2678
J. Chem. Eng. Data 2009, 54, 2678–2688
Measurements of the Thermal Conductivity of HFC-134a in the Supercritical Region† B. Le Neindre,*,‡ Y. Garrabos,§ F. Gumerov,| and A. Sabirzianov| L.I.M.H.P.-C.N.R.S., Institut Galile´e, Universite´ Paris Nord, 99 Av. J. B. Cle´ment, 93430 Villetaneuse, France, and ICMCB-CNRS UPR 9048, Universite´ Bordeaux I, 87 Av du Dr A. Schweitzer, F 33608 PESSAC Cedex, France, and Kazan State Technological University, 68 K. Marx Street, Kazan, 420015 Tatarstan, Russia
Measurements of the thermal conductivity of HFC-134a made in a coaxial cylinder cell operating in steady state are reported. Measurements were performed along several quasi-isotherms at temperatures ranging from the critical temperature Tc (∼374 K) to Tc + 150 K and in a pressure range from (0.1 to 40) MPa. Parameters of a background equation were determined from experimental data to analyze the critical enhancement of the thermal conductivity as a function of temperature and density. An analysis of the various sources of errors leads to an estimated uncertainty of the thermal conductivity data of ( 3 %.
Table 1. Thermal Conductivity of HFC-134a Near Atmospheric Pressure
Introduction In the past years, there has been a great industrial interest in the determination of the thermophysical properties of alternative refrigerants. In our laboratory, we have carried out a series of measurements of the thermal conductivity of several alternative refrigerants over a large range of temperature (T) and pressure (p), including the subcritical and supercritical regions.1,2 Among them, HFC-134a (1,1,1,2-tetrafluoroethane) was considered for a long time as an environmentally acceptable alternative refrigerant to CFC-12 (dichlorodifluoromethane). Some previous measurements3 of the thermal conductivity of HFC-134a have been already performed, along several quasi-isotherms between (300 and 530) K and from (0.1 to 50) MPa. The present thermal conductivity measurements of HFC-134a were carried out as a function of temperature between Tc (∼374 K) and Tc + 150 K and pressures up to 40 MPa, including then the homogeneous critical region, using vertical coaxial cylinders, operating in the steady state mode. This method of measurement and the applied corrections were previously described.4 During the experiments, the stability of the temperature was kept better than 0.05 K, and the precision of temperature measurements was ( 0.05 K. The pressure was measured with a precision pressure transducer with accuracy of 0.02 %. The consistency of pressure and temperature measurements was checked along the saturation pressure curve with vapor pressures calculated from the equation of state developed by Tillner-Roth and Baehr.5 The purity of the sample was estimated to be better than 99.9 % by the manufacturer’s analysis and appears to conform with the purity specifications (especially for water contamination) of the RoundRobin samples of R134a.6 New experimental results are reported in Table 1 for the dilute gas state (near 1 bar pressure) and in Table 3 to Table 17 for the fluid dense state along 15 quasiisotherms, up to 21 MPa. The nominal temperature is the temperature at the critical density. The density (F) was calculated with the equation of state developed by Tillner-Roth and Baehr,5 * Corresponding author. E-mail:
[email protected]. † Part of the “William A. Wakeham Festschrift”. ‡ Universite´ Paris Nord. § Universite´ Bordeaux I. | Kazan State Technological University.
T
p
λ0(exp)
100 · [(λ0(cal) - λ0(exp)]/λ0(exp)
K
MPa
mW · m-1 · K-1
[λ0(cal) from eq 5]
299.29 299.63 318.97 318.97 319.14 345.58 346.36 348.54 351.84 363.88 365.00 365.41 365.69 366.52 366.80 372.42 372.43 372.43 378.07 378.63 378.74 378.90 379.17 379.43 379.72 392.65 426.86 436.44 446.25 446.25 451.17 455.69 455.92 466.31 476.24 496.18 515.97
0.17 0.17 0.74 0.74 0.16 0.18 0.18 0.18 0.64 0.18 0.18 0.18 0.18 0.18 0.18 0.68 0.17 0.68 0.18 0.44 0.18 0.33 0.30 0.15 0.16 0.18 0.16 0.18 0.20 0.76 0.18 0.10 0.68 0.65 0.10 0.10 0.10
12.87 12.79 14.89 14.86 14.60 17.11 17.39 17.32 17.63 18.91 18.91 18.93 18.93 18.92 18.93 19.75 19.68 19.65 20.03 20.17 20.12 20.26 20.23 20.35 19.96 21.36 24.69 25.42 26.65 26.61 26.83 26.80 27.19 28.46 29.22 31.10 33.04
–0.27 0.59 –1.64 –1.44 0.42 –0.09 –1.28 0.27 0.23 –0.69 –0.15 –0.05 0.08 0.54 0.62 –0.94 –0.58 –0.43 0.28 –0.16 0.13 –0.49 –0.21 –0.69 1.39 0.31 –0.46 0.15 –1.09 –0.94 –0.06 1.60 0.22 –0.89 –0.34 –0.46 –0.80
with accuracy of the order of ( 0.2 % in the critical region as stated by the authors. In fact, the accuracy can be much lower, close to the critical point, and can reach ( 2 %. The critical parameters together with the estimated uncertainties are given as follows
10.1021/je900210h CCC: $40.75 2009 American Chemical Society Published on Web 06/12/2009
Journal of Chemical & Engineering Data, Vol. 54, No. 9, 2009 2679
Tc ) (374.21 ( 0.06) K, pc ) (4.05928 ( 0.016) MPa, -3
and Fc ) (511.9 ( 4) kg·m
(1)
The measurements were carried out to make an analysis of the data based on the residual method. The thermal conductivity data are covering the supercritical region (T > TC) of the p, F, T phase surface, corresponding to three main parts, i.e., the gaseous state (F ∼ 0) at p ) 1.01325 bar, the moderate (0 < F e 1.2Fc), and the high (F > 1.2Fc) dense states, respectively. Consequently, the thermal conductivity expressed as a function of temperature and density was separated into three parts
λ(T, F) ) λ0(T) + δλ(F) + ∆λc(T, F)
(2)
where λ0(T) is the dilute gas thermal conductivity which is only a function of temperature at F ) 0, δλ(F) is the residual thermal conductivity which is only a function of density, and ∆λc(T,F) is the critical enhancement approaching the liquid-gas critical point. In that description, the two first contributions of eq 2 represent the background thermal conductivity
λb(T, F) ) λ0(T) + δλ(F)
in Table 1. As a theoretical calculation of the thermal conductivity of a polar fluid in the dilute-gas region requires several parameters which are difficult to evaluate with a good accuracy,3 these experimental data were fit to an empirical linear equation
λ0 ) a0 + a1T
(4)
where λ0 and T are expressed in mW · m-1 · K-1 and K, respectively. Fitting the complete set of the 37 data of Table 1 and the 32 data given in ref 3, the related values of two adjustable parameters are a0 ) -14.7107 mW · m-1 · K-1 and a1 ) 0.0920389 mW · m-1 · K-2. The relative deviation between the experimental data and the theoretical values calculated using eq 4 are reported in Table 1 and shown in Figure 1 for the complete data set. The related 1σ-standard deviation of 0.70 % is also illustrated in Figure 1, which appears within our estimated experimental uncertainty of 1.5 % (i.e., approximately 2σ, see ref 3) in this range of measurement.
(3)
To determine λb(T,F), we have considered the data in the high dense phase and in the gas phase at atmospheric pressure and far from the critical temperature (T > 385 K) region, including then the related parts of our previous measurements reported in ref 3. The resulting critical enhancement ∆λc(T,F) of the thermal conductivity is given in column 5 of Tables 3 to 17. The uncertainties of the thermal conductivity measurements were estimated to be of the order of ( 3 %. The main error is due to the measurement of the temperature difference between the concentric cylinders. We note that several other measurements of the thermal conductivity of R134a were performed under pressure, for instance those of Assael et al.,7 with an estimated accuracy of 0.5 %. At supercritical temperatures from (390 to 450) K and at pressures to 70 MPa, Perkins et al. have reported several sets of data obtained with a transient hot-wire cell.8,9
Dilute-Gas Thermal Conductivity The thermal conductivity data measured at atmospheric pressure in the temperature range (299 to 516) K are presented
Figure 1. Fractional deviation 100 · ∆λ0/λ0 ) 100 · [λ0(cal) - λ0(exp)]/λ0(exp) of the dilute gas thermal conductivity of R134a as a function of temperature. This work: 9 (Table 1) and [ Le Neindre and Garrabos (ref 3), where the calculated values were obtained from eq 4. From experimental values of Table 1 and calculated values obtained from the analytical correlations reported in Table 2: ×, Perkins et al.;8,9 *, Soldner et al.;10 -, Krauss et al.;;11 4, Hammerschmidt;12 0, Tsvetkov et al.;13 +, Gross et al.;14 ], Tanaka et al.;15 O, Yata et al.16 The dashed lines are one standard deviation of the fit of our complete experimental data set using eq 4 (see text).
In Table 2 are listed similar analytic equations (up to the third order) that have been reported in the literature8–16 to
Table 2. Representative Equations of the Thermal Conductivity λ0 of R134a, at Atmospheric Pressure equation λ0 ) a0 + a1T λ0 ) a0 + a1T + a2T2 λ0 ) a0 + a1T + a2T2 λ0 ) a0 + a1T + a2T2 λ0 ) a0 + a1T λ0 ) a0 + a1T + a2T2 + a3T3
λ0 ) a0 + a1ta λ0 ) a0 + a1T λ0 ) a0 + a1(T/Tc) + a2(T/Tc)2b
a
t ) T - 273.15 K. b Tc ) 374.27 K.
coefficients a0 a1 a0 a1 a2 a0 a1 a2 a0 a1 a2 a0 a1 a0 a1 a2 a3 a0 a1 a0 a1 a0 a1 a2
) -14.7107 ) 0.0920389 ) 5.79393 ) -0.0283907 ) 0.176461 · 10-3 ) -17.314 ) 0.12048 ) -0.5926 · 10-4 ) -16.5744 ) 0.124286 ) -0.761796 · 10-4 )-12.68 ) 0.087 ) -1.203289 ) 0.020169658 ) 0.113395 · 10-4 ) -0.5232105 · 10-7 )11.3853 ) 0.0754957 ) -13.39 ) 0.0872932 ) -6.3643 ) 19.099 ) 6.8704
T-range/K
sd (σ) (%)
ref
299 to 516
0.70 (69 data)
this work
200 to 450
2.43 (52 data)
8, 9
294 to 363
1.89 (19 data)
10
240 to 410
3.12 (41 data)
11
298 to 463
1.43 (58 data)
12
240 to 400
2.33 (40 data)
13
250 to 360
2.42 (18 data)
14
293 to 363
0.85 (19 data)
15
200 to 600
1.25 (69 data)
16
2680
Journal of Chemical & Engineering Data, Vol. 54, No. 9, 2009 Table 4. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 374.96 K
Table 3. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 374.37 K T
p
K
MPa
379.43 375.04 376.38 376.23 375.93 375.76 375.59 375.42 375.10 374.92 374.76 375.34 375.14 375.10 375.18 374.89 374.72 374.66 374.61 374.57 374.54 375.24 374.47 374.39 374.37 374.37 374.38 374.38 374.28 374.31 374.32 374.34 374.36 374.38 374.40 374.18 374.19 374.21 374.22 374.22 374.23 374.37 374.37 374.37 374.50 374.51 374.52 374.52
0.154 0.888 2.200 2.350 2.700 3.000 3.196 3.349 3.544 3.675 3.717 3.828 3.928 3.981 4.030 4.035 4.049 4.063 4.068 4.073 4.072 4.120 4.071 4.070 4.073 4.074 4.078 4.083 4.085 4.090 4.093 4.097 4.100 4.105 4.110 4.090 4.100 4.120 4.142 4.149 4.187 4.245 4.270 4.358 4.533 4.621 4.710 4.759
F
λ -3
kg · m
5.0 31.6 91.9 100.6 123.4 146.4 164.1 180.1 205.0 225.8 234.2 252.4 280.5 299.7 320.4 332.3 350.4 369.7 382.1 397.9 400.5 403.9 408.6 436.4 529.9 554.1 584.3 605.8 632.7 635.3 637.9 640.2 641.2 644.4 647.4 652.3 660.1 672.4 683.7 686.9 701.3 713.2 719.6 737.9 761.2 771.3 780.4 785.0
-1
mW · m · K 23.1 23.5 25.2 25.6 26.5 27.4 28.7 29.8 32.1 34.2 35.2 36.7 40.9 42.8 47.9 49.4 52.9 59.3 67.4 76.4 83.1 101.6 109.6 164.4 197.4 197.4 182.8 174.2 141.0 123.3 113.9 105.7 99.3 91.4 85.1 72.9 70.8 68.0 65.9 65.4 64.2 64.3 64.3 64.1 63.7 63.5 61.7 61.0
T
∆λc -1
-1
-1
mW · m · K 0.1 0.2 0.2 0.3 0.6 0.8 1.5 2.2 3.7 5.1 5.8 6.6 9.9 11.1 15.3 16.3 19.1 24.4 31.7 39.8 46.1 63.6 71.1 122.7 150.6 149.5 133.9 124.5 91.1 74.1 64.9 57.0 50.8 43.0 36.8 25.0 22.6 19.2 16.6 15.8 13.9 13.3 12.9 11.6 9.7 8.8 6.5 5.5
F
p
λ -3
kg · m
K
MPa
376.96 376.51 376.33 375.81 375.46 375.43 375.34 375.30 375.11 375.07 375.03 374.99 374.97 374.96 374.96 374.96 374.97 375.00 375.03 375.04 375.06 375.10 375.13 375.14 375.16 375.17 375.18 375.19 375.20 375.19 375.19 375.19 375.18 375.18 375.03 375.01 374.99
2.027 2.527 3.030 3.534 4.029 4.051 4.085 4.089 4.099 4.097 4.099 4.107 4.112 4.117 4.119 4.126 4.132 4.149 4.155 4.160 4.168 4.185 4.199 4.216 4.244 4.289 4.340 4.379 4.456 4.458 4.551 4.557 5.054 5.057 6.045 8.053 10.033
∆λc
-1
-1
mW · m · K
82.2 111.3 148.0 201.0 312.9 324.8 350.9 356.4 381.3 382.1 389.6 413.7 437.0 470.7 486.3 543.3 571.9 611.8 616.9 623.0 630.7 643.8 652.6 663.2 676.8 694.1 709.0 711.3 733.5 734.1 748.9 749.8 799.6 799.6 854.6 915.5 954.6
21.8 22.9 24.2 27.9 39.7 42.5 51.7 57.3 67.1 77.3 91.0 110.6 117.6 123.2 124.4 124.4 119.8 101.0 91.0 86.1 79.6 70.4 64.3 61.2 58.6 56.5 55.0 54.6 53.0 54.1 54.5 53.8 55.7 55.0 57.6 61.5 64.0
mW · m-1 · K-1 -0.2 0.2 0.5 2.8 11.2 13.7 22.0 27.5 36.4 46.6 60.0 78.7 84.9 89.2 89.7 87.2 81.2 60.4 50.2 44.9 38.1 28.1 21.6 17.8 14.5 11.4 9.0 8.4 5.4 6.5 6.0 5.3 3.9 3.1 1.8 1.1 0.4
function that represents λ0 (see eqs 4 to 6 of ref 3). In spite of the difficulty to calculate accurately the contributions from internal degrees of freedom, this theoretical estimation gives a 1σ-standard deviation of 0.94 % from our λ0(exp) data set. However, we note that the correct CPo temperature exponent of the right-hand side of eq 6 given in ref 3 is i, and not i - 1.
Dense Fluid Thermal Conductivity represent the temperature variation of the thermal conductivity of R134a at atmospheric pressure. The values of their adjustable coefficients are reported in column 2, while their respective temperature range of validity is given in column 3, covering then the complete temperature range from (200 to 600) K. The corresponding 1σ-standard deviation from our experimental results given in Table 1 is reported in column 4, which clearly shows the weakness of some analytic equations.8,9,11,13 These systematic deviations of the temperature dependence of the thermal conductivity of the dilute gas are also illustrated in Figure 1, while this figure simultaneously reveals the noticeable agreement between the experimental data obtained by static methods.12,15 As was mentioned in ref 17, the uncertainty at low densities is much larger than that at higher densities due to the large inner and outer boundary corrections in transient hotwire measurements. Therefore, Figure 1 indicates that the experimental data generally agree with the linear correlation of eq 4 within an uncertainty of 2σ, especially for the temperature range (300 to 600) K. Such a linear form can be of practical interest to complement our previous theoretical dilute gas
To determine the residual density term δλ(F) of the thermal conductivity, we have used our present measurements in the high dense phase and in the gas phase outside the critical region, in a temperature range from (300 to 550) K and pressure up to 50 MPa, including some related data already reported in ref 3. This density-dependent term of the thermal conductivity was then represented by the following six-order polynomial dimensionless form
δλ(F) ) ΛC
6
()
∑ li FFc i-1
i
(5)
where ΛC is the reference value of the critical thermal conductivity defined as
ΛC )
R5/6P2/3 c 1/2 1/3 T1/6 c M NA
) 2.053 mW·m-1·K-1
(6)
By fitting experimental data with eq 5, we have obtained the following values of the adjustable parameters
Journal of Chemical & Engineering Data, Vol. 54, No. 9, 2009 2681
l1 ) 6.02829, l2 ) 2.36295, l3 ) -4.43933, l4 ) 6.20601, l5 ) -2.80918, and l6 ) 0.451252 with Fc ) 511.9 kg · m-3. The 1σ-standard deviations between calculated and experimental thermal conductivity values are within the uncertainty of the measurements, which was estimated to ( 1.5 %. We note that similar dimensionless correlations of eq 5 with different magnitude of σ were reported by Krauss et al.11 and Yata et al.16
Critical Enhancement The following analysis of the thermal conductivity and diffusivity of R134a in the critical region was carried out in terms of the effective values of the critical exponents, using a similar approach of the crossover modeling initially proposed by Luettmer-Strathmann and Sengers18 to describe the singular behavior of the thermal diffusivity DT ) λ/(FCp). Cp is the isobaric specific heat capacity. As noted previously, the value of the critical enhancement term ∆λc(T, F) of the thermal Table 5. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 375.14 K T
p
K
MPa
376.23 376.21 376.01 375.99 375.95 375.65 375.61 375.60 375.53 375.51 375.46 375.40 375.37 375.33 375.32 375.15 375.14 375.14 375.14 375.14 375.14 375.14 375.14 375.15 375.15 375.16 375.16 375.17 375.18 375.19 375.20 375.21 375.21 375.23 375.23 375.24 375.28 375.30 375.31 375.31 375.31 375.31 375.31 375.32 375.32 375.32 375.46 375.48
3.734 3.800 3.908 3.936 3.993 4.023 4.059 4.065 4.097 4.107 4.118 4.126 4.128 4.129 4.130 4.130 4.132 4.134 4.136 4.138 4.140 4.142 4.144 4.146 4.148 4.150 4.153 4.157 4.160 4.163 4.166 4.170 4.171 4.178 4.181 4.189 4.208 4.229 4.254 4.265 4.294 4.307 4.341 4.389 4.450 4.490 4.735 4.848
F
λ -3
kg · m
228.5 240.3 265.0 272.4 289.9 306.1 324.0 327.6 351.8 361.8 377.4 395.1 402.6 410.6 414.8 460.7 477.2 490.4 504.2 517.8 530.9 543.0 553.8 559.1 567.9 572.1 582.6 591.3 596.4 600.9 605.0 610.6 612.3 619.9 624.1 632.4 646.2 659.6 672.8 677.9 689.6 694.1 704.5 716.4 729.1 736.3 766.6 777.8
-1
∆λc -1
mW · m · K 30.1 31.2 34.0 34.9 37.4 39.3 42.5 43.2 49.2 52.1 59.1 68.3 76.3 87.0 91.6 105.5 107.2 108.0 108.0 108.0 108.0 107.2 106.3 104.7 102.4 100.9 98.8 94.1 91.6 88.1 84.9 82.3 81.4 77.7 76.0 73.1 66.7 62.7 61.3 61.0 60.5 60.3 60.0 59.2 58.5 58.8 58.2 56.6
mW · m-1 · K-1 4.1 4.9 7.0 7.7 9.6 11.0 13.7 14.3 19.5 22.0 28.6 37.1 44.9 55.2 59.7 71.9 72.9 73.1 72.6 72.0 71.4 70.0 68.7 66.8 64.0 62.3 59.7 54.6 51.9 48.1 44.6 41.8 40.8 36.7 34.7 31.4 24.3 19.5 17.4 16.8 15.6 15.1 14.2 12.7 11.2 11.0 8.5 6.1
Table 6. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 375.28 K T
p
K
MPa
376.07 376.06 376.05 376.04 375.95 375.94 375.90 375.87 375.87 375.80 375.76 375.73 375.69 375.66 375.63 375.61 375.58 375.56 375.52 375.49 375.46 375.42 375.36 375.34 375.32 375.29 375.28 375.28 375.28 375.28 375.28 375.29 375.30 375.31 375.32 375.33 375.36 375.38 375.39 375.40 375.42 375.43 375.43 375.45 375.46 375.46 375.46 375.46 375.18
0.200 0.524 0.773 1.048 2.429 2.662 2.970 3.165 3.264 3.504 3.658 3.772 3.874 3.917 3.965 3.999 4.040 4.055 4.086 4.097 4.107 4.115 4.121 4.126 4.134 4.139 4.145 4.150 4.156 4.162 4.167 4.174 4.181 4.189 4.197 4.204 4.216 4.225 4.235 4.244 4.265 4.280 4.292 4.349 4.473 4.527 4.625 4.681 4.942
F
λ -3
kg · m
-1
∆λc -1
mW · m · K
6.6 17.9 27.1 37.8 105.6 120.7 143.7 160.6 170.1 197.2 218.7 238.2 259.9 271.0 285.4 297.4 315.2 323.3 343.8 353.8 364.8 376.7 390.9 402.6 426.0 452.9 488.8 518.5 550.4 574.4 589.2 602.8 613.4 623.4 631.6 637.7 645.4 650.6 656.7 661.6 671.6 678.0 683.0 701.4 729.6 739.0 753.3 760.3 790.8
20.5 20.8 21.2 21.4 23.5 24.0 25.1 26.1 26.4 29.0 30.6 31.9 34.3 36.1 37.8 39.5 42.1 43.4 47.5 50.8 54.5 61.1 72.1 77.3 85.6 94.3 98.6 100.2 100.2 99.4 98.4 95.5 92.7 88.9 84.8 81.6 74.6 68.6 66.8 64.5 60.8 59.6 58.8 56.7 55.5 55.0 55.0 55.2 55.9
mW · m-1 · K-1 0.5 0.5 0.7 0.6 1.0 1.1 1.5 2.1 2.1 4.0 4.9 5.7 7.5 8.9 10.2 11.6 13.6 14.6 18.1 21.0 24.4 30.5 41.1 45.9 53.3 61.0 63.8 64.1 62.6 60.7 58.9 55.4 52.1 47.7 43.2 39.7 32.2 25.9 23.8 21.1 16.9 15.4 14.3 11.1 8.1 7.1 6.1 5.9 4.6
conductivity was estimated by subtracting the background thermal conductivity term λb(T, F) of eq 3 from the experimental data. The above separation of the thermal conductivity into critical and background contributions implies a corresponding separation of the thermal diffusivity DT into a critical contribution ∆Dc ) (∆λc)/(FCp) and a background contribution Db ) λb/(FCp), with
DT ) ∆Dc + Db
(7)
The mode coupling of critical dynamics theory19 predicts that ∆Dc of a critical pure fluid in the hydrodynamic limit can be written asymptotically near its critical point as
∆Dc ) RD
kBT 6πηξ
(8)
where η is the shear viscosity; and ξ is the long-range correlation length of the density fluctuations. The prefactor RD is a universal amplitude combination whose value is close to unity.19 Along
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Journal of Chemical & Engineering Data, Vol. 54, No. 9, 2009
χ*T ) κT pcZc
Table 7. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 375.72 K T
F
p
K
MPa
376.61 376.60 376.59 376.56 376.55 376.53 376.51 376.50 376.48 376.46 376.44 376.42 376.41 376.40 376.11 376.10 376.08 376.06 376.04 375.99 375.92 375.87 375.83 375.80 375.77 375.76 375.72 375.73 375.72 375.73 375.74 375.76 375.79 375.82 375.84 375.85 375.86 375.86 375.87 375.87 375.88 375.88 375.89 375.88 375.88 375.89 376.16 376.15 376.15 376.15 376.15 376.15 376.14 376.14 376.14
3.210 3.210 3.210 3.210 3.302 3.468 3.504 3.593 3.675 3.783 3.783 3.872 3.872 3.872 3.913 3.941 3.975 4.011 4.039 4.072 4.120 4.132 4.141 4.150 4.161 4.161 4.178 4.179 4.187 4.199 4.210 4.223 4.241 4.254 4.279 4.296 4.316 4.334 4.349 4.363 4.387 4.432 4.451 4.461 4.461 4.630 4.754 4.789 5.037 5.092 5.216 5.296 5.520 5.651 5.740
λ -3
kg · m
-1
∆λc -1
-1
mW · m · K
163.3 163.4 163.4 163.4 172.4 190.4 194.8 206.2 217.9 235.6 235.7 253.1 253.2 253.3 265.2 272.5 282.4 294.3 305.0 320.6 352.4 365.3 377.5 391.9 415.1 416.6 479.0 479.9 516.7 557.7 584.4 604.9 624.3 634.0 651.4 660.9 670.4 678.3 683.6 688.6 695.9 708.2 712.4 715.0 715.0 744.7 755.6 759.9 784.4 788.9 798.4 803.9 818.0 825.3 830.0
-1
mW · m · K
24.4 24.6 24.8 25.7 26.3 26.9 27.5 28.1 28.8 29.7 30.3 31.5 31.7 32.3 33.3 34.2 35.2 36.2 37.9 41.7 48.5 54.8 60.0 65.6 73.1 75.2 85.7 83.0 88.1 84.1 79.9 75.2 67.3 62.7 58.9 57.3 55.8 55.1 54.6 53.9 53.5 52.6 52.1 52.6 52.4 52.0 52.3 52.7 52.9 53.1 53.4 53.6 54.0 54.3 54.5
kBT FCc 6πηξ p
t)
2
F
T
T - Tc Tc
(13)
is the usual reduced temperature distance to Tc. In eqs 11 and 12
Zc )
mp pc kBTcFc
(14)
where mp is the molecular mass and kB is the Boltzmann constant and
Yc )
[( ) ( )] ∂p ∂T
Fc,Tc
Tc pc
-1
(15)
(∂p/∂T)Fc,Tc is the slope of the critical isochore line at the critical point. κ*T ) κTpc is the usual reduced form of the isothermal compressibility. A similar analysis21 has also shown that the rescaled correlation length Table 8. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 376.02 K T
(9)
( TF )( ∂T∂p ) κ
(12)
where
where Cpc is defined as the specific heat difference C pc ) Cp CV. This latter critical property can be then calculated from the isothermal compressibility κT by using the thermodynamic relation
Ccp ) Cp - CV )
lies on a single curve when it is plotted as a function of the rescaled temperature
τ ) Yct
0.3 0.5 0.7 1.6 1.9 2.0 2.5 2.8 3.1 3.5 4.1 4.8 5.0 5.6 6.3 7.0 7.7 8.3 9.7 13.0 18.8 24.7 29.4 34.5 41.1 43.2 51.3 48.5 52.0 46.1 40.7 35.0 26.0 20.9 16.2 14.0 12.0 10.8 10.0 9.0 8.1 6.5 5.8 6.1 5.9 3.7 3.2 3.3 2.0 1.9 1.4 1.3 0.8 0.5 0.4
the critical isochore, eq 8 can be also written as the following approximated form.20
∆λc ) RD
(11)
(10)
On the other hand, it was found that along the critical isochore over an extended temperature range of any pure fluid21 the rescaled isothermal compressibility χ*T
p
K
MPa
376.98 376.95 376.92 376.90 376.87 376.81 376.78 376.73 376.69 376.21 376.19 376.10 376.07 376.05 376.04 376.03 376.02 376.02 376.02 376.02 376.02 376.03 376.03 376.04 376.04 376.06 376.07 376.08 376.09 376.09 376.10 376.11 376.26 376.26 376.54 376.54 376.68 376.82 376.82
3.510 3.650 3.740 3.800 3.880 3.990 4.040 4.100 4.140 4.150 4.160 4.180 4.190 4.198 4.200 4.203 4.205 4.207 4.209 4.213 4.217 4.222 4.225 4.230 4.233 4.242 4.250 4.260 4.270 4.280 4.300 4.320 4.360 4.390 4.470 4.580 4.680 4.820 4.860
F
λ -3
kg · m
194.0 212.2 225.8 235.9 251.3 277.7 292.9 316.3 337.1 363.3 374.0 408.5 433.5 459.6 468.3 481.0 490.7 497.7 504.7 518.6 531.9 544.7 553.3 564.0 571.2 585.9 598.0 610.6 621.0 630.8 645.6 657.3 668.4 680.6 694.8 720.4 734.3 750.8 755.6
-1
∆λc -1
mW · m · K 27.4 28.5 29.5 30.4 31.8 34.6 36.4 39.5 42.7 48.0 50.3 62.2 69.3 75.4 78.4 80.7 81.7 82.2 82.6 82.6 82.1 80.6 79.7 77.4 76.6 73.7 70.8 68.6 66.9 65.6 63.8 62.4 60.8 60.0 59.0 58.6 58.3 57.7 57.5
mW · m-1 · K-1 2.3 3.0 3.6 4.2 5.1 7.0 8.4 10.8 13.2 17.9 19.7 30.4 36.7 41.7 44.4 46.2 46.7 46.9 47.1 46.5 45.3 43.3 41.9 39.2 38.0 34.3 30.8 28.0 25.8 24.0 21.3 19.3 17.0 15.5 13.7 11.7 10.6 8.9 8.4
Journal of Chemical & Engineering Data, Vol. 54, No. 9, 2009 2683
ξ* ) ξ/a
(16)
where
T
( )
kBTc a) pc
1/3
(17)
is a single function of τ. Consequently, from eq 9, the rescaled quantity22
R*c )
Table 10. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 377.01 K
∆λcη RD χ*T ) PcZca 6π ξ* kB(T(∂P/∂T)Fc)2
(18)
should be also a single function of τ. The validity of eq 18 along the critical isochore can be controlled in the R134a case by using the critical parameters of eq 1 and (∂p/∂T)Fc,Tc ) 0.0831 MPa · K-1. The critical enhancement of the thermal conductivity data and the shear viscosity data are given in columns 2 and 3 of Table 18, respectively. The values of ∆λc were calculated at each temperature value reported in column 1 of Table 18, by Table 9. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 376.27 K T
p
F
λ
∆λc
K
MPa
kg · m-3
mW · m-1 · K-1
mW · m-1 · K-1
376.87 376.82 376.68 376.65 376.61 376.58 376.55 376.51 376.48 376.44 376.41 376.38 376.34 376.31 376.29 376.28 376.27 376.26 376.28 376.29 376.34 376.44 376.52 376.57 376.87 376.89 377.73 377.93 377.93 377.86 378.11 377.96 378.07 378.05 378.18 378.16 378.29 378.27 378.26 378.25 378.24 378.37 378.36 378.34 378.33 378.31 378.29 378.27 378.25
4.101 4.117 4.141 4.147 4.150 4.158 4.165 4.168 4.174 4.178 4.184 4.192 4.199 4.210 4.214 4.218 4.230 4.264 4.267 4.279 4.305 4.354 4.404 4.451 4.504 4.548 4.701 4.800 4.851 4.997 5.504 6.000 6.502 7.003 7.450 8.004 8.408 9.008 9.506 9.919 10.507 10.981 11.376 12.420 13.441 14.505 16.500 18.500 21.005
313.7 321.8 338.0 342.6 345.9 352.6 359.1 363.5 370.4 376.7 385.4 398.1 413.3 440.0 453.7 466.9 535.1 590.0 591.7 607.2 628.8 654.7 673.6 688.0 692.2 703.0 711.3 724.0 731.9 752.3 793.3 824.4 846.1 864.9 878.2 893.9 903.3 917.1 927.5 935.4 946.0 953.2 959.5 974.7 988.1 1000.8 1021.8 1040.2 1060.4
45.2 47.0 52.4 54.5 56.2 57.9 59.8 61.8 63.9 66.1 68.4 71.2 74.0 77.0 78.0 78.9 81.1 81.0 80.3 78.6 74.6 66.3 61.2 58.7 57.1 56.2 55.0 54.7 54.7 54.8 56.0 56.9 58.0 58.9 59.7 60.5 61.2 62.0 62.7 63.2 64.0 64.6 65.0 66.3 67.4 68.3 70.1 71.6 73.6
16.7 18.2 23.0 25.0 26.6 28.1 29.7 31.6 33.5 35.5 37.5 39.8 42.0 44.1 44.5 44.9 44.2 41.4 40.6 38.2 33.0 23.2 17.1 13.7 11.9 10.3 8.6 7.5 7.0 5.8 4.2 3.0 2.5 2.1 1.8 1.5 1.4 1.1 1.0 0.9 0.8 0.8 0.7 0.7 0.6 0.5 0.4 0.2 0.2
p
K
MPa
377.58 377.55 377.51 377.32 377.29 377.27 377.26 377.24 377.19 377.18 377.17 377.01 377.01 377.01 377.01 377.01 377.02 377.02 377.03 377.04 377.05 377.05 377.06 377.06 377.07 377.07 377.08 377.08 377.36 377.36 377.36 377.36 377.37 377.37
4.079 4.126 4.164 4.211 4.226 4.235 4.241 4.249 4.266 4.270 4.275 4.289 4.294 4.307 4.309 4.312 4.322 4.332 4.343 4.356 4.380 4.384 4.397 4.417 4.439 4.473 4.495 4.538 4.613 4.653 4.697 4.760 4.836 4.907
F
λ -3
kg · m
293.7 310.3 327.1 359.9 372.9 382.2 388.7 399.0 426.9 434.6 444.7 499.9 511.1 538.7 542.6 548.4 564.6 580.1 593.4 606.8 627.0 630.0 638.1 649.8 660.1 674.0 681.2 694.0 703.3 712.0 720.4 731.0 741.8 750.8
-1
∆λc -1
mW · m · K
mW · m-1 · K-1
36.6 39.0 41.5 46.6 49.4 51.3 52.9 55.5 63.0 64.8 67.1 72.0 72.4 72.4 72.0 71.7 70.6 69.1 67.4 66.1 63.6 63.4 62.5 61.7 60.6 60.0 59.5 58.7 58.3 57.8 57.8 57.6 56.9 56.2
8.5 10.4 12.5 16.5 18.8 20.4 21.8 24.0 30.5 32.0 33.9 36.6 36.5 35.2 34.7 34.1 32.2 30.0 27.6 25.6 22.1 21.7 20.4 18.9 17.2 15.9 14.9 13.4 12.4 11.4 10.9 10.0 8.7 7.4
Table 11. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 377.83 K T
p
K
MPa
379.10 377.97 377.95 377.93 377.91 377.89 377.87 377.85 377.84 377.83 377.83 377.83 377.83 377.84 377.84 377.85 377.85 377.86 377.86 377.87 377.87 378.02 377.89 377.89 377.90 377.90 377.90 377.90
4.331 4.302 4.301 4.309 4.321 4.330 4.339 4.351 4.360 4.369 4.381 4.390 4.400 4.411 4.420 4.432 4.441 4.451 4.460 4.470 4.479 4.500 4.518 4.531 4.541 4.551 4.561 4.582
F
λ -3
kg · m
369.9 404.5 404.9 415.6 432.9 448.2 464.8 487.7 504.5 521.1 540.2 553.5 567.0 579.2 589.1 599.8 607.7 614.9 621.5 627.4 633.0 635.5 652.1 657.7 661.3 665.2 668.8 675.9
-1
∆λc -1
mW · m · K 44.1 48.7 50.3 52.1 55.7 58.3 61.7 65.2 66.8 67.8 68.1 68.1 67.8 66.8 65.8 65.2 64.0 63.1 62.3 61.4 60.6 60.1 58.7 58.2 57.5 57.0 56.6 56.3
mW · m-1 · K-1 13.5 18.0 18.5 19.9 22.8 24.9 27.6 30.2 31.1 31.4 30.8 30.2 29.2 27.6 26.2 25.0 23.4 22.1 20.9 19.8 18.7 18.0 15.8 14.9 14.0 13.3 12.6 12.0
interpolating at the critical density the values of the experimental data measured at the closest density of Fc. The values of η at each temperature were obtained from the NIST Tables23 for F
2684
Journal of Chemical & Engineering Data, Vol. 54, No. 9, 2009 Table 13. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 379.09 K
Table 12. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 378.25 K T
p
K
MPa
378.90 378.85 378.83 378.81 378.54 378.53 378.53 378.50 378.49 378.47 378.45 378.43 378.41 378.40 378.38 378.36 378.35 378.33 378.31 378.29 378.28 378.27 378.26 378.26 378.25 378.25 378.26 378.26 378.26 378.27 378.27 378.27 378.28 378.28 378.29 378.29 378.29 378.29 378.29 378.30 378.30 378.30 378.30 378.31 378.31 378.31 378.31 378.31 378.32 378.59 378.59
3.950 4.050 4.090 4.120 4.130 4.140 4.150 4.180 4.200 4.220 4.240 4.260 4.280 4.290 4.300 4.310 4.320 4.330 4.340 4.350 4.360 4.370 4.380 4.390 4.395 4.400 4.410 4.420 4.430 4.440 4.450 4.460 4.470 4.480 4.490 4.500 4.510 4.520 4.530 4.540 4.550 4.570 4.590 4.620 4.626 4.650 4.688 4.730 4.770 4.860 4.900
F
λ -3
kg · m
250.2 271.4 281.3 289.5 295.9 299.1 302.3 313.1 320.9 329.7 339.5 350.6 363.4 370.5 378.8 388.0 397.5 408.8 421.3 435.4 449.9 465.4 481.7 497.0 506.0 513.5 527.0 541.0 554.1 565.1 576.2 586.4 594.7 603.2 610.1 617.2 623.8 629.8 635.4 640.0 644.9 653.8 661.7 671.7 673.6 680.9 691.0 700.7 708.7 718.1 724.6
-1
mW · m · K 31.7 33.9 35.1 36.0 36.4 36.8 37.1 38.5 39.5 40.8 42.2 43.8 46.1 47.2 48.8 50.6 52.4 54.7 57.0 59.7 62.2 64.5 65.7 66.4 66.7 66.7 66.7 66.1 65.5 64.5 63.9 63.3 62.5 61.6 61.1 60.8 60.3 60.1 59.8 59.5 59.3 58.8 58.5 58.1 57.8 57.3 57.1 56.9 56.7 56.5 56.5
T
∆λc -1
-1
-1
mW · m · K 4.9 6.5 7.3 8.0 8.2 8.5 8.7 9.8 10.6 11.5 12.6 13.9 15.7 16.6 17.9 19.4 20.8 22.7 24.6 26.8 28.7 30.4 31.0 30.9 30.9 30.6 30.0 28.7 27.5 26.0 24.9 23.8 22.5 21.2 20.3 19.7 18.8 18.2 17.7 17.2 16.6 15.6 14.9 13.9 13.6 12.7 11.8 11.1 10.4 9.6 9.2
) Fc. The corresponding calculated values of R*c in the rescaled temperature range (2.6 · 10-3 < τ ) Yc((T - Tc)/Tc) < 5 · 10-1) covered by the experiments are shown in column 4 of Table 18. The fit of the complete data set by an effective pure power law with adjustable exponent and adjustable amplitude gives R*c ) 0.0164τ-0.6162. We have then eliminated in the following fitting procedures the data point corresponding to the closest temperature distance to Tc, due to the large uncertainty in our estimation of ∆λc at F ) Fc and T ) 374.26 K. The best fit of the remaining data set of R*, c was thus
R*c ) 0.01441τ-0.6703
(19)
The latter power law compares favorably (deviations lower than 2 %) with the single power law R*c ) 0.0139τ-0.655 given in ref 22, which was obtained in a similar rescaled temperature range by the analysis of 14 other pure fluids. The calculated values of R*c by using eq 19 are given in column
p
K
MPa
379.50 379.50 379.48 379.47 379.45 379.43 379.42 379.40 379.40 379.39 379.37 379.36 379.34 379.33 379.32 379.30 379.29 379.28 379.27 379.26 379.25 379.23 379.23 379.09 379.09 379.09 379.09 379.09 379.09 379.22 379.09 379.23 379.23 379.24 379.24 379.24 379.24 379.25 379.25 379.25 379.26 379.26 379.26 379.26 379.26 379.26 379.40 379.40 379.41 379.41 379.41
4.165 4.175 4.200 4.224 4.254 4.282 4.288 4.309 4.317 4.330 4.346 4.355 4.369 4.376 4.386 4.396 4.404 4.414 4.424 4.431 4.442 4.452 4.453 4.452 4.460 4.464 4.470 4.475 4.484 4.504 4.500 4.520 4.530 4.540 4.547 4.560 4.570 4.600 4.620 4.642 4.662 4.680 4.703 4.739 4.766 4.806 4.838 4.888 4.932 5.000 5.051
F
λ -3
kg · m
293.4 296.2 304.0 311.9 322.9 334.6 337.5 347.8 351.8 359.1 369.2 375.3 385.8 391.5 400.0 409.7 417.7 428.3 439.6 448.2 461.8 475.7 476.9 489.5 499.6 504.6 512.0 518.2 528.9 538.6 547.0 554.5 564.3 572.6 578.7 589.3 596.7 615.6 626.7 637.3 645.5 652.5 660.7 672.0 679.5 689.4 692.4 702.7 710.5 721.6 729.0
-1
∆λc -1
mW · m · K
mW · m-1 · K-1
35.5 35.8 36.7 37.6 38.9 40.5 40.8 42.2 42.7 43.7 45.3 46.2 47.8 48.7 50.1 51.6 53.0 54.5 55.9 57.3 59.3 60.8 61.4 62.5 62.7 63.0 63.0 63.0 63.0 62.8 62.7 62.0 61.5 60.9 60.6 60.1 59.9 59.1 58.7 58.4 57.9 57.7 57.5 57.3 57.3 57.0 56.7 56.2 55.8 55.4 55.0
7.4 7.6 8.2 8.9 9.8 11.0 11.2 12.3 12.7 13.4 14.7 15.4 16.6 17.3 18.4 19.5 20.6 21.7 22.7 23.8 25.2 26.2 26.7 27.3 27.1 27.2 26.9 26.6 26.1 25.5 25.0 23.9 22.9 21.9 21.4 20.3 19.7 17.9 16.9 16.1 15.2 14.6 13.9 13.0 12.6 11.8 11.2 10.2 9.3 8.2 7.3
5 of Table 18. The residuals are lower than 5 % with the present experimental estimation of this quantity (except the 25 % residual of the eliminated point). The equivalent best fit of the critical thermal conductivity enhancement in the reduced temperature range (4.3 · 10-4 < t ) ((T - Tc)/Tc) < 8 · 10-2) gives
∆λc (mW·m-1·K-1) ) 1.5742t-0.6437
(20)
In eqs 19 and 20, the effective values (0.6703 and 0.6437, respectively) of the critical exponents are significantly different from their expected asymptotic Ising values (0.61 and 0.57, respectively). As already noted in ref 22, these non-Ising values of the exponent means that the experimental data obtained in our present temperature range can be described only accounting for the classical-to-critical crossover theories. An alternative empirical approach consists thus to explicit directly the effective pure power laws that are only valid in this restricted temperature
Journal of Chemical & Engineering Data, Vol. 54, No. 9, 2009 2685 Table 14. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 381.53 K
∆λc (mW·m-1·K-1) ) 1.4954
T
p
F
λ
∆λc
K 381.63 381.63 381.63 381.62 381.61 381.61 381.61 381.60 381.58 381.56 381.55 381.53 381.53 381.53 381.53 381.53 381.53 381.54 381.54 381.55 381.55 381.55 381.56 381.56 381.56 381.56 381.56 381.56 381.56 381.56 381.57 381.57 381.57 381.57 381.57
MPa 4.490 4.500 4.500 4.510 4.516 4.520 4.525 4.530 4.569 4.600 4.612 4.627 4.640 4.650 4.660 4.670 4.680 4.700 4.760 4.780 4.800 4.815 4.830 4.860 4.900 4.920 4.950 4.982 5.016 5.068 5.092 5.125 5.136 5.169 5.201
kg · m-3 373.9 379.1 379.1 384.9 388.6 390.9 393.8 397.2 423.8 448.0 458.1 471.5 482.1 490.4 498.6 506.7 514.7 529.5 571.0 582.3 593.2 600.8 607.4 620.5 635.6 642.3 651.6 660.6 669.2 680.9 685.6 691.9 693.9 699.7 705.0
mW · m-1 · K-1 46.4 47.2 46.8 47.8 48.3 48.7 49.0 49.5 51.8 53.6 54.2 55.1 55.5 56.0 56.2 56.5 56.5 56.2 56.0 55.8 55.5 55.3 55.1 54.9 54.2 54.2 54.3 54.3 54.3 54.3 53.9 53.8 53.8 53.6 53.6
mW · m-1 · K-1 15.4 16.0 15.7 16.4 16.8 17.1 17.3 17.7 19.0 19.9 20.1 20.4 20.4 20.5 20.4 20.3 20.0 19.1 16.9 16.1 15.3 14.7 14.2 13.3 11.8 11.4 11.0 10.5 10.0 9.4 8.7 8.2 8.1 7.5 7.2
(23)
range. That allows us to obtain more easily each “theoretical” power law which is equivalent to the experimental power law of eq 20, in conformity with the single power law that governs the master behavior of eq 18 for any pure fluid. In the following, we illustrate this “theoretical” estimation of ∆λc when we assume that the knowledge of the specific heat data and the correlation length data replace the knowledge of the isothermal compressibility and the slope of the critical isochore in our experimental temperature range. We have thus rewritten eq 18 as follows
∆λc ) RD,eff
kBT F (C - CV) 6πηξ c p
(21)
In eq 21, we have introduced an effective amplitude term RD,eff to account for the effective values of the exponents that describe the observed temperature behavior of η, Cp - CV, and ξ along the critical isochore. The viscosity data calculated from NIST tables23 (column 3, Table 18) were fit by the linear equation η(µPa · s-1) ) 34.686 (1 + 0.4731t) in agreement with a small singular behavior of this transport property (which can be neglected in our temperature range). From the values of Cp and CV given in columns 6 and 7 of Table 18, respectively, the values of Fc(Cp - CV) were fit by the power law equation Fc(Cp - CV) (kJ · m-3 · K-1) ) 73.161t-1.03222. In the absence of measurements of the correlation length of R134a, we have assumed that the theoretical estimation24,25 of the leading Ising power law ξ ) ξ0t-0.63, with ξ0 ) 1.8785 · 10-10 m,22 remains valid in our experimental range. Accordingly, the fit of the prefactor term RD,eff leads to the following power law
RD,eff ) 0.4859t-0.25057
1+t t-0.6528 1 + 0.4731t
(22)
Introducing all these previous temperature behaviors in eq 21, the “theoretical” power law equation of the critical enhancement of the thermal conductivity reads as follows
The small difference of the exponent values in eqs 20 and 23 is due to the small explicit linear temperature dependences of eq 23. The calculated values of ∆λc by using eq 23 (column 8, Table 18) compare favorably with the experimental values (column 2, Table 18) and with the calculated values using Table 15. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 385.18 K T
p
F
λ
∆λc
K
MPa
kg · m-3
mW · m-1 · K-1
mW · m-1 · K-1
385.78 385.75 385.61 385.59 385.57 385.42 385.41 385.39 385.37 385.36 385.34 385.33 385.31 385.30 385.29 385.28 385.26 385.25 385.23 385.22 385.21 385.20 385.19 385.19 385.18 385.18 385.18 385.18 385.18 385.18 385.18 385.18 385.18 385.18 385.18 385.18 385.18 385.18 385.19 385.19 385.19 385.19 385.19 385.19 385.72 385.72 387.25 387.23 387.22 387.20 387.19 387.30 387.27 387.37 387.37 387.34 387.33 387.33 387.32 387.30 387.29 387.28
3.920 4.060 4.120 4.200 4.260 4.310 4.360 4.410 4.460 4.500 4.540 4.580 4.610 4.640 4.670 4.700 4.730 4.760 4.800 4.820 4.840 4.880 4.900 4.918 4.940 4.959 4.978 5.000 5.022 5.042 5.058 5.080 5.100 5.120 5.151 5.179 5.224 5.254 5.296 5.322 5.358 5.402 5.454 5.513 5.700 5.714 6.371 6.650 7.002 7.501 7.800 8.468 9.502 11.008 11.503 13.000 13.500 14.002 14.910 16.290 17.000 18.006
213.2 230.5 239.3 251.1 260.7 270.2 279.5 289.6 300.5 309.8 320.1 331.0 340.1 349.6 359.8 370.7 382.7 395.3 413.8 423.6 433.8 454.7 465.6 475.2 487.3 497.3 507.2 518.5 529.4 539.0 546.4 556.3 564.8 573.0 585.0 595.1 609.8 618.8 630.1 636.7 645.3 654.9 665.3 675.9 693.8 695.7 740.5 761.5 783.2 808.2 820.9 844.2 874.3 907.7 917.1 942.4 949.9 957.1 969.1 985.7 993.6 1004.0
29.3 30.5 31.1 32.0 32.8 33.5 34.3 35.2 36.3 37.1 38.2 39.3 40.1 41.1 42.1 43.1 44.6 45.8 47.1 48.0 48.8 50.5 51.2 51.8 52.2 52.6 52.8 53.0 53.0 53.0 53.0 53.0 53.0 53.0 52.8 52.6 52.4 52.2 51.8 51.8 51.6 51.6 51.4 51.2 51.4 51.6 52.7 53.6 54.5 55.9 56.4 57.8 59.7 61.9 62.5 64.2 64.8 65.3 66.3 67.6 68.2 69.1
2.9 3.6 4.0 4.6 5.1 5.5 6.0 6.6 7.4 7.9 8.6 9.4 9.9 10.6 11.3 11.8 12.9 13.7 14.4 14.8 15.3 16.1 16.5 16.6 16.5 16.5 16.3 16.0 15.5 15.1 14.7 14.3 13.9 13.5 12.7 11.9 11.0 10.3 9.3 9.0 8.3 7.8 7.0 6.2 5.3 5.4 3.6 3.1 2.7 2.3 1.9 1.7 1.3 0.9 0.8 0.5 0.5 0.4 0.3 0.3 0.1 0.1
2686
Journal of Chemical & Engineering Data, Vol. 54, No. 9, 2009
Table 16. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 394.47 K
∆F* )
T
p
F
λ
∆λc
K
MPa
kg · m-3
mW · m-1 · K-1
mW · m-1 · K-1
395.24 395.23 394.95 394.94 394.93 394.91 394.89 394.88 394.88 394.60 394.57 394.56 394.55 394.54 394.53 394.52 394.50 394.49 394.49 394.49 394.47 394.47 394.47 394.47 394.47 394.47 394.46 394.46 394.46 394.46 394.45 394.45 394.45 394.45 394.45 394.44 394.44 394.44 394.57 394.57 394.57 394.57 394.56 394.56 394.69 394.69 394.68 394.68 394.68 394.81 394.81
4.405 4.495 4.583 4.649 4.722 4.821 4.890 4.950 4.978 5.055 5.176 5.246 5.300 5.314 5.366 5.428 5.527 5.583 5.626 5.641 5.760 5.767 5.800 5.811 5.851 5.886 5.940 5.980 6.044 6.100 6.140 6.200 6.260 6.300 6.399 6.457 6.523 6.557 6.665 6.720 6.754 6.800 6.908 7.067 7.159 7.321 7.457 7.521 7.635 7.706 7.827
234.9 245.1 256.9 265.5 275.4 289.8 300.6 310.5 315.3 331.7 356.2 371.5 384.1 387.6 400.3 416.2 442.6 457.9 469.5 473.6 505.6 507.5 515.9 518.7 528.6 537.1 549.9 558.9 572.5 583.8 591.6 602.5 612.8 619.3 634.2 642.5 651.2 655.5 665.1 672.4 676.0 680.8 691.5 705.6 711.5 723.8 733.4 737.6 744.8 747.8 754.8
30.9 31.5 32.2 32.8 33.5 34.6 35.3 36.0 36.3 37.4 39.0 40.0 40.9 41.1 41.9 43.0 44.4 45.3 45.7 46.0 47.2 47.2 47.6 47.6 48.0 48.0 48.3 48.7 49.1 49.2 49.4 49.6 49.8 50.0 50.3 50.7 50.9 51.1 51.3 51.5 51.5 51.5 51.9 52.3 52.6 52.8 53.3 53.5 53.7 53.8 54.1
3.0 3.4 3.7 4.1 4.5 5.1 5.5 5.9 6.1 6.6 7.4 8.0 8.4 8.5 8.8 9.3 9.7 10.0 10.0 10.1 9.9 9.8 9.9 9.8 9.7 9.3 9.1 9.0 8.7 8.3 8.1 7.7 7.4 7.2 6.8 6.7 6.5 6.4 6.1 5.9 5.7 5.4 5.2 4.7 4.7 4.1 3.9 3.9 3.6 3.6 3.4
(25)
Now the effective function RD,eff(t, ∆F*) is an empirical function which characterizes our extended temperature and density range measured by t and ∆F*. Equation 24 is then similar to the crossover semiempirical equation ∆λc ) RD[(kBT)/(6πηξ)]FCpc F(t, ∆F*) proposed by Hanley et al. in ref 20 where F(t, ∆F*) is an empirical damping function used in their extended critical region. The values of Cp(t, ∆F*), CV(t, ∆F*) are calculated from the equation of state5 and η(t, ∆F*) from the NIST tabulated Table 17. Thermal Conductivity of HFC-134a along the Quasi-Isotherm 404.36 K T
eq 20, except for the closest temperature to Tc. It is essential to note that a similar approach that starts with the single power law R*c ) 0.0139τ-0.655 of ref 21 in place of our above eq 19 also provides eq 23 with only minor differences in the values of the effective amplitude and the effective exponent. To calculate ∆λc outside the critical isochore, we have then generalized eq 21 as the following form
kBT × 6πη(t, ∆F*)ξ(t, ∆F*) F[Cp(t, ∆F*) - CV(t, ∆F*)]
F - Fc Fc
∆λc(t, ∆F*) ) RD,eff(t, ∆F*)
(24)
where the temperature and density dependences of the different quantities are accounted for introducing the reduced quantity t of eq 13 and the reduced density difference
p
K
MPa
406.09 405.81 405.27 405.26 405.25 405.25 405.34 405.07 405.06 405.05 404.77 404.76 404.76 404.73 404.73 404.73 404.71 404.70 404.68 404.67 404.65 404.51 404.50 404.49 404.48 404.47 404.46 404.45 404.44 404.43 404.43 404.42 404.40 404.40 404.38 404.34 404.34 404.36 404.36 404.23 404.23 403.83 403.82 403.69 403.68 405.49 403.69 405.48 405.47 405.46 407.01 407.00 406.99 406.96 406.94 406.93 406.91 406.90 406.88
1.196 1.930 2.343 2.524 2.700 2.863 3.411 3.698 3.790 3.914 4.070 4.201 4.278 4.451 4.467 4.523 4.641 4.754 4.909 5.039 5.162 5.242 5.330 5.419 5.494 5.579 5.668 5.750 5.814 5.860 5.908 5.967 6.039 6.080 6.342 6.434 6.516 6.559 6.640 6.723 6.754 6.852 6.973 7.105 7.203 7.506 7.364 7.814 8.558 8.835 10.638 10.911 11.426 12.915 13.921 14.350 15.679 16.345 18.500
F
λ -3
kg · m
39.3 67.5 85.3 93.6 101.9 109.9 139.4 157.1 163.0 171.3 182.6 192.3 198.2 212.2 213.5 218.2 228.6 240.1 254.4 268.0 281.7 291.8 302.6 314.0 324.1 335.9 348.9 361.3 371.3 378.7 386.4 396.2 408.5 415.4 461.1 477.5 491.5 498.4 511.7 526.8 531.6 552.1 569.4 588.6 600.8 611.9 619.0 642.0 697.9 714.6 782.0 791.2 807.2 845.6 866.9 875.1 898.1 908.5 938.0
-1
∆λc -1
mW · m · K 23.6 24.3 24.7 24.9 25.2 25.4 26.4 27.1 27.4 27.8 28.3 28.7 29.0 29.8 29.8 30.1 30.7 31.2 32.1 32.9 33.6 34.2 34.9 35.5 36.1 36.8 37.5 38.3 38.8 39.3 39.6 40.2 41.0 41.3 43.3 44.0 44.6 44.9 45.3 46.0 46.1 46.8 47.5 48.0 48.4 49.0 49.0 49.9 51.5 52.1 55.7 56.2 56.9 59.1 60.3 60.8 62.3 63.0 65.2
mW · m-1 · K-1 0.0 0.0 0.0 0.0 0.1 0.1 0.3 0.5 0.6 0.8 1.0 1.2 1.4 1.7 1.7 1.8 2.1 2.4 2.8 3.2 3.5 3.8 4.1 4.4 4.7 5.0 5.3 5.7 5.9 6.1 6.1 6.4 6.7 6.7 7.0 7.0 7.0 7.0 6.9 6.9 6.8 6.6 6.4 6.0 5.8 5.6 5.4 4.9 3.4 3.0 2.1 1.9 1.6 1.1 0.7 0.6 0.3 0.3 0.1
Journal of Chemical & Engineering Data, Vol. 54, No. 9, 2009 2687 Table 18. Variation of ∆λc, η, R*, c Cp, and CW as a Function of Temperature T along the Critical Isochore T
∆λc(exp)
η
R*(exp) c
R*c (cal)
Cp
CV
∆λc(cal)
K
mW · m-1 · K-1
µPa · s-1
eq 18
eq 19
J/g · K
J/g · K
mW · m-1 · K-1 eq 23
374.37 374.96 375.14 375.28 375.72 376.02 376.27 377.01 377.83 378.25 379.08 381.53 385.18 394.47 404.36
151.9 89.7 73.1 64.1 52.0 47.5 45.0 36.6 31.4 30.9 27.2 20.5 16.5 10.0 7.3
34.693 34.719 34.727 34.733 34.752 34.766 34.777 34.809 34.845 34.863 34.900 35.005 35.167 35.575 36.008
0.4534 0.2671 0.2175 0.1906 0.1544 0.1408 0.1333 0.1081 0.0924 0.0908 0.0797 0.05945 0.0472 0.02757 0.01939
0.7332 0.2603 0.2253 0.2051 0.1628 0.1442 0.1322 0.1077 0.0906 0.0842 0.0743 0.0565 0.0431 0.02857 0.02189
409.68 87.058 70.299 61.17 43.516 36.407 32.067 23.769 18.543 16.689 13.968 9.5471 6.6389 3.9986 3.002
1.2403 1.2343 1.2325 1.2311 1.2268 1.2240 1.2217 1.2151 1.2081 1.2048 1.1983 1.1815 1.1612 1.1279 1.1102
236.66 86.397 75.097 68.541 54.773 48.682 44.755 36.667 31.042 28.913 25.622 19.704 15.206 10.314 8.058
data.23 The values of the correlation length ξ(t, ∆F*) were calculated in terms of the cubic restricted model, i.e., ξ(t, ∆F*) ) ξ(r, θ) ) r-0.63ξ0+ R(θ),26 as proposed by Sengers and Levelt-Sengers.27 In the latter equation, the (r, θ) parameters are related to (t, ∆F*) by t ) r(1 - b2θ2) and ∆F* ) r0.325kθ(1 + cθ2), while R(θ) ) 1 + 0.16θ2.26 The corresponding values of the model parameters are b2 ≈ 1.2766 and c ≈ 0.055,26 while the values k ) 1.1629 and a ) 23.45627,28 of the two system-dependent parameters are obtained for the R134a case.25,28 Then the functional form of RD,eff(t, ∆F*) was written as a product of three main contributions RD,eff(t, ∆F*) ) RD,eff(t) · RS,eff(|∆F*|) · RA,eff(∆F*), where the temperature contribution RD,eff(t) is accounted for by our previous eq 22, while the density contribution was split into a symmetrical function RS,eff(|∆F*|) of the cubic equation
RS,eff(|∆F*|) ) 16.0|∆F*| 3 + 1
(26)
and an asymmetrical function RA,eff(∆F*) of the cubic equation
RA,eff(∆F*) ) 0.9(∆F*)3 + 1
(27)
Accordingly, the calculated values of the critical enhancement ∆λc of the thermal conductivity using eqs 24 to 27 are in satisfactory agreement (within the experimental uncertainty) with the experimental values calculated in the whole density range. As expected, we have observed that the residuals increase (overpassing then 50 %) only when ∆λc decrease significantly to reach a very small part of the total thermal conductivity (i.e., when the temperature and density distance to the critical point increase). Finally, after estimation of the total thermal conductivity λ(T, F) by using eqs 2, 4, 5, and 24, we have reported the related residuals rλ(%) ) 100 · (λcal/λexp - 1) (expressed in %) in Figure 2. The standard average deviation between the thermal conductivity calculated by eqs 2, 4, 5, and 24 and the experimental data is 4.5 % and reduces to 3.73 % when ignoring the isotherm 374.37 K closest to Tc. Though some more important deviations are observed between calculated and experimental values in limited density ranges, we estimate that they are mainly due to the fact that the experimental curves of the critical enhancement of the thermal conductivity and the tabulated values of Fc(Cp CV) are not exactly symmetrical with respect to Fc.
Conclusion New measurements of the thermal conductivity of HFC-134a are presented in the supercritical region, at temperatures from
(370 to 405) K, along 15 quasi-isotherms and at pressures up to 40 MPa with an estimated uncertainty of ( 3 %. As expected, the (background + critical enhancement) additive form of the thermal conductivity data implies that the determination of the critical enhancement term is very sensitive to the analytical form of the background term. After a careful analysis of the temperature and density dependences of the background term, we have then shown that the magnitude of the critical enhancement is very large and follows a behavior which can be predicted from theory along the critical isochore. In this critical isochoric region, a correlation was developed to represent the variation of the thermal conductivity in terms of reduced parameters, which can be thus applied to any fluid. The variation of the thermal conductivity as a function of density was estimated by an empirical relation. The average deviations between calculated and experimental thermal conductivity data are less than 4.5 %, in the whole temperature range from (300 to 550) K and pressure range from (0.1 to 40) MPa. In the critical region, the temperature range is restricted to T > Tc + 1 K. However, to correlate the critical enhancement of R134a as a function of density in an extended critical region, a critical evaluation of other experimental data of R134a, reported in the literature, is necessary. Moreover, further comparisons with other fluids are required to develop a universal equation to represent the thermal
Figure 2. Fractional deviation 100 · ∆λ/λ ) 100 · [λ(cal) - λ(exp)]/λ(exp) of the thermal conductivity data of R134a, as a function of density. The calculated values are obtained using eqs 2, 4, 5, and 24 (see text for details). ], 374.37 K (not selected in the fitting procedure, see text); 0, 377.83 K; 4, 375.14K; 9, 381.53 K; *, 379.08 K; O, 374.96 K; +, 375.14 K; -, 375.28 K; -, 375.72 K; b, 376.02 K; +, 376.27; 4, 377.01 K; ×, 385.18 K; 2, 394.47 K; [, 404.36 K. The dashed-dotted lines are one standard deviation of the fit of our experimental data set of Tables 4 to 17.
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Journal of Chemical & Engineering Data, Vol. 54, No. 9, 2009
conductivity of any fluid, in liquid and vapor states, including the extended critical region.
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