Determination of the Relative Permittivity, εr, of Octane at

Mar 24, 2014 - The relative permittivity, εr(T, p), of liquid octane has been determined with an estimated expanded relative uncertainty of ± 0.018 ...
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Determination of the Relative Permittivity, εr, of Octane at Temperatures between (303 and 393) K and Pressures below 25 MPa with a Concentric Cylinder Capacitor at a Frequency of 1 kHz Laurent Pirolli,† Anthony R. H. Goodwin,*,†,‡ Kenneth N. Marsh,‡ and Eric F. May‡ †

Schlumberger Technology Corporation, 150 Gillingham Lane, Sugar Land, Texas 77478, United States Centre for Energy, School of Mechanical & Chemical Engineering, The University of Western Australia, Crawley WA 6009, Australia

‡

ABSTRACT: The relative permittivity, εr(T, p), of liquid octane has been determined with an estimated expanded relative uncertainty of ± 0.018 % from measurements of the complex capacitance of a concentric cylinder capacitor at temperatures between (293 and 393) K and pressures below 25 MPa. The measurements were corrected for the capacitor’s isothermal compressibility that was determined by comparison of measurements of εr(He, 303 K and 393 K, p) with ab initio results from quantum mechanics. The measured εr(C8H18, T, p) were combined with the amount-of-substance density, ρ, which was obtained from the equation of state reported by Span and Wagner (Int. J. Thermophys. 2003, 24, 41−109), to calculate the molar polarizability, P, that was fit with an expanded (k = 2) relative uncertainty of ± 0.01 % to a correlation with three parameters. The εr(T, p) obtained from the literature were converted to P·ρ and compared with our correlation and found to differ by < ± 0.2 % at temperatures that overlap ours. by Shen et al.,12 for the determination of εr(C7H8, T, p)12 and εr(C7H16, T, p).13 In this work, εr(T, p) was determined from measurements of capacitance C(C8H18, T, p), using eq 1, with C(T, p = 0) from eq 3 of ref 12 and κT that was obtained by comparing measurements of εr(He, T, p) with ab initio value results from quantum mechanics as reported in ref 12. The uncertainties of each of the measured quantities were as stated in refs 12 and 13 and are provided as follows for completeness: (1) the capacitance was measured with a relative standard uncertainty of ± 6·10−6 with a ratio transformer bridge at a frequency of 1 kHz; (2) the temperature was determined on the International Temperature Scale of 1990 with a nominal 100 Ω platinum resistance thermometer with an expanded uncertainty of ± 0.01 K; and (3) pressures were measured with a resonant quartz transducer with an uncertainty of δp/MPa = {0.0001·(p/MPa) + 0.022}. The uncertainty arising from items 2 and 3 resulted in a negligible additional uncertainty in the determination of εr(T, p). The expanded relative uncertainty of the relative permittivity, including items 1 through 3 and the calibration with He, is δεr/εr ≈ 1.8·10−4. 1.2. Materials. The octane was obtained from Fluka Chemie GmbH with a stated mole fraction purity greater than 0.995 as listed in Table 1. The manufacturer also specified the octane contained a measured mole fraction of water less than 0.0005; this x(H2O) is equivalent to the recommended mole fraction solubility of water in octane of 5·10−4 at T =

1. INTRODUCTION In this article, the relative permittivities of liquid octane, εr(C8H18, T, p), are reported at temperatures T between (303 and 393) K and pressures p below 25 MPa as determined from measurements of the capacitance C(C8H18, T, p) between two coaxial cylinders at a frequency of 1 kHz. The εr(C8H18, T, p) were determined from C(C8H18, T, p) with εr(C8H18, T , p) =

C(C8H18, T , p) (1 + κTp /3) C(T , p = 0)

(1)

where C(T, p = 0) is the capacitance of the concentric cylinder under vacuum as a function of temperature and κT the isothermal compressibility {κT = −(∂V/∂p)T/V} of the material used to form the capacitor. The εr(C8H18, T, p) were combined with the amount-of-substance density ρ to give the total polarizability P(ρ, T) ⎛ ε − 1⎞1 P(ρ , T ) = ⎜ r ⎟ = A εr + Aμ /T + Bεr ρ ⎝ εr + 2 ⎠ ρ

(2)

and were represented by three parameters. The εr(C8H18, T, p) reported by Scaife and Lyons1 and Brazier and Freeman2 were also converted to P(ρ, T)·ρ and compared with our correlation. The εr(T, pr = 0.1 MPa) obtained by extrapolating the εr(T, p) to pr were compared with the results reported by other workers.1−11 1.1. Apparatus, Experimental Procedures, and Calibration. The apparatus, which was used without alteration, and experimental procedures, including the method used to fill the capacitor with liquid samples, have been described previously © 2014 American Chemical Society

Received: January 21, 2014 Accepted: March 11, 2014 Published: March 24, 2014 1609

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were obtained at a frequency of 1 kHz at temperatures between (303 and 393) K and pressures below 25 MPa. The uncertainties, listed in Table 2, are at a confidence interval of 0.95 (k = 2) and were obtained by combining in quadrature uncertainties arising from the uncertainty of each capacitance measurement, the calibration with He to determine the isothermal compressibility, δT(dεr/dT) and δp(dεr/dp). The contributions of these major sources of the expanded uncertainty were discussed previously in ref 12. The contribution to the uncertainty arising from the uncertainty in composition was assumed to be zero. The εr(T, pr = 0.1 MPa) were obtained by extrapolating the εr(T, p) of Table 2 for each isotherm with a linear equation in pressure to pr = 0.1 MPa. The values of εr(T, pr) so determined are listed in Table 3 and shown in Figure 1 as deviations from

Table 1. Chemical Substance, B, Supplier, and Stated Mole Fraction Purity, xi, for Octanea B

supplier

xi

octane

Fluka Chemie GmbH

0.995

a

The sample was degassed and dried prior to the measurements. No other purification was performed.

298.15 K reported by Mac̨ zyński et al.14 that was based on the measurements reported by Polak and Lu.15 The sample was degassed by vacuum sublimation and dried as specified in ref 12. To estimate the potential systematic error in εr arising from the presence of the mole fraction of water in octane as given in Table 1, which is equal to both the solubility and the contamination cited by the sample supplier, two mixing rules16,17 were used with εr(H2O, 298 K, 0.1 MPa) as reported by Fernandez et al.18 and with the results listed in Table 2 that

1

εr(T , pr ) =

∑ Ai{(T /K) − 273.15}i

(3)

i=0

Table 2. Relative Permittivity, εr, for Octane Determined at a Frequency of 1 kHz at the Temperature T and Pressure p Listed with the Expanded Uncertainty (k = 2) T/K 303.070 ± 0.010

333.250 ± 0.01

373.160 ± 0.01

398.170 ± 0.01

εr

p/MPa 5.010 10.050 20.060 25.030 5.040 6.780 10.020 20.050 25.050 5.040 10.030 20.030 25.060 5.010 10.010 20.050 25.040

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.023 0.023 0.024 0.025 0.023 0.023 0.023 0.024 0.025 0.023 0.023 0.024 0.025 0.023 0.023 0.024 0.025

Table 3. Relative Permittivity εr(T, pr = 0.1 MPa) for Octane as a Function of Temperature T at a Frequency of 1 kHza

1.94148 1.94875 1.96184 1.96820 1.90899 1.91211 1.91777 1.93287 1.93989 1.85755 1.86840 1.88737 1.89562 1.82430 1.83740 1.85955 1.87023

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

εr

T/K 303.070 333.250 373.160 398.170

0.00035 0.00035 0.00036 0.00036 0.00035 0.00035 0.00035 0.00035 0.00035 0.00034 0.00034 0.00034 0.00035 0.00033 0.00033 0.00034 0.00034

± ± ± ±

0.010 0.010 0.010 0.010

1.93522 1.90189 1.84884 1.81390

± ± ± ±

0.00035 0.00035 0.00034 0.00033

a The εr(T, pr = 0.1 MPa) were obtained by extrapolating the εr(T, p) from Table 2 for each isotherm with a linear polynominal in pressure to pr = 0.1 MPa. The expanded (k = 2) uncertainties are cited.

that fit the values of Table 3 with a relative standard deviation of the mean σ(⟨ε′⟩) of 100·σ(⟨εr⟩)/εr = ± 0.07 when the parameters were as follows: A0 = 1.975912 and A1 = −0.001281682. The σ(⟨ε′⟩) is taken as an estimate of the

were assumed to be exact for pure octane. For x(H2O) = 5·10−4 at a temperature of 298 K and a pressure of 0.1 MPa both mixing rules gave rise to δεr/εr ≈ 2·10−4 that is equivalent to the expanded relative uncertainty of the measurements reported here. It is also plausible that there were hydrocarbons of similar normal boiling temperature to those of the major constituent in our samples which include isomers of heptane and octane. As stated in ref 12, a mole fraction x ≈ 0.001 of those plausible chemicals would, based on estimates of εr obtained from REFPROP19 with the correlation of Harvey and Lemmon,20 introduce a relative uncertainty of δεr/εr ≈ 1.8·10−4 in the measured εr that is equal the expanded uncertainty of our measurements. We conclude that hydrocarbon impurities are insignificant for this work and have assumed the chemical composition was invariant from that cited by the supplier.

Figure 1. Fractional deviations Δεr = εr(expt) − εr(calc) of the measured relative permittivity εr(expt) of octane as deviations from the calculated values εr(calc) of eq 3 at as a function of temperature T for p = 0.1 MPa. ●, this work, Table 3; ◇, ref 1; ×, ref 10, □, ref 11; △, ref 4; gray filled circle with black outline, ref 6; +, ref 5; ∗, ref 7; −, ref 8; circle with gray outline, ref 3; and square with gray outline, ref 9. The dashed lines at 0 indicates an extrapolation of eq 3, while the solid line represents the temperature range of the fit to eq 3. The dashed lines at ± 0.07 represent 100·σ(εr)/⟨εr⟩ of the fit to eq 3.

2. RESULTS AND DISCUSSION The relative permittivity εr for octane obtained from eq 1 with measurements of C(T, p) combined with C(T, p = 0) from eq 3 and κT both of ref 12 are listed in Table 2. The values reported 1610

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deviations from eq 3 at temperatures that overlap ours between −(0.26 and 0.52) %; these differences increase with increasing temperature with a slope remarkably similar to that described, and shown in Figure 1, for the measurements of Scaife and Lyons.1 At two temperatures of (283 and 293) K the results of Champion et al.3 differ from (the extrapolation of) eq 3 within the assigned expanded uncertainty of our results. The values reported by Dornte and Smyth4 and at temperatures between (223 and 323) K lie below eq 3 but within the estimated uncertainty despite the 80 K extrapolation to their lowest temperature, while the result reported by Richards and Shipley5 at T = 293 K lies 0.6 % above eq 3. The recommendations of Maryott and Smith6 were based on the results reported in refs 4 and 5 and lie between −0.12 % at T = 293 K and −0.98 % at T = 348 K with differences that show a trend similar to those of Scaife and Lyons.1 The εr(T = 293.15 K, p = 0.1 MPa) reported by Hermiz,7 from measurements at a frequency of 2 kHz, and of Rosenberg et al.,8 which also which were at microwave frequencies between (1 and 34.8) GHz, are, as shown in Figure 1, both lower than but within the estimated uncertainty of eq 3. The measurement of εr(T = 298.15 K, p = 0.1 MPa) reported by de Cominges et al.9 lies 0.32 % above eq 3. The εr(T, p = 0.1 MPa) reported by Alonso et al.10 deviate from eq 3 by between −0.05 % and 1.8 %, while those of Alonso et al.11 differ systematically by about 0.9 %; the two sets of measurements reported by Alonso et al.10,11 concur at a temperature of 303 K, and both exhibit differences from eq 3 of 0.85 %. Our results lie between the values reported in the literature. No measurements have been performed by us specifically to eliminate plausible sources of the differences between the various studies shown in Figure 1, which include unaccounted for errors in the measurement of temperature, pressure, and capacitance as well as variations in chemical composition of the fluid, in particular the presence of water and for εr(T, p = 0.1 MPa) the extrapolation from 5 MPa to pr. However, based on the discussion in refs 12 and 13 and herein, it is reasonable to assume all, except for the variation in the mole fraction of water for which no measurements were performed, have been eliminated for our measurements. If we assume our measurements were obtained with a sample contaminated with water of mole fraction equal the solubility limit at a temperature reported in ref 14, we can estimate, with the mixing rules of refs 16 and 17, in the worst case, the variation of the εr(T, p = 0.1 MPa) arising from the removal of water. At T = 303 K removal of the contribution of x(H2O) ≈ 5·10−4 to εr results in a relative decrease in εr(303, p = 0.1 MPa) of 0.02 %, which is equal to the claimed uncertainty, from that shown in Figure 1, while at T = 373 K where the solubility of water is x(H2O) ≈ 7· 10−3 it results in a relative decrease in εr(373, p = 0.1 MPa) of 0.15 % which is about 4 times less than the difference shown in Figure 1 between eq 3 and the results reported in refs 1 and 3. No independent experiments have been performed to verify this conjecture, and in their absence we cannot exclude this source of error in our measurements. We will return to the discussion of other plausible sources of error in our measurements as well as those of Scaife and Lyons1 after comparing values of {εr(T, p) − 1}/{εr(T, p) + 2}. The {εr(T, p) − 1}/{εr(T, p) + 2} obtained from εr(T, p) of Table 2 are shown in Figure 3 as deviations from the P·ρ calculated with eq 2 and ref 21 along with measurements of εr(T, p) reported by other workers1,2 also converted to {εr(T, p) − 1}/{εr(T, p) + 2}. The measurements of Brazier and Freeman2 at a temperature of 303.14 K and pressures between

additional uncertainty, not reported in Table 3, that arose from extrapolation of the results listed in Table 2, where the minimum pressure is about 5 MPa, to pr. The measurements of εr(T, p) of Table 2 were combined with the amount-of-substance density ρ obtained from REFPROP19 using the equation of state published by Span and Wagner21 to give the total polarization P(ρ, T) of eq 2. The P(ρ, T) were fit by Aεr = 34.455 cm3·mol−1, Aμ = −248.713 K· cm3·mol−1, and Bεr = −278.189 cm6·mol−2 in eq 2 with a relative standard deviation of 100·σ(P)/⟨P⟩ = ± 0.12. This functional form is similar to that used by Harvey and Lemmon,20 although the parameters of eq 2 were assumed to be independent of temperature. The quantity Aμ determined in this work can only be interpreted as an adjustable parameter used to correlate the results listed in Table 2. Figure 2 shows

Figure 2. Deviations ΔP·ρ = P(expt)·ρ − P(calc)·ρ of the total polarizability P(expt) determined by combination of the measured relative permittivity εr(expt) of octane listed in Table 2 with the amount-of-substance density ρ of ref 21 (that was implemented within ref 19) as deviations from the calculated values P(calc) obtained from eq 2 at pressure p. ○, T = 303.07 K; □, T = 333.25 K; △, T = 373.16 K; ◇, T = 398.17 K; ·, the standard uncertainty as determined from the standard deviation of eq 2. The ordinate error bars represent the relative expanded (k = 2) uncertainty of P·ρ(expt, T, p) listed in Table 2.

the deviations of the measured εr(T, p) of Table 2 converted to {εr(T, p) − 1}/{εr(T, p) + 2} from the dimensionless quantities P·ρ calculated from a combination of eq 2 and ref 21 as implemented within REFPROP.19 The {εr(T, p) − 1}/{εr(T, p) + 2} of Table 2 differ from eq 2 within three times the assigned estimated expanded uncertainty. 2.1. Comparison with Literature Data. The εr(T, p = 0.1 MPa) reported by other workers1−11 are shown as deviations from eq 3 in Figure 1. The εr(T, p = 0.1 MPa) of Scaife and Lyons,1 which were obtained with a concentric cylinder capacitor, lie within the estimated uncertainty of eq 3 at temperatures below 318 K and so include an extrapolation of about 53 K below the lowest temperature of our measurements. However, at temperatures between 318 K and the upper temperature reported by Scaife and Lyons1 of 373 K, the relative differences increase with increasing temperature to be −0.6 % at T = 373 K that is within 10 times the estimated uncertainty of eq 3. The εr(T, p = 0.1 MPa) reported by Champion et al.3 that were obtained with a parallel plate capacitor at temperatures between (283 and 363) K and have a stated estimated relative uncertainty of ± 0.02 % exhibit 1611

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to T = 273 K gives differences between (4 and 7)·10−4 that are within the uncertainty assigned by Scaife and Lyons; the agreement at temperatures of (248 and 273) K is considered anomalous. Scaife and Lyons23 report a correlation of measurements of εr(T, p) not shown in Figure 3. We can reasonably speculate that there are three plausible sources for the differences between ΔP·ρ of our measurements and those of Scaife and Lyons1 in the overlapping temperature range. The first is the variation of C(T, p = 0), the second, the value of κT used in ref 1 that is required to account for the decrease in measured capacitance arising from increased hydrostatic pressure that would, if incorrect, lead to a pressure-dependent systematic error in εr(T, p). The third arises from variation of the purity of the octane used in ref 1. In our work,12 we have measured κT and effectively eliminated this source of error, and thus, it is to a discussion of the work of Scaife and Lyons1 that we now turn. The coaxial cylinder capacitor developed by Scaife and Lyons24 was used for the measurements reported in ref 1 and was similar in construction to that used in this work. The uncertainty of the measurement of pressure and temperature were cited as ± 0.1 MPa and ± 0.15 K, respectively. Scaife and Lyons reported the relative variation of C(T, p = 0) of 10−4 after an isotherm and of 10−3 after reassembly which are equivalent to the observed differences between isotherms. In ref 24, reliance was placed on the calculation of the variation of capacitance with pressure utilizing a value of κT = 2.6·10−6 MPa−1 that was estimated from the literature values of κT for the materials used and assumed independent of temperature; in our measurement of κT we did not observe a significant variation with temperature.12 Based solely on these comparisons we conclude the increase of ΔP·ρ at a temperature shown in Figure 3 might arise from the method used by Scaife and Lyons24 to determine κT. The observed differences in ΔP·ρ at a fixed temperatures can be eliminated with a κT an order of magnitude greater than used in ref 1, which is implausible. Thus the observed variations of ΔP·ρ with temperature cannot be attributed to κT. The same result can be obtained with an error in the pressure measurement of −0.04·(p/MPa) which also appears implausible. Neither experiments nor detailed calculations have been performed to confirm this speculative conjecture. The octane used by Scaife and Lyons1,22 has a cited mole fraction purity of 0.9981 and was used as provided by the manufacturer without recourse to further purification, including either degassing or drying, and analysis of the chemical composition. The presence of a significant chemical impurity in either our measurements or those reported by Scaife and Lyons1,22 is plausible. The temperature dependence of the results would be significantly greater than observed if a significant mole fraction of water, which is the most significant impurity, were present, and it is indeed plausible that water inexcess of the solubility limit existed as droplets and increased the measured capacitance. However, this must, in the absence of independent experiments, remain purely conjecture.

Figure 3. Deviations ΔP·ρ = P(expt)·ρ − P(calc)·ρ of the total polarizability P(expt) determined by combination of the measured relative permittivity εr(expt) of octane listed in Table 2 with the amount-of-substance density ρ of ref 21 (that was implemented within ref 19) as deviations from the calculated values P(calc) obtained from eq 2 at pressure p. ○, this work, T = 303.07 K; □, this work, T = 333.25 K; △, this work, T = 373.16 K; ◇, this work, T = 398.17 K; +, T = 248 K, ref 1; ×, T = 258 K, ref 1; ●, T = 273 K, ref 1; ■, T = 278 K, ref 1; ◆, T = 298 K, ref 1; ▲, T = 313 K, ref 1; gray filled circle with black outline, T = 318 K, ref 1; gray filled square with black outline, T = 333.034 K, ref 1; black square with gray outline, T = 333.134 K, ref 1; gray filled triangle with black outline, T = 338 K, ref 1; gray filled diamond with black outline, T = 353 K, ref 1; gray filled circle with black outline, T = 358 K, ref 1; ∗, T = 363 K, ref 1; −, T = 373 K, ref 1; ·, is the expanded uncertainty of eq 2; - - - -, at an ordinate of 0 indicates an extrapolation of eq 2; and , at an ordinate of 0 the temperature range of the fit to eq 2.

(0.1 and 400) MPa show, at pressures of (0.1, 50, and 100) MPa, relative deviations from the P·ρ calculated with eq 2 and ref 21 of ≈ 2400·10−4 and are, consequently, not shown on the ordinate axis scale of Figure 3. Scaife and co-workers1,22 have reported two sets of measurements of εr(T, p). Those of Scaife and Lyons1 were at temperatures between (248 and 373) K and pressures below 196 MPa, while in ref 22 the εr(T, p) were determined at temperatures between (273 and 373) K and pressures below 294 MPa; in the latter case, however, solely a correlation of measurements of εr(T, p) was reported and not the measured values. The measurements of εr(T, p) reported by Scaife and Lyons1 were converted to P·ρ and, as Figure 3 shows, lie between −25·10−4 at T = 373 K and 15·10−4 at T = 248 K, vary systematically from the highest to lowest temperature, from a combination of eq 2 and ref 21, which is a span twice that observed for heptane;13 at each temperature the differences ΔP·ρ increase with increasing density typically by