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Application of M4 cubic equation of state for refrigerants

Hossein Rezaei a, Hamid Modarress b,*, Mohsen Mohsen-Nia c, Mohsen Amiri b

aDepartment of Chemical Engineering, Amir Kabir University of Technology, Mahshahr, IranbDepartment of Chemical Engineering, Amir Kabir University of Technology, Tehran, IrancThermodynamic Research Laboratory, Kashan University, Kashan, Iran

a r t i c l e i n f o

Article history:

Received 6 January 2010

Received in revised form

6 April 2010

Accepted 2 June 2010

Available online 9 June 2010

Keywords:

Refrigerant

Equation of state

Comparison

Calculation

Saturation

Equilibrium

Abbreviation: VLE, Vaporeliquid equilibriuRedlich-Kwang EOS (Soave, 1972); PR, Peng-Rof Data Points; Ref, Reference.* Corresponding author. Tel.: þ98 21 6454317E-mail addresses: h.rezaei@aut.ac.ir (H. R

0140-7007/$ e see front matter ª 2010 Elsevdoi:10.1016/j.ijrefrig.2010.06.005

a b s t r a c t

In this work, M4 cubic equation of state (EOS) (proposed by Mohsen-Nia et al., 2003) is

applied to calculate the saturated properties of refrigerants. A wide range of different types

of refrigerants (CFCs, HCFCs, HFCS, etc.) is examined by calculating their saturated pres-

sures, saturated liquid and vapor molar volumes. The comparison between the VLE

calculation results and the experimental data, shows that the agreement of this EOS is

better than two frequently-used (SRK and PR) EOSs for 42 refrigerants and some of their

mixtures.

ª 2010 Elsevier Ltd and IIR. All rights reserved.

Application d’une equation cubique d’etat M4 aux frigorigenes

Mots cles : Frigorigene ; Equation d’etat ; Comparaison ; Calcul ; Saturation ; Equilibre

m; EOS, Equation of state; M4, An EOS proposed by Mohsen-Nia et al. (Eq.1); SRK, Soave-obinson EOS (Peng and Robinson, 1976); AAD, Average Absolute Deviation (%); NDP, Number

6; fax: þ98 21 66405847.ezaei), hmodares@aut.ac.ir (H. Modarress).ier Ltd and IIR. All rights reserved.

Nomenclature

z compressibility factor

N constant (2)

R gas constant (82.06 atm.cm�3 mol�1 K�1)

T temperature

P pressure

a EOS attractive parameter

b EOS co-volume parameter

x mole fraction

kij binary interaction parameter of mixture

Greek letters

y molar volume (cm�3 mol�1)

a constant (1.3191)

4 fugacity

Subscript

mix mixture

P pressure

Lv liquid molar volume

Vv vapor molar volume

R reduced

i, j indices for component

Superscript

v vapor

l liquid

Exp experimental

Cal calculated

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 3 5 0e1 3 5 5 1351

1. Introduction

Accurate knowledge of the thermodynamic properties of

refrigerants and their hazardous effects on atmosphere is

required to design the economically and environmentally

acceptable refrigeration cycles. Actually the most funda-

mental tool in providing a basis to predict the properties of

refrigerants is the equation of state (EOS). So the develop-

ment and/or modification of available equations for accu-

rately prediction of properties of alternative refrigerants are

more pronounced (Nasrifar and Moshfeghian, 1998). Equa-

tions of state can be applied to pure substances as well as to

mixtures, and therefore a very large number of publications

deal with the development or improvement of equations of

state (Goharshadi and Moosavi, 2006). Nasrifar and

Moshfeghian (1999) give a review on the comparison of

fourteen correlations and four equations of state for

calculating the liquid densities of 15 refrigerants. There are

also other correlation schemes such as the corresponding-

states liquid densities by Hankinson and Thomson (1979)

and the modified Racket correlation by Spencer and

Danner (1973) for the accurate prediction of the saturated

liquid densities of refrigerants (Eslami, 2004). The correla-

tion developed by Iglesias-Silva and Hall (1997) has been

compared with the corresponding-states liquid densities

(Hankinson and Thomson, 1979) and the modified Rackett

correlation (Spencer and Danner, 1973) by Nasrifar et al.

(1999) and is extended to multicomponent mixtures.

Feroiu and Geana (2003) applied three EOSs to predict the

volumetric and thermodynamic properties of three pure

refrigerants (R32, R125 and R134a) as well as their mixtures.

The density of 11 refrigerants has been calculated using

GoharshadieMorsalieAbbaspour equation of state (GMA

EOS) by Goharshadi and Moosavi (2007). Eslami and

Farrokhnia (2005) employed a modified perturbed hard-

sphere-chain equation of state to halogenate organic

compounds. In recent years, the thermodynamic properties

of refrigerant mixtures have been investigated by several

researchers (Moshfeghian et al., 1992; Kiselev et al., 1999;

Ahlers and Gmehling, 2001; Quinones-Cisneros et al., 2005;

Eslami et al., 2006). Today, refrigerants are divided to three

specific types: Chlorofluorocarbons (CFCs),

Hydrochlorofluorocarbons (HCFCs) and Hydrofluorocarbons

(HFCs). In 1987, the modification of the Montreal protocol

has prohibited the use and the production of CFCs in

industrialized nations (Feroiu and Geana, 2003). The HCFCs

are less stable in the lower atmosphere, enabling them to

break down before reaching the ozone layer, so they are of

promising substitutes for CFCs at present (Goharshadi and

Moosavi, 2005). Later alternative refrigerants (HFCs) are

lacking the chlorine and have an even shorter life times in

the lower atmosphere. The VLE calculations for all of above

types of refrigerants have been done in this work using the

general M4 EOS (Mohsen-Nia et al., 2003). The results are

compared with Soave-Redlich-Kwong (SRK) (Soave, 1972)

and PengeRobinson (PR) (Peng and Robinson, 1976) EOSs for

42 pure refrigerants and their mixtures.

2. Theoretical basis and calculations

A general cubic equation of state has been recently proposed

by Mohsen-Nia et al. (2003). “M4 EOS”, is a two parameter EOS

in the following form:

z ¼ vþ abv� b

� a

RT1:5ðvþNabÞ (1)

where z, v, T and R respectively are compressibility factor,

molar volume, temperature and gas constant. Mohsen-Nia

et al. (2003) found the values of 2 and 1.3191 respectively for

N and a. Parameters a and b is expressed as:

a ¼ aC

�1þm

�1� T0:5

r

��2(2)

b ¼ bC

�1þ n1

�1� T0:5

r

�þ n2

�1� T0:75

r

��2for Tr < 1 (3)

where

aC ¼ 0:47312�R2T2:5

C =PC

�(4)

bC ¼ 0:04616ðRTC=PCÞ (5)

and

m ¼ 0:32ð1þ 2uÞ (6)

n1 ¼ 3:270572� 6:4127uþ 10:6821u2 (7)

n2 ¼ �1:72192þ 3:85288u� 7:202286u2 (8)

Table 1 e Absolute average deviations percent of saturated properties.

N Comp. Tr NDPa AADpð%Þ AADlvð%Þ AADvvð%Þ AADðpþlvÞð%Þ AADðpþlvþvvÞð%ÞM4 SRK PR M4 SRK PR M4 SRK PR M4 SRK PR M4 SRK PR

1 R11 0.34e0.98 27 0.71 1.17 5.19 4.71 9.03 5.06 1.27 1.32 4.32 5.42 10.20 10.26 6.68 11.54 14.58

2 R12 0.38e0.98 24 1.82 1.21 3.31 4.41 9.47 5.38 4.77 3.11 4.75 6.23 10.68 8.69 11.01 13.81 13.44

3 R13 0.37e0.96 19 2.32 1.66 4.49 4.56 6.60 6.64 3.25 1.32 3.79 6.88 8.26 11.13 10.13 9.58 14.91

4 R14 0.43e0.96 13 0.61 1.84 1.35 4.57 5.75 7.35 2.01 1.98 2.11 5.18 7.59 8.68 7.19 9.58 10.79

5 R21 0.55e0.95 19 0.95 0.65 0.22 3.92 14.10 4.76 1.89 1.2 0.89 4.87 14.74 4.98 6.77 16.04 5.88

6 R22 0.36e0.99 24 1.17 1.16 4.18 5.19 14.02 3.53 1.43 0.10 3.91 6.31 15.18 7.73 7.75 16.18 11.64

7 R23 0.39e0.96 18 3.50 3.47 1.09 7.45 19.93 6.25 3.91 4.76 2.54 10.95 23.41 7.35 14.87 28.17 9.89

8 R32 0.41e0.93 19 5.06 5.64 1.18 15.91 31.96 17.04 6.48 8.72 4.18 20.97 37.61 18.22 27.45 46.33 22.41

9 R41 0.55e0.96 14 1.47 2.15 0.22 16.16 29.65 14.67 3.74 5.87 4.95 17.63 31.81 14.91 21.37 37.68 19.87

10 R50 0.48e0.95 10 0.99 2.77 0.30 3.06 4.33 8.29 1.53 3.08 0.77 4.06 7.10 8.60 5.59 10.18 9.37

11 R113 0.49e0.98 25 0.74 0.72 0.73 4.14 11.26 4.38 1.93 0.59 1.15 4.88 11.97 5.11 6.81 12.58 6.27

12 R114 0.45e0.98 23 0.87 1.27 1.14 6.68 7.95 7.16 2.01 1.73 1.81 7.54 9.22 8.29 9.56 10.95 10.10

13 R115 0.49e0.97 18 0.77 1.33 0.60 4.34 9.93 4.54 1.60 1.43 0.95 5.11 11.27 5.14 6.71 12.70 6.10

14 R116 0.59e0.97 12 0.61 1.01 0.15 6.34 8.59 5.81 1.96 0.87 0.67 6.96 9.61 5.96 8.92 10.48 6.64

15 R123 0.38e0.98 28 0.67 1.51 4.74 4.26 13.42 2.87 1.05 1.47 4.11 4.92 14.93 7.61 5.98 16.41 11.72

16 R124 0.38e0.99 24 0.73 1.23 4.71 4.72 12.82 3.49 0.98 1.50 4.11 5.45 14.05 8.20 6.43 15.55 12.31

17 R125 0.54e0.98 16 0.46 0.73 0.35 4.61 14.34 3.54 0.90 1.98 0.95 5.07 15.08 3.90 5.97 17.07 4.84

18 R134a 0.45e0.96 21 1.23 1.40 1.59 4.41 18.46 5.00 1.19 2.27 2.39 5.63 19.86 6.52 6.83 22.14 8.91

19 R141b 0.35e0.98 31 1.92 2.57 6.87 4.89 12.54 3.52 2.67 2.45 5.497 6.81 15.11 10.40 9.48 17.57 15.89

20 R142b 0.37e0.98 26 0.74 1.16 4.11 4.94 14.41 3.02 1.16 1.37 3.73 5.68 15.58 7.12 6.85 16.96 10.85

21 R143 0.46e0.98 20 2.38 2.87 0.28 6.02 19.45 5.82 2.81 4.64 2.32 8.40 22.31 6.11 11.22 26.96 8.43

22 R143a 0.46e0.98 19 1.94 2.42 0.34 8.40 22.10 8.23 2.10 3.92 2.12 10.34 24.52 8.57 12.44 28.45 10.70

23 R152a 0.39e0.97 23 3.65 3.62 0.75 9.81 23.26 9.21 3.93 4.84 2.29 13.45 26.89 9.96 17.39 31.73 12.25

24 R170 0.31e0.96 21 1.51 3.94 3.66 6.23 7.467 5.99 2.00 4.30 3.64 7.74 11.40 9.66 9.74 15.71 13.30

25 R218 0.39e0.97 21 1.13 1.06 4.08 6.03 8.763 5.23 1.60 0.98 3.62 7.16 9.82 9.31 8.77 10.81 12.93

26 R227ea 0.39e0.97 23 1.02 1.98 5.69 6.33 9.50 4.60 0.99 1.91 4.95 7.35 11.48 10.29 8.34 13.38 15.24

27 R236ea 0.58e0.97 17 2.07 2.27 0.16 8.27 10.82 5.45 8.60 7.63 6.60 10.38 13.09 5.61 18.95 20.73 12.21

28 R236fa 0.45e0.98 22 1.45 1.24 2.67 5.16 13.63 2.87 1.32 1.71 3.30 6.60 14.87 5.55 7.93 16.58 8.85

29 R245ca 0.45e0.98 25 1.28 1.44 1.86 5.29 11.62 3.37 1.66 3.13 3.70 6.57 13.07 5.23 8.23 16.20 8.94

30 R245fa 0.47e0.98 23 1.63 1.36 2.12 5.37 14.42 3.23 2.01 1.12 2.24 7.01 15.78 5.36 9.01 16.91 7.60

31 R290 0.31e0.96 17 1.84 2.44 5.49 5.65 8.77 5.05 2.73 2.47 4.66 7.50 11.21 10.54 10.23 13.69 15.20

32 RC318 0.60e0.99 16 0.80 0.67 0.17 8.03 10.68 5.53 2.29 0.72 0.89 8.82 11.35 5.71 11.12 12.06 6.60

33 R600 0.31e0.99 20 3.58 2.83 4.74 5.74 11.04 4.62 4.80 3.05 4.11 9.33 13.87 9.37 14.13 16.92 13.48

34 R600a 0.31e0.98 19 1.51 1.30 6.61 5.56 10.07 4.71 2.31 1.37 5.37 7.06 11.38 11.40 9.38 12.75 16.71

35 R702 0.42e0.99 19 5.57 6.45 0.11 6.09 10.92 14.98 5.36 9.06 1.09 11.66 17.37 15.10 17.03 26.44 16.19

36 R704 0.42e0.99 31 9.68 8.80 0.07 13.53 13.61 19.48 8.98 14.64 1.55 23.21 22.41 19.55 32.19 37.05 21.11

37 R717 0.48e0.92 13 2.54 3.11 0.59 13.85 28.38 13.54 3.88 5.52 3.51 16.39 31.49 14.14 20.27 37.01 17.64

38 R720 0.55e0.98 20 0.77 2.08 0.11 3.83 4.85 11.91 2.47 2.18 0.57 4.60 6.93 12.01 7.08 9.12 12.59

39 R728 0.5e0.99 13 0.98 1.39 0.19 3.39 5.18 8.91 2.04 1.71 1.04 4.38 6.57 9.10 6.43 8.28 10.1

40 R732 0.35e0.97 20 2.01 3.94 1.81 5.43 3.58 8.88 2.50 4.38 2.16 7.44 7.52 10.69 9.951 11.90 12.8

41 R744 0.71e0.98 21 0.29 0.51 0.06 4.82 14.47 4.59 1.28 1.27 0.55 5.11 14.99 4.65 6.40 16.26 5.20

42 R764 0.50e0.98 13 0.86 1.09 0.29 3.10 7.09 6.80 1.39 1.61 0.54 3.958 8.18 7.093 5.35 9.79 7.64

Average 20.2 1.81 2.18 2.11 6.32 12.82 6.79 2.69 3.11 2.83 8.12 14.98 8.902 10.81 18.10 11.72

N: Number, NDP: Number of data.

a Data are from NIST Chemistry webbook. AADp¼1=NDPPNDP

i

jPexpi �Pcali j=Pexpi ; AADlv¼1=NDPPNDP

i

jvl;expi �vl;cali j=vl;expi ; AADvap¼1=NDPPNDP

i

jvv;expi �vv;cali j=vv;expi ; AADðpþlvÞ ¼ðAADpþAADlvÞ; AADðpþlvþvvÞ¼ðAADpþAADlvþAADvvÞ.

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 3 5 0e1 3 5 51352

where u, TC and PC are respectively acentric factor, critical

temperatures and critical pressure which are used as the only

initial parameters.

The saturation pressure and saturation molar volume for

eachtemperaturewerecalculatedbyusingequalityof fugacities

at vaporeliquid equilibrium for pure substances (Walas, 1984):

4v ¼ 4l (9)

The final expression for the fugacity coefficient of pure

substanceswithoriginalN ¼ 2anda ¼ 1:3191will beobtainedas:

ln4 ¼ z� 1� lnz� 2:3191� lnð1� b=vÞ � a2:6382RT1:5b

� lnð1þ 2:6382b=vÞ (10)

In order to use the M4 EOS for mixtures, van der Waals

mixing rules was used:

amix ¼Xk

i¼1

Xk

j¼1

xixj

�aiaj

�0:5�1� kij

�(11)

bmix ¼Xk

i¼1

xibi (12)

where xi is the mole fraction and ai and bi are the AEOS

parameters of component i in pure state. kij is the binary

interaction coefficient between components i and j. k is the

number of components. The fugacity coefficient of compo-

nent i in the mixture is as follow:

100

102

104

106

108 100

200

300

400

500

10-4

10-2

100

102

T (K)

ν (cm3.mol -1)

P (a

tm)

Fig. 1 e Three-dimensional saturated diagram of molar volume-temperatureepressure for (C: R11, -: R22 and :: R23)

respectively as the samples of CFCs, HCFCs and HFCs. Lines are calculated by M4 EOS (dd: R11, ------: R22 and $$$$$$$: R23).

Table 2 e Absolute average deviation percent in saturated pressure (AADpð%Þ) and in vapor mole fraction (AADpð%Þ) of VLEcalculations of refrigerants binary systems with kij[0.

System T(k) NDP AADpð%Þ AADpð%Þ Ref.

M4 SRK PR M4 SRK PR

R134a þ R124 307.25 8 1.81 1.91 2.15 3.31 3.33 3.28 Lee et al. (1996)

302.25 9 1.95 2.09 2.35 3.01 3.07 3.01 Lee et al. (1996)

296.45 6 1.81 2.00 2.83 1.51 1.50 1.51 Lee et al. (1996)

R744 þ R227ea 323.15 13 7.27 8.81 7.13 14.71 13.55 16.72 Valtz et al. (2003)

R134a þ R236fa 303.68 10 1.74 1.26 0.29 0.58 0.65 0.50 Bobbo et al. (1998)

283.62 9 1.41 0.72 0.32 0.65 0.68 0.43 Bobbo et al. (1998)

R116 þ R134a 273.32 9 16.74 17.56 18.3 10.83 9.55 10.05 Madani et al. (2008)

263.43 10 22.51 24.09 21.91 12.13 11.24 11.98 Madani et al. (2008)

R744 þ R600a 394.26 5 0.78 0.68 0.30 37.91 37.94 37.97 Besserer and Robinson (1973)

377.59 7 1.53 1.64 0.89 32.00 32.04 32.22 Besserer and Robinson (1973)

310.92 8 8.98 13.15 12.12 10.33 7.91 6.39 Besserer and Robinson (1973)

R143a þ R600a 333.15 10 9.44 9.17 7.06 12.92 12.57 12.43 Yun et al. (2008)

318.15 10 7.97 9.58 8.49 14.35 13.81 13.03 Yun et al. (2008)

303.15 11 9.67 9.44 8.79 12.92 12.65 12.41 Yun et al. (2008)

R125 � R152a 293.15 9 2.51 3.47 2.29 3.11 4.26 4.21 Nishiumi et al. (1997)

273.15 5 8.39 8.29 7.70 8.13 9.56 7.53 Nishiumi et al. (1997)

268.15 8 8.13 9.82 7.53 9.01 9.82 10.00 Nishiumi et al. (1997)

R32 � R123 313.95 10 6.97 9.69 9.72 2.61 2.81 1.35 Lee et al. (1998)

304.55 8 8.43 11.91 12.09 1.92 2.23 0.63 Lee et al. (1998)

R32 � R142b 314.95 8 3.16 4.02 3.63 1.68 2.01 1.73 Lee et al. (1998)

304.55 8 3.63 4.58 4.34 1.62 2.00 1.60 Lee et al. (1998)

R32 � R125 323.15 8 1.44 7.52 0.28 1.05 2.03 0.95 Lee et al. (1999)

303.15 9 1.07 1.11 0.18 0.86 0.82 0.83 Lee et al. (1999)

R32 � R143a 323.15 7 1.18 1.06 0.27 1.52 1.47 1.50 Lee et al. (1999)

303.15 8 1.42 1.56 1.41 1.31 1.44 1.32 Lee et al. (1999)

R134a � R600 303.68 11 20.55 20.67 22.41 18.86 19.04 17.83 Bobbo et al. (1998)

293.66 16 25.24 26.43 26.44 27.68 27.96 27.84 Bobbo et al. (1998)

R600a � R236fa 303.68 15 21.34 21.24 23.52 19.877 19.74 19.24 Bobbo et al. (1998)

NDP: Number of data points, Ref.: Reference.

AADp ¼ 1=NDPPNDP

i

jPexpi � Pcali j=Pexpi ; AADy ¼ 1=NDPPNDP

i

jyexpi � ycali j=yexpi .

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 3 5 0e1 3 5 5 1353

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

2

3

4

5

6

7

8

x1 , y 1

P (a

tm)

Fig. 2 e VLE calculation result for R134a (1)/R236fa (2). Lines are calculated by M4 EOS (dd: kij [ 0, ------: kij [ L0.008 and

$$$$$$$: kij [ L0.005) points are experimental data (Bobbo et al., 1998) at temperatures (C: 283.62 K and -: 303.68 K).

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 3 ( 2 0 1 0 ) 1 3 5 0e1 3 5 51354

ln4i ¼B

bmixðz� 1Þ � lnz� 2:3191� lnð1� bmix=vÞ

þ amixB=bmix �A2:6382RT1:5bmix

� lnð1þ 2:6382bmix=vÞ (13)

By using of Eqs. (11) and (12) for amix and bmix, A and B in Eq.

(13) will be obtained (Walas, 1984):

A ¼ v�n2amix

��nvni ¼ 2

Xk

l

xlail (14)

B ¼ vðnbmixÞ=vni ¼ bi (15)

where ai and bi are given by Eqs. (2) and (3).

3. Results and discussion

The ability of M4 EOS to predict the saturated properties for all

pure refrigerants and their mixtures may be evaluated by

absolute average deviation (AAD). The percent of AAD values

between calculated and experimental saturated pressures

(AADp%), saturated liquid molar volumes (AADlv%) and satu-

ratedvapormolarvolumes (AADvv%)andalsothesummationof

these deviations are reported in Table 1 for 42 refrigerants. The

values of AAD in comparison with those of SRK and PR EOSs

confirm the ability ofM4 EOS inVLE calculations of refrigerants.

It isworthmentioning that theparametersofM4EOSused in the

calculations are the general parameters of the EOS were not

evaluatedbyfitting therefrigerants. Fig.1 isa three-dimensional

saturated diagram of temperatureepressure molar volume

calculated by M4 EOS for R11, R22 and R23 respectively as

examples of CFCs, HCFCs and HFCs. In the calculations, the

binary interaction parameter kij ¼ 0 which indicates the real

abilityofEOSswithout theneedofkij asanadjustableparameter

in the mixing rules (Eq. (11)). The results for several binary

mixtures of refrigerants are represented inTable 2. Also theVLE

calculations of R134a (1)/R236fa (2) system using M4 EOS is

shown in Fig. 2 with zero and non-zero binary interaction

coefficients to demonstrate its ability with kij ¼ 0.

4. Conclusions

The saturated properties of a wide range of refrigerants were

calculated by recently developed M4 cubic EOS (Mohsen-Nia

et al., 2003) and the results were compared with SRK and PR

EOSs. The calculated AAD (%) values for 42 pure refrigerants

demonstrate the capability of M4 EOS. The three-dimensional

temperatureepressure molar volume diagram of the most

popular refrigerants (CFCs, HCFCs and HFCs) shows the good

agreement with experimental data by M4 EOS. Also the VLE

calculations for mixtures were done using M4 EOS and the

agreement of the result with experimental datawas compared

with those of SRK and PR EOSs.

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