An analytical equation of state for refrigerant mixtures

International Journal of Refrigeration 29 (2006) 150–154 www.elsevier.com/locate/ijrefrig

An analytical equation of state for refrigerant mixtures Hossein Eslamia,*, Nargess Mehdipourb, Ali Boushehrib a

Department of Chemistry, College of Sciences, Persian Gulf University, Boushehr 75168, Iran b Department of Chemistry, College of Sciences, Shiraz University, Shiraz 71454, Iran

Received 25 March 2004; received in revised form 29 October 2004; accepted 15 December 2004 Available online 18 October 2005

Abstract We have extended our previous work on the equation of state for refrigerants to their mixtures successfully. The temperaturedependent parameters of the equation of state have been calculated using our previous corresponding-states correlation based on the normal boiling point temperature and the liquid density at the normal boiling point. We have applied a simple combining rule for the normal boiling point constants to extend our previously proposed equation of state to mixtures of refrigerants. In this work the liquid densities of a large number of refrigerant mixtures have been calculated and the results are compared both with experimental data and a recent correlation by Nasrifar et al. (1999). The agreement is good. q 2005 Elsevier Ltd and IIR. All rights reserved. Keywords: Refrigerant; Mixture; Calculation; Equation

Equation d’e´tat analytique pour les me´langes de frigorige`nes Mots cle´s : Frigorige`ne ; Me´lange ; Calcul ; E´quation

1. Introduction There are a great number of equations of state for refrigerants and their mixtures in the literature. Of the most accurate presented equations of state for pure refrigerants we can address to the international equations of state reported for R-134a [1], R-152a [2], R-32 [3], and R-125 [4]. For mixtures of refrigerants, perhaps, the equations of state reported by Lemmon and Jacobsen [5,6] and one proposed by Miyamoto and Watanabe [7] are the most accurate among the others. Generally, these accurate equations of state are empirical with so many numbers of parameters to predict the whole PVT surface of refrigerants.

* Tel.: C98 771 4541494; fax: C98 771 4545188. E-mail address: [email protected] (H. Eslami).

0140-7007/$35.00 q 2005 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2004.12.012

In this work we have developed previous works [8–11], on the prediction of the equation of state for compressed liquids and their mixtures from fairly easy and simple ways, to incorporate refrigerant mixtures. In the previous paper [12] we have developed an analytical equation of state for refrigerants using the normal boiling point temperature and the liquid density at the normal boiling point as input data. The equation of state has been originally developed by Song and Mason [13] based on the statistical–mechanical perturbation theory in the perturbation scheme of Weeks– Chandler–Andersen [14]. This equation of state reads as: P ðaKB2 Þr ar Z 1K C rkT 1 C 0:22lbr 1Klbr

(1)

where P is the pressure, r is the molar (number) density, B2 is the second virial coefficient, a is the contribution of the repulsive forces to the second virial coefficient, b is a

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151

Nomenclature a1, a2, c1, c2 constants in Eqs. (7) and (8) AAD average absolute deviation (%) b an effective hard-sphere diameter (m3 molK1) B2 second virial coefficient (m3 molK1) d molecular diameter (m) F a dimensionless parameter defined by Eq. (3) G a dimensionless parameter defined by Eq. (4) k Boltzmann’s constant (J KK1) MD maximum deviation (%) NP the number of data points P pressure (bar) r interparticle distance (m) T temperature (K) V molar volume (m3 molK1) x mole fraction

Greek symbols a a scaling factor (m3 molK1) r molar density (mol mK3) l an adjustable parameter d 0.22l 3 potential well-depth (J) z3 a dimensionless parameter defined by Eq. (5) Subscripts B Boyle bp boiling point i substance i j substance j ij pairwise ij m minimum

temperature-dependent parameter analogous to the van der Waals covolume, T is the absolute temperature, k is the Boltzmann’s constant, and l is an adjustable parameter. Eq. (1) for single substances has been developed to the mixtures of any number of components by Song [15]. The result can be expressed as:

purpose of this work to predict PVT properties of refrigerant mixtures without the knowledge of the intermolecular potential energy curve.

X X P Z 1 Cr xi xj ½ðB2 Þij Kaij Fij C r xi xj aij Gij rkT ij ij

2. Simplified method for the determination of temperature-dependent parameters

(2)

with Fij Z

P   di dj 1 ð1=6Þpr k xk dk2 ð4dk C 1Þ   P K dij ð1Kz3 Þ 1 C 23 pr k xk dk3 dk 1Kz3

(3)

Gij Z

P   di dj 1 ð1=6Þpr k xk dk2 ð4lk K1Þ   P C 1Kz3 dij ð1Kz3 Þ 1K 23 pr k xk dk3 lk

(4)

and X 1 xk dk3 z3 h pr 6 k

(5)

where xi and xj are mole fractions of components, dkZ 0.22lk, Gij is the pair distribution function, di is the hard sphere diameter, and the summation runs over all components of the mixture. Knowledge of the intermolecular potential energy curve plays an important role in the equation of state, Eq. (2), in that all the temperature-dependent parameters, (B2)ij, aij, and bij, are related to the intermolecular pair potential. Song and Mason [13,16,17] used these equations for the prediction of computer simulation data on fluids over a wide range from perfect gas to compressed liquids. Also, they have developed the method to the case of simple systems that have accurate potential energy curves, reported in the literature. However, for many systems, an accurate intermolecular potential energy curve is unknown. It is the

If the intermolecular potential energy function is known, the temperature-dependent parameters of the equation of state, (B2)ij, aij, and bij, can be calculated. For many systems of interest no such accurate information is known. It is worth mentioning that the parameter B2 is dependent on the details of the potential energy curve and is strongly temperaturedependent, while the other two parameters do no depend on the details of the potential energy curve and their temperature-dependency is weak. Therefore, if the experimental second virial coefficients are known, they can be fitted with a mean-spherical potential to obtain the range parameter, rm, and the strength parameter, 3. It is shown [17] that reduced a and b are universal functions of the reduced temperature, when they are reduced with the Boyle parameters. In fact the Boyle temperature is used to reduce temperature and the Boyle volume is used to reduce a and b. Thus, the experimental second virial coefficients can be used to calculate the Boyle parameters, and hence, a and b. It is clear that the second virial coefficient plays an important role in the equation of state. It is used both directly and indirectly, for calculating a and b, in the equation of state. We have reviewed several correspondingstates correlations in the previous work [12] for the calculation of the second virial coefficients. The simplest one, which has been developed by Eslami to predict the equation of state for nonpolar fluids [8,9], LNG [10], and refrigerants [11] reads as:

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   2 Tbp Tbp K10:588 B2 rbp Z 1:033K3:0069 T T  3  4 Tbp Tbp C 13:096 K9:8968 T T

(6)

where rbp is the liquid density at the normal boiling temperature, Tbp. Knowing the second virial coefficient from Eq. (6), one can determine the Boyle temperature, TB, and the Boyle volume, VB, for scaling a and b. If the Boyle parameters are known, the empirical formulas [18] and the tabulated results [17] for a/VB and b/VB as a function of T/ TB are available in the literature. We have rescaled the empirical formulas by Song and Mason [18] in terms of normal boiling point constants to obtain [8]:   

T arbp Z a1 exp Kc1 Tbp   1=4 

T C a2 1Kexp Kc2 Tbp

(7)

and 



brbp Z a1 1Kc1

T Tbp

    T exp Kc1 Tbp

 1=4    1=4 

 Tbp Tbp Ca2 1K 1 C 0:25c2 exp Kc2 T T (8)

where a1 ZK0:0860

c1 Z 0:5624

a2 ZK2:3988

c2 Z 1:4267

These correlations for B2, a, and b, have been employed to predict the equation of state of a large number of nonpolar fluids including noble gases, diatomic molecules, saturated hydrocarbons, and a number of aliphatic, aromatic, and cyclic hydrocarbons over a wide range of temperatures and pressures within an accuracy of a few percent [8]. Determination of the temperature-dependent parameters of the equation of state by this procedure also self-adjusts the parameter l to 0.495 [8]. While the present method uses less input information than the other reported ones in the previous work [19–21] and is easier to apply, it is shown that it has nearly the same predictive power as the previous methods [19–21]. Employing Eqs. (6)–(8) in Eq. (1) we have obtained: p ðaKB2 Þr ar Z 1K C rkT 1 C 0:11br 1K0:495br

(9)

which is the most simplified version of Eq. (1). Our corresponding-states correlation for the calculation of the second virial coefficient [8] and the other two temperature-dependent parameters can be extended to mixtures by applying simple mixing rules for parameters rbp and Tbp. To mix the liquid density at the normal boiling point, a quadratic relation proposed by Nasrifar et al. [22] is used, i.e. ðTbp Þij Z ½ðTbp Þi ðTbp Þj 1=2 ð1KKij Þ

(10)

where Kij Z 1K

2½ðrbp Þi ðrbp Þj K1=2 1 K1 ðrbp ÞK i C ðrbp Þj

(11)

and

Table 1 The calculated orthobaric liquid densities for refrigerant mixtures DT (K)

Refrigerant mixture

Azeotrope of R-12 and R-152a (R-500) Azeotrope of R-12 and R-115 (R-502) Azeotrope of R-23 and R-13 (R-503) Azeotrope of R-32 and R-115 (R-504) Azeotrope of R-12 and R-31 (R-505) Azeotrope of R-31 and R-114 (R-506) Azeotrope of R-152a and R-218 (R-507) R-32CR-134a R-125CR-134a R-32CR-125CR-134a 0.6548 R-22C0.1547 R-152aC0.1905 R-124 (R-401) 0.5787 R-22C0.186 R-152aC0.2353 R-124 (R-401a) 0.5077 R-125C0.0462 R-290C0.4461 R-22 (R-402) 0.2999 R-125C0.0431 R-290C0.657 R-22 (R-402-38/2/60) 0.3577 R-125C0.604 R-143aC0.0383 R-134a (R-404a) a b

MDZmaximum value of ((jri,cal.Kri,Exp.j)/(ri,Exp.))!100. NP P AADZ ð100=NPÞ jri;cal:  ri;Exp: j=ri;Exp: . iZ1

200–350 200–330 150–260 190–310 172–360 227–380 230.5–320 220–340 280–330 280–340 233–323 233–373 223–293 223–303 220–303

NP

16 14 12 13 20 16 10 43 30 20 10 15 8 9 9

MDa (%)

2.93 3.02 2.86 3.51 3.38 3.08 3.22 2.53 4.20 3.55 2.98 3.32 3.53 3.90 3.71

AADb (%)

Ref.

This work

Ref. [22]

0.90 1.06 0.79 1.17 1.09 0.98 1.04 0.82 1.26 1.09 1.41 1.19 1.50 1.65 1.33

0.13 0.08 0.42 0.11 0.14 0.29 1.17 0.36 1.59

[23] [23] [23] [23] [24] [24] [23] [25] [25] [25] [23] [23] [23] [23] [26]

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Table 2 Calculated results on the density of refrigerants and their mixtures in the compressed state compared with experiment Refrigerant

DT (K)

DP (bar)

NP

AAD (%)

MD (%)

Ref.

0.0866 R-125C0.9134 R-134a 0.2674 R-125C0.7326 R-134a 0.4603 R-125C0.5397 R-134a 0.6653 R-125C0.3347 R-134a 0.9231 R-125C0.0769 R-134a 0.3955 R-32C0.645 R-134a 0.3819 R-32C0.1792 R-125C0.4389 R-134a 0.4639 R-32C0.0669 R-125C0.4692 R-134a 0.3577 R-125C0.604 R-143aC0.0383 R-134a (R-404a)

280–320 280–310 280–320 280–300 280–300 250–330 280–330 280–320 263–323

4.25–30.12 5.12–30.11 6.00–30.16 6.52–30.07 7.92–30.08 5.02–30.06 7.24–30.02 7.30–30.02 15–150

28 22 27 16 16 36 26 26 45

1.45 1.75 2.01 2.29 2.52 0.86 0.89 0.58 1.91

3.68 3.15 2.61 3.28 4.10 1.28 4.73 1.62 4.91

[25] [25] [25] [25] [25] [25] [25] [25] [26]

1 ðrbp Þij Z ½ðrbp Þi C ðrbp Þj  2

(12)

Knowing the parameters (rbp)ij and (Tbp)ij from Eqs. (10)–(12), the equation of state, Eq. (2), can be used to predict the PVT properties of refrigerant mixtures.

3. Results and discussion Using Eqs. (10)–(12) for the calculation of (Tbp)ij and(rbp)ij, we have employed correlation Eqs. (6)–(8) for the prediction of the equation of state, Eq. (2), for refrigerant mixtures. The calculated saturated liquid density of refrigerant mixtures, including several binary and ternary, mixtures are given in Table 1 and are compared with experimental data and a new correlation by Nasrifar et al. [22]. The equation of state is also examined to check the compressed liquid density of several refrigerant mixtures, for which accurate experimental data do exist in the literature [23–26], in Table 2. The predicted results are accurate over a wide range of temperatures and pressures. The present equation of state is also compared with one based on the heat of vaporization [11] in Fig. 1. As a typical example, we have selected 0.3953 R-32C0.6047 R-134a, a mixture of environmentally acceptable refrigerants, which has replaced the ozone-depleting ones, for which accurate experimental data are available in the literature [25]. Comparison of the predicted results in Fig. 1 shows that the present equation of state is more accurate and covers a much wider range of temperatures and pressures than the previous one [11]. Also the present equation of state is simpler than the previous one [11] in that the input data (normal boiling point parameters) are more readily available and the parameter l is self-adjusted, thus there is no need for it to be calculated for each substance. It is shown in this work that without knowing the detailed shape of the potential energy curve, the analytical equation of state by Song [15] can be applied successfully to refrigerant mixtures. A minimum input information, namely the normal boiling point constants, are sufficient for this purpose. Although compounds studied in this work have a variety of complexities, and hence, different intermolecular

forces, Eqs. (7) and (8) for a and b, respectively, which have been obtained using the results of a simple Lennard–Jones potential [8,18], can still well reproduce the experimental data. The reason is that the parameters a and b depend only on the repulsive branch of the potential energy function and are insensitive to the details of the intermolecular forces [17]. As it is shown by Song and Mason [16,17], the statistical-mechanical perturbation theory can be applied to determine the equation of state of real fluids if the intermolecular forces are known. For real fluids, however, the equation of state can be used with much less input information than the full potential, because the temperaturedependent parameters of the equation of state that depend only on the repulsion, a and b, are insensitive to the detailed shape of the potential and can be scaled with two fixed constants, TB and VB, or in our procedure by Tbp and rbp. In addition, the present method does not need the critical parameters [19] and such properties as the heat of vaporization [20] or surface tension [21]. It should be noted that the critical; parameters are scarce for many systems and are subject to a greater experimental

Fig. 1. Deviation plot for the liquid density of 0.3953 R-32C0.6047 R-134a at saturation state (%) and at 100 bar (C) compared with experiment [25]. The corresponding open markers represent the results from our previous work [11].

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H. Eslami et al. / International Journal of Refrigeration 29 (2006) 150–154

uncertainty in comparison to the boiling point parameters. Furthermore, the parameters li are all equal and selfadjusted to 0.495, while in the previous methods [19–21] they are substance-dependent properties. It is worth considering that there are more accurate equations of state in the literature for the prediction of the PVT properties of refrigerant mixtures, such as those reported my Lemmon and Jacobsen [5,6], but the present equation of state presents a fairly easy way to compute refrigerant mixture properties.

Acknowledgements We are indebted to the Research Council of Persian Gulf University for the Support of this work.

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