Fluid Phase Equilibria 178 (2001) 73–85

Speed of sound, ideal-gas heat capacity at constant pressure, andsecond virial coefficients of HFC-227ea

Chang Zhang1, Yuan-Yuan Duan∗, Lin Shi, Ming-Shan Zhu, Li-Zhong HanDepartment of Thermal Engineering, Tsinghua University, Beijing 100084, PR China

Received 5 April 2000; accepted 3 October 2000

Abstract

The speed of sound of the gaseous 1,1,1,2,3,3,3-heptafluoropropane (HFC-227ea) was measured for tempera-tures from 273 to 333 K and pressures from 26 to 315 kPa with a cylindrical, variable-path acoustic interferometeroperating at 156.252 kHz. The uncertainty of the speed of sound was less than±0.05%. The ideal-gas heat capacityat constant pressure and the second acoustic virial coefficients were determined over the temperature range from thespeed of sound measurements. The uncertainty of the ideal-gas heat capacity at constant pressure was estimated tobe less than±0.5%. The ideal-gas heat capacity at constant pressure results and second virial coefficients calculatedfrom the present speed of sound measurements were compared with the available data. © 2001 Elsevier ScienceB.V. All rights reserved.

Keywords:Ideal state function; Data; Speed of sound; Second virial coefficients; Heat capacity;1,1,1,2,3,3,3-Heptafluoropropane; HFC-227ea

1. Introduction

1,1,1,2,3,3,3-Heptafluoropropane (HFC-227ea) is a recently introduced, commercially availablehydrofluorocarbon (HFC) and is useful in fire suppression, refrigeration, sterilization and propellantapplications. It can be used as an alternative to halon, and blends containing HFC-227ea are potentialalternatives to HCFC-22 and R502. Effective use of HFC-227ea requires that the thermodynamic andtransport properties be accurately measured, but there are few data available, especially no available speedof sound data. Wirbser et al. [1] measured the specific heat capacity and Joule–Thomson coefficient ofHFC-227ea; Salvi-Narkhede et al. [2] measured the vapor pressure, liquid molar volumes and criticalproperties; Park [3] measured the gaseous PVT properties with a Burnett apparatus at five temperatures;Klomfar et al. [4] measured the liquid PVT properties; Robin [5] listed the thermophysical properties of

∗ Corresponding author. Tel.:+86-10-62788608; fax:+86-10-62770209.E-mail address:yyduan@te.tsinghua.edu.cn (Y.-Y. Duan).

1 Visiting Scholar from Wuhan Institute of Science and Technology, Wuhan 430073, PR China.

0378-3812/01/$20.00 © 2001 Elsevier Science B.V. All rights reserved.PII: S0378-3812(00)00477-5

74 C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85

HFC-227ea including estimated transport properties; Defibaugh and Moldover [6] measured the liquidPVT behavior and the saturated liquid density; Weber [7] measured the vapor pressure of HFC-227ea;Laesecke and Hafer [8] measured the viscosity of HFC-227ea with a coiled capillary viscometer atlow temperature and a straight capillary viscometer at high temperature; Pátek et al. [9] measured PVTproperties of HFC-227ea with a Burnett apparatus at temperatures of 393 and 423 K; Shi et al. [10] mea-sured the vapor pressure; Liu et al. [11,12] measured the saturated liquid viscosity and gaseous thermalconductivity; Shi et al. [13] measured PVT properties of HFC-227ea.

This paper reports the experimental results of the speed of sound of the gaseous HFC-227ea mea-sured for temperatures from 273.15 to 333.215 K and pressures from 26 to 315 kPa, with a cylindrical,variable-path acoustic interferometer operating at 156.252 kHz. The ideal-gas heat capacity at constantpressure and the second acoustic virial coefficients were determined over the temperature range fromthe speed of sound measurements. The present ideal-gas heat capacity data at constant pressure werecompared with the available data [1]. Second virial coefficients calculated from the present speed ofsound measurements were compared with results from the literature determined from PVT measurements[3,9,13].

The sample of HFC-227ea was obtained from Shanghai Huiyou Chemical Corp., China and was usedwithout further purification. The manufacturer stated that the water content was less than 20 ppm. Fromthe gas chromatographic analysis, the purity of the sample was better than 99.9 mol%.

2. Working equation

The relationship between the speed of soundW and the isoentropic compressibility is given by

W 2 =(

∂p

∂ρ

)s

(1)

From thermodynamics, the acoustic virial expansion is given by

W 2 = γ0RT

M

[1 +

(βa

RT

)p +

( γa

RT

)p2 + · · ·

](2)

The subscripts in Eq. (1) refers to an isoentropic process,M is the molar mass of the sample gas,p thegas pressure,T the gas temperature,R the universal gas constant,γ 0 the zero-pressure limit of the heatcapacity ratio andβa andγ a the second and third acoustic virial coefficients of the gas.

The speed of sound was measured as a function of pressure at constant temperatures and at low pressures.The measured speed of sound results along each isotherm were correlated as a function of pressure withthe following function

W 2 = A0 + A1p + A2p2 (3)

whereA0, A1 andA2 are numerical constants for each isotherm. If bothT andM are known, the heatcapacity ratioγ 0 can be obtained. From Eqs. (2) and (3), the heat capacity ratioγ 0 can be determinedfrom

γ0 = A0M/RT (4)

C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85 75

The ideal-gas heat capacity at constant pressureC0p can be determined from

C0p = Rγ0

γ0 − 1(5)

The second acoustic virial coefficients of the gasβa can be determined from

βa = A1M

γ0(6)

3. Experimental instrument

The experimental instrument, which was described previously [14–16], will be introduced again brieflyhere. The schematic of the entire measuring system is shown in Fig. 1. A steel pressure vessel was used inthe instrument to withstand the pressure. The vessel consisted of a cylinder with two pistons at oppositeends of the cylinder. One piston equipped with an emitting transducer was fixed, while the other one

Fig. 1. Schematic of the speed of sound measuring system: (1) main body; (2) thermostat; (3 and 4) displacement measuringsystem; (5) generator; (6) amplifier; (7) frequency meter; (8) wave indicator; (9) phase detector; (10) temperature acquisitionunit; (11) three-way valve; (12) gas sample bottle; (13) temperature controller; (14) vacuum pump; (15) vacuum meter; (16)differential pressure detector; (17) digital pressure detector; (18) dead weight tester.

76 C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85

equipped with a reflector, could slide freely in the cylinder. The reflector also operated as a detector.The operating frequency of the emitting transducer was determined by a frequency generator made ofpiezoelectric crystal, which is placed outside of the thermostat, so the frequency is a constant and doesnot change with the experimental temperature, pressure and fluid property. The operating frequency is156.252 kHz with an uncertainty of±1 Hz. The vessel was suspended in a stirred fluid bath during thecourse of the experiment. The temperature uncertainty was less than±10 mK. The pressures of the gassample were measured with a dead weight tester, a digital pressure gauge and a differential pressuretransducer with an uncertainty of±200 Pa. During the experiments, the movable transducer was slidrelative to the fixed transducer. The wave emitted by the fixed transducer and the wave reflected by thefree transducer will interfere with each other, when the distance between the two transducers is sameinteger multiple of half the wavelength. Once the changed distancel of the movable transducer andthe number of the interferenceN are measured, the wavelengthλ can be determined according to theprinciple of ultrasonic interference. Then the speed of sound in the test gas sample can be determinedwith the wavelengthλ and the sound frequencyf. The frequency of the sound wave emitted from thepiezoelectric crystal transducer is essentially constant, because the resonating frequency of the crystalis nearly independent of the environment [14]. Thus, the precision of the determination of the speed ofsound depends mainly on the precision of the wavelength measurement. The precision of the wavelengthmeasurements was improved by moving the piston more than 30 wavelengths in this study. A discussionof the uncertainty in the experiment has been described in detail in the previous publication [14]. Theinstrument was checked with argon and nitrogen before HFC-227ea speed of sound measurements; themeasured results showed that the uncertainty of the speed of sound measured with this instrument is lessthan±0.05% and the uncertainty of the ideal-gas heat capacity at constant pressure determined with themeasured speed of sound is estimated less than±0.5%.

4. Results and analysis

Wavelength measurements for HFC-227ea were made along 10 isotherms between 273.15 and 333.215 K.The maximum pressure along the isotherms was about 315 kPa. The speed of sound of HFC-227ea wasobtained from the corrected wavelengths together with the fixed frequency. The measurement values werecorrected for diffraction and guided mode dispersion [17] using the empirical equation

1λdg = λ

[α2

(λ

D

)4

+ α3

(λ

D

)6]

(7)

whereλ is the wavelength,D the diameter of the resonance tube (75.04 mm in this study),α2 = 0.3806andα3 = 79.74 [14].

The measured speed of sound was corrected for absorption dispersion using the Kirchhoff Helmholtz(boundary layer) absorption coefficientαKH and the classical absorption coefficientαCL. According tothe boundary layer theory [17]

αKH = 2

DW

[ν0.5 + (γ − 1)

(κ

ρCp

)0.5]

0.5ω0.5 (8)

C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85 77

whereD is the tube diameter,ν the kinematic viscosity of the sample gas,κ the thermal conductivity; thevalues ofν andκ are calculated from literature [18],Cp the heat capacity at constant pressure providedby Wirbser et al. [1],ρ the density,γ the heat capacity ratio andω the angular frequency.

The classical absorption coefficientαCL is given by two parts: the thermal conductivity and the shearviscosity [17]

αCL = κ(1 − γ )ω2

2ρW 3Cv+ 2ω2η

3W 3ρ(9)

whereCv is the heat capacity at constant volume and the other parameters are the same as for the lastequation. The shear viscosityη required to determine the absorption was calculated from literature [18].

The correction is then

1λab = λ2(αKH + αCL)1 + λ(αKH + αCL)/π

2π(10)

For polyatomic molecules, vibrational relaxation has the most important impact on the speed of soundmeasurements. Its correction can be calculated as

1λvib = λCv0(Cv0 + r) + Cv1(Cv1 + r)(ωτ)2

2[C2v0 + C2

v1(ωτ)2](1 + r/Cv0)(11)

whereCv0 is the heat capacity at zero frequency,r = Cp −Cv1, Cv1 = Cv0−Cvib andCvib (C0p −4R), the

vibrational contribution to the heat capacity. The three corrections for the wavelength are then combinedas

λc = λ − (1λdg + 1λab + 1λvib) (12)

The corrected speed of sound results for gaseous HFC-227ea are listed in Table 1 . Fig. 2 shows thespeed of sound versus pressure along each isotherm. The data for the ideal-gas heat capacity at constantpressure,C0

p and the corresponding second acoustic virial coefficientβa are listed in Table 2. Figs. 3and 4 show the data forC0

p andβa as a function of temperature. The ideal-gas heat capacity at constantpressure and the second acoustic virial coefficient were derived from regression analysis of the presentspeed of sound measurements. TheC0

p data was correlated by the following equation

C0p

R= −3.00623+ 0.09753

(T

K

)− 1.10336× 10−4

(T

K

)2

(13)

whereR is the universal gas constant. Fig. 5 shows the deviations of the present ideal-gas heat capacityat constant pressure results and the data of Wirbser et al. [1] from Eq. (13), which reproduces our datavery well with a maximum deviation of 0.31% and the RMS deviation is 0.23%. The data of Wirbseret al. [1] are 0.22–1.54% higher than Eq. (13). The difference between the acoustic and flow calorimetricmeasurements of ideal-gas heat capacity at constant pressure may be caused by the chemical impuritiesespecially water vapor.

Theβa data were correlated with

βa(dm3 mol−1) = −6.22999+ 0.02579

(T

K

)− 2.88544× 10−5

(T

K

)2

(14)

78 C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85

Table 1Speed of sound results for HFC-227ea

T (K) p (kPa) W (m s−1)

273.150 35.47 118.24250.03 117.70465.78 117.111

107.31 115.513122.20 114.911145.00 113.987160.64 113.345

293.555 28.59 122.88565.42 121.768

115.41 120.226155.20 118.958187.88 117.852236.15 116.207270.24 114.985

298.230 47.37 123.32585.26 122.212

132.78 120.790176.86 119.427225.28 117.882255.64 116.856311.41 114.935

303.171 38.16 124.61372.51 123.677

126.76 122.181182.13 120.577216.43 119.489237.73 118.908287.79 117.366

308.225 30.17 125.89068.10 124.903

124.71 123.401162.05 122.349197.86 121.363235.91 120.258291.20 118.636

313.255 30.42 126.90462.48 126.107

121.09 124.621153.58 123.772209.20 122.298258.68 120.944293.45 119.976

318.165 26.26 128.00263.76 127.116

C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85 79

Table 1 (Continued).

T (K) p (kPa) W (m s−1)

113.80 125.909165.27 124.629212.41 123.452244.17 122.619298.18 121.203

323.195 29.66 128.91765.81 128.092

121.53 126.842151.44 126.146182.45 125.393226.90 124.282255.72 123.605301.30 122.485

328.205 26.01 130.00357.09 129.324

115.95 128.072150.62 127.316181.50 126.596238.20 125.261269.32 124.572315.59 123.463

333.215 42.27 130.69473.64 130.045

125.38 128.961172.24 127.979199.87 127.379259.75 126.055294.07 125.255

Table 2Ideal-gas heat capacity at constant pressure and second acoustic virial coefficient for HFC-227ea

T (K) C0p/R βa (dm3 mol−1)

273.150 15.396 −1.3437293.555 16.119 −1.1334298.230 16.314 −1.0920303.171 16.413 −1.0681308.225 16.521 −1.0244313.255 16.699 −0.9900318.165 16.825 −0.9524323.195 17.039 −0.9123328.205 17.172 −0.8698333.215 17.195 −0.8325

80 C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85

Fig. 2. The obtained speed of sound vs. pressure of different temperatures for HFC-227ea.

Fig. 3. The obtained ideal-gas heat capacity at constant pressure vs. temperature.

Fig. 4. The obtained second acoustic virial coefficientβa vs. temperatures.

C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85 81

Fig. 5. Deviation of ideal-gas heat capacity at constant pressure from Eq. (13): (h) present study from Table 2; (4) Wirbseret al. [1].

The maximum deviation and RMS deviation of the present data from Eq. (14) are 1.2 and 0.76%. Fig. 6shows the deviations of the presentβa data from Eq. (14).

The thermodynamic relation betweenβa and second virial coefficientB is given by

βa = 2B + 2(γ0 − 1)TdB

dT+ (γ0 − 1)2

γ0T 2 d2B

dT 2(15)

whereγ 0 is the ideal-gas heat capacity ratio. We adopted a semi-empirical method to solve this secondorder differential equation. This procedure is similar to that described in detail by Ewing et al. [19–21].

For the second virial coefficient, the square-well model leads to the simple equation [22]

B(T ) = b0[1 − (r3 − 1)∆] (16)

Fig. 6. Deviation of the experimental results of second acoustic virial coefficientβa from Eq. (14).

82 C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85

Fig. 7. The second virial coefficientB vs. temperature: (——) Eq. (19); (- - -) Shi et al. [13]; (e) Park [3]; (4) Patek et al. [9].

with

∆ = exp( ε

kT

)− 1 (17)

Eqs. (15)–(17) include three parameters to be fit to the data: the co-volumeb0, the scaled well depthε/k and the ratio of the radius of the well to the radius of the hard corer. From Eq. (15), the correspondingexpression forβa is

βa

2b0= r3 +

[−1 + γ0 − 1

γ0

ε

kT− (γ0 − 1)2

2γ0

( ε

kT

)2]

(r3 − 1) exp( ε

kT

)(18)

The numerical constants in this equation were determined by trial-and-error analysis using the presentβa values and Eq. (13) forC0

p. The regression analysis led to the parameters:b0 = 0.11411 dm3 mol−1,ε/k = 510.0 K andr = 1.3497. The second virial coefficientB was correlated as

B(dm3 mol−1) = 0.11411

{1 − 1.4588

[exp

(510.0

T

)− 1

]}(19)

Fig. 8. Deviations of the present second acoustic virial coefficient data and Eq. (14) from Eq. (20): (h) present data.

C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85 83

Fig. 7 shows the second virial coefficientBversus the temperature calculated with Eq. (19) in this study.The calculated values ofB from Eq. (19) for gaseous HFC-227ea are compared with available correlationand experimental data [3,9,13]. The differences between the Eq. (19) by this study and literature data forB are less than 3% for a temperature range from 270 to 450 K.

From Eqs. (19) and (15), we have

βa(dm3 mol−1) = 0.561147

+[−0.332927+ γ0 − 1

γ0

169.793

T− (γ0 − 1)2

γ0

43 297.20

T 2

]exp

(510.0

T

)(20)

whereγ0 = C0p/(C0

p − R) is ideal-gas heat capacity ratio that can be calculated from Eq. (13). Fig. 8shows that the maximum deviation of the presentβa data and Eq. (14) from Eq. (20) is less than 2%.

5. Conclusion

The speed of sound in gaseous HFC-227ea has been measured with an accuracy of approximately±0.05% in the temperature range from 273.15 to 333.215 K and in pressure range from 26 to 315 kPa.The ideal-gas heat capacity at constant pressure and the second acoustic virial coefficients were determinedover the temperature range from the speed of sound measurements. The ideal-gas heat capacity at constantpressure results were compared with the available data. The second virial coefficients calculated from thepresent speed of sound measurements were compared with results from the literature determined fromPVT measurements; the deviations are within 3% for a temperature range from 270 to 450 K.

List of symbolsA0 adjustable parameterA1 adjustable parameterA2 adjustable parameterb0 co-volume in the square-well modelB second virial coefficientCp heat capacity at constant pressureC0

p ideal-gas heat capacity at constant pressureCv heat capacity at constant volumeCv0 isochoric heat capacity at zero frequencyCvib vibrational contribution to the heat capacityD diameter of the resonance tubef sound frequencyk Boltzmann constantl changed distance of the movable transducerM molar massN number of the interferencep pressurer ratio of the radius of the well to the radius of the hard core in the square-well modelR universal gas constant (8.314471 J mol K−1)T thermodynamic temperature

84 C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85

Greek lettersα absorption coefficientβa second acoustic virial coefficient∆ differenceε well depth in the square-well modelγ a third acoustic virial coefficientγ 0 zero-pressure limit of the heat capacity ratioη shear viscosityκ thermal conductivityλ wavelengthν kinematic viscosityπ ratio of the circumference of a circle to its diameter (3.1415926)ρ densityω angular frequency

Subscriptab absorptionc correctedCL classicaldg diffraction and guided mode dispersionKH Kirchhoff–Helmholtzvib vibrational relaxation

Acknowledgements

This study was supported by the National Natural Science Foundation of China (No. 59906006) andpart of this study was supported by Education Commission of Hubei, China (No. 99B029). We are gratefulto Shanghai Huiyou Chemical Corp. for providing the HFC-227 sample.

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