqs and 6 a2 ) constant coefficient in eqs and 6 b ) parameter analogous to the van der Waals covolume, kg/m3 B ) second virial coefficient, m3/mol bij ) interaction parameter analogous to the van der Waals covolume, kg/m3 Bij ) interaction second virial coefficient, m3/mol d1 ) constant coefficient in eqs and 6 d2 ) constant coefficient in eqs and 6

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G ) pair distribution function at contact k ) Boltzmann constant P ) pressure, Pa r ) intermolecular distance, m rm ) separation at which the potential energy has a minimum value u ) intermolecular pair potential, J u0 ) unperturbed pair potential, J x ) mole fraction Greek Letters R ) correction factor for the softness of the repulsive force, m3/mol Rij ) interaction parameter of the correction factor for the softness of the repulsive force, m3/mol ∆Hvap ) heat of vaporization, J/mol η ) packing factor κ ) function defined by eq 3 λ ) adjustable parameter ξ ) parameter in eq 14 F ) mass density, kg/m3 Fnb ) normal boiling density ω ) Pitzer acentric factor Subscripts 0 ) reference state c ) critical mix ) mixture nb ) normal boiling point vap ) vaporization

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ReceiVed for reView November 2, 2008 ReVised manuscript receiVed March 6, 2009 Accepted March 13, 2009 IE8016658