Cellular Ceramics / p4

4.5 Acoustic Properties 391

r

which was also derived by Helmholtz [11]. This approach forms the basis of many theoretical analyses of porous materials. By far the most common configuration considered is propagation along hollow cylindrical tubes, which is considered variously by Arnott [12], Biot [13,14], Lambert [15–19], Stinson and Champoux [4], Selamet et al. [20], and Hovem and Ingram [21]. The technique is, however, general, and can be used to model the behavior of sound waves propagating along the outside of material elements in a cellular material.

4.5.6.2

Acoustic Damping by an Array of Elements Perpendicular to the Propagation Direction

Real porous materials contain elements which are not only parallel to the direction of propagation, but also elements which are perpendicular to the propagation direction. Since in most cases the magnitude of the acoustic velocity is very small, and the dimensions of the material elements are also generally very small, the Reynolds number associated with these cross-elements is on the order of unity, as pointed out by Attenborough [22]. In this regime the flow is very viscous and was first considered by Stokes [23], who derived a solution in the limit that the inertial terms in the momentum equation are negligible. For a sphere, an exact solution exists for which the drag coefficient CD is related to the the Reynolds number Re by CD = 24/Re, which is valid for steady flow with Reynolds numbers less than about 10. The drag observed due to a combination of spheres is discussed by Datta and Dea [24]. Urick [25] uses the Stokes–Lamb model as the basis for a model of the attenuation of sound by a spherical particle. He shows that attenuation derived by Lamb for the dissipation of sound by fog [26] can be attributed to a combination of viscous loss described by Stokes flow and scattering by the spherical object itself [22]. This model was extended by Umnova and Attenborough [27], who consider the effect of neighboring particles in acoustic propagation through suspensions by including crude outer boundary conditions to describe the interaction between the particles. For flow perpendicular to the axis of a cylinder, Stokes noted that no solution exists which satisfies all the boundary conditions. However, an approximate solution was obtained by Lamb [28] by partially including the inertial terms. In the limit of low Reynolds number this reduces to CD = 15.98/Re. This was extended by Cheung et al.

[28]to consider the interaction caused by two closely spaced cylinders, and by Deo

[29]to describe the effect of multiple cylinders. While some authors have modeled the acoustic behavior of the elements by introducing a time-dependent drag using

the formulas derived by Stokes and Lamb, it is also possible (see Sect. 4.5.6.5) to

!0

extend Stokes’ work by including the unsteady term, @ u =@t, still ignoring the inertial term. When this term is included a solution exists for both propagation past a sphere and past a cylinder without having to include the inertial term at all.

392Part 4 Properties

4.5.6.3

Generalized Models

A general model was given by Attenborough [22], who described the flow via the modified one-dimensional momentum and mass continuity equations:

where eff is the effective density, g the porosity, reff the effective resistivity, and u is taken to be the average particle velocity. The porosity g used here is a geometrical factor and should include only nonsolid volume for which there is a path for the fluid to pass through the solid. The effective resistivity and density reff and eff are phenomenological parameters. Heat conduction from the walls can also be included by using the relationship:

c2 0 ¼ ceff p0

where ceff is the effective ratio of specific heats, which describes thermal conduction and has a value of 1 for adiabatic propagation and 1.4 for isothermal propagation. In practice, a value between these two extremes should be taken.

4.5.6.4

Complex Viscosity and Complex Density Models

For the simple case of a narrow cylindrical pore, Rayleigh’s approach has a simple form. Beginning with the axisymmetric Navier–Stokes equation [30]:

The second term on the right-hand side is typically small and, when neglected, gives the solution (when the boundary conditions of zero velocity on the pore walls and zero shear stress at r = 0 are applied) [22]:

where r is the local radius, r0 is the radius of the cylindrical pore, and J0 is the zeroth order Bessel function of the first kind. This solution can been expressed more sim-

4.5 Acoustic Properties 393

ply by defining a complex viscosity leff, which is defined by comparing the wall drag due to the unsteady velocity profile given by the above equation with wall drag for steady Poiseuille flow. The complex viscosity is then given by:

by Biot [13,14] for sound propagation along pores having arbitrary cross sections by including a dynamic shape factor.

Alternatively, the same result can be expressed by using the velocity averaged over the cross section < u0 > and writing a simplified momentum equation in terms of a

ðr0 ðx=mÞ1=2 ð1þiÞÞF

This approach is favored by Wilson [31] and Brennan [32].

4.5.6.5

Direct Models

Rather than describing the acoustic behavior in terms of modified physical parameters, such as complex density, it is also possible to take a more direct approach in which the waves are solved directly in the limit of very low Reynolds number. This involves two terms: propagation parallel to the axes of the material elements, and perpendicular to the axes. This can be seen by considering the situation illustrated in Fig. 5 which shows both parallel and normal propagation.

For propagation parallel to the axes (i.e., in direction 2), the momentum equation

This equation can be solved for a variety of different boundary conditions in terms of Kelvin functions. For the situation illustrated in Fig. 5, the boundary conditions are:

@uz ¼ 0 on hn ¼ 0

@h

u2 and derivatives continuous on hn ¼ – Np uz ðr; hÞ ¼ 0 on r ¼ r0

@uz ¼ 0 on x ¼ xn

@xn

394 Part 4 Properties

Fig. 5 Array of cylinders.

h takes values between –p/N and p/N and is zero at the center of each face), and xn the distance normal to the periodic boundary (i.e. x = xn on the boundary and xn is normal to the boundary). The general solution which satisfies the first three boundary conditions is:

where Ak and Bk are coefficients and Be2Nk and Ke2Nk are Kelvin functions of order 2Nk. The remaining coefficients can be determined by applying the final boundary condition and using colocation along the periodic boundary shown.

When the propagation is normal to the axes, the flow field can be solved by following Stokes’ procedure but including the inertial term. Taking the curl of the momentum equation we have:

!

where x is the unsteady vorticity. This has a boundary condition of zero vorticity on the periodic boundaries and must result from a velocity field with no flow on the cylinder surfaces. The pressure field is similarly solved by taking the divergence of the momentum equation to give:

4.5 Acoustic Properties 395

Combining these equations; the drag on the acoustic wave is given by:

"#

0

f1 ðR0 Þ R0 f10 ðR0 Þ D1 eixt f1 ðR0 ÞþR0 f1 ðR0 Þ

where:

These two models can be combined to give a model for a wave of arbitrary incidence. Since the acoustic properties are primarily dependent on the geometry, rather than the material, the acoustic properties of cellular ceramics are similar to the acoustic properties of cellular metal foams. As can be seen from Fig. 6 [33] (which compares prediction of this model and experiment for a 100 ppi FeCrAlY metal foam with a rigid backing and a 0.8 m cavity), the absorption, defined here in terms of the reflection coefficient, can be extremely high.

Fig. 6 Comparison between theory (solid line) and experiment

(+) for the absorption coefficient for a metal foam block with rigid backing and a 0.8 m cavity at high frequency.

4.5.6.6

Biot’s Model

The problem of waves propagating through an elastic porous isotropic solid was considered by Biot [13,14], both with and without viscous dissipation of the waves prop-

396 Part 4 Properties

agating through the fluid. With no viscous dissipation, Biot considered an elemental cube which is assumed large in comparison with the pore size, but small in comparison with the wavelength of the elastic waves in the fluid–solid aggregate. The equations of motion for the solid and the fluid are coupled by assuming that the system is conservative (i.e., the oscillation is about a state of minimum potential energy). The solution is then found in terms of four elastic constants which are determined empirically. Viscous dissipation of low-frequency sound waves is dealt with [13] by assuming Poiseuille-type flow. A similar approach was used by Morse and Ingard [34]. This assumption breaks down when the frequency of oscillation is sufficiently high that the size of the acoustic boundary layer is small in comparison with the pore size. To quantify this, Biot assumes that the assumption breaks down when pore diameter 2r is equal to the quarter wavelength of a viscous boundary layer of an acoustic wave above an infinite plane as considered by Rayleigh [7] and Lighthill [10]. This gives an upper frequency limit of pv/16 r.

With Poiseuille flow in the pores, the stresses are recalculated with the new viscous shear stress giving rise to a wave which decays exponentially as it propagates through the porous material.

Higher frequency sound waves were treated by Biot in a later paper [14]. Here the limit on the frequency is less restrictive and set by the assumption that the wavelength is much larger than the characteristic pore size. The frequency-dependent dissipation is then calculated using a complex viscosity approach with a dynamic shape factor as described above. Biot’s model has been used by a number of authors to characterize the acoustic properties of a variety of porous materials, including saturated sand (Hovem and Ingram [21]), rectangular and triangular pores (Stinson and Champoux [4]) and catalytic converters (Selamet et al [20]). Stinson and Champoux [4] also show that the shape factor used to account for the arbitrary shape should be frequency-dependent.

4.5.6.7

Lambert’s Model

Biot’s model was extended by Lambert [15–19] to include thermal effects. In the absence of heat flow the acoustic waves propagate isentropically and the temperature fluctuations are given by the equation:

T0 ¼ ðc 1Þ 0

where T0 is the temperature fluctuation, c is the ratio of specific heats, and T is the mean temperature in the pore. Lambert assumes that the temperature fluctuations are related to the density fluctuations via a modified form of this equation:

4.5 Acoustic Properties 397

where xT is the reciprocal of the thermal time constant describing the conduction within the pore. Lambert then predicts xT using the first law of thermodynamics and Fourier’s law for the heat conduction within a cell:

where Q is the heat flux from a unit cell, N the average number of material ele-

T 0

ments per cell, kT is the thermal conductivity of the fluid in the pore, @@Tn the temperature gradient normal to the material elements, Vol the volume of the unit cell, and cv the specific heat capacity at constant volume for the fluid. The temperature gradi-

number by Nu = d/LT, where d is a characteristic length scale for the material element. The integral then depends only on the geometry of the material elements and can be evaluated to give:

A similar approach was used by Arnott et al [12] to predict the properties of stacks of arbitrarily shaped pores.

4.5.7

Acoustic Applications of Cellular Ceramics

The good sound-absorbing properties exhibited by cellular ceramics make them a good choice for a number of different applications. On highways, cellular ceramics are currently being used to acoustically insulate road tunnels and as noise barriers. Cellular glass materials have also been used in the building industry for sound absorption. Since ceramics have very high melting points, cellular ceramics may also be suitable in the future as acoustic liners inside combustion chambers, particularly in low-NOx combustors, where combustion oscillations are often a problem. A specific example of this type of application is presented in Chapters 5.5 and 5.6.

398Part 4 Properties

4.5.8

Summary

This chapter discussed the propagation of acoustic waves both in free space and through cellular materials. While it is traditional to ignore direct viscous effects, the small cell sizes which are present in cellular materials give rise to significant differences in the propagation characteristics. In particular, the propagation speed is reduced and the drag induced by the viscous effects also gives rise to attenuation of the acoustic waves.

A number of different ways in which these effects can be quantified, such as by defining an acoustic impedance, have been discussed. These can be measured empirically, and methods employed to measure the acoustic properties are discussed. We included a comparison between a theoretical model and experimental measurements for a porous metal.

It is also possible to model the acoustic properties empirically or theoretically. A number of different models exist and are described in Sections 4.5.5 and 4.5.6.

Finally, some examples of applications are given.

References