Acoustic attenuation, phase and group velocities in liquid-filled pipes: Theory, experiment, and examples of water and mercury

Abstract

Del Grosso's [Acustica 24, 299-311 (1971)] formulation, which predicts the phase speed of propagating axisymmetric modes inside a liquid-filled tube, is here extended to the complex domain in order to predict the attenuation, as well as the sound speed, of the modes as a function of frequency. Measurements of the sound speeds and the attenuations of the modes were performed in a water-filled Poly (methyl methacrylate) (PMMA) tube of internal radius, b=4.445 cm, in the range of the wavenumber-radius product, k(1)b, from 2 to 10. Parts of three or four modes were investigated and the measured sound speeds and the damping of the modes were compared with the theoretical predictions. The theory was then used to estimate the modal sound speeds and attenuations in a stainless-steel pipe filled with mercury having the same dimensions as are used in the Spallation Neutron Source at Oak Ridge National Laboratory, Tennessee.

The use of acoustic inversion to estimate the bubble size distribution in pipelines

“…It is assumed that the bubbly liquid axisymmetrically exerts on the tube wall an average pressure which is established by equation (2.1). Therefore, when a bubbly liquid with a known BSD fills an elastic pipe, the axisymmetric modes within the pipe are calculated by substituting the expression of k 1 from equation (2.1) into eqn (6) (with the matrix elements of appendix B) of Baik et al (2010). This is done in figure 1.…”

“…These changes are attributed wholly to the presence of bubbles. When modal propagation occurs in pipes where the liquid is coupled to the walls, even in bubble-free conditions each mode propagates with its own frequency-dependent longitudinal phase velocity and attenuation (Baik et al 2010). The differences between these phase speeds and attenuations, and those assumed for infinite volumes of bubble-free liquid, would be attributed by the PWFF inversion as being caused by bubble presence.…”

“…For pipes with boundary conditions resembling those of the mercury-filled steel pipes of ORNL or the water-filled PMMA pipes of §3, Baik et al (2010) showed how the complex axial wavenumber, q 0m of the axisymmetric coupled modes that propagate in the axial direction is obtained by their eqn (6) with the matrix elements of their appendix B (the '0' in the subscript '0m' refers to the mode being axisymmetric and 'm' refers to the mode index). Eqn (6) of that paper gives the infinite number of the axisymmetric modes (denoted as ETm where m is mode index related to the radial motion) and the real and the imaginary parts of the complex solution, q 0m , which give the phase speed and the attenuation of the modes respectively in a pipe containing a bubble-free liquid.…”

“…The result predicts subsonic sound speeds as a function of the gas volume void fraction, but it is not sufficient for the current study because acoustic coupling between the elastic solid and the bubbly liquid will in practice also give an infinite number of supersonic modes. To encompass such modes in this study, the characteristic equation for the axisymmetric modes in a tube filled with bubbly liquid is implemented by combining the description of acoustic modes in a pipe containing bubble-free liquid (Baik et al 2009(Baik et al , 2010 with an analysis which predicts the bubble-induced changes in phase speed and attenuation that occur in an infinite bubbly liquid (Commander & Prosperetti 1989). Wilson et al (2005) applied the formulation of Commander & Prosperetti (1989) in order to measure the phase speed and attenuation that occurred in an impedance tube made of heavy stainless steel, where the phase speed of the lowest axisymmetric mode at low frequencies does not differ significantly from the intrinsic sound speed of the liquid within the tube.…”

“…This inversion is validated against simulated data from pipes and is shown to give similar levels of accuracy to those obtained when the PWFF inversion is applied to data taken in PWFF conditions. Section 3 describes experimental tests on water-filled poly methylmethacrylate (PMMA) pipes, which had previously been shown to mimic the acoustic coupling that would occur in the mercuryfilled steel pipes of ORNL's SNS TTF (Baik et al 2010;Jiang et al 2011). Although these data show that the new inversion technique would provide accurate BSDs if the frequency range of the original sensor design were to be used, budget cuts forced by the 2008 global financial crisis dramatically reduced the frequency range that could be afforded.…”