Speed of Sound, Ideal-Gas Heat Capacity at Constant Pressure, And

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Fluid Phase Equilibria 178 (2001) 73–85

Speed of sound, ideal-gas heat capacity at constant pressure, and second virial coefﬁcients of HFC-227ea Chang Zhang1, Yuan-Yuan Duan∗, Lin Shi, Ming-Shan Zhu, Li-Zhong Han Department 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 temperatures from 273 to 333 K and pressures from 26 to 315 kPa with a cylindrical, variable-path acoustic interferometer operating at 156.252 kHz. The uncertainty of the speed of sound was less than 0.05%. The ideal-gas heat capacity at constant pressure and the second acoustic virial coefficients were determined over the temperature range from the speed of sound measurements. The uncertainty of the ideal-gas heat capacity at constant pressure was estimated to be less than 0.5%. The ideal-gas heat capacity at constant pressure results and second virial coefficients calculated from the present speed of sound measurements were compared with the available data. © 2001 Elsevier Science B.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 available hydrofluorocarbon (HFC) and is useful in fire suppression, refrigeration, sterilization and propellant applications. It can be used as an alternative to halon, and blends containing HFC-227ea are potential alternatives to HCFC-22 and R502. Effective use of HFC-227ea requires that the thermodynamic and transport properties be accurately measured, but there are few data available, especially no available speed of sound data. Wirbser et al. [1] measured the specific heat capacity and Joule–Thomson coefficient of HFC-227ea; Salvi-Narkhede et al. [2] measured the vapor pressure, liquid molar volumes and critical properties; 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: [email protected] (Y.-Y. Duan). 1 Visiting Scholar from Wuhan Institute of Science and Technology, Wuhan 430073, PR China.

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74 C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85 HFC-227ea including estimated transport properties; Defibaugh and Moldover [6] measured the liquid PVT 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 at low temperature and a straight capillary viscometer at high temperature; Pátek et al. [9] measured PVT properties of HFC-227ea with a Burnett apparatus at temperatures of 393 and 423 K; Shi et al. [10] measured the vapor pressure; Liu et al. [11,12] measured the saturated liquid viscosity and gaseous thermal conductivity; 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 measured 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 constant pressure and the second acoustic virial coefficients were determined over the temperature range from the speed of sound measurements. The present ideal-gas heat capacity data at constant pressure were compared with the available data [1]. Second virial coefficients calculated from the present speed of sound 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 used without further purification. The manufacturer stated that the water content was less than 20 ppm. From the gas chromatographic analysis, the purity of the sample was better than 99.9 mol%.

2. Working equation

The relationship between the speed of sound W and the isoentropic compressibility is given by ∂p W 2 = (1) ∂ρ s From thermodynamics, the acoustic virial expansion is given by γ RT β γ W 2 = 0 1 + a p + a p2 +··· (2) M RT RT The subscript s in Eq. (1) refers to an isoentropic process, M is the molar mass of the sample gas, p the gas pressure, T the gas temperature, R the universal gas constant, γ 0 the zero-pressure limit of the heat capacity ratio and βa and γ 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 with the following function

2 2 W = A0 + A1p + A2p (3) where A0, A1 and A2 are numerical constants for each isotherm. If both T and M are known, the heat capacity ratio γ 0 can be obtained. From Eqs. (2) and (3), the heat capacity ratio γ 0 can be determined from

γ0 = A0M/RT (4) 中国科技论文在线 http://www.paper.edu.cn

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

0 The ideal-gas heat capacity at constant pressure Cp can be determined from Rγ 0 0 Cp = (5) γ0 − 1

The second acoustic virial coefﬁcients of the gas βa can be determined from

A1M βa = (6) γ0

3. Experimental instrument

The experimental instrument, which was described previously [14–16], will be introduced again brieﬂy here. The schematic of the entire measuring system is shown in Fig. 1. A steel pressure vessel was used in the instrument to withstand the pressure. The vessel consisted of a cylinder with two pistons at opposite ends 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 measuring system; (5) generator; (6) ampliﬁer; (7) frequency meter; (8) wave indicator; (9) phase detector; (10) temperature acquisition unit; (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. 中国科技论文在线 http://www.paper.edu.cn

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 of piezoelectric crystal, which is placed outside of the thermostat, so the frequency is a constant and does not change with the experimental temperature, pressure and fluid property. The operating frequency is 156.252 kHz with an uncertainty of 1 Hz. The vessel was suspended in a stirred fluid bath during the course of the experiment. The temperature uncertainty was less than 10 mK. The pressures of the gas sample were measured with a dead weight tester, a digital pressure gauge and a differential pressure transducer with an uncertainty of 200 Pa. During the experiments, the movable transducer was slid relative to the fixed transducer. The wave emitted by the fixed transducer and the wave reflected by the free transducer will interfere with each other, when the distance between the two transducers is same integer multiple of half the wavelength. Once the changed distance l of the movable transducer and the number of the interference N are measured, the wavelength λ can be determined according to the principle of ultrasonic interference. Then the speed of sound in the test gas sample can be determined with the wavelength λ and the sound frequency f. The frequency of the sound wave emitted from the piezoelectric crystal transducer is essentially constant, because the resonating frequency of the crystal is nearly independent of the environment [14]. Thus, the precision of the determination of the speed of sound depends mainly on the precision of the wavelength measurement. The precision of the wavelength measurements was improved by moving the piston more than 30 wavelengths in this study. A discussion of the uncertainty in the experiment has been described in detail in the previous publication [14]. The instrument was checked with argon and nitrogen before HFC-227ea speed of sound measurements; the measured results showed that the uncertainty of the speed of sound measured with this instrument is less than 0.05% and the uncertainty of the ideal-gas heat capacity at constant pressure determined with the measured 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 was obtained from the corrected wavelengths together with the fixed frequency. The measurement values were corrected for diffraction and guided mode dispersion [17] using the empirical equation " # λ 4 λ 6 1λ = λ α + α dg 2 D 3 D (7) where λ is the wavelength, D the diameter of the resonance tube (75.04 mm in this study), α2 = 0.3806 and α3 = 79.74 [14]. The measured speed of sound was corrected for absorption dispersion using the Kirchhoff Helmholtz (boundary layer) absorption coefficient αKH and the classical absorption coefficient αCL. According to the boundary layer theory [17] " # κ 0.5 2 0.5 0.5 αKH = ν + (γ − 1) 0.5ω (8) DW ρCp 中国科技论文在线 http://www.paper.edu.cn

C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85 77 where D is the tube diameter, ν the kinematic viscosity of the sample gas, κ the thermal conductivity; the values of ν and κ are calculated from literature [18], Cp the heat capacity at constant pressure provided by Wirbser et al. [1], ρ the density, γ the heat capacity ratio and ω the angular frequency. The classical absorption coefﬁcient αCL is given by two parts: the thermal conductivity and the shear viscosity [17] κ(1 − γ)ω2 2ω2η α = + (9) CL 3 3 2ρW Cv 3W ρ where Cv is the heat capacity at constant volume and the other parameters are the same as for the last equation. The shear viscosity η required to determine the absorption was calculated from literature [18]. The correction is then 1 + λ(α + α )/π 1λ = λ2(α + α ) KH CL (10) ab KH CL 2π For polyatomic molecules, vibrational relaxation has the most important impact on the speed of sound measurements. Its correction can be calculated as C (C + r) + C (C + r)(ωτ)2 1λ = λ v0 v0 v1 v1 (11) vib C2 + C2 (ωτ)2 ( + r/C ) 2[ v0 v1 ] 1 v0 0 where Cv0 is the heat capacity at zero frequency, r = Cp −Cv1, Cv1 = Cv0 −Cvib and Cvib (Cp −4R), the vibrational contribution to the heat capacity. The three corrections for the wavelength are then combined as

λc = λ − (1λdg + 1λab + 1λvib) (12) The corrected speed of sound results for gaseous HFC-227ea are listed in Table 1 . Fig. 2 shows the speed of sound versus pressure along each isotherm. The data for the ideal-gas heat capacity at constant 0 pressure, Cp and the corresponding second acoustic virial coefficient βa are listed in Table 2. Figs. 3 0 and 4 show the data for Cp and βa as a function of temperature. The ideal-gas heat capacity at constant pressure and the second acoustic virial coefficient were derived from regression analysis of the present 0 speed of sound measurements. The Cp data was correlated by the following equation C0 T T 2 p =− . + . − . × −4 R 3 00623 0 09753 K 1 10336 10 K (13) where R is the universal gas constant. Fig. 5 shows the deviations of the present ideal-gas heat capacity at constant pressure results and the data of Wirbser et al. [1] from Eq. (13), which reproduces our data very well with a maximum deviation of 0.31% and the RMS deviation is 0.23%. The data of Wirbser et al. [1] are 0.22–1.54% higher than Eq. (13). The difference between the acoustic and flow calorimetric measurements of ideal-gas heat capacity at constant pressure may be caused by the chemical impurities especially water vapor. The βa data were correlated with T T 2 β ( 3 −1) =− . + . − . × −5 a dm mol 6 22999 0 02579 K 2 88544 10 K (14) 中国科技论文在线 http://www.paper.edu.cn

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

Table 1 Speed of sound results for HFC-227ea T (K) p (kPa) W (m s−1)

273.150 35.47 118.242 50.03 117.704 65.78 117.111 107.31 115.513 122.20 114.911 145.00 113.987 160.64 113.345 293.555 28.59 122.885 65.42 121.768 115.41 120.226 155.20 118.958 187.88 117.852 236.15 116.207 270.24 114.985 298.230 47.37 123.325 85.26 122.212 132.78 120.790 176.86 119.427 225.28 117.882 255.64 116.856 311.41 114.935 303.171 38.16 124.613 72.51 123.677 126.76 122.181 182.13 120.577 216.43 119.489 237.73 118.908 287.79 117.366 308.225 30.17 125.890 68.10 124.903 124.71 123.401 162.05 122.349 197.86 121.363 235.91 120.258 291.20 118.636 313.255 30.42 126.904 62.48 126.107 121.09 124.621 153.58 123.772 209.20 122.298 258.68 120.944 293.45 119.976 318.165 26.26 128.002 63.76 127.116 中国科技论文在线 http://www.paper.edu.cn

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

Table 1 (Continued). T (K) p (kPa) W (m s−1)

113.80 125.909 165.27 124.629 212.41 123.452 244.17 122.619 298.18 121.203 323.195 29.66 128.917 65.81 128.092 121.53 126.842 151.44 126.146 182.45 125.393 226.90 124.282 255.72 123.605 301.30 122.485 328.205 26.01 130.003 57.09 129.324 115.95 128.072 150.62 127.316 181.50 126.596 238.20 125.261 269.32 124.572 315.59 123.463 333.215 42.27 130.694 73.64 130.045 125.38 128.961 172.24 127.979 199.87 127.379 259.75 126.055 294.07 125.255

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

0 3 −1 T (K) Cp/R βa (dm mol ) 273.150 15.396 −1.3437 293.555 16.119 −1.1334 298.230 16.314 −1.0920 303.171 16.413 −1.0681 308.225 16.521 −1.0244 313.255 16.699 −0.9900 318.165 16.825 −0.9524 323.195 17.039 −0.9123 328.205 17.172 −0.8698 333.215 17.195 −0.8325 中国科技论文在线 http://www.paper.edu.cn

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 coefﬁcient βa vs. temperatures. 中国科技论文在线 http://www.paper.edu.cn

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): (ᮀ) present study from Table 2; (4) Wirbser et al. [1].

The maximum deviation and RMS deviation of the present data from Eq. (14) are 1.2 and 0.76%. Fig. 6 shows the deviations of the present βa data from Eq. (14). The thermodynamic relation between βa and second virial coefﬁcient B is given by dB (γ − 1)2 d2B β = 2B + 2(γ − 1)T + 0 T 2 (15) a 0 2 dT γ0 dT where γ 0 is the ideal-gas heat capacity ratio. We adopted a semi-empirical method to solve this second order 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]

3 B(T ) = b0[1 − (r − 1)∆] (16)

Fig. 6. Deviation of the experimental results of second acoustic virial coefﬁcient βa from Eq. (14). 中国科技论文在线 http://www.paper.edu.cn

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

Fig. 7. The second virial coefﬁcient B vs. temperature: (——) Eq. (19); (- - -) Shi et al. [13]; (᭛) Park [3]; (4)Patek´ et al. [9]. with ε ∆ = exp − 1 (17) kT

Eqs. (15)–(17) include three parameters to be ﬁt to the data: the co-volume b0, the scaled well depth ε/k and the ratio of the radius of the well to the radius of the hard core r. From Eq. (15), the corresponding expression for βa is β γ − 1 ε (γ − 1)2 ε 2 ε a = r3 + −1 + 0 − 0 (r3 − 1) exp (18) 2b0 γ0 kT 2γ0 kT kT The numerical constants in this equation were determined by trial-and-error analysis using the present 0 3 −1 βa values and Eq. (13) for Cp. The regression analysis led to the parameters: b0 = 0.11411 dm mol , ε/k = 510.0 K and r = 1.3497. The second virial coefﬁcient B was correlated as 510.0 B( 3 −1) = . − . − dm mol 0 11411 1 1 4588 exp T 1 (19)

Fig. 8. Deviations of the present second acoustic virial coefﬁcient data and Eq. (14) from Eq. (20): (ᮀ) present data. 中国科技论文在线 http://www.paper.edu.cn

C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85 83 Fig. 7 shows the second virial coefﬁcient B versus the temperature calculated with Eq. (19) in this study. The calculated values of B from Eq. (19) for gaseous HFC-227ea are compared with available correlation and experimental data [3,9,13]. The differences between the Eq. (19) by this study and literature data for B are less than 3% for a temperature range from 270 to 450 K. From Eqs. (19) and (15), we have β ( 3 −1) = . a dm mol 0 561147 γ − 1 169.793 (γ − 1)2 43 297.20 510.0 + −0.332927 + 0 − 0 exp (20) 2 γ0 T γ0 T T 0 0 where γ0 = Cp/(Cp − R) is ideal-gas heat capacity ratio that can be calculated from Eq. (13). Fig. 8 shows that the maximum deviation of the present βa 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 determined over the temperature range from the speed of sound measurements. The ideal-gas heat capacity at constant pressure results were compared with the available data. The second virial coefficients calculated from the present speed of sound measurements were compared with results from the literature determined from PVT measurements; the deviations are within 3% for a temperature range from 270 to 450 K. List of symbols A0 adjustable parameter A1 adjustable parameter A2 adjustable parameter b0 co-volume in the square-well model B second virial coefficient Cp heat capacity at constant pressure 0 Cp ideal-gas heat capacity at constant pressure Cv heat capacity at constant volume Cv0 isochoric heat capacity at zero frequency Cvib vibrational contribution to the heat capacity D diameter of the resonance tube f sound frequency k Boltzmann constant l changed distance of the movable transducer M molar mass N number of the interference p pressure r ratio of the radius of the well to the radius of the hard core in the square-well model R universal gas constant (8.314471 J mol K−1) T thermodynamic temperature 中国科技论文在线 http://www.paper.edu.cn

84 C. Zhang et al. / Fluid Phase Equilibria 178 (2001) 73–85 Greek letters α absorption coefficient βa second acoustic virial coefficient ∆ difference ε well depth in the square-well model γ a third acoustic virial coefficient γ 0 zero-pressure limit of the heat capacity ratio η shear viscosity κ thermal conductivity λ wavelength ν kinematic viscosity π ratio of the circumference of a circle to its diameter (3.1415926) ρ density ω angular frequency

Subscript ab absorption c corrected CL classical dg diffraction and guided mode dispersion KH Kirchhoff–Helmholtz vib vibrational relaxation

Acknowledgements

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

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