Vapor phase acoustic measurements for R125 and development of a Helmholtz free energy equation

Fluid Phase Equilibria 174 (2000) 69–79

Vapor phase acoustic measurements for R125 and development of a Helmholtz free energy equation M. Grigiante a , G. Scalabrin a,∗ , G. Benedetto b , R.M. Gavioso b , R. Spagnolo b a

b

Dipartimento di Fisica Tecnica, Università di Padova, via Venezia 1, I-35131 Padova, Italy Istituto Elettrotecnico Nazionale Galileo Ferraris, Strada delle Cacce 91, I-10135 Torino, Italy

Abstract This paper presents speed of sound measurements on pentafluoroethane (R125) in the vapor phase. The measurements were performed in a stainless steel spherical resonator of ∼900 cm3 at temperatures in the range 260–360 K and pressures up to 500 kPa. Acoustic virial coefficients and ideal gas heat capacities are deduced directly from the data. The whole set of speed of sound measurements and the ideal gas heat capacities are then correlated in the forms u2 (T, p) and cp0 (T), respectively. Analytical expressions for the temperature dependencies of the thermodynamic virial coefficients, based on a hard-core square-well potential, are then assumed and the model is fitted to the acoustic data, obtaining a virial equation of state for the vapor phase. A highly accurate Helmholtz equation of state a(ρ, T) is established on the basis of the measured data, representing the (pρT) surface of the vapor phase in the same temperature and pressure ranges. The ideal gas Helmholtz equation a0 (ρ, T) is obtained from the former cp0 (T ) correlation. Given the high accuracy of the equation form, only very precise experimental data, such as acoustic measurements, are suitable for fitting the a R (ρ, T ) equation parameters. Both the thermodynamic models are validated on available density data. The good level of consistency reached by the Helmholtz equation, shows its form to be very reliable. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Acoustic virial coefficient; Ideal gas heat capacity; Speed of sound; Vapor density; Helmholtz energy equation of state

1. Introduction Highly accurate speed of sound measurements have become available in the last few years, thanks to the method developed by Moldover and coworkers [1]. This technique has been used to determine the ideal gas heat capacities and acoustic virial coefficients for several proposed alternative refrigerants [2–6]. As a consequence, there is renewed interest in using acoustic data to obtain thermodynamic equations which enable the calculation of other thermodynamic properties in the vapor phase of the fluid of interest. This work presents new speed of sound measurements obtained in the vapor phase of R125 and a simple and ∗

Corresponding author.

0378-3812/00/$20.00 © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 0 ) 0 0 4 1 8 - 0

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original method for converting these data into a Helmholtz energy aR equation. The proposed equation has been validated on vapor phase experimental density data and calculated against values from a dedicated EoS. The validation shows that this kind of procedure is capable of yielding reliable results beyond the range in which they were fitted to the experimental data.

2. Apparatus, materials and procedures The acoustic apparatus consisted of a stainless steel spherical resonator of ∼900 cm3 . Details of the construction and performance of the resonator and its calibration with argon have been described in detail elsewhere [7]. The R125 sample was produced by Ausimont SpA. The manufacturer’s analysis of the sample stated that the two most abundant impurities were R115 (3200 ppm) and R22 (700 ppm). A preliminary speed of sound measurement near 320 K and comparison with previously published results [3], revealed the presence of a consistent amount of air which was removed by repeated freeze–pump–thaw cycles. A comparison of results before and after the purification process is shown in Table 1. Each isotherm included measurements of the speed of sound at several pressures. Before the first round of measurements, the resonator was evacuated and degased at 380 K, while it was maintained at a minimum pressure of 5 × 10−1 Pa for 48 h. At each state point, the speed of sound was deduced from the measured resonance frequencies and calculated half-widths of the four lower-order, purely radial modes. Experimental frequencies were corrected according to the model developed by Moldover [1] to allow for thermal boundary layer losses, effects of coupling with shell motion and the presence of the gas-inlet port. For the present measurements, the thermal boundary layer correction is the most important. This was calculated using thermal conductivity data predicted by the REFPROP database [8]. While it is difficult to assess a measure of the uncertainty associated with these data, it is possible to estimate its effect on the corrected frequencies; an error of 20% in the value of thermal conductivity would change the corrected frequencies of ∼10 ppm. All the remaining corrections were very small and they were included for the sake of completeness. All the thermodynamic and transport properties of R125 required for their calculation were obtained from REFPROP. The molar mass of the sample Mm was calculated by Mm = (1 − x1 − x2 )M125 + x1 M115 + x2 M22

(1)

where M125 = 120.022 g/mol is the molar mass of pure R125, and M115 = 154.465 g/mol, M22 = 86.468 g/mol and x1 , x2 are the molar masses and concentrations of R115 and R22, respectively. The speed of sound measured in the mixture um and the corrected values for the pure component at each state point are given in Table 2, together with the corresponding deviations from the surface fit, which Table 1 Perfect gas heat capacity, second and third acoustic virial coefficients of R125 determined from isotherm fits

Before purification After purification

T (K)

cp0 /R

β a (cm3 /mol)

γ a (cm3 /mol kPa)

320.006 320.006

11.594 11.839

−496.5 −496.6

−0.018 −0.019

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71

Table 2 Mean speed of sound values determined from N radial modes and deviations δ in R125 T (K)

p (kPa)

um a (m/s)

u125 b (m/s)

N

105 ×δ

259.996

261.3 224.6 199.7 174.5 149.5 124.8 100.1 74.7 49.7

133.5766 134.7062 135.4603 136.2088 136.9353 137.6468 138.3442 139.0519 139.7341

133.6247 134.7547 135.5091 136.2579 136.9847 137.6964 138.3940 139.1021 139.7845

4 4 4 4 4 4 4 3 3

−6.1 −1.5 −0.7 −0.5 −0.3 −0.7 −2.6 −3.4 −2.5

280.002

499.9 451.0 400.1 350.1 299.7 249.7 199.5 149.4 100.0 49.7

134.0715 135.3876 136.7119 137.9824 139.2274 140.4335 141.6172 142.7672 143.8798 144.9862

134.1198 135.4364 136.7612 138.0321 139.2776 140.4842 141.6683 142.8187 143.9317 145.0385

4 4 4 4 4 4 4 4 4 2

−4.3 −2.9 1.7 3.6 4.2 3.7 3.2 3.3 1.1 1.6

300.029

500.5 450.8 400.2 349.8 299.8 249.6 199.5 149.4 100.4 49.8

141.3845 142.4012 143.4176 144.4094 145.3798 146.3362 147.2749 148.1974 149.0897 149.9948

141.4354 142.4525 143.4692 144.4614 145.4322 146.3889 147.3280 148.2507 149.1434 150.0488

4 4 4 4 4 4 4 4 4 2

5.3 3.8 2.7 2.0 0.7 −0.7 −1.1 −0.5 −1.9 −0.7

310.020

501.0 450.5 400.1 349.6 299.8 249.4 199.5 149.7 99.6 49.6

144.6844 145.6013 146.5028 147.3891 148.2523 149.1145 149.9559 150.7832 151.6078 152.4186

144.7366 145.6538 146.5556 147.4422 148.3058 149.1682 150.0099 150.8375 151.6624 152.4736

4 4 4 4 4 4 3 4 4 4

11.1 7.4 4.8 3.3 2.2 1.0 0.9 1.2 0.3 0.6

320.006

500.1 450.3 400.1 349.8 299.6 249.4

148.0599 148.8619 149.6603 150.4510 151.2296 151.9999

147.9639 148.7654 149.5633 150.3534 151.1315 151.9013

4 4 4 4 4 4

12.0 −12.7 −12.6 −11.4 −10.0 −8.5

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Table 2 (Continued) p (kPa)

um a (m/s)

u125 b (m/s)

N

105 ×δ

199.3 149.5 99.6 49.9

152.7592 153.5069 154.2481 154.9795

152.6601 153.4073 154.1480 154.8790

2 4 4 4

−6.5 −4.0 −2.9 −1.0

339.989

504.0 450.3 399.5 349.1 299.2 249.2 199.7 149.5 99.8 49.8

153.7150 154.4095 155.0612 155.7031 156.3362 156.9653 157.5856 158.2097 158.8255 159.4410

153.7655 154.4602 155.1121 155.7542 156.3875 157.0168 157.6373 158.2616 158.8776 159.4933

4 4 4 4 4 4 3 3 3 2

−9.4 −6.5 −4.0 −2.1 −0.5 0.7 1.2 2.0 1.6 1.1

360.013

502.2 450.4 400.2 349.6 299.5 249.3 199.4 149.5 99.3 49.7

159.2392 159.7871 160.3156 160.8479 161.3729 161.8979 162.4181 162.9374 163.4593 163.9726

159.2917 159.8398 160.3685 160.9010 161.4261 161.9513 162.4717 162.9911 163.5132 164.0267

4 4 4 4 4 3 3 3 3 3

−1.0 1.7 3.8 5.6 6.7 7.3 7.2 6.9 4.8 3.2

T (K)

a b

Measured values. Corrected values.

is described below. The correction applied to um was merely a temperature-dependent multiplication factor: u2125 = u2m

0 γ125 Mm M125 γm0

which corrected to 0(x2 ) becomes   0 0 0 0 0 0 − γ125 ) + (x2 /x1 )cv22 (γ125 − γ22 ) cv115 (γ115 u2125 ≈ 1 + 0 0 2 um cv125 γ125   (1 + (x2 /x1 ))M − M115 − (x2 /x1 )M22 − x1 M

(2)

(3)

The speeds of sound determined from the individual modes were averaged together, then weighted by the statistical variance between the resonance fits and the data. The resonance frequency measurements for R125 were taken along seven isotherms between 260–360 K and spanned pressures ranging from 50 to 500 kPa. At 260 K, the maximum pressure was 0.6 times the vapor pressure to avoid pre-condensation

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effects. The speed of sound values were obtained from the corrected resonance frequencies, together with the value of the resonator’s radius determined from a calibration with argon. As usual, the speed of sound was represented by means of an acoustic virial expansion around the ideal gas limit in powers of the pressure:       γ  βa RTγ 0 δa a 2 2 3 1+ p+ p + p + ··· (4) u (p, T ) = M RT RT RT where β a , γ a and δ a are the second, third and fourth virial coefficients, which are functions of temperature alone. In this equation, R = 8.314471 J/mol/K is the universal gas constant as reported by Moldover et al. [9], and γ 0 = cp0 /cv0 is the ratio of the isobaric to the isochoric ideal gas heat capacity. Eq. (4) was fitted to acoustic data on each isotherm by adjusting the first three or four acoustic virial coefficients (up to γ a or δ a ), depending on the statistical significance of the fit. All the isotherms were combined together to define a surface u(p, T), whose deviations from a fitted correlation were a measure of the internal consistency between the isotherms. The correlation yielded estimates of the first two density virial coefficients, B and C, from the analysis of the acoustic data and suitable parameterized expressions for the acoustic virial coefficients defined by a hard-core square-well (HCSW) approximation of the intermolecular potential [3].

3. Experimental results Table 2 shows the mean speed of sound values at each temperature and pressure, together with deviations δ = 105 [huexp i − ufit ]/ufit from the surface fit. Table 3 shows the results of the isotherm fits, together with the values determined for the second, B, and third, C, virial coefficients obtained from the surface fit. The heat capacity values in Table 3 can be represented by cp0 R

= 2.2792 + 3.801 × 10−2 T − 25.2 × 10−6 T 2

(5)

with a standard deviation δ = 0.023R. The results of comparison with cp0 data in the literature are shown in Fig. 1 in the form of deviations from Eq. (5), calculated as 1cp0 /cp0 = (cp0 exp − cp0 calc )/cp0 calc . The Table 3 Perfect gas heat capacity, associated 2σ uncertainties, and second and third acoustic virial coefficients of R125 determined from isotherm fits T (K)

cp0 /R

2σ [cp0 /R]

β a (cm3 /mol)

γ a (cm3 /mol kPa)

B (cm3 /mol)

10−4 C (cm6 /mol2 )

259.996 280.002 300.029 310.020 320.006 339.989 360.013

10.468 10.933 11.420 11.655 11.839 12.321 12.687

0.012 0.015 0.011 0.004 0.014 0.008 0.003

−806.1 −682.8 −579.2 −537.5 −496.6 −431.0 −376.0

−0.180 −0.074 −0.047 −0.029 −0.019 −0.008 0.004

−517.5 −430.8 −363.3 −335.0 −309.1 −266.2 −230.1

2.7 3.7 3.6 3.4 3.1 2.7 2.3

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Fig. 1. Deviations of ideal gas heat capacities of experimental origin for R125 with respect to cp0 calc from Eq. (5).

data are divided according to the experimental source: spectroscopic or acoustic. The spectroscopic data [3–6,10,11] enable a continuous evaluation of the ideal gas heat capacity, but with a rather lower accuracy than acoustic data. In Yokozeki et al. [4], the spectroscopic data were also processed using the RRHO model. The cp0 values emerging from the present study are consistent with acoustic data values in the literature (with the Yokozeki’s values from spectroscopic data). Clearly, the scatter in the data considerably exceeds the imprecision in fitting any single isotherm. We suspect that minor variations in the sample’s air content at different subsequent resonator fillings are responsible for this effect. The surface u(p, T ) was fit with eight adjustable parameters, while Eq. (5) was used to represent the experimental γ 0 (T ) values, required to calculate the temperature dependence of the acoustic virial coefficients. The regression led to the parameters: b0B = 117.75 cm3 /mol; ε/k B = 422.5 K; r B = 1.3243; b0C = 277.07 cm3 /mol; ε/k C = 531.51 K;b0D = 198.65 cm3 /mol; ε/k D = 162.70 K with r D = 2. Here, b0i , ε/ki and ri are the co-volume, the scaled well depth and the ratio of the width of the well to that of the hard-core for the second B, third C and fourth D virial coefficient, respectively. A comparison of estimated temperature dependence B(T ) and C(T ) from this work with those obtained from p–V–T measurements [12,13] and other acoustic data [3], shows an agreement which is in the order of 0.5% for B(T ) and within 20% for C(T ) over the experimental temperature range. 4. Gas density determination from speed of sound 4.1. Reduction procedure The present work proposes a new and simplified approach to the set-up of a non-conventional equation, from speed of sound data alone, which can determine gas phase densities with a high degree of accuracy.

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The problem of deriving easy and accurate thermodynamic models from speed of sound data alone has aroused much interest in recent years, since precise speed of sound data have become available particularly for the new refrigerant fluids. The procedures involved can be summarized as follows. Among the presently available data reduction methods, one possibility [14] consists in the solution of the analytical relations between density and speed of sound. The solution is obtained using methods based on the numerical integration of the differential equations, linking density and speed of sound with the predictor–corrector method [15]. The integration method assumes no specific form for the equation of state, but it does require initial conditions, usually specified values of an isotherm close to the critical temperature. The method was used to reduce speed of sound data for the vapor phase of ethane [15]. The second method [3,4] aims to estimate the density of gases in the range of temperatures and pressures spanned by speed of sound data: 0.6 ≤ T r ≤ 1.0; 0.05 ≤ p ≤ 1.0 MPa. The gas density is deduced from a virial equation of state in which the temperature dependencies of the first three density virial coefficients are calculated using the HCSW potentials model. This method was applied to convert speed of sound data on a group of refrigerants [3].Whatever the method, the conventional equation assumed is usually a virial-type equation. In the present work, a new approach is followed that aims to obtain a fundamental Helmholtz free energy equation specifically for the gas phase from speed of sound data alone. For this purpose, the Helmholtz equation assumed is expressed in the following dimensionless form: φ(τ, δ) =

Am = φ 0 (τ, δ) + φ r (τ, δ) RT

(6)

where Am is the molar free energy, R the universal gas constant, τ = T c /T the inverse reduced temperature and δ = ρ/ρ c the reduced density. The critical parameters T c = 339.33 K and ρc = 571.29 kg/m3 were used for temperature and density reduction. The dimensionless form φ of Eq. (6) is split into an ideal part φ 0 , describing the ideal gas properties, and a residual part φ r , taking the behavior of the real fluid into account. From the ideal gas heat capacity, Eq. (5), the dimensionless ideal part φ 0 can be expressed through the Helmholtz energy of the ideal gas A0 as (Z )    Z T 0 T 0 C − R A 1 ρ p − Ts00 C 0 dT + h00 − RT − T = dT − RT ln (6a) φ0 = RT RT T0 p T ρ0 T0 where h00 and s00 are the enthalpy and entropy of the ideal gas in a reference state. From the fundamental Eq. (6), the speed of sound u is determined by the following relation r 2 (1 + δφδr − δτ φδτ u2 (τ, δ) ) r + = 1 + 2δφδr + δ 2 φδδ RT cv /R

(7)

r r are the first and second derivatives of the residual part φ r with respect to δ, while φδτ where φδr and φδδ r is the cross-derivative of the residual part φ with respect to δ and τ . cv is the isochoric heat capacity, which is related to the fundamental Eq. (6) by the relation

cv (τ, δ) = −τ 2 (φτ0τ + φτr τ ) R

(8)

where φτ0τ and φτr τ are the second derivatives, with respect to τ , of the ideal part φ 0 and of the residual part φ r , respectively.

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Table 4 Coefficients for the residual part φ r regressed from speed of sound data n

an

tn

dn

1 2 3 4 5 6 7 8

6.582354133 × 10−1 5.650210429 × 10−2 −3.45848603 × 10−1 8.934278078 × 10−4 1.561128939 × 10−3 8.380824041 × 10−1 −2.20543052 × 10+0 −7.81026567 × 10−2

−0.5 0.0 0.0 0.0 1.5 1.5 2.0 2.0

2.0 1.0 3.0 6.0 6.0 1.0 1.0 2.0

In this work, the cp0 function is obtained from the same speed of sound measurements and enables the derivative φτ0τ required to calculate the cv value, Eq. (8), to be determined directly from the known ideal gas relation: cp0 − cv0 = R

(9)

All the other derivatives are easily calculated from the following assumed form of the residual part φ r : φ (τ, δ) = r

8 X

ai τ ti δ di

(10)

i=1

The structure of the present equation has been found by employing the optimization method proposed by Wagner and coworkers [16,17], also known as Schmidt–Wagner (SW) technique, through which all the modern high precision dedicated EoS are nowadays obtained. This optimization procedure allows to select from a so-called bank of Fi (T,ρ) terms, the combination which yields the smallest weighted least-squares sum after fitting the proposed equations to the experimental available data. Due to the assumed form of the φ r term, no temperature dependences are provided for the coefficients ai , ti and di in Eq. (10), as is the case with a virial equation, instead they are obtained directly from fitting on speed of sound data without considering any potential energy model. Unlike the method proposed by Trusler [14], this procedure does not require the solution of a strict algorithm with specific initial conditions. Using a classical regression procedure, the coefficients for the residual part φ r are obtained and shown in Table 4. Table 5 Density validation for R125 Tr range

pr range

AAD % aR model

AAD %HCSW model

AAD %Dedic. EoS

NPT

Ref.

0.91–1.41 0.82–1.39 0.81–1.07 Overall

0.039–0.55 0.225–0.54 0.091–0.54

0.230 0.985 0.241 0.355

0.092 0.860 0.282 0.304

0.058 0.691 0.064 0.162

46 20 59 125

[14] [15] [18]

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77

Fig. 2. Density validation for the aR Eq. (6) vs. Tr .

4.2. Model validation The proposed Helmholtz free energy equation has been used to deduce gas density data from available vapor phase speed of sound data [12,18,19]. As shown in Table 5, the available density data cover a Tr –pr range of 0.91 ≤ T r ≤ 1.41; 0.039 ≤ pr ≤ 0.55, which differs significantly from the speed of sound measurements range, i.e. 0.766 ≤ T r ≤ 1.061; 0.0136 ≤ p r ≤ 0.138, on which the parameters have

Fig. 3. Density validation for the aR Eq. (6) vs. Pr .

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been fitted. The results obtained should consequently be regarded as having the accuracy reached by the proposed equation in an extrapolation test. The accuracy reached in terms of AAD% is comparable, however, with that of the R125 dedicated EoS [20] and, as shown in Fig. 3, on a similar level particularly at low pressure values. Its behavior depends mainly on pressure, with a general decline in the results at high pressures outside the range covered by the speed of sound measurements. With respect to the experimental data examined, a reliable conclusion cannot be drawn because the dedicated EoS also fails at high pressures. In Fig. 2, the error deviation results versus Tr are plotted for the proposed aR equation, the dedicated EoS and the HCSW potential model. The datum-line is set on the experimental data available [12,18,19]. The same analysis is represented in Fig. 3, where the values are plotted versus pr . The results may be of some interest, suggesting that a consistent prediction accuracy can be obtained for the whole pρT surface when speed of sound data become available on a wider Tr –pr range.

5. Conclusions Speed of sound measurements in the vapor phase of pentafluoroethane (R125) were performed in a stainless steel spherical resonator in a temperature range from 260 to 360 K and at pressures up to 500 kPa. Ideal gas heat capacities and acoustic virial coefficients are deduced directly from the data. Accurate gas density values can be deduced from the speed of sound data. A new procedure for density equation development has been proposed, aiming to simplify the data reduction by comparison with the two conventional methods. In this case, the diluted-gas speed of sound data are not needed and the temperature dependences of the equation coefficients are fitted directly on the experimental data. The density predictions are compared with experimental data in similar Tr –pr ranges and with a highly accurate dedicated equation. The results obtained are highly accurate and are always comparable with those of the dedicated EoS. List of symbols B(T ) second virial coefficient for the thermodynamic model cp isobaric heat capacity isochoric heat capacity cv C(T) third virial coefficient for the thermodynamic model M molar mass R universal gas constant T temperature u speed of sound x molar fraction Greek letters βa sound virial coefficient γ = cp /cv heat capacities ratio sound virial coefficient γa δ deviation of experimental data from fit δ = ρ/ρ c reduced density φ0 ideal part of Helmholtz free energy

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φr residual part of Helmholtz free energy τ = T c /T inverse reduced temperature Subscripts c critical state r reduced state exp experimental calc calculated Superscripts 0 ideal state r residual part

Acknowledgements The contribution of the Ausimont SpA (Italy) Meforex laboratory is gratefully acknowledged. In particular, Dr. Musso and Mr. Girolomoni have been of considerable help with advice, the provision of the sample and its GC analysis. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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