Measurements of the Thermal Conductivity of Propane in the

Article pubs.acs.org/jced

Measurements of the Thermal Conductivity of Propane in the Supercritical Region B. Le Neindre,*,† Y. Garrabos,‡ and M. Nikravech† †

Université Paris 13, Sorbonne Paris Cité, Laboratoire des Sciences des Procédés et des Matériaux, CNRS, (UPR 3407), F-93430, Villetaneuse, France ‡ ICMCB-CNRS UPR 9048, Université Bordeaux I, 87 Av du Dr A. Schweitzer, PESSAC Cedex F 33608, France ABSTRACT: Measurements of the thermal conductivity of propane performed in a coaxial cylinder cell operating in steady state conditions are reported. The measurements of the thermal conductivity of propane were carried out along eight quasi-isotherms above the critical temperature. The present data cover the temperature range from (370.83 to 381.00) K, and the pressure range (0.1 to 15) MPa. An analysis of the various sources of error leads to an estimated uncertainty of approximately 3 %. The parameters of a background equation were determined from experimental data in order to analyze the critical enhancement of the thermal conductivity as a function of temperature and density. On the basis of the measurement of more than 500 experimental points, a phenomenological equation is provided to describe the thermal conductivity of propane, from the critical temperature 369.825 K to 600 K and densities up to 550 kg·m−3. This equation can be generalized to calculate the supercritical behavior of the thermal conductivity for any fluid.

■

INTRODUCTION Recently there was a renewed interest in the thermophysical properties of hydrocarbons that are largely used in the chemical industry, and some of them can be also considered as substitutes for the often criticized halocarbons in refrigeration and heat pumps. Moreover, to design efficient chemical processes and equipment, the chemical industry requires accurate transport property data. However, large uncertainties are associated with the theoretical prediction of transport properties, especially in the case of thermal conductivity which is an important property in the development of molecular kinetic theories of fluids. Indeed, the thermal conductivity of fluids varies drastically from the vapor phase to the liquid phase at temperatures below the critical temperature and from the dilute gas to the compressed supercritical fluid at temperatures above the critical temperature. In addition, one of the most important characteristics of supercritical fluids near the critical point is that their physical properties display very large variations with a slight change in temperature in the vicinity of the critical point. For example, the heat diffusive transport shows a critical slowing down while the thermal conductivity diverges when the critical point is approached. It remains therefore, impossible to estimate the behavior of the thermal conductivity of supercritical fluids without a complete experimental and theoretical study. Unfortunately, thermal conductivity measurements of gases at elevated densities that include the critical region are scarce. In most of the experimental works, either the critical region is not included at all, or, if it is, the observations are affected by the presence of © 2014 American Chemical Society

convective heat transfer. Moreover, due to the singularities of the equation of state in the critical region, the density becomes very sensitive to small errors in either the pressure or the temperature. As there are very few published thermal conductivity data in the critical region, it is extremely difficult to test the corresponding theoretical approaches. Therefore, empirical correlations were generally proposed to represent the transport properties as functions of temperature and pressure (or density). Up to now, there were very few measurements of the thermal conductivity of propane by comparison to other industrial fluids, while several empirical correlations to represent the thermal conductivity of propane have already been published. The first previous measurement in the critical region was performed by one of us.1 More recently, Marsh et al.2 reported steady-state and transient measurements of the thermal conductivity of propane from the triple point to 600 K with pressures to 70 MPa. In 1979, Holland et al.3 reported a correlation for the thermal conductivity of propane at temperatures from (140 to 500) K and at pressures up to 50 MPa, which has an estimated uncertainty4,5 of 8 % outside the critical region and 15 % near the critical point. In 1987, Younglove and Ely4 reported a correlation for the thermal conductivity of propane at temperatures from (86 to 600) K and at pressures up to 100 MPa that has an estimated uncertainty4 of 5 % outside the Received: May 12, 2014 Accepted: October 3, 2014 Published: October 22, 2014 3422

dx.doi.org/10.1021/je500395f | J. Chem. Eng. Data 2014, 59, 3422−3433

Journal of Chemical & Engineering Data

Article

Table 1. Thermal Conductivity of Propane at Atmospheric Pressure and in the Zero Density Limit (Using a Linear Fit To Calculate the Ideal Isobaric Heat Capacity−See Text) Ta K 302.89 315.20 315.99 326.24 326.40 326.70 326.85 341.88 342.61 343.34 346.81 347.10 347.10 373.28 374.63 386.17 387.29 433.89 480.64 528.03 579.11

λ01 (exp) −1

mW·m ·K 18.87 20.14 20.15 21.45 21.43 21.45 21.45 23.19 23.30 23.38 23.78 23.92 23.85 26.91 27.20 28.68 28.83 35.22 42.28 50.22 59.27

λ01 (cal) −1

−1

−1

mW·m ·K

18.84 20.17 20.26 21.40 21.41 21.45 21.46 23.19 23.28 23.37 23.78 23.81 23.81 27.02 27.19 28.68 28.82 35.23 42.31 50.13 59.31

(eq 4)

λ01b (cal)

λ00

mW·m−1·K−1 (Marsh)

mW·m−1·K−1 (eq6)

C0p/R

mW·m−1·K−1 (eq7)

λ00 (cal)

18.81 20.19 20.28 21.47 21.49 21.52 21.54 23.34 23.43 23.52 23.94 23.98 23.98 27.30 27.48 29.02 29.17 35.78 43.04 51.06 60.43

18.82 20.09 20.10 21.40 21.38 21.40 21.40 23.15 23.26 23.34 23.74 23.88 23.81 26.87 27.16 28.64 28.79 35.19 42.25 50.19 59.24

8.711 8.972 8.988 9.206 9.209 9.215 9.219 9.537 9.553 9.568 9.642 9.648 9.648 10.202 10.231 10.476 10.499 11.487 12.477 13.481 14.564

18.68 20.14 20.23 21.48 21.50 21.54 21.56 23.44 23.53 23.63 24.07 24.11 24.11 27.58 27.74 29.31 29.46 36.14 43.29 50.98 59.73

Uncertainties (0.95 level of confidence): T, 0.05 K; λ01 (exp), 1.5 %; λ01 (cal), 1.5 %; λ00, 1.5 %; C0p/R, 5 %; λ00 (cal), 5 %. bλ01 (Marsh) = −0.00124778 + 0.00816371(T/Tc) + 0.0199374(T/Tc)2.

a

the dilute gas state in the temperature range (300 to 580) K, near atmospheric pressure, and in Table 2 to Table 9 for the fluid dense state along eight supercritical isotherms, then covering the density range from (50 to 550) kg·m−3. The equation-of-state of Miyamoto and Watanabe was used to calculate the density ρ, with an accuracy of the order of 0.2 %.8 The critical parameters together with the estimated uncertainties are reported as follows:

critical region and 10 % near the critical point. In 1996, Ramires et al.5 reported a correlation for the thermal conductivity of propane at temperatures from (192 to 725) K and at densities up to 17 mol·L−1. This correlation has an estimated uncertainty of 5 % outside the critical region and an uncertainty of 10 % near the critical point.3 In 2001, Yata et al.,6 published a correlation to represent the thermal conductivity of several refrigerants including propane in the liquid phase. The average deviation of the experimental data with their equation was of the order of 1.9 %. In 2002, Marsh et al.2 developed improved correlations for the thermal conductivity of propane, from 85.5 K (triple point) to 600 K and in the pressure range from (0.1 to 70) MPa, with an uncertainty of about 3 % at a 95 % confidence level, with the exception of state points near the critical point and the dilute gas, where the uncertainty of the correlation increases to 5 %.

Tc = (369.825 ± 0.06) K, pc = (4.24766 ± 0.016) MPa

■

■

ρc = (220.00 ± 4) kg·m−3

(1)

RESULTS AND DISCUSSION Correlations. The measurements were carried out to make an analysis of the data based on the residual concept, for which the thermal conductivity is represented as a sum of three contributions

EXPERIMENTAL RESULTS The present thermal conductivity measurements of propane were carried out as a function of temperature between Tc and Tc + 10 K and pressures up to 15 MPa, in the homogeneous critical region, using vertical coaxial cylinders, operating in the steady-state mode. This method of measurement and the related corrections were previously described in several papers.7 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. Indeed, during the experiments, the stability of the temperature was better than 0.05 K and the precision of the temperature measurements was within ± 0.05 K. The pressure was measured with a precision pressure transducer with accuracy of 0.02 %. The sample propane was provided by Air Liquide and its purity was estimated to be better than 0.9995 initial mole fraction by the manufacturer’s analysis. New experimental results are reported in Table 1 for

λ(T , ρ) = λ 00(T ) + δλ(ρ) + Δλc(T , ρ)

(2)

where λ00(T) is the dilute-gas thermal conductivity for the zero density limit, which is dependent only on the temperature, δλ(ρ) is the residual thermal conductivity, which is dependent only of the density, and Δλc(T,ρ) is the thermal conductivity critical enhancement approaching the liquid−gas critical point, which depends on both density and temperature. This representation allows the theoretically based analysis of each contribution to be considered separately. In that description, the two first contributions of eq 2 represent the background thermal conductivity λb(T , ρ) = λ 00(T ) + δλ(ρ) 3423

(3)



Article

Table 2. Thermal Conductivity of Propane along the Isotherm 370.45 K pa MPa 15.100 11.300 11.200 11.000 10.500 10.000 9.500 9.000 8.500 8.000 7.500 7.000 6.500 6.000 5.760 5.550 5.500 5.000 4.800 4.580 4.560 4.420 4.400 4.377 4.360 4.355 4.350 4.345 4.340 4.335 4.330 4.325 4.320 a

ρ kg·m

λ −3

431.26 411.89 411.28 410.03 406.78 403.31 399.60 395.60 391.24 386.46 381.13 375.10 368.11 359.74 355.00 350.32 349.12 334.13 325.60 312.19 310.58 295.15 291.82 287.24 283.06 281.65 280.12 278.46 276.64 274.62 272.35 269.74 266.66

Δλc −1

−1

λ/mW·m ·K 80.94 76.45 77.35 76.77 76.24 75.72 75.45 74.21 73.81 72.77 71.78 70.91 69.97 68.74 68.12 67.91 67.84 67.17 66.54 67.69 68.14 70.14 70.99 72.17 73.93 75.28 76.69 78.14 81.22 88.09 92.01 96.28 100.97

−1

mW·m ·K

pa −1

MPa

0.14 1.21 2.11 1.84 2.09 2.41 3.02 2.70 3.29 3.30 3.44 3.84 4.30 4.65 4.87 5.49 5.63 7.50 8.23 11.38 12.07 16.22 17.52 19.32 21.63 23.17 24.77 26.43 29.74 36.88 41.08 45.68 50.74

4.315 4.310 4.305 4.300 4.295 4.290 4.285 4.280 4.275 4.270 4.265 4.260 4.255 4.250 4.245 4.244 4.241 4.232 4.227 4.224 4.217 4.205 4.200 4.190 4.170 4.105 4.042 3.986 3.856 3.770 3.487 3.452

ρ kg·m

λ −3

262.89 257.97 250.83 238.19 213.91 192.98 182.38 175.78 170.95 167.12 163.92 161.15 158.71 156.51 154.51 154.13 153.02 149.99 148.46 147.58 145.64 142.63 141.46 139.27 135.34 125.23 117.70 112.10 101.52 95.72 80.53 78.93

Δλc −1

λ/mW·m ·K 106.12 118.13 129.08 145.20 156.04 142.27 125.20 111.75 96.22 88.04 81.06 77.40 74.30 71.21 68.79 68.56 67.23 63.83 62.49 61.33 59.02 55.77 55.00 53.29 51.02 46.23 43.85 42.04 39.38 38.05 35.90 35.58

−1

mW·m−1·K−1 56.35 68.95 80.75 98.32 111.82 100.24 84.26 71.47 56.41 48.60 41.93 38.54 35.69 32.81 30.59 30.39 29.15 26.03 24.84 23.75 21.63 18.65 17.99 16.49 14.56 10.67 8.94 7.60 5.85 4.93 4.01 3.76

Uncertainties (0.95 level of confidence): p, 0.02 %; ρ, 0.2 %; λ, 3 %; Δλc, 4 %.

To determine λ00(T) and δλ(ρ), we have considered the λ(T,ρ) data in the high dense phase and in the moderate dense gas phase far from the critical temperature (T ≫ 369.8 K) region, comprising mostly parts of our previous measurements reported in ref 1. The calculated thermal conductivity data from the background contribution of eq 3 are assumed to have accuracies better than 1.5 %, as was observed from the deviations of the calculated values by the correlation and experimental data. The resulting critical enhancement Δλc (T,ρ) of the thermal conductivity is reported in column 4 of Tables 2 to 9. Dilute Gas Thermal Conductivity at 0.1 MPa. The complete set of experimental data of the dilute gas thermal conductivity λ01 at 0.1 MPa (recalled by the index 1) given in column 2 of Table 1 was fit to an empirical quadratic equation λ 01 = a0 + a1T + a 2T 2

The λ01 values calculated by using eq 4 are reported in column 3 of Table 1. In column 4 of Table 1 are given the calculated values of λ01 by an equation reported by Marsh et al.2 In Figure 1 are shown the deviations between both equations from reference to the experimental data. The agreement is reasonable, the main deviations occur at high temperatures but they are always less than 2 %. In this figure a comparison is also shown with the steady state measurements of λ01 by Marsh et al. near atmospheric pressure. Here also the mean deviation is less than 2 % at the exception of some points near room temperature. Dense Fluid Thermal Conductivity. There are some accurate measurements of the thermal conductivity of propane at high pressure. In 1982, Roder and Nieto de Castro9 measured the thermal conductivity of liquid propane in the temperature range from (110 to 300) K and at pressure from (0.1 to 70) MPa, by a transient hot wire method, with an estimated accuracy of 1.5 %. In 1989, Prasad et al.10 performed measurements of the thermal conductivity of propane in gas and liquid phases in the temperature range from (192 to 320) K, and at pressures from (0.2 to 70) MPa, by a transient hot wire method with an estimated uncertainty of 1.5 %. In 1996, Yata et al.11 reported a series of 16 data points in the temperature range from (254 to 315.1) K and pressure range from (1 to 30) MPa. Unfortunately all these measurements were performed below the critical temperature and

(4)

where λ01 and T are expressed respectively in mW·m−1·K−1 and K, leading to the following values of the ai adjustable parameters a0 = 0.3099542 mW·m−1·K−1, a1 = 1.65763·10−2 mW·m−1·K−2 a 2 = 1.473017·10−4 mW·m−1·K−3 3424



Article

Table 3. Thermal Conductivity of Propane along the Isotherm 371.05 K

a

pa

ρ

λ

Δλc

pa

ρ

λ

Δλc

MPa

kg·m−3

λ/mW·m−1·K−1

mW·m−1·K−1

MPa

kg·m−3

λ/mW·m−1·K−1

mW·m−1·K−1

11.600 9.500 8.900 8.500 7.500 6.800 6.200 6.100 5.672 5.350 5.200 5.000 4.840 4.720 4.715 4.590 4.560 4.555 4.485 4.440 4.420 4.414 4.408 4.402 4.396 4.392 4.387 4.383 4.375 4.370 4.367 4.365 4.361 4.360 4.358 4.356 4.354 4.351 4.349 4.347

412.62 398.35 393.43 389.85 379.53 370.62 361.19 359.39 350.59 342.31 337.70 330.40 323.11 316.24 315.91 306.32 303.41 302.90 294.25 286.40 281.79 280.18 278.45 276.56 274.47 272.95 270.88 269.04 264.77 261.52 259.27 257.62 253.83 252.76 250.45 247.85 244.91 239.74 235.71 231.16

77.56 74.98 74.24 73.39 71.47 69.96 69.27 69.17 68.29 67.73 67.47 67.07 66.68 67.05 67.18 67.82 68.24 68.55 69.21 71.05 72.31 73.63 74.94 76.33 77.76 78.65 80.15 81.41 84.06 86.16 88.36 89.89 93.11 93.95 96.57 99.34 102.77 108.09 112.15 115.89

1.75 2.64 3.04 2.98 3.29 3.58 4.71 4.95 5.65 6.53 7.03 7.81 8.55 9.98 10.16 12.17 12.99 13.38 15.24 18.12 19.99 21.53 23.05 24.68 26.38 27.46 29.24 30.72 33.89 36.38 38.86 40.58 44.26 45.22 48.11 51.18 54.95 60.86 65.38 69.62

4.346 4.345 4.344 4.343 4.342 4.340 4.339 4.338 4.337 4.336 4.334 4.332 4.331 4.329 4.328 4.326 4.325 4.324 4.322 4.319 4.318 4.313 4.308 4.301 4.295 4.289 4.284 4.276 4.265 4.252 4.235 4.215 4.155 4.095 3.900 3.655 3.375 3.100 2.710

228.72 226.16 223.52 220.81 218.06 212.55 209.84 207.21 204.66 202.22 197.73 193.74 191.93 188.64 187.14 184.40 183.14 181.95 179.74 176.79 175.89 171.87 168.47 164.44 161.48 158.85 156.86 153.97 150.47 146.86 142.76 138.58 128.66 121.01 103.34 88.13 75.06 64.69 52.52

117.84 118.51 119.19 119.19 119.19 118.51 117.84 117.18 115.89 113.99 110.38 107.53 105.36 101.27 99.34 95.68 93.95 91.07 88.36 84.06 81.41 76.59 73.32 68.67 66.02 63.57 61.21 58.54 56.69 53.94 51.90 49.15 46.33 44.17 40.56 37.45 34.63 33.44 32.05

71.85 72.80 73.76 74.06 74.35 74.27 73.89 73.51 72.47 70.83 67.69 65.25 63.27 59.51 57.73 54.35 52.75 49.98 47.49 43.48 40.93 36.50 33.56 29.31 26.94 24.74 22.57 20.18 18.66 16.24 14.57 12.21 10.30 8.79 6.72 4.84 2.98 2.54 1.93


where δλ(ρ) and ρ are expressed in mW·m−1·K−1 and kg·m−3, respectively. By fitting selected experimental data with eq 5, we have obtained the following values of the ci adjustable parameters (with ci in mW·K−1·m3i−1·kg−i):

are not sufficient to determine the residual density term δλ(ρ). More recently, Marsh et al.2 published extensive measurements of the thermal conductivity of propane determined by both steady state and transient methods, in large ranges of temperature and pressure, including the supercritical region. However, these two sets of data do not appear to have the same accuracies and will not be used to establish δλ(ρ). Moreover, in the correlation developed by Marsh et al.2 this term depends not only on the density but also on the temperature. Then, in order to determine the residual density term δλ(ρ), of the thermal conductivity, we have used only our present data at high pressure and the data in the gas phase outside the critical region, mostly along isotherms 528 K and 579 K reported in ref 1, at pressure up to 70 MPa. This densitydependent term of the thermal conductivity was represented by the following five-order polynomial equation:

c1 = 2.70048341·10−2 c 2 = 5.45040468·10−4 c3 = −2.01805738·10−6 c4 = 3.34577853·10−9 c5 = −8.8286258·10−13

For interest regarding the related corresponding state analyses, eq 5 can be rewritten in the convenient form δλ(ρ) = Λc∑i 5= 1bi(ρ/ρc)i, where Λc = 18.21544 mW·m−1·K−1 is the residual thermal conductivity at the critical density, and

5

δλ(ρ) =

∑ ciρi i=1

(5) 3425



Article

Table 4. Thermal Conductivity of Propane along the Isotherm 371.97 K ρ

pa MPa 15.100 8.810 7.600 6.800 6.400 5.900 5.200 4.850 4.740 4.650 4.570 4.550 4.530 4.505 4.500 4.490 4.475 4.465 4.460 4.455 4.450 4.445 4.440 4.435 4.430 4.425 4.420 4.418 4.416 4.414 4.412 4.410 4.408 a

kg·m

λ −3

429.01 390.57 378.23 367.78 361.44 351.86 332.48 316.27 308.78 300.76 290.85 287.64 283.94 278.36 277.07 274.24 269.24 265.18 262.85 260.26 257.37 254.09 250.33 245.98 240.92 235.06 228.38 225.52 222.56 219.54 216.48 213.41 210.37

Δλc −1

−1

λ/mW·m ·K 80.67 72.69 71.52 70.02 69.23 68.01 66.43 66.02 66.18 66.36 66.65 67.59 68.77 69.97 70.21 72.26 73.22 75.05 76.42 77.87 80.92 84.23 87.82 91.61 93.69 96.76 99.65 100.13 100.13 99.99 99.61 99.18 98.54

−1

mW·m ·K

pa −1

MPa

0.31 2.03 3.48 4.10 4.51 5.03 6.71 8.81 10.08 11.39 13.04 14.39 16.07 17.98 18.39 20.80 22.38 24.70 26.36 28.12 31.53 35.23 39.26 43.56 46.22 49.95 53.58 54.38 54.72 54.90 54.85 54.75 54.43

4.406 4.404 4.402 4.400 4.398 4.396 4.394 4.392 4.390 4.388 4.386 4.384 4.382 4.380 4.378 4.374 4.370 4.368 4.366 4.362 4.354 4.348 4.342 4.336 4.324 4.312 4.294 4.286 3.803 3.701 3.070 3.025

ρ kg·m

λ −3

Δλc −1

λ/mW·m ·K

207.37 204.45 201.63 198.93 196.36 193.92 191.62 189.45 187.41 185.49 183.67 181.96 180.35 178.82 177.37 174.67 172.22 171.08 169.98 167.90 164.18 161.70 159.42 157.31 153.53 150.19 145.80 144.03 95.08 89.31 63.14 61.66

−1

mW·m−1·K−1

98.07 97.39 96.04 94.29 93.12 91.77 89.68 87.74 86.62 84.81 83.08 81.74 79.86 79.23 77.49 74.88 72.74 71.49 70.25 67.60 63.61 61.24 59.15 57.24 54.59 52.67 50.55 49.83 37.83 36.55 33.13 33.01

54.28 53.90 52.85 51.38 50.47 49.37 47.53 45.82 44.91 43.29 41.74 40.58 38.86 38.38 36.79 34.44 32.55 31.40 30.28 27.83 24.18 22.05 20.18 18.47 16.16 14.56 12.83 12.27 4.52 3.66 2.24 2.15


By fitting the results with the following quadratic polynomial form

b1 = 0.32615538 b2 = 1.4482197

λ 00 = d0 + d1T + d 2T 2

b3 = −1.1796737

(6) −1

−1

where λ00 and T are expressed in mW·m ·K and K, respectively, we have obtained the following values of di adjustable parameters.

b4 = 0.43027713 b5 = −2.4978529·10−2

d0 = 0.1978423 mW·m−1·K−1

Thermal Conductivity in Zero Density Limit. A theoretical analysis of the available experimental data for the thermal conductivity of propane in the limit of zero density is difficult to perform, because the thermal conductivity of polyatomic molecules is strongly influenced by inelastic collisions, and the exchange of energy between translational and internal modes (rotation, vibration, electronic) during collisions. Then, we have used two ways introducing approximate functional forms where the very small differences from λ01 values calculated by eq 4 can be easily controlled. The first way to estimate the thermal conductivity in the zero density limit can be performed by combining the experimental values of the thermal conductivity at atmospheric pressure with the corresponding values of eq 3, then using eq 5 to calculate the residual thermal conductivity for the gas density at p = 0.101325 MPa.

d1 = 1.684868 ·10−2 mW·m−1·K−2 d 2 = 1.470885 ·10−4 mW·m−1·K−3

λ00 values calculated from eq 6 are reported in column 5 of Table 1. We note that, introducing the ideal gas equation of state and the linear approximation of eq 5, δλ(ρ) = c1((Mp)/(RT)) (with p = 101325 Pa), the analytic equation λ00 = a0 + a1T +a2T2 − c1((Mp)/(RT)) inferred by eq 4 is equivalent to eq 6, with d0 = 0.20353108 mW·m−1·K−1 d1 = 1.683062·10−2 mW·m−1·K−2 d 2 = 1.4710418· 10−4 mW· m−1· K−3

M is the molar mass and R is the ideal gas constant. 3426



Article


pa MPa

kg·m

11.900 10.500 8.900 7.780 7.400 6.700 6.300 5.540 5.100 4.900 4.740 4.680 4.660 4.640 4.610 4.605 4.590 4.574 4.568 4.564 4.556 4.550 4.544 4.536 4.530 4.524 4.518 4.510 4.508 4.506 4.504 a

λ −3

Δλc −1

−1

λ/mW·m ·K

410.61 401.46 388.58 377.04 372.38 362.29 355.25 337.10 320.25 308.48 294.11 286.11 282.83 279.10 272.32 271.00 266.62 261.02 258.60 256.87 253.09 249.93 246.44 241.23 236.87 232.12 227.00 219.73 217.86 215.98 214.10

78.18 76.40 73.50 71.52 70.86 69.32 68.78 66.98 66.15 66.13 67.07 67.97 68.87 69.75 71.19 71.68 72.40 74.50 75.32 76.43 78.15 79.94 81.85 84.17 85.52 86.58 87.68 87.30 86.93 86.85 86.56

−1

mW·m ·K

ρ

pa −1

MPa

2.62 3.07 3.13 3.60 3.88 4.28 5.04 6.35 8.19 9.92 12.84 14.80 16.13 17.51 19.80 20.46 21.73 24.51 25.62 26.93 29.10 31.27 33.58 36.49 38.33 39.93 41.59 42.00 41.84 41.95 41.87

4.502 4.500 4.498 4.496 4.494 4.486 4.484 4.482 4.476 4.470 4.466 4.462 4.452 4.450 4.444 4.436 4.428 4.416 4.404 4.384 4.346 4.270 4.230 4.150 4.060 4.030 3.820 3.780 3.450 3.103

kg·m

λ −3

Δλc −1

λ/mW·m ·K

212.22 210.34 208.48 206.63 204.81 197.83 196.18 194.58 190.02 185.86 183.30 180.90 175.50 174.51 171.71 168.31 165.24 161.11 157.46 152.16 143.95 131.75 126.65 118.09 110.17 107.83 94.04 91.80 76.28 63.55

86.38 86.20 85.84 85.47 85.12 83.06 82.40 81.75 79.58 77.19 75.45 73.88 70.37 69.64 67.54 64.99 62.60 59.96 57.88 54.98 51.56 47.56 46.11 43.61 41.49 40.85 37.88 37.46 34.88 33.19

−1

mW·m−1·K−1 41.89 41.91 41.75 41.59 41.43 40.09 39.60 39.12 37.40 35.43 33.94 32.62 29.66 29.03 27.21 24.99 22.90 20.64 18.91 16.52 13.87 10.98 9.99 8.25 6.79 6.35 4.49 4.25 2.87 2.11


where T* = kBT/ε, and

The second way refers only to approximate relations to apply the kinetic theory of gases, in which the thermal conductivity of propane is then related to the reduced effective collision cross section Ω*λ which contains all the contributions from translational, rotational, vibrational, and electronic degrees of freedom of the molecule. However, to calculate the thermal conductivity in the zero-density limit in a lack of reliable data on these contributions in the case of propane molecule, the following practical engineering form12 was used: 0.177568(T /M )0.5 C p0/R σ 2 Ω*λ

λ 00 =

A 0 = 0.444358;

(7)

where is the ideal isobaric heat capacity, and where the above practical form only needs to use the scaling factors σ and ε/kB, which are, respectively, the length and the energy parameters of a 12−6 Lennard-Jones potential (kB is the Boltzmann constant). Note that, in eq 7 the numerical prefactor accounts for dimensional aspects when λ00 and T are expressed respectively in mW·m−1·K−1 and K, respectively, while M is expressed in g/mol and σ is expressed in nm. The reduced effective collision cross section Ω*λ for thermal conductivity was calculated with the following equation:13 2

∑ Ai /T *i i=0

A 2 = 0.1936835

The scaling factors σ and ε/k were determined by Wilhelm and Vogel,14 from a fit of viscosity data of propane (ε/k = 263.88 K; σ = 0.49748 nm). The reduced ideal specific heat at constant pressure C0p/R, reported in column 6 of Table 1, was estimated from the linear fit (C0p/R) = 2.292885 + 0.02118846 × T of the experimental values reported by Beeck15and Esper et al.16 λ00 values calculated from eq 7 are reported in column 7 of Table 1 and compare favorably with values of column 5. Critical Enhancement. To represent the critical enhancement Marsh et al.2 have tested several crossover equations-ofstate for propane. One is an empirical formulation, the second is a simplified crossover model, and the third is a full crossover model. With both last correlations, they were able to reproduce their data within ± 5 % at a level of uncertainty of 95 %. In this work, the analysis of the thermal conductivity and diffusivity of propane 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 Sengers17 to describe the singular behavior of the thermal diffusivity DT = (λ/(ρCp)), where Cp is the isobaric specific heat capacity. As noted previously, the value of the critical enhancement term Δλc(T,ρ) of the thermal conductivity, was estimated by subtracting the background

C0p

Ω*λ =

A1 = 0.327867;

(8) 3427



Article

thermal conductivity term λb(T,ρ) 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

Table 6. Thermal Conductivity of Propane along the Isotherm 374.28 K pa

ρ

λ

Δλc

MPa

kg·m−3

λ/mW·m−1·K−1

mW·m−1·K−1

6.760 6.323 5.798 5.500 5.200 5.000 4.900 4.820 4.780 4.760 4.740 4.720 4.710 4.685 4.670 4.650 4.640 4.630 4.625 4.620 4.615 4.610 4.605 4.600 4.595 4.590 4.585 4.580 4.575 4.570 4.565 4.560 4.550 4.545 4.535 4.520 4.518 4.515 4.510 4.485 4.475 4.460 4.450 4.440 4.425 4.380 4.320 4.280 4.200 4.080 3.975 3.945 3.424 3.087 2.900

359.68 351.65 339.29 330.01 317.41 305.36 297.08 288.36 282.82 279.61 276.01 271.92 269.64 263.07 258.36 250.84 246.43 241.50 238.83 236.03 233.08 230.01 226.81 223.51 220.13 216.70 213.23 209.77 206.34 202.97 199.70 196.53 190.58 187.80 182.67 175.91 175.08 173.88 171.95 163.54 160.65 156.71 154.30 152.04 148.90 140.80 132.14 127.27 118.96 108.86 101.54 99.63 74.34 62.45 56.71

68.67 68.29 67.07 66.45 65.84 66.03 66.93 67.78 68.68 69.09 69.79 70.26 70.96 72.40 73.34 74.77 75.99 77.55 78.54 79.61 80.36 81.17 81.60 81.48 81.12 80.76 80.16 79.54 78.75 77.86 76.64 75.25 72.67 71.23 68.74 65.51 65.02 64.24 63.24 59.60 58.23 56.26 54.83 53.78 52.38 49.61 47.19 45.58 43.76 41.20 39.18 38.87 34.53 32.71 31.64

4.01 5.05 5.94 6.84 8.16 10.09 12.18 14.20 15.78 16.64 17.80 18.78 19.76 22.00 23.50 25.83 27.57 29.68 30.99 32.38 33.46 34.60 35.39 35.63 35.67 35.66 35.41 35.17 34.75 34.21 33.32 32.25 30.30 29.14 27.17 24.61 24.20 23.54 22.75 19.94 18.85 17.25 16.03 15.20 14.08 12.07 10.45 9.28 8.18 6.48 5.07 4.91 2.53 1.54 0.84

Δλc ρCp

ΔDc =

(9)

and a background contribution

λb ρCp

(10)

DT = ΔDc + D b

(11)

Db = with

The mode coupling of critical dynamics theory predicts that ΔDc of a critical pure fluid in the hydrodynamic limit can be written asymptotically near its critical point as

ΔDc = RD

kBT 6πηξ

(12)

where η is the shear viscosity, ξ is the long-range correlation length of the density fluctuations, and RD ≈ 1.05 is a universal amplitude combination.18,19 Critical Enhancement along the Critical Isochore. Along the critical isochore, eq 12 can be also rewritten following the approximated form20 Δλcc = RD

kBT ρC pc 6πηξ

(13)

where Ccp is defined along the critical isochore, as the specific heat difference Ccp = Cp − Cv, and is calculated from the isothermal compressibility κT, using the thermodynamic relation ⎛ T ⎞⎛ ∂p ⎞2 C pc = Cp − Cv = ⎜ ⎟⎜ ⎟ κT ⎝ ρ ⎠⎝ ∂T ⎠ ρ

(14)

On the other hand, it was found along the critical isochore over an extended temperature range of any pure fluid,21 that the rescaled isothermal compressibility χ*T

χT* = κTpc Zc

(15)

lies on a single curve when it is plotted as a function of the rescaled temperature τ = Yct

(16)

where t t=

T − Tc Tc

(17)

is the usual reduced temperature distance to Tc. In eq 15, pc Zc = RTcρc

(18)

and in eq 16, ⎤ ⎡ ⎛ ∂p ⎞ ⎛ T ⎞ Yc = ⎢⎜ ⎟ ⎜⎜ c ⎟⎟⎥ − 1 ⎢⎝ ∂T ⎠ ρ ,T ⎝ p ⎠⎥ ⎣ c ⎦ c c

Uncertainties (0.95 level of confidence): p, 0.02 %; ρ, 0.2 %; λ, 3 %; Δλc, 4 %. a

3428

(19)



Article


pa MPa

kg·m

10.360 9.700 9.500 8.950 8.020 7.140 6.540 6.480 5.830 5.580 5.322 5.226 5.160 5.140 5.040 4.950 4.920 4.890 4.850 4.800 4.790 4.770 4.750 4.740 4.720 4.710 4.700 4.695 4.685 4.680 4.675 4.670 4.665 4.660 4.655 a

λ −3

396.66 391.39 389.68 384.66 374.80 363.09 352.87 351.70 336.28 328.24 317.59 312.63 308.76 307.50 300.32 292.16 288.90 285.26 279.65 270.87 268.78 264.16 258.81 255.81 248.98 245.09 240.86 238.61 233.84 231.32 228.72 226.04 223.30 220.51 217.67

Δλc −1

−1

λ/mW·m ·K 75.00 73.90 73.55 72.71 71.14 69.58 68.46 68.42 66.91 66.44 65.76 65.25 65.29 65.30 65.33 65.33 65.54 65.99 66.76 68.07 68.52 69.65 70.62 71.11 72.61 73.46 74.12 74.66 75.44 75.69 75.97 76.25 76.53 76.39 76.11

−1

pa

mW·m ·K

−1

MPa

2.54 2.65 2.70 2.95 3.45 4.16 4.92 5.10 6.21 7.02 7.97 8.20 8.76 8.95 9.99 11.10 11.74 12.67 14.18 16.61 17.32 19.01 20.63 21.47 23.77 25.06 26.22 27.01 28.32 28.87 29.44 30.01 30.60 30.76 30.78

4.650 4.645 4.640 4.635 4.630 4.625 4.620 4.610 4.605 4.600 4.595 4.590 4.585 4.580 4.575 4.570 4.565 4.560 4.555 4.550 4.545 4.535 4.525 4.520 4.510 4.450 4.400 4.350 4.315 4.300 4.210 3.794 3.790 3.780

ρ kg·m

λ −3

214.81 211.94 209.07 206.22 203.42 200.66 197.97 192.81 190.35 187.99 185.71 183.52 181.42 179.41 177.47 175.61 173.83 172.12 170.47 168.88 167.36 164.47 161.77 160.49 158.04 145.90 138.04 131.45 127.38 125.75 117.07 89.88 89.68 89.17

Δλc −1

λ/mW·m ·K

−1

75.83 75.42 75.07 74.53 73.71 72.91 72.15 70.49 69.65 68.77 67.95 67.06 66.11 65.09 64.22 63.25 62.57 61.95 61.15 60.52 59.74 58.45 57.20 56.54 55.52 51.49 49.12 47.39 45.87 45.26 42.76 37.55 37.32 36.95

mW·m−1·K−1 30.81 30.70 30.66 30.41 29.89 29.40 28.91 27.76 27.18 26.54 25.95 25.28 24.54 23.72 23.06 22.26 21.76 21.30 20.67 20.19 19.56 18.55 17.57 17.02 16.24 13.37 11.72 10.60 9.46 8.99 7.21 4.31 4.09 3.73


respectively. The values of Δλcc were calculated at each temperature value reported in column 1 of Table 10, by interpolating at the critical density the values of the experimental data measured at the closest density of ρc. The values of η at each temperature are obtained from the NIST tables23 for ρ = ρc. The corresponding calculated values of Rc* in the rescaled temperature range (0.00964 < τ = Yc((T − Tc)/Tc) < 0.172) covered by the experiments are shown in column 4 of Table 10. The fit of the complete data set by an effective pure power law with adjustable exponent and adjustable amplitude gives

where (∂p/∂T)ρc,Tc is the slope of the vapor pressure curve at the critical point. κT* = κTpc is the usual reduced form of the isothermal compressibility. A similar analysis21 has also shown that the rescaled correlation length ξ* = ξ / a

(20)

where ⎛ k T ⎞1/3 a = ⎜⎜ B c ⎟⎟ ⎝ pc ⎠

(21)

is a single function of τ. Consequently, from eq 13, the rescaled quantity22 Rc* =

Δλccη RD χT* = p Zca 6π ξ* kB(T (∂P /∂T )ρc )2 c

R c* = 0.01676686τ −0.61

(23)

The calculated values of R*c by using eq 23 are given in column 5 of Table 10. The residuals are lower than 3 % with the present experimental estimation of this quantity. The equivalent best fit of the critical thermal conductivity enhancement in the reduced temperature range

(22)

should be also a single function of τ. For propane, the validity of eq 22, along the critical isochore was checked using the critical parameters of eq 1, (∂p/∂T)ρc,Tc = 0.077035 MPa·K−1, the critical enhancement of the thermal conductivity data and the shear viscosity data given in columns 2 and 3 of Table 10,

⎛ T − Tc ⎞ 1.69·10−3 < t = ⎜ ⎟ < 3.0·10−2 ⎝ Tc ⎠ 3429



Article


pa MPa 13.000 12.000 11.500 11.028 11.000 10.000 9.806 9.190 9.000 8.500 7.800 7.200 6.800 6.200 6.000 5.900 5.820 5.700 5.550 5.500 5.250 5.200 5.100 5.050 4.950 4.880 4.840 4.820 4.800 4.780 4.760 4.740 a

λ

kg·m

−3

Δλc −1

−1

λ/mW·m ·K

412.36 406.43 403.20 399.97 399.77 392.18 390.57 385.09 383.28 378.16 369.93 361.50 354.85 342.42 337.28 334.44 332.00 328.04 322.42 320.34 307.65 304.45 296.99 292.53 281.29 270.29 261.96 256.95 251.22 244.61 237.03 228.45

78.45 77.24 76.50 75.97 75.91 74.39 74.05 73.14 72.88 72.00 70.56 69.60 68.84 67.46 67.14 66.96 66.82 66.71 66.31 66.34 65.88 66.03 66.20 66.21 67.08 68.35 69.36 69.95 70.50 71.06 72.06 72.57

−1

pa

mW·m ·K

−1

MPa

2.02 2.28 2.36 2.56 2.59 2.81 2.82 3.10 3.23 3.44 3.66 4.32 4.78 5.56 6.11 6.40 6.63 7.17 7.61 7.99 9.41 10.01 11.21 11.83 14.16 16.83 18.86 20.05 21.28 22.60 24.48 25.94

4.720 4.700 4.690 4.680 4.670 4.660 4.650 4.640 4.630 4.620 4.600 4.600 4.590 4.580 4.570 4.560 4.540 4.510 4.450 4.400 4.350 4.250 4.125 4.050 4.000 3.800 3.600 3.550 3.300 3.250 3.134 3.110

ρ kg·m

λ −3

Δλc −1

λ/mW·m ·K

219.07 209.30 204.45 199.71 195.15 190.81 186.70 182.84 179.22 175.84 169.70 169.70 166.92 164.30 161.82 159.49 155.17 149.43 139.89 133.31 127.59 117.97 108.18 103.13 100.02 89.14 80.05 77.99 68.62 66.90 63.09 62.33

72.83 71.56 70.58 69.67 68.35 67.01 65.72 64.48 62.78 61.28 59.49 59.49 58.26 57.14 56.07 55.03 53.50 51.75 48.77 47.29 45.89 43.45 40.91 39.86 39.34 37.47 35.95 35.63 34.33 34.12 33.80 33.73

−1

mW·m−1·K−1 27.22 27.00 26.52 26.11 25.26 24.36 23.49 22.64 21.31 20.13 18.95 18.95 18.00 17.12 16.30 15.48 14.35 13.15 11.04 10.19 9.31 7.73 5.98 5.36 5.10 4.11 3.28 3.11 2.50 2.42 2.36 2.33


In eq 25 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 the NIST tables23 (column 3, Table 10) were fit by the linear equation η(μPas−1) = 22.8357(1 + 0.4949t) 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 10, respectively, the values of ρc(Cp− CV) were fit by the power law equation ρc(Cp −CV) (MJ m−3 K−1) = 77.0227t−1.03174. In the absence of measurements of the correlation length of propane, we have assumed that the theoretical estimation24,25 of the leading Ising power law ξ = ξ0t−0.63, with ξ0 = 2.0332·10−10 m,21 remains valid in our experimental range. Accordingly, the fit of the prefactor term RD,eff leads to the following power law

gives Δλcc

−1

−1

(mW· m · K ) = 2.41175t

−0.60157

(24)

The corresponding calculated values of the critical thermal conductivity excess are given in column 8 of Table 10. The standard deviation is of the order of 1.1 with experimental data. In eq 23 the effective value (0.61) of the critical exponents is equal to the asymptotic Ising values (0.61). In eq 22, the effective value (0.60157) of the critical exponents is slightly higher than the asymptotic Ising values (0.57). An alternative empirical approach consists to explicitly direct the effective pure power laws that are only valid in this restricted temperature range. This allows a more easy acquisition of each “theoretical” power law which is equivalent to the experimental power law of eq 24, in conformity with the single power law that governs the master behavior of eq 22 for any pure fluid. In the following, we illustrate this “theoretical” estimation of Δλc 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 22 as follows Δλcc

kT = RD,eff B ρc (Cp − CV ) 6πηξ

RD,eff = 0.560553t −0.19155

(26)

Introducing all these previous singular behaviors in eq 22, the phenomenological power law equation of the critical enhancement of the thermal conductivity reads as follows Δλcc (mW m−1 K−1) = 2.5210

(25)

(1 + t ) t −0.5933 (1 + 0.4949t ) (27)

3430



Article

Table 9. Thermal Conductivity of Propane along the Isotherm 381.00 K pa

ρ

λ

Δλc

MPa

kg·m−3

λ/mW·m−1·K−1

mW·m−1·K−1

14.750 12.000 11.000 10.110 9.036 7.583 7.300 7.162 6.900 6.500 6.200 5.900 5.700 5.550 5.450 5.390 5.350 5.300 5.250 5.200 5.150 5.100 5.060 5.040 5.020 5.000 4.960 4.920 4.900 4.880 4.860 4.800 4.780 4.740 4.660 4.600 4.560 4.520 4.220 4.100 4.040 3.880 3.820 3.580 3.428 3.320 3.260 3.253

413.57 397.14 389.75 382.20 371.34 351.42 346.33 343.63 338.02 327.77 318.09 305.48 294.26 283.25 273.90 267.16 262.05 254.81 246.46 236.85 225.97 214.13 204.38 199.55 194.82 190.21 181.51 173.56 169.88 166.38 163.05 154.02 151.29 146.19 137.19 131.28 127.66 124.27 103.78 97.26 94.24 86.87 84.32 75.02 69.75 66.24 64.37 64.15

78.55 75.31 73.71 72.35 70.35 67.67 67.14 66.99 66.40 65.54 65.03 64.23 64.35 64.36 64.35 64.27 64.39 64.44 64.75 64.48 64.48 63.74 62.68 61.98 61.24 60.46 58.66 56.85 55.94 55.06 54.18 51.87 51.26 49.88 47.56 46.18 45.39 44.59 40.27 38.90 38.25 37.05 36.65 35.58 34.98 34.51 34.29 34.26

1.18 1.97 2.04 2.31 2.54 3.58 3.95 4.26 4.61 5.39 6.35 7.40 9.07 10.53 11.72 12.47 13.22 14.14 15.42 16.27 17.47 18.01 17.98 17.79 17.55 17.25 16.32 15.31 14.76 14.22 13.65 12.21 11.85 10.96 9.48 8.65 8.15 7.65 5.09 4.25 3.83 3.23 3.02 2.65 2.43 2.21 2.12 2.10

Figure 1. Fractional deviation 100*[λ01(cal) − λ01(exp)]/λ01(exp) of the dilute gas thermal conductivity of propane at atmospheric pressure and in the zero density limit as a function of temperature, where λ01(exp) is reported in column 2 of Table 1: ⧫, eq 4 ; ■, eq 3; ▲, correlation of Marsh et al.2 (see footnote 2 in Table 1); ●, correlation of Ramires;5 and ×, λ00 values calculated with eqs 6 and 7.5 λ 01 (Ramires) = − 0.00125 + 8.45· 10−5T + 6.02· 10−8T 2 .

Critical Enhancement in the Supercritical Domain. To calculate Δλc outside the critical isochore, we have then generalized eq 27 as the following form Δλc(t , Δρ*) = RD,eff (t , Δρ*)

kBT 6πη(T , ρ)ξ(t , Δρ*)

× ρ[Cp(T , ρ) − CV (T , ρ)]

(28)

where the temperature and density dependences of the correlation length and the amplitude term RD,eff(t,Δρ*), are accounted for by introducing the reduced quantity t of eq 17 and the reduced density ρ − ρc Δρ* = ρc (29) Now the effective function RD,eff(t,Δρ*) is an empirical function which characterizes our extended temperature and density range measured by t and Δρ*. Then eq 28 is similar to the crossover semiempirical equation Δλc = RD((kBT)/(6πηξ))ρCcpF(t,Δρ*) proposed by Hanley et al. in ref 20, where F(t,Δρ*) is an empirical damping function used in their extended critical region.4 The values of Cp(T,ρ), CV(T,ρ) and η(T,ρ) are calculated from the NIST tabulated data.23 The values of the correlation length ξ(t,Δρ*) are given by ⎡ P ρ ⎤−0.509⎡ ∂ρ(T , ρ) ⎤−0.509 ξ(t , Δρ*) = ξ0 Ψ⎢ c2 ⎥ ⎢ ⎥ ⎣ ∂P ⎦ ⎢⎣ ρc ⎥⎦

(30)

where Ψ = 238.85042 The values of the amplitude Ψ and the exponent (0.59) were determined along the critical isochore from a fitting with the theoretical relation

Uncertainties (0.95 level of confidence): p, 0.02 %; ρ, 0.2 %; λ, 3 %; Δλc, 4 %. a

The small difference of the exponent values in eq 27 and eq 24 is due to the small explicit linear temperature dependences of eq 27. The calculated values of Δλcc with eq 27 (column 8, Table 10) compare favorably with the experimental values (column 2, Table 10) (standard deviation 1.3) and with the calculated values using eq 24 (standard deviation 1.1).

ξ(t ) = ξ0t −0.63

(31)

The distribution of the critical enhancement of the thermal conductivity of propane as a function of density along 8 isotherms is shown in Figure 2. The variation of the correlation length as a function of the reduced density is shown in Figure 3. The functional form of RD,ef f(t,Δρ*) was written as the product 3431



Article

Table 10. Variation of Δλc, η, Rc*, Cp and CV as a Function of Temperature T along the Critical Isochore T4

Δλc (exp)

η

Cp

CV

Δλc (cal)

Δλc (cal)

K

mW·m−1·K−1

μPa·sec−1

R*c (exp) (eq 22)

R*c (cal) (eq 23)

J/g·K

J/g·K

mW·m−1·K−1 (eq 24)

mW·m−1·K−1 (eq 27)

370.45 371.05 371.97 373.20 374.28 375.10 375.90 381.00 381.00

111.82 74.35 54.89 41.98 35.75 30.75 27.22 18.40 18.74

22.855 22.873 22.901 22.939 22.972 22.997 23.021 23.177 23.177

0.2842 0.1885 0.1387 0.1055 0.0895 0.0767 0.0677 0.0448 0.0457

0.2856 0.1893 0.1345 0.1019 0.0860 0.0776 0.0712 0.0491 0.0491

255.450 131.140 74.310 46.871 35.315 29.728 25.663 14.131 14.131

2.546 2.522 2.491 2.451 2.420 2.397 2.380 2.285 2.285

112.19 74.84 53.43 40.68 34.42 31.10 28.56 19.80 19.80

113.30 74.80 52.96 40.09 33.81 30.49 27.97 19.32 19.32

Uncertainties (0.95 level of confidence): T, 0.05 K; Δλc, 4 %; η, 2.5 %; Rc* (exp), 4 %; Rc* (cal), 4 %; Cp, 2 %; CV, 2 %; Δλc (cal), 5 %.

4

Figure 4. Variation of RD,eff (Δρ*) of propane as a function of density along eight isotherms at ◆, T = 370.45 K; ■, T = 371.05 K; ▲, T = 371.97 K; ×, T = 373.20 K; ∗, T = 374.28 K; ●, T = 375.10 K; +, T = 375.90 K; and ○, T = 381.00 K.

Figure 2. Distribution of the critical enhancement of the thermal conductivity of propane as a function of density along eight isotherms at ◆, T = 370.45 K; ■, T = 371.05 K; ▲, T = 371.97 K; ×, T = 373.20 K; ∗, T = 374.28 K; ●, T = 375.10 K; +, T = 375.90 K; and ○, T = 381.00 K.

Figure 5. Fractional deviation of the thermal conductivity of propane, as a function of density along eight isotherms at ◆, T = 370.45 K; ■, T = 371.05 K; ▲, T = 371.97 K; ×, T = 373.20 K; ∗, T = 374.28 K; ●, T = 375.10 K; +, T = 375.90 K; and ○, T = 381.00 K.

Figure 3. Variation of the correlation length of propane as a function of reduced density along eight isotherms at ⧫, T = 370.45 K; ■, T = 371.05 K; ▲, T = 371.97 K; ×, T = 373.20 K; ∗, T = 374.28 K; ●, T = 375.10 K; +, T = 375.90 K; ○, T = 381.00 K.

with the experimental values calculated in the whole density range. Finally, the total thermal conductivity λ(T,ρ) was estimated by using eqs 2, 3, 5, 6 and 28. The residuals rλ (%) = 100 × ((λcal./λexpt) − 1) are shown in Figure 5. The standard deviation is 1.26.

of two main contributions, RD,ef f(t,Δρ*) = RD,ef f(t)·RD,ef f(Δρ*) separating then the respective temperature and density contributions.RD,ef f(t) is given by eq 26. A Lorentzian functional form was used to express RD,eff(Δρ*), shown in Figure 4, as a function of density. RD , eff (Δρ*) = 1/(1 + 3.3380Δρ2 )

■

CONCLUSION New measurements of the thermal conductivity of propane are presented in the supercritical region, at temperatures from (370.45 to 381.00) K along eight isotherms and at pressures up to 15 MPa with an estimated uncertainty of 3 %. As expected,

(32)

Accordingly, the calculated values of the critical enhancement Δλc of the thermal conductivity using eqs 28 to 32 are in satisfactory agreement (within the experimental uncertainty) 3432



Article

methane (HCFC-22) in the Temperature Range from 300 to 515 K and at Pressures up to 55 MPa. J. Chem. Eng. Data 2001, 46, 193−201. (13) In ref 12, the correct j-index variation for the sum in the r.h.s. member of eq 6 is j = 0 to j = 2. (14) Vogel, E.; Kuchenmeister, C.; Bich, E.; Laesecke, A. Reference Correlation of the Viscosity of Propane. J. Phys. Chem. Ref. Data 1998, 27, 947−970. (15) Beeck, O. J. J. The Exchange of Energy between Organic Molecules and Solid. Chem. Phys. 1936, 4, 680−689. (16) Esper, G.; Lemming, W.; Beckermann, W.; Kholer, F. Acoustic Determination of Ideal Gas Heat Capacity and Second Virial Coefficient of Small Hydrocarbons. Fluid Phase Equilib. 1995, 105, 173−192. (17) Luettmer-Strathmann, J.; Sengers, J. V. The Thermal Conductivity of R134a in the Critical Region. High Temp. High Press. 1994, 26, 673−682. (18) Hohenberg, P. C.; Halperin, B. I. Theory of Dynamic Critical Phenomena. Rev. Mod. Phys. 1977, 49, 435−479. (19) Sengers, J. V.; Perkins, R. A.; Huber, M. L.; Le Neindre, B. Thermal Diffusivity of H2O near the Critical Point. Int. J. Thermophys. 2009, 30, 1463−1465. (20) Hanley, H. J. M.; Sengers, J. V.; Ely, J. F. On Estimating Thermal Conductivity Coefficients in the Critical Region of Gases. In Thermal Conductivity 14; Klemens, P. G., Chu, T. K., Eds.; Plenum Press: New York, 1976; pp 383−407. (21) Garrabos, Y. Phenomenological Scale Factors for the Liquid− Vapor Critical Transition of Pure Fluids. J. Phys. (Paris) 1985, 46, 281−291. (22) Le Neindre, B.; Garrabos, Y.; Tufeu, R. The Critical Thermal− Conductivity Enhancement along the Critical Isochore. Int. J. Thermophys. 1991, 12, 307−321. (23) McLinden, M. O.; Klein, S. A.; Lemmon, E. W.; Peskin, A. W.; NIST Standard Database 23, REFPROP, version 7; Nat. Inst. Stand. Technol.: Boulder, CO, 2004. (24) Garrabos, Y.; Palencia, F.; Lecoutre, C.; Erkey, C.; Le Neindre, B. Master Singular Behavior from Correlation Length Measurements for Seven One-Component Fluids near Their Gas-Liquid Critical Point. Phys. Rev. E 2006, 73, 026125. (25) Garrabos, Y.; Lecoutre, C.; Palencia, F.; Broseta, D.; Le Neindre, B. Master Singular Behaviour for the Sugden Factor of OneComponent Fluids near Their Gas-Liquid Critical Point. Phys. Rev. E 2007, 76, 061109. (26) Lemmon, E. W.; McLinden, M. O.; Wagner, W. Thermodynamic Properties of Propane. III. A Reference Equation of State for Temperatures from the Melting Line to 650 K and Pressures up to 1000 MPa. J. Chem. Eng. Data 2009, 54, 3141−3180.

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 concluded that the residual thermal conductivity was independent of the temperature. In this critical isochoric region simple correlations were developed to represent the variation of the thermal conductivity in terms of reduced parameters, which can be thus applied to any fluid. Using a Lorentzian equation for the variation of the thermal conductivity as a function of density, the standard deviations between calculated and experimental thermal conductivity were estimated to be 1.26. Further comparisons with some other fluids, like pentane and hexane, have shown that our method of analysis can be generalized to represent the thermal conductivity of any fluid, in the extended critical region. A new equation-of-state was developed by Lemmon et al. to calculate the density of propane.26 With this correlation, which is considered actually as the best available, the accuracy of densities can be slightly improved.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Tufeu, R.; Le Neindre, B. Thermal Conductivity of Propane in the Temperature Range 25−305 °C and Pressure Range 1−70 MPa. Int. J. Thermophys. 1987, 8, 27−38. (2) Marsh, K. N.; Perkins, R. A.; Ramires, M. L. V. Measurement and Correlation of the Thermal Conductivity of Propane from 86 to 600 K at Pressures to 70 MPa. J. Chem. Eng. Data 2002, 47, 932−940. (3) Holland, M.; Hanley, H. J. M.; Gubbins, K. E.; Haile, J. M. A. Correlation of the Viscosity and Thermal Conductivity Data of Gaseous and Liquid Propane. J. Phys. Chem. Ref. Data 1979, 8, 559− 575. (4) Younglove, B. A.; Ely, J. F. Thermophysical Properties of Fluids. II. Methane, Ethane, Propane, Isobutane and Normal Butane. J. Phys. Chem. Ref. Data 1987, 16, 577−798. (5) Ramires, M. L. V.; Nieto de Castro, C. A.; Cusco, L.; Perkins, R. A. Improved Correlations for the Thermal Conductivity of Propane and n-Butane. In Thermal Conductivity 23; Wilkes, K. E., Dinwiddie, R. B., Graves, R. S., Eds.; Technomic Publishing Company Inc.: Lancaster, PA, 1996; pp 13−25. (6) Yata, J.; Hori, M.; Isono, Y.; Ueda, Y. Equations for Thermal Conductivity of Natural Refrigerants. Int. J. Thermophys. 2001, 22, 1369−1381. (7) Le Neindre, B.; Tufeu, R. Measurement of the Transport Properties of Fluids, Experimental Thermodynamics Vol. III; Wakeham, W. A., Nagashima, A., Sengers, J. V., Eds.; Blackwell Scientific Publications: Oxford, 1991; pp 111−142. (8) Miyamoto, H.; Watanabe, K. A Thermodynamic Property Model for Fluid Phase Propane. Int. J. Thermophys. 2000, 21, 1045−1072. (9) Roder, H. M.; Nieto de Castro, C. A. Thermal Conductivity of Liquid Propane. J. Chem. Eng. Data 1982, 27, 12−15. (10) Prasad, R. C.; Wang, G.; Venart, J. E. S. The Thermal Conductivity of Propane. Int. J. Thermophys. 1989, 10, 1013−1027. (11) Yata, J.; Hori, M.; Hagiwara, T. Thermal Conductivity of Propane and Butane in the Liquid Phase. Fluid Phase Equilib. 1996, 125, 267−274. (12) Le Neindre, B.; Garrabos, Y.; Sabirzianov, A.; Goumerov, F. Measurements of the Thermal Conductivity of Chlorodifluoro3433


Measurements of the Thermal Conductivity of Propane in the

Recommend Documents