ISSN 0036�0244, Russian Journal of Physical Chemistry A, 2014, Vol. 88, No. 6, pp. 978–983. © Pleiades Publishing, Ltd., 2014.

978

1 INTRODUCTION

The energy economy and emission regulationshave become more rigorous and the need for highenergy efficiency and low emission technologies hasemerged. In the thermal cycles such as refrigerationcycles or heat pump cycles, organic Rankine cycles(ORC) the chosen of the suitable working fluids canhelp not only to reduce energy consumption but alsoto decrease the global warming and ozone depletionproblems. For the reasons, the researches and studieson the looking for the next�generation refrigerants andworking fluids are still attracted attentions of scien�tists, companies, and government organizations. Oneof typical groups of the promising working fluids forthe ORC and refrigerants is hydrocarbon because ithas a very low global warming potential (GWP) andthe ozone depletion potential (ODP) of zero.

Cyclopropane has excellent environmental charac�teristics. In detail, its ODP and GWP are zero and 20[1], respectively. The cyclopropane is one of thepotential hydrocarbon working fluids for the ORC andrefrigerants and it has received many attentions fromresearchers. The primary thermodynamic propertiesof the cyclopropane have been reported in differentrefereed sources [2–14]. Lin et al. [7, 8] and Ruehr�wein and Powell [9] carried experimental studies andpublished the experimental vapor pressures in thetemperature range of from 183 to 398 K. Lin et al. [8]and Costa Gomes et al. [10, 11] published experimen�

1 The article is published in the original.

tal saturated liquid densities of the cyclopropane in thetemperature range of from 152 to 398 K. The ideal gasheat capacity, important property for the calculationof caloric properties such as enthalpy, heat capacity,entropy, and internal energy is also published by vari�ous authors for the temperature range of from 100 to6000 K [12–14]. Besides the above important primarythermodynamic properties, the critical properties ofthe cyclopropane have also been well published [2–6].

In the investigation of the possibility of the applica�tion of the potential working fluids and alternativerefrigerants for organic Rankine cycles and heat pumpcycles, [1, 15, 16], the mentioned above primary ther�modynamic properties can not be used because thereare no experimental data for entropy. The latter canonly be determined by equation of state.

The equations of state models can be divided intodifferent categories such as cubic equation of state,Bender�type equation of state, multi�parameter equa�tion of state (EOS), PC�SAFT EOS, BACKONEEOS. The most typical cubic equations of state areequations of Van der Waals [17], Redlich and Kwong[18], Soave [19], Peng and Robinson [20], Stryjek andVera [21]. The cubic equations of state are well�know forlimitation of accuracy. Jacobsen et al. [22] tried with var�ious experimental p–ρ–T data for R�134a and found outthat an average absolute error in the liquid region is of 5to 10% and in the vapor region is of 2 to 5%.

Bender modified the Webb and Rubin EOS [23],which was published in 1940, to improve the accuracyof the EOS in the 1970s [24]. The Bender�type EOS

Description of Cyclopropane with Backone Equation of State1

Ngoc Anh Lai* and Thai Son PhamHeat Engineering Department, Hanoi University of Science and Technology, Hanoi, Vietnam

*e�mail: anhngoclai@yahoo.comReceived April 25, 2013

Abstract—This paper aims to accurately describe the thermodynamic properties of Cyclopropane with amolecular based BACKONE equation of state. The parameters of the BACKONE equation of state found byfitting to experimental vapor pressures and liquid densities are the characteristic temperature T0, character�istic density ρ0, anisotropy factor α, and reduced quadrupolar moment Q*2. The values of these parametersare 393.9583 K, 6.076139 mol/L, 1.295445, and 0.699483, respectively. The average absolute deviationbetween experimental values and those derived from BACKONE EOS is 0.29% for vapor pressures, 0.75% forsaturated liquid densities. The prediction power of the BACKONE equation of state are investigated. It isshown that the uncertainties of values derived from the BACKONE equation of state are within 0.90% for iso�baric densities in the liquid phase and 2.0% for enthalpy of evaporation.

Keywords: alternative refrigerants, cyclopropane (RC270), BACKONE, equation of state (EOS), thermody�namic properties.

DOI: 10.1134/S003602441406020X

STRUCTURE OF MATTERAND QUANTUM CHEMISTRY

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 6 2014

DESCRIPTION OF CYCLOPROPANE WITH BACKONE EQUATION OF STATE 979

have been used to describe thermodynamic propertiesof the cyclopropane [4]. The estimated uncertaintiesof the Bender�type EOS for the cyclopropane for den�sity, vapor pressure, and heat capacities are 1, 2, and5%, respectively.

The multi�parameter equations require a largenumber of accurate experimental data in large spacefor the construction of the equations [25, 26]. Theextension from multi�parameter equations of state forpure fluids to mixtures is a big problem. Up to now, theextension to mixtures has only been applied for naturalgas which has a large number of experimental data formixtures. PC�SAFT EOS and BACKONE EOS aremolecular based EOS which have only three to fivesubstance specific parameters, [27, 28]. The mostadvantages of these EOS are that they are accurate andcan be easily extended to mixtures with only oneparameter for each binary. The PC�SAFT EOS ismore suitable for chain�like molecule and the BACK�ONE is more suitable for compact molecule.

The cyclopropane is a compact molecular and is apotential alternative refrigerant [15–17]. The currentcommon equation of state for the cyclopropane isBender�type EOS with the uncertainties of 1, 2, and5% for density, vapor pressure, and heat capacities,respectively, [4]. Unfortunately, the extension of thebender�type EOS to mixture is rather difficult. Thus,in order to investigate the possibility of the applicationof the mixture of the cyclopropane with other sub�stances other types of EOS should be considered

instead of the Bender�type EOS. The BACKONEEOS currently proves to be an accurate EOS for smalland compact molecules and can be extended to mix�ture easily, [28–32]. Thus in this study, the BACK�ONE EOS is used to describe thermodynamic proper�ties of the cyclopropane.

AVAILABLE EXPERIMENTAL DATA

Typical critical properties of the cyclopropane arecompiled in Table 1 including the critical temperature(Tc), the critical pressure (pc), and the critical density(ρc). As seen in the Table 1, almost all the critical prop�erties from different sources are nearly identical.Liessmann et al. [2] published a large number ofexperimental data, thus the critical data from [2] werechosen in this study.

In order to develop the BACKONE EOS for cer�tain fluids, vapor pressure and saturated liquid densityare used in the fitting for finding the substance charac�teristic specific parameters. The p–V–T data in thesingle phase and/or caloric properties are usually usedin the evaluation of the possibility of the prediction ofthe EOS for the studied substance.

There are various available experimental data setsfor the vapor pressure and saturated liquid density ofthe cyclopropane. Table 2 presents the vapor pressuresused in the construction of BACKONE EOS for thecyclopropane. In order to distribute the equal weightfor the experimental data fitting range and to evaluateand eliminate the unusual data points, the auxiliaryequations for vapor pressure and saturated liquid den�sity are employed. In this study, the Wagner equation,for vapor pressure [33] is used for the evaluation of theavailable experimental data:

(1)

where T is temperature in K, τ = 1 – T/Tc. We first fit�ted data in the Table 2 to the above Wagner equation.We found out that data points at 313.889 in [7] and393.15 K in [8] have strange behaviour thus wedropped these two data points and refitted. The finalparameters of the Wagner equation are A = –6.60088,B = 1.27594, C = –1.78063, and D = –2.07475. TheTable 2 includes temperature ranges, vapor pressureranges, experimental data sources, the average abso�lute deviation (AAD) between experimental data andthose from Wagner equation, and the maximum devi�ation (MD) between experimental data and thosefrom Wagner equation. The Table 2 shows that the datasources [7, 8] have the same AAD of 0.04% and thedata source [9] has the highest AAD of 0.15%. TheAAD for all the data from the three data sources is0.06%.

The available experimental data for saturated liquiddensity [8, 10] are in the range from 293.15 to397.55 K and from 152.89 to 175.56 K, respectively.The experimental data range is from reduce tempera�

p/pc( )ln Aτ Bτ1.5

Cτ3

D6

+ + +( )/ T/Tc( ),=

Table 1. Experimental critical temperature (Tc), criticalpressure (pc), critical density (ρc), and sources of the cyclo�propane

Tc, K pc, kPa ρc, mol dm–3 Sources

398 5540 6.154987 [3]

398.250 5574.90 6.143105 [2]

398.300 5579.70 6.143105 [4]

398.300 5579.00 6.142883 [5]

398.300 5579.00 6.142883 [6]

Table 2. Experimental temperature range (T), and vaporpressure range ps, data sources, the AAD and MD from theWagner equation for the cyclopropane

T, K ps, kPa Sources AAD, %

MD, %

291.667–397.223 605.2–5479.2 [7] 0.04 0.18

293.15–393.15 631.5–5565.5 [8] 0.04 0.11

183.121–241.067 3.5–104.7 [9] 0.15 0.64

980

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 6 2014

NGOC ANH LAI, THAI SON PHAM

ture of 0.38 to 1 thus in order to evaluate experimentaldata of saturated liquid density, the generalized Rack�ett equation, recommended by Daubert [34], withfour parameters is employed:

(2)

where T is temperature in K. The experimental data ofsaturated liquid densities of [8, 10] were fitted to thegeneralized Rackett equation. The parameters of thegeneralized Rackett equation are A = 1.42272,B = ⎯1.63206, C = 399.266, and D = 0.187411. TheAADs between experimental data and those fromEq. (2) for the data of Lin et al. [8] and Costa Gomeset al. [10] are 0.04 and 0.21%, respectively. The AADfor two data sources is 0.15%. Experimental data of theisobaric density in the liquid phase, [11], are used inthe evaluation of the predictive capability of theBACKONE for the cyclopropane.

In order to calculate the caloric properties withBACKONE EOS such as enthalpy, entropy, heat

capacity, the ideal gas heat capacity is required.There are various data sources for the ideal gas heatcapacity of the cyclopropane [12–14]. In order toevaluate the ideal gas heat capacity from differentsources the equation (3) is used.

(3)

where R is the ideal gas constant, R =8.314472 J mol–1 K–1 and k1j, k2j are parameterswhich are found by fitting with available data. In thisstudy, the reliable temperature range for BACKONEfor cyclopropane is from ~150 to 630 K. Thus the tem�perature range in fitting and evaluation of the ideal gasheat capacity is from 150 to 630 K. The parameters ofthe equation for ideal gas heat capacity are as follows:k11 = 4.50339, k21 = –161.647, k12 = 140.29, k22 =7056.34, k13 = 13.696, k23 = 1513.92. The AADsbetween data from the above equation for ideal gasheat capacity and those from Dorofeeva et al. [12],Burcat [13], and Linnett [14] are 0.73, 0.76, and0.25%, respectively.

THE BACKONE EOSFOR THE CYCLOPROPANE

The molecular based BACKONE EOS was devel�oped by Fischer and co�workers [28, 29] on the basicof the original BACK (Boublik–Alder–Chen–Kre�glewski) equation [35–37]. The BACKONE expres�sion for the Helmholtz free energy, F, is given by the

ρ'( )ln A B 1 T/C–( )D

,+=

cp0

cp0/R

= k1 j k2 j/T( )2

k2 j/T( )/ 1 k2 j/T( )exp–{ }2

exp[ ],

j 1–3=

∑

sum of characteristic intermolecular interactions:F = FH + FA + FD + FQ, where FH is the contributionof the hard�sphere [28]:

(4)

Here, ξ is the packing fraction and α is the anisotropyparameter, Müller et al. [28]. In the BACKONE, α isassumed to be state independence but the packingfraction ξ is state dependent. The packing fraction is afunction of characteristic critical density ρ0, charac�teristic critical temperature T0.

FA stands for the contribution of the attractive dis�persion forces [28],

(5)

All the exponents and coefficients of Eq. (5) werefound by fitting to experimental data of ethane, meth�ane and oxygen. The values of all the exponents andcoefficients of the FA were given in Müller et al. [28].FD and FQ are defined as dipolar contribution and qua�drupolar contribution, respectively [28],

(6)

and

(7)

where μ* is a reduced dipole moment, and Q* is areduced quadrupole moment. The exponents and coef�ficients of Eqs. (6) and (7) were determined by fitting tosimulation data and were given in Müller et al. [28].

Depending on the molecular structure of sub�stances, the BACKONE EOS contains three to fivespecific parameters as characteristic temperature T0,characteristic density ρ0, anisotropy factor α, reduceddipolar moment μ*2, and reduced quadrupolar momentQ*2. These specific parameters can be determined byfitting to experimental data of vapor pressures and sat�urated liquid densities.

The four parameters BACKONE EOS for thecyclopropane were found, T0 = 393.76 K,ρ0 = 6.0646 mol dm–3, α = 1.2959, Q*2 = 0.7077, by fit�ting with experimental data ranges of from 3.50 to5.575 MPa for vapor pressure, 9.5963 to18.402 mol dm–3 for saturated liquid density, and

FH/RT α2

1–( ) 1 ξ–( )ln=

+ α2

3α+( )ξ 3αξ2

–{ }/ 1 ξ–( )2.

FA/RT

= ci ρ/ρ0( )mj

T/T0( )ni/2

αj i

oi ρ/ρ0( )li

–{ }exp[ ].

i

∑

FD/RT ci ρ/ρ0( )mi/2

T/1.13T0( )ni/2

μ*2

( )ki/4

[

i

∑=

× oi ρ/ρ0( )2

–{ }exp ]

FQ/RT ci ρ/ρ0( )mi/2

T/1.13T0( )ni/2

Q*2

( )ki/4

[

i

∑=

× oi ρ/ρ0( )2

–{ }exp ],

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 6 2014

DESCRIPTION OF CYCLOPROPANE WITH BACKONE EQUATION OF STATE 981

152.9 to 397.2 K for temperature. These parameterscan be used to calculate thermodynamic properties ofthe cyclopropane not only in two phase region but alsoin single phase region.

Next we turn to the results derived from theBACKONE EOS for the cyclopropane. Figure 1compares the calculated values from the BACKONEEOS and experimental data of vapor pressure. As wecan see in this figure, the relative differences betweenthe values from the BACKONE EOS and the experi�mental values range from –0.48 to +0.60%. The AADbetween experimental vapor pressures of Lin and

McKetta [7], Lin et al. [8], Ruehrwein and Powell [9]and those from BACKONE EOS is 0.22, 0.22, and0.50%, respectively. The AAD for all the data from thethree data sources is 0.29%.

The deviations between experimental saturated liq�uid densities and those from BACKONE EOS aregiven in Fig. 2. The AAD between experimental datafrom Lin et al. [8], Costa�Gomes et al. [10] and thosefrom BACKONE EOS is 1.52 and 1.04%, respectively.The AAD between the data from BACKONE EOS andall experimental vapor pressures and saturated liquiddensities of the cyclopropane is 0.29 and 0.75%,respectively. These values prove that BACKONE EOSis good for representing saturated data of the cyclopro�pane.

In order to evaluate the predictive power of theBACKONE EOS for the cyclopropane, densities inthe isobaric liquid phase from [11] and enthalpy ofevaporation from Liessmann et al. [2] are used. As wecan see in the Fig. 2, the uncertainties are mostlywithin 1.0% for densities in the isobaric liquid phaseand the AAD is only of 0.90%. Table 3 presents thecomparison between values of enthalpy of evaporationfrom Liessmann et al. [2] and those from the BACK�ONE EOS. The AAD between values of enthalpy ofevaporation from Liessmann et al. and those fromBACKONE EOS is 2.0%. If one drops the point at393.15 K the AAD is only of 1.3%. These results provethat BACKONE EOS is a good tool for prediction ofnot only p–V–T data but also caloric properties.

Finally, for the convenience of the reader, thermody�namic properties of cyclopropane are graphically pre�sented in Fig. 3. The reference state of the data in this

−0.4

280200 360T, K

0

0.4

0.8

[ps, exp − ps, cal)/ps, exp] × 102

123

Fig. 1. Deviation of BACKONE vapor pressures andexperimental data: (1) [7], (2) [8], and (3) [9].

−1

160 200 280 320 360 T, K

~ ~

0

1

[ρ'exp − ρ'cal)/ρ'exp] × 102

Fig. 2. Deviation of the BACKONE liquid densities andexperimental data: (1) [8], (2) [10]).

Table 3. Comparison between values of enthalpy of evapo�ration (ΔH, kJ/kg) from Liessmann et al. [2] (I) and thosefrom the BACKONE EOS (II)

T, K I II

193.15 523 517.0

213.15 507 500.4

233.15 486 482.7

253.15 463 463.3

273.15 438 441.4

293.15 410 416.2

313.15 379 386.6

333.15 345 350.7

353.15 301 305.3

373.15 238 243.2

393.15 128 140.2

2

1

982

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 6 2014

NGOC ANH LAI, THAI SON PHAM

table is selected at T0 = 298.15 K, p0 = 0.101325 MPawith H0 = 0.0 and S0 = 0.0.

CONCLUSIONS

Thermodynamic properties of cyclopropane weremodelled with the BACKONE EOS. The specificcharacteristic parameters of the BACKONE EOS forthe cyclopropane were found by using the vapor pres�sure and saturated liquid density. The comparisonbetween the data from the BACKONE EOS and avail�able data proves that the BACKONE is not only goodfor the describing data, but also good for the predic�tion. In details, in the describing data, the uncertain�ties of calculated values from the BACKONE EOS are0.29% for vapor pressures and 0.75% for saturated liq�uid densities. In the prediction, the uncertainties ofcalculated values from the BACKONE EOS are 0.90%for isobaric densities in liquid phase and 2.0% forenthalpies of vaporization.

The BACKONE EOS for the cyclopropane can beused to evaluate thermodynamic cycles, such as refrig�eration cycles [38, 39] and organic Rankine cycles[40–42] using cyclopropane as a working fluid as wellas to study mixtures of cyclopropane with other sub�stances.

The readers can contact the author to get free soft�ware for calculation of the thermodynamic propertiesof cyclopropane.

ACKNOWLEDGMENTS

This research is funded by Vietnam NationalFoundation for Science and Technology Development(NAFOSTED) under grant no. 107.01�2012.29.

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200120 80 40 0

−s, J/(mol K)

300

400

T, K

3.5 MPa

10 m

ol/L

1 m

ol/L

0.1

mol

/L

0.05

mol

/L0.

001

mol

/L

2.2 MPa

1.4 MPa

0.8 MPa

0.4 MPa

0.2 MPa

0.1 MPa

Fig. 3. T–S diagram of the cyclopropane from theBACKONE EOS.

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 88 No. 6 2014

DESCRIPTION OF CYCLOPROPANE WITH BACKONE EQUATION OF STATE 983

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