Acoustic Techniques for Measuring Transport Properties of Gases

KEITH A. GILLIS AND MICHAEL R. MOLDOVER

1.1 Introduction: Acoustic Measurements of Gas Properties

The acoustic resonance frequencies f and the resonance half-widths g of a gas-filled cavity are functions of the cavity's size and shape, the speed of sound in the gas c, and the thermophysical properties of the gas. In a first approximation, the resonance frequencies f depend on the speed of sound; in contrast, the resonance half-widths g are sums of terms that account for energy dissipated by the gas's thermal conductivity λ, shear viscosity η, bulk (or second) viscosity ζ, and the term gmech, where gmech accounts for energy losses from mechanical effects such as friction in joints, transducer losses, and acoustic radiation outside the cavity. Cavity resonators used for measuring the speed of sound are designed to have narrow resonance peaks so that the resonance frequencies can be determined precisely and so that the frequencies are comparatively insensitive to the transport properties λ, η, and ζ. In contrast, cavity resonators used to measure these transport properties are designed so that the energy dissipated by λ, η, and ζ is much larger than gmech. Consequently, resonators that are optimized to measure transport properties have broad resonance peaks. With reasonable precautions, the measurements of the resonance frequencies and half-widths have very low uncertainties; the uncertainty of acoustic determinations of transport properties is dominated by imperfect modeling/understanding of cavity resonators and gmech.

The sections that follow describe the design and performance of three acoustic resonators that we developed for measurements in gases: one resonator was optimized for measurements of shear viscosity, one was optimized for thermal conductivity measurements, and one was optimized for bulk viscosity measurements. The three resonators are small so that they require only small gas samples whose temperature and pressure are easily controlled over wide ranges, and they are rugged with no moving parts. Using the first resonator, we accurately measured the shear viscosity of hazardous gases at temperatures between 200 K and 400 K and pressures up to 3.4 MPa. With the third resonator, we accurately measured the critical-fluctuation-driven bulk viscosity of xenon on the critical isochore at reduced temperatures 100 times closer to the critical point (290 K and 5.8 MPa) and at frequencies 3000 times lower than ever before.

Of the three transport properties, the shear viscosity is easiest to measure at low densities. Accurate acoustic measurements of the thermal conductivity are more difficult because the dissipation from thermal conduction is usually smaller than that from viscosity by a factor on the order of (γ - 1), and (γ - 1) [much less than] 1 for polyatomic molecules of interest. (Here γ [equivalent to] Cp/Cv is the heat-capacity ratio.) (In Section 1.3, we discuss an exception to this generalization. In a spherical cavity, the acoustic velocity of the radially-symmetric modes is perpendicular to the cavity's walls; therefore these modes are not damped by the shear viscosity.) In low-density gases, the bulk viscosity ζ [varies] ρ2, which vanishes as ρ [right arrow]0; therefore, it is too small to measure. Near the critical point, ζ diverges more strongly than the other transport coefficients; therefore, it dominates acoustic losses and is easy to measure accurately.

1.2 Shear Viscosity Measurements: The Greenspan Viscometer

1.2.1 Description

In 1953, Greenspan and Wimenitz attempted to determine the viscosity of air from measurements of f and g in a two-chambered Helmholtz resonator somewhat like the resonator shown in Figure 1.1. Their results deviated relatively from literature data by as much as 38 %. In 1996, we began a program to develop an accurate viscometer based on Greenspan's concept, which we call the Greenspan acoustic viscometer. Since then, we have improved the theory and the resonator design to achieve significantly better results; our measurements with the Greenspan viscometer deviate relatively from reference data by less than [+ or -] 0.5 %.

The Greenspan acoustic viscometer is a double Helmholtz resonator composed of two gas-filled chambers connected by a tube (or duct) (see Figure 1.1). The fundamental acoustic mode is a low-frequency, low-Q mode in which the gas oscillates between the two chambers through the duct. As a characteristic of a Helmholtz acoustic mode, the wavelength of the mode is much longer than the internal dimensions of the resonator. The frequency response of the Helmholtz mode is easy to measure because the mode is non-degenerate and isolated; its frequency is far below the other acoustic modes of the enclosed gas and below the elastic modes of the resonator body. The low frequency leads to a thick boundary layer that reduces the requirement for a fine surface finish compared to the moderate-frequency resonators used for sound speed measurements. The low Q reduces the relative importance of the difficult-to-estimate contributions to the measured half-width gmeas and also reduces the need to maintain very high temperature stability. The only requirement is that the transducers have a smooth, well-behaved response (no peaks) in the range fmeas [+ or -] 4gmeas that can be described by a low-order polynomial. The Greenspan viscometer is rugged; it has no moving parts (aside from the minute motion of the transducers) and can be made from corrosion-resistant alloys. In our work, the acoustic transducers, and the "dirty" materials of which they are made, are separated from the test gas by thin metal diaphragms that are machined into the chamber's walls.

In the lowest-order approximation, neglecting dissipation, the gas in the duct and just outside the duct ends moves like a rigid plug with mass ρAd (Ld + 2δi) that oscillates back and forth between two identical springs. Here, ρ is the gas density, Ad = π rd2 is the cross-sectional area of the duct, rd and Ld are the duct radius and length, and the length δi ≈ 0.655 rd is an inertial end correction that accounts for diverging flow at the end of the duct. As gas flows into a chamber, the pressure in that chamber increases and provides a restoring force, like a spring. The combined stiffness 2ρc2Ad2/Vc of the two springs is a result of the compression and rarefaction of the gas in the chambers. In this level of approximation, the resonance frequency is

f0 = 1/2π [square root of stiffness/mass] = c [square root of (r2d/2πVc (Ld + 2δi))], (1.1)

where c is the speed of sound in the gas, and Vc is the volume of each chamber. The wavelength of sound (c/f0) determined by geometric factors and is independent of the properties of the gas. For the resonator described in ref. 3, eqn (1.1) predicts f0 = (0.9299 m-1) x c, which is a few hundred hertz for most gases near ambient temperature. However, the corresponding wavelength (approximately 1.075 m) is the same for all gases. Heat and momentum diffusion near the cavity walls decrease the measured resonance frequency fmeas from the estimate in eqn (1.1) by a few percent, but eqn (1.1) is sufficient as a design aid.

For the Helmholtz mode, the acoustic velocity is greatest in the duct; therefore, shear in the gas flow near the duct's wall dissipates most of the acoustic energy. If this were the only dissipation mechanism, the gas's shear viscosity η could be determined from a measurement of the resonance frequency fmeas and the quality factor Qmeasusing the expression

η ≈ ρπr2dfmeas Q2meas (1 + 2δi/Ld/1 + 2 εr0rd/Ld)2. (1.2)

The numerically-calculated parameters δi and εr0 describe, respectively, the inertial and dissipative effects of the duct ends. However, heat transport between the oscillating gas and the metal wall of the resonator causes significant acoustic energy dissipation; therefore eqn (1.2) overestimates η. The fractional overestimate is 0.44 (γ - 1) Pr-1/2 for the viscometer described in ref. 3, where Pr = η Cp/λ] is the Prandtl number, and λ is the thermal conductivity of the gas. This overestimate ranges from 36 %, for monatomic gases such as argon and helium, to as little as 5% for typical polyatomic gases. Fortunately, the design of the Greenspan viscometer is such that, in most cases, the uncertainty in the thermal conductivity of the gas is a small contribution to the uncertainty of the deduced viscosity. For example, the thermal conductivities of helium and argon have such low relative uncertainties, [+ or -] 0.002% and [+ or -] 0.02% respectively, that their contribution to the uncertainty of the viscosity measurement is negligible, despite the rather large effect of heat conduction. For a typical polyatomic molecule with γ ≈ 1.1 and Pr ≈ 0.7, a relative uncertainty in the thermal conductivity of [+ or -] 10% contributes only about [+ or -] 0.2 % to the relative un-certainty of the deduced viscosity.

The next level of approximation includes the most important dissipation mechanisms that contribute to the half-width gmeas and the resonance line shape. In this approximation, the inverse of the resonance quality factor Q-1 [equivalent to] 2gmeas /fmeas is the sum of three terms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

The first term in eqn (1.3) is the damping that occurs in the viscous boundary layer at the wall and near the ends of the main duct. Here, δv = [Dv/(π fmeas)]1/2 is the thickness of the viscous boundary layer in oscillating flow at frequency fmeas, where Dv = η/ρ is the viscous diffusivity (also called the kinematic viscosity). The second term is the damping from heat conduction that occurs near the wall of the chambers, which have surface area Sc. The length δt = [Dt/(π fmeas)]1/2 is the thickness of the thermal boundary layer, where Dt = λ/(ρCp)‡ is the thermal diffusivity. Although δv and δv differ only by the factor Pr-1/2, the second term is significantly reduced by the smaller surface area-to-volume ratio Sc/Vc in the chamber and the factor (γ - 1), which is small for polyatomic molecules. The third term in eqn (1.3) is important for certain gases (e.g. CH4, CO2) that have symmetries such that many intermolecular collisions are required for their internal degrees of freedom to adjust to the temperature change associated with the acoustic oscillation. In such gases, the acoustic dissipation is characterized by the product Crelaxτrelax, where Crelax is the heat capacity associated with the slowly relaxing degrees of freedom and τrelax is the relaxation time, which is proportional to ρ-1.

The next section describes the model for acoustic response of the Greenspan viscometer and how its measurement is used to determine the gas viscosity. Ref. 3 contains more details and a derivation of the model.

1.2.2 Basic Theory

In a Greenspan viscometer, a continuous sound source located in one chamber generates an acoustic wave at frequency f that is reflected back and forth between the chambers through the duct. We assume the time dependence is eiωt with ω = 2π f. When the frequency is such that a reflected wave arrives back at the source in phase with the wave being generated there, resonance occurs. Acoustic waves in long gas-filled ducts are governed by the equations first proposed by Kirchhoff, whose classic paper includes a description of the effects of the coupled acoustic, thermal, and vorticity waves in ducts of circular cross section. The low-frequency (long-wavelength) limit of Kirchhoff's solutions is generally attributed to Crandall. Below the cutoff frequency for transverse modes in a duct, only plane waves can propagate. In a duct with a circular cross section, this limit corresponds to wavelengths greater than about 1.7 times the duct diameter, i.e. about 8 mm for the duct in Figure 1.1, which is much shorter than the wavelength of the natural mode of the Greenspan viscometer.

Low-frequency sound wave propagation in a duct is described accurately by a set of differential equations for lossy transmission lines that relate the acoustic pressure p and volume velocity U. When there is an acoustic source with frequency f and volume velocity U0 in one chamber (chamber 1), a steady-state acoustic pressure p2 develops in the other chamber (chamber 2). Finite-length transmission lines are conveniently described by an equivalent circuit containing lumped acoustic impedances in a T-network. The complete equivalent circuit we use to model the Greenspan viscometer, shown in Figure 1.2, contains a T-network for each half of the central duct (split by the symmetrically-located fill capillary) with additional impedances that model the effects of the duct ends (Zend), the chambers (ZV), and the fill capillary (Zc). The acoustic response of the Greenspan viscometer p2/U0 is a complex-valued function of frequency that contains the line shape of the Helmholtz resonance.

The acoustic response based on the equivalent circuit is

p2/U0 = Z20ZcZ2V, D1D2 (1.4a)

where the factors in the denominator are

D1 = (Zb + Za)(ZV + Z'a) + ZbZa (1.4b)

D2 = ZbZV + 2Zc(Zb + ZV) + Z'a (Zb + 2Zc) + Za(Zb + ZV + Z'a. (1.4c)

Eqn (1.4a) is the basis of the complex resonance function used in the data analysis. The Helmholtz resonance condition is defined as the complex frequency FH = fH + ig for which D1 = 0, where fH is the measured resonance frequency and g is the resonance half width. [2g is defined as the full-width at half-maximum of the energy response function. The quality factor Q is defined as Q = fH/(2g).] The acoustic pressure for the Helmholtz mode is an odd function of distance from the resonator's mid-plane, so there is a pressure node at the entrance to the capillary; therefore the resonance frequency and the line shape for the Helmholtz mode are independent of Zc. The effect that Zc has on the total acoustic response in eqn (1.4) in the vicinity of the Helmholtz mode is to change the "background" due to the tails of other modes, which is weakly dependent on frequency.

Within the chambers, the gas compressibility in the long wavelength limit considered here is isothermal near the walls and adiabatic far from the walls. The acoustic impedance of each chamber calculated from the average compressibility within the chamber is given by the expression

ZV = ρc2/iωVc 1/[square root of (1 + 1/2 (1 - i)(γ - 1)Scδt/Vc (1.5)

which includes the effect of heat conduction over the chamber's surface area Sc.