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J. Chem. Eng. Data 2009, 54, 2758–2771
Thermophysical Properties of Natural Gas Components: Apparatus and Speed of Sound in Argon †
Anthony R. H. Goodwin*,‡ and James A. Hill Center for Applied Thermodynamic Studies, College of Engineering, University of Idaho, Moscow, Idaho 83843
An apparatus consisting of two spherical resonators of nominal internal radius (25 and 45) mm was constructed and used to determine the speed of sound in argon at temperatures between (250 and 350) K at pressures below 7 MPa. The estimated expanded uncertainty in the speed of sound determined was less than (0.03 m · s-1, that is, 1 MPa the shell motion dominates corrections to the resonance frequency that are determined from elastic theory for a thick walled isotropic spherical shell; resonators are neither spherical nor isotropic when machined and fitted with acoustic transducers. The variation with respect to the temperature of the sound speed in a gas is greater than of the elastic constants of the sphere wall, and not surprisingly, an acoustic mode may be perturbed by shell motion at one temperature but not another. Thus, observations of the difference between (u/a) obtained from each resonance frequency from the mean 〈u/a〉 combined with the excess fractional resonance line widths at a temperature and pressure formed the basis for omitting modes from the regression analysis with eq 3; the procedure adopted is similar to that described elsewhere and elucidated here.39,40,45,71-73 The systematic departure of (u/a) obtained with a mode from the 〈u/a〉 was interpreted in terms of coupling between gas and shell modes. For all isotherms the values of (u/a) obtained from the selected modes span 10-5 · u at a temperature and pressure over the full pressure range, and this internal consistency is comparable with that obtained in other resonators.39,40,44,71-73 The fractional excess line widths, defined by ∆g/f ) {g(expt) g(calc)}/f, where g(expt) is the measured line width and g(calc) values determined from the acoustic model including bulk dissipation are another means of determining otherwise unaccounted energy losses and were also considered. The following three paragraphs are a digression from the discussion regarding the resonators radii and second acoustic virial coefficients determined from the sound speed measurements to consider in further detail the modes omitted. For the 45 mm radius resonator the difference between the (u/a) obtained from mode (0, 5) and the mean 〈u/a〉 increased
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Journal of Chemical & Engineering Data, Vol. 54, No. 9, 2009
Table 1. Resonator Radius a(T, p ) 0), the Second Acoustic Virial Coefficient βa, and the Relative Standard Deviation of the Fit s(u/a)2/{u(p ) 0)/a)}2 Where s(u/a)2 Is the Standard Deviation at Temperature Ta T
a(T, p ) 0)
βa
K
mm
cm3 · mol-1
106s/u
320.000 310.000 300.000 290.000 280.000 270.000 260.000 250.000
October 1995 44.4442 ( 0.0028 15.79 ( 0.18 44.4382 ( 0.0013 13.91 ( 0.16 44.4306 ( 0.0013 12.12 ( 0.16 44.42407 ( 0.00072 9.533 ( 0.084 44.4181 ( 0.0018 7.51 ( 0.22 44.41095 ( 0.00060 4.656 ( 0.094 44.4041 ( 0.0016 1.72 ( 0.19 44.39817 ( 0.00072 -1.27 ( 0.11
94 30 39 20 37 15 33 18
349.868 325.046 300.117 275.103 250.161
June 1996 44.46975 ( 0.00064 21.028 ( 0.092 44.45085 ( 0.00082 17.34 ( 0.16 44.4331 ( 0.0015 12.35 ( 0.28 44.41576 ( 0.00051 6.38 ( 0.15 44.3979 ( 0.0011 -1.12 ( 0.18
14 14 26 18 19
320.000 300.111 280.121
May 1997 44.4477 ( 0.0030 44.4327 ( 0.0024 44.41807 ( 0.00080
16.06 ( 0.56 12.12 ( 0.58 7.80 ( 0.18
41 31 9
349.868 325.046 300.117 275.103 250.161
June 1996 25.35386 ( 0.00054 21.92 ( 0.21 25.34225 ( 0.00058 17.19 ( 0.12 25.33223 ( 0.00042 12.65 ( 0.25 25.32191 ( 0.00064 6.51 ( 0.18 25.31168 ( 0.00056 -0.84 ( 0.15
16 16 17 15 18
320.000 300.111 280.112
May 1997 25.3392 ( 0.0011 25.33100 ( 0.00062 25.3221 ( 0.0012
22 14 9
16.45 ( 0.32 12.92 ( 0.26 7.28 ( 0.18
The expanded (k ) 2) uncertainties are listed for both a(T, p ) 0) and βa. a
with increasing pressure consistent with coupling of the radial gas modes with a predicted nonradial shell resonances.38,39 The excess line width also increased with increasing pressure and confirmed the correct selection of mode (0, 5) for exclusion from the analysis. At T ) 300 K in the 45 mm sphere ∆g/f for mode (0, 5) was 350 · 10-6 at p ) 4 MPa. For the 45 mm resonator the (u/a) obtained from mode (0, 2) was lower than the 〈u/a〉, and the excess half widths were at all pressures about 200 · 10-6, that is, a factor of 5 greater than for any other mode studied. At T ) 300 K ∆g/f for the selected modes increased from about 3 · 10-5 to 6 · 10-5, and these variations with density were anticipated on the basis of results reported elsewhere.39,40,44,45,71-73 Although no measurements were conducted to determine the source of these discrepancies, it is tempting to speculate that the excess losses and dispersion observed for (0, 2) arose from the slot around the equator terminated by a volume partially occupied by an O-ring that acted as a Helmholtz resonator, and based solely on this argument mode (0, 2) was eliminated from the regression. Nevertheless, (u/a) that span
