Application of the amended Coriolis flowmeter "bubble theory" to sound propagation and attenuation in aerosols and hydrosols
aa r X i v : . [ phy s i c s . g e n - ph ] D ec Propagation and attenuation of sound in isothermalsuspensions: An extension of the viscous andincompressible theory
Nils T. Basse Elsas v¨ag 23, 423 38 Torslanda, SwedenDecember 17, 2019
Abstract
The existing viscous and incompressible theory of isothermal sound prop-agation and attenuation in suspensions considers solid particles which areinfinitely viscous. We extend the theory by applying the Coriolis flowme-ter ”bubble theory”. Here, the drag force is a function of both the Stokesnumber and the particle-to-fluid ratio of the dynamic viscosity [S.-M.Yangand L.G.Leal, A note on the memory-integral contributions to the force on anaccelerating spherical drop at low Reynolds number, Phys. Fluids A , 1822-1824 (1991)]. Aerosol and hydrosol examples are presented and differencesbetween the original and extended theories are discussed.
1. Introduction
When sound propagates through a suspension, the sound speed is mod-ified and the sound is attenuated; this can have important practical impli-cations, e.g. for jet engines and rocket motors [1]. In this paper, we definea suspension to be any combination of particles entrained in a fluid. Theparticle can be either a fluid or a solid. Specifically, for aerosols (hydrosols),we define the fluid to be air (water), respectively.The linear theory of isothermal sound propagation and attenuation ina suspension has been presented in [2, 3] for solid, i.e. infinitely viscous,particles. We will name this theory the ”solid particle” (SP) theory.Another linear theory of suspensions considers the reaction force on anoscillating fluid-filled container due to entrained particles [4]. This theory [email protected]
2. Particle-to-fluid velocity ratio
The particle-to-fluid velocity ratio is given as: V = u p u f , (1)where u p is the particle velocity and u f is the fluid velocity. For both theories, there are a number of common assumptions: • Both the particles and fluid are considered to be incompressible. • The particles are rigid, i.e. they are spherical and do not deform. • Only translational effects are treated, thermal and pulsational effects[3] are disregarded. • The theories are valid for low Reynolds number. • The drag force for both theories is a function of the Stokes number.2imultaneously, the main conceptual and physical differences are: • The angular frequency ω has a different physical meaning for the twotheories: For the SP theory, it is the acoustic wave frequency and forthe VP theory, it is the frequency of the container oscillation. However,mathematically they are completely equivalent. • The drag force constitutes the physical difference between the theories;for the VP theory, the drag force depends on the dynamic viscosity ofthe particle, which is not the case for the SP theory.
The particle-to-fluid velocity ratio is: V SP = 3 δ β (2 β + 3) + 3i(1 + β )2 β (2 + δ ) + 9 βδ + 9i δ (1 + β ) , (2)where β is the Stokes number and δ = ρ f /ρ p . Here, ρ f is the fluid densityand ρ p is the particle density. The Stokes number is: β = a s ωρ f µ f , (3)where a is the particle radius and µ f is the dynamic viscosity of the fluid.From Eq. (2), the amplitude of the velocity ratio is: | V SP | = 3 δ vuut β + 12 β + 18 β + 18 β + 94(2 + δ ) β + 36 δβ (2 + δ ) + 81 δ (2 β + 2 β + 1) , (4)and the phase angle of velocity ratio is:tan η SP = − β ) β (1 − δ )4 β (2 + δ ) + 12 β (1 + 2 δ ) + 27 δ (2 β + 2 β + 1) , (5)where we have changed sign to follow the bubble theory convention [6] that apositive (negative) η means that the particles are leading (lagging) the fluid,respectively. The particle-to-fluid velocity ratio is: V VP = 1 + 4(1 − τ )4 τ − (9i G/β ) , (6)3here τ = 1 /δ = ρ p ρ f (7)The quantities below are defined in [9]: G = 1 + λ + λ − (1 + λ ) f ( λ ) κ [ λ − λ tanh λ − f ( λ )] + ( λ + 3) f ( λ ) , (8)where λ = (1 + i) β (9)and f ( λ ) = λ tanh λ − λ + 3 tanh λ (10)The viscosity ratio is: κ = µ p µ f , (11)where µ p is the dynamic viscosity of the particle.As for the SP theory, we write the amplitude of the velocity ratio: | V VP | = q Re ( V VP ) + Im ( V VP ) , (12)and the phase angle of velocity ratio:tan η VP = Im ( V VP ) Re ( V VP ) (13) Density and viscosity ratios for the aerosols are collected in Table 1. Cor-responding amplitudes and phases of the velocity ratio are shown in Figure1. For small Stokes numbers, the amplitude ratio is close to one, meaningthat the particles are moving at the same velocity as the fluid. As the Stokesnumber increases, the particle velocity decreases with respect to the fluidvelocity.The phase of the velocity ratio becomes negative which means that theparticles are lagging the fluid. This is mainly because the density of theparticles is much higher than the density of air.Results from the SP and VP theories are almost identical.4 able 1: Density and viscosity ratios for the aerosols treated. τ κ
Water-air mixture 831.7 50Oil-air mixture 723.3 2500Sand-air mixture 1833.3 5e16 ( ∞ ) -2 | V | Water-air mixture (SP)Water-air mixture (VP)Oil-air mixture (SP)Oil-air mixture (VP)Sand-air mixture (SP)Sand-air mixture (VP) -2 -80-60-40-200 Water-air mixture (SP)Water-air mixture (VP)Oil-air mixture (SP)Oil-air mixture (VP)Sand-air mixture (SP)Sand-air mixture (VP)
Figure 1: Velocity ratio for aerosols: Left: Amplitude, right: Phase. The SP theory ismarked by solid lines and the VP theory is marked by dashed lines.
Density and viscosity ratios for the hydrosols selected are collected inTable 2. Corresponding amplitudes and phases of the velocity ratio are shownin Figure 2.
Table 2: Density and viscosity ratios for the hydrosols treated. τ κ
Air-water mixture 1.2e-3 2e-2Oil-water mixture 0.87 50Sand-water mixture 2.2 1e15 ( ∞ )As for aerosols, the amplitude of the velocity ratio is approximately onefor small Stokes numbers; however, it can become both larger and smallerthan one for large Stokes numbers. This is mainly a density effect, but aswe see for the air-water mixture, there is also an additional effect due to theparticle viscosity which is not captured by the SP theory.The two theories also differ for the phase of the velocity ratio of the air-water mixture, both in the position and maximum value of the peak; the5 -2 | V | Air-water mixture (SP)Air-water mixture (VP)Oil-water mixture (SP)Oil-water mixture (VP)Sand-water mixture (SP)Sand-water mixture (VP) -2 -100102030 Air-water mixture (SP)Air-water mixture (VP)Oil-water mixture (SP)Oil-water mixture (VP)Sand-water mixture (SP)Sand-water mixture (VP)
Figure 2: Velocity ratio for hydrosols: Left: Amplitude, right: Phase. The SP theory ismarked by solid lines and the VP theory is marked by dashed lines. phase is positive, meaning that the particles are leading the fluid.For the oil-water and sand-water mixtures, the results from the SP andVP theories are almost identical.What is different for the air-water case is the small value of κ combinedwith a small value of τ , see Table 2; for this case we have derived the followingapproximation in [8]: V VP , air − water mixture ≈ β β + 9 (cid:16) β β +6 β +9 + κβ (cid:17) − i9 (cid:16) β +8 β +62 β +6 β +9 + κβ (cid:17) (14)
3. Propagation and attenuation of sound
We present the dispersion relation following analysis as presented in Sec-tion 9.4 of [3], but only keeping the force source and disregarding the volumeand heat sources: k k = 1 + φ v ( τ V − , (15)where k is the complex wavenumber, k is the equilibrium wavenumber and φ v is the volumetric particle fraction. Separating this into real and imaginaryparts: 6 ( ω ) = Re k k ! = 1 + φ v ( τ Re ( V ) −
1) (16) Y ( ω ) = Im k k ! = φ v τ Im ( V ) (17)To obtain phase velocity and attenuation, we note that the wavenumberratio can also be expressed as: k k = c sf c s ( ω ) − ˆ α + 2i ˆ α c sf c s ( ω ) , (18)where c sf is the isentropic fluid sound speed, c s ( ω ) is the nonequilibriumisentropic sound speed, ˆ α = αc sf /ω is the nondimensional attenuation basedon c sf and α is the attenuation coefficient.Phase velocity and attenuation can now be specified as: c sf c s ( ω ) = 12 X (cid:20) q Y /X ) (cid:21) (19)ˆ α = q X/ (cid:20)q Y /X ) − (cid:21) / (20)We note that there is a typo in the equation for c sf /c s ( ω ) in [3] (Equation(9.4.19a)). For low damping, i.e. ˆ α ≪
1, we can write: c sf c s ( ω ) ≈ X (21)and: ˆ α ≈ | Y | = 12 φ v τ | Im ( V ) | (22)For this case we can write the scaled attenuation as:ˆ α/φ v ≈ τ | Im ( V ) | (23)7 -2 -1 c s () / c s f Water-air mixture, v = 0.01 SP (exact)SP (simplified)VP (exact)VP (simplified) -2 -6 -4 -2 Water-air mixture, v = 0.01 SP (exact)SP (simplified)VP (exact)VP (simplified)
Figure 3: Water-air mixture, left: Normalized sound speed, right: Nondimensional atten-uation. Exact expressions are marked by solid lines and simplified expressions are markedby dashed lines.
For all examples in this section, we plot both the exact and simplifiedexpressions for the normalized sound speed and the nondimensional attenu-ation.Also, we note that the volumetric particle fraction φ v is set to 0.01 (1%)for the examples. For the aerosols presented in Figures 3-5, the sound speed reduces signif-icantly for small Stokes numbers, with ratios in the range of 0.2-0.4.The peak nondimensional attenuation is of order one, meaning that thesimplified expression is not accurate for small Stokes numbers; here, the exactexpression should be employed.Generally, the SP and VP theories agree well for all three aerosol exam-ples.
Only very small sound speed changes are occurring for the hydrosols asshown in Figures 6-8.The peak nondimensional attenuation is several orders of magnitude be-low one, meaning that the simplified expression can be used instead of theexact expression.We observe that the nondimensional damping for the air-water mixtureis different for the SP and VP theories; we will return to this finding in theDiscussion.For the oil-water and sand-water mixtures, the results from the SP andVP theories are almost identical. 8 -2 -1 c s () / c s f Oil-air mixture, v = 0.01 SP (exact)SP (simplified)VP (exact)VP (simplified) -2 -6 -4 -2 Oil-air mixture, v = 0.01 SP (exact)SP (simplified)VP (exact)VP (simplified)
Figure 4: Oil-air mixture, left: Normalized sound speed, right: Nondimensional attenua-tion. Exact expressions are marked by solid lines and simplified expressions are markedby dashed lines. -2 -1 c s () / c s f Sand-air mixture, v = 0.01 SP (exact)SP (simplified)VP (exact)VP (simplified) -2 -6 -4 -2 Sand-air mixture, v = 0.01 SP (exact)SP (simplified)VP (exact)VP (simplified)
Figure 5: Sand-air mixture, left: Normalized sound speed, right: Nondimensional attenu-ation. Exact expressions are marked by solid lines and simplified expressions are markedby dashed lines. -2 c s () / c s f Air-water mixture, v = 0.01 SP (exact)SP (simplified)VP (exact)VP (simplified) -2 -10 -5 Air-water mixture, v = 0.01 SP (exact)SP (simplified)VP (exact)VP (simplified)
Figure 6: Air-water mixture, left: Normalized sound speed, right: Nondimensional atten-uation. Exact expressions are marked by solid lines and simplified expressions are markedby dashed lines. -2 c s () / c s f Oil-water mixture, v = 0.01 SP (exact)SP (simplified)VP (exact)VP (simplified) -2 -10 -5 Oil-water mixture, v = 0.01 SP (exact)SP (simplified)VP (exact)VP (simplified)
Figure 7: Oil-water mixture, left: Normalized sound speed, right: Nondimensional atten-uation. Exact expressions are marked by solid lines and simplified expressions are markedby dashed lines. -2 c s () / c s f Sand-water mixture, v = 0.01 SP (exact)SP (simplified)VP (exact)VP (simplified) -2 -10 -5 Sand-water mixture, v = 0.01 SP (exact)SP (simplified)VP (exact)VP (simplified)
Figure 8: Sand-water mixture, left: Normalized sound speed, right: Nondimensional at-tenuation. Exact expressions are marked by solid lines and simplified expressions aremarked by dashed lines. . Discussion We have seen one case where the VP theory is quite different from theSP theory: The air-water mixture, i.e. air bubbles entrained in water. Thisis an important case because the mixture (or similar ones) is quite common.In terms of measurements we refer to Figures 1.1.2 and 1.1.3 in [3]; the SPcompressible [10] theory agrees well with the measurements for the speed ofsound, but has deviations for the attenuation: The VP theory may help toexplain this discrepancy.
The incompressible theories are valid for Stokes numbers below values ofaround 10-100; for larger Stokes numbers, compressibility effects have to betaken into account.Compressible theories exist for both acoustics [10] and Coriolis flowmeters[11]: • Acoustics: Compressible viscous fluid, solid particles • Coriolis flowmeter: Compressible inviscid fluid, no particlesSince only the VP theory includes the particle viscosity, a next step wouldbe to combine that theory with the compressible acoustics theory to describe:Compressible viscous fluid with viscous particles. A further step beyond thatwould be to consider effects due to compressible particles.
5. Conclusions
We have extended the linear theory of isothermal sound propagation andattenuation in suspensions by applying the Coriolis flowmeter ”bubble the-ory”: Here, the drag force is a function of both the Stokes number and theparticle-to-fluid ratio of the dynamic viscosity.Aerosol and hydrosol examples are presented.Our main result is that we have established a significant difference indamping between the theories for the air-water mixture, i.e. air bubblesentrained in water.
Acknowledgements
The author is grateful to Dr. John Hemp for creating, providing andexplaining/discussing the Coriolis flowmeter bubble theory [4].11 eferences [1] M.S.Howe, Acoustics of Fluid-Structure Interactions, Cambridge Uni-versity Press, 1998.[2] S.Temkin, Viscous attenuation of sound in dilute suspensions of rigidparticles, J. Acoust. Soc. Am. , 825-831 (1996).[3] S.Temkin, Suspension Acoustics, An Introduction to the Physics ofSuspensions, Cambridge University Press, 2005.[4] J.Hemp, Reaction force of a bubble (or droplet) in a liquid undergoingsimple harmonic motion, Unpublished, 1-13 (2003).[5] N.T.Basse, A review of the theory of Coriolis flowmeter measurementerrors due to entrained particles, Flow. Meas. Instrum. , 107-118(2014).[6] N.T.Basse, Coriolis flowmeter damping for two-phase flow due to de-coupling, Flow. Meas. Instrum. , 40-52 (2016).[7] M.S.Howe, On the theory of unsteady high Reynolds number flowthrough a circular aperture, Proc. R. Soc. Lond. A. , 205-223(1979).[8] N.T.Basse, On the analogy between the bias flow aperture theoryand the Coriolis flowmeter ”bubble theory”, Flow. Meas. Instrum. ,101663 (2020).[9] S.-M.Yang and L.G.Leal, A note on the memory-integral contributionsto the force on an accelerating spherical drop at low Reynolds number,Phys. Fluids A , 1822-1824 (1991).[10] S.Temkin and C.-M.Leung, On the velocity of a rigid sphere in a soundwave, J. Sound Vibration , 75-92 (1976).[11] J.Hemp and J.Kutin, Theory of errors in Coriolis flowmeter readingsdue to compressibility of the fluid being metered, Flow Meas. Instrum.17