A New Empirical Model

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of noise barriers, factory spaces, sports halls and auditoria. A large selection of

acoustic materials is currently offered for these uses, the majority of which are

fibrous layers and reticulated foams.

Recent work suggests that granular porous materials can be regarded as an alter-native to many existing fibrous and foam absorbers in many indoor and outdoor

applications[1]. In granular materials good absorption can be combined with good

mechanical strength and very low manufacturing costs which is important in many

acoustic applications. There are many theoretical and empirical models which are

used to model sound propagation in granular porous media. However, recent results

for rubber granulates[2]suggest that some existing models for sound propagation in

porous media can fail due to the complexity of the porous structure of these mate-

rials. In this previous work four models[36]have been tested to predict the surface

impedance of loose and consolidated mixes of recycled rubber granulates. It has

been shown that in order to provide a tolerable fit to the experimental data the

values of the flow resistivity and porosity in these models need adjusting by up to

70%. These results are of considerable concern, and suggest the need for developing

an improved model for the acoustic properties of granular media.

On the other hand, there is a general lack of experimental data on the character-

istic impedance and propagation constant in practical granular mixes. It appears

that many researchers report on routinely measurable acoustic surface impedance

and normal incidence absorption coefficient, which, are in many cases, not particu-

larly helpful to obtain a clear insight into the physical mechanisms of the acoustic

absorption in the porous media. In this respect, the availability of independentlymeasured data on the acoustic characteristic impedance and propagation constant is

of great importance for developers of new models, which provide the basis for

benchmarks and validation of their work.

The purpose of this paper is to investigate experimentally the fundamental

acoustic properties of granular media and to use the experimental results to develop

a practical acoustic model, which is robust in a broad range of acoustic frequencies

and particle sizes. The following materials were investigated in this work: vermiculite

(phyllosilicate mica), granulated rubber from automotive tyres, perlite (expanded

silicone glass) and granulated nitrile foam.

The paper is organised in the following manner. Section 2discusses the physicalparameters, which are required for modelling the acoustic properties of granular

media.Section 3provides the methodology for the experimental investigation, which

was conducted on a representative selection of loose granular mixes. Section 4pre-

sents the new empirical expression for the structural characteristic, which then is

used inSection 5to predict the acoustic characteristic impedance and propagation

constant.Section 6presents the conclusions.

2. Physical parameters of porous materials

Granular materials are often modelled as a rigid frame porous medium formed by

the rigid, interconnected particles and voids in which a slow compressional wave can

416 N.N. Voronina, K.V. Horoshenkov / Applied Acoustics 64 (2003) 415432


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propagate. The sound speed and the attenuation of the compressional wave in the

porous medium are, therefore, functions of the size of pores and the proportion of

the open pores. These properties are largely related to the size, shape and the degree

of compaction of the particles, which constitute the rigid frame. The degree ofcompaction is determined by the density of the granular mix and affects its porosity

and permeability (e.g. [7,8]). It is common to expect a considerable variation in the

acoustic properties of granular mixes, which are composed of differently compacted

grains although the grains are identical in size [1]. The shape of particles largely

influences the degree of compaction and, therefore, the porosity of the granular mix

[7]. As an example, particles of granulated rubber tend to take the form of irregular

parallelepipeds, particles of perlite are close to ellipses and particles of vermiculite

appear in the form of plates. In this respect, the definition of the characteristic

dimension can be confusing. A more general way to characterise granular mixes

with particles of different shape is to assume that the characteristic particle dimen-

sion is the diameter of a sphere which volume is equivalent to the mean volume of

the particles in a given mix. The way in which the characteristic dimension is defined

is of importance, because it is convenient to relate the acoustic properties of a loose

granular mix the to the characteristic dimension of its particles, D, and to the por-

osity, H 1 m=g, both of which are routinely measurable characteristics. Herem is the density of the granular mix and g is the specific density of the grain

material. In many cases, the value of the latter parameter is significantly influenced

by the presence of the cracks and micro-pores, which also affect the acoustic per-

formance of the loose particle mix. The characteristic dimension of the particles canbe found provided that the number of the particles in a unit volume, Vg, of the

granular mix is known, in which case D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVg=0:5233p

. This method yields a 25%

accuracy for large grain mixes. For small grain mixes the accuracy of this method

deteriorates to around 20%, which is still an acceptable value for many applications

of engineering acoustics. The specific density of the granular mix is easily determined

from the principle of Archimedes.

When an acoustic wave is incident on a porous layer, the thermo-viscous effects in

the fluid filling the voids between the particles are responsible for the energy loss in

the oscillating acoustic flow [913]. It has been shown that the thermal dissipation

effects in commercial porous materials are typically small (e.g. [14]). The viscous effectsare important only inside the viscous boundary layer and the viscous energy loss in the

acoustic model can be accounted for by introducing a dimensionless parameter

D0c

104 1

whereis the dynamic viscosity of air,0is the equilibrium density of air andcis the

sound speed in air. The parametercan be used to classify granular mixes according

to the particle characteristic dimension, i.e. one can refer to large grain mixes for

5 2, medium grain mixes for 1 <


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This can be related to two separate phenomena: mechanical friction between the

elements of the non-rigid frame in porous materials with low specific density and to

the absorption in the frame micro-pores. The friction phenomenon can be of

importance in the low frequency range, where the inertial effects are small and vis-cous drag is relatively large. In this regime the shape of the particles is of relatively

small importance and the effect can be modelled using the expressions for the oscil-

lating flow drag experienced by a stack of identical spherical beads (e.g. [16]). The

influence of the particle micro-porosity is likely to be pronounced in the higher fre-

quency range [17], where additional dissipation can take place due to the thermal

non-equilibrium in micro-pores. These two effects can be accounted for phenomen-

ologically by introducing a dimensionless parameter

M

g

103

0

;

2

where the factor of 103 is a dimensionless normalisation factor. The measured values

of the density, porosity, characteristic particle dimension and the dimensionless

parameters andMare provided inTable 1for the eight granular mixes.

In addition to the above parameters, it is also common to include in an acoustic

model the measured values of flow resistivity, r, and tortuosity, q, which are closely

related to the macrostructure of the porous mix[9]. In this work the flow resistivity

have been measured at Bradford University using the standard method detailed in

[18]. The results for six material samples are provided in Table 2. In the special case

of an oscillatory flow past a stack of spherical beads, the tortuosity can be predictedfrom the approximate expression[16]

q2 1 1 H2H1 3However, for realistic mixes with irregular particle shapes expression (3) is rather

idealised. In this particular work the tortuosity was experimentally deduced using

Table 1

Basic physical parameters used in the empirical model for granular media

Material Density, m(kg/m3)

Porosity,H Characteristic particlesize,D (mm)

Parameter

Vermiculite,g 1200 kg/m3,M 0:9751 385 0.68 1.4 3.1

2 370 0.69 0.5 1.1

3 420 0.65 0.4 0.89

Rubber crumb, g 1050 kg/m3,M 0:9254 590 0.44 3.5 7.1

5 520 0.54 1.6 3.6

Perlite,g 200kg/m3,M 0:1636 80 0.60 2.2 4.9

7 44 0.78 0.5 1.1Nitrile foam granulate, g 165kg/m3, M 0:3588 15 0.91 1.2 2.29



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two independent methods: from the upper-frequency data on the real part of the

refraction index[19]and the ultrasonic time of flight method[20]. The results, whichare provided in Table 2 suggest that the experimentally determined values of the

tortuosity can differ considerably from those predicted by expression (3). In the case

of vermiculite, the predicted values are consistently lower that the experimental

results (see Table 2), which can be attributed to the deviation of the shape of ver-

miculite particles from the assumed spherical shape.

The values of the tortuosity can also be deduced from the behaviour of the real

part of the characteristic impedance. At frequencies above some critical frequency,

fcr, the real part of characteristic impedance approaches asymptotically to its higher-

frequency limit[9]

Wcra q=H; 4and is relatively independent of the frequency. The larger the size of the grain base,

the lower the value of the critical frequency. It can be shown experimentally that for

large grain mixes (5 2) fcr 4000 Hz), above which the reliable impedance data are

usually unavailable.

Table 2

Values of non-acoustic parameters used for modelling of the acoustic properties of porous media. The

experimental values of the tortuosity were deduced using the method [19] and are compared with the

bracketed values from the method[20]

Material Flow resistivity (Pa s m2) r Tortuosity,q

Exp. (14) Experiment Exp. (3) Exp. (4) Experiment

Vermiculite

1 7610 7580 1.11 1.72 1.58 (1.51)

2 56800 61000 1.11 1.78 1.63 (1.76)

3 107000 135000 1.13 1.48 (1.45)

Rubber crumb

4 3190 2800 1.27 1.11 1.13 (1.19)

5 9800 13600 1.21 1.29 1.26 (1.23)

Perlite6 4280 1.50 1.48

7 32800 1.08

Nitrile foam granulate

8 1163 2800 1.03 1.32 1.31 ()

N.N. Voronina, K.V. Horoshenkov/ Applied Acoustics 64 (2003) 415432 419


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3. Methodology

An experimental investigation of the acoustic properties has been carried out

using the impedance tube method in the frequency range of 2504000 Hz using thestandard procedure detailed in ISO 10534-1:1996 and ISO 10534-2:1998. Samples of

granular materials have been tested in two independent laboratories to ensure the

reproducibility of the experimental data. The impedance tube in the University of

Bradford (BK 4206) is installed in the vertical position to allow the acoustic prop-

erties of non-consolidated mixes to be easily measured. The impedance tube in the

Institute of Building Physics in Moscow (BK 4002) is installed in the horizontal

position. In the case of the horizontally installed tube the investigated materials were

packed in a special attachment container and the front surface of the samples was

covered with a nylon mesh to hold loose granules together and to ensure that the

surface was flat. Loose samples were compacted sufficiently to ensure good friction

between individual particles and the walls of the impedance tube. A separate set of

measurements has been carried out to confirm that the mesh has a negligible effect

on the acoustic properties of the investigated samples. The density of the granular

mixes was measured and kept constant in all the experiments to ensure the same

degree of compaction between different experimental set-ups.

The surface impedance Ws of each sample has been measured for the doubled

thickness of the porous layer W2s, so that the characteristic impedance,

W Wa iWi, and propagation constant, i, in the porous samples can bedetermined from the expressions Ws W

cothd and W2s W

coth2d . HereWa, Wi, are real and imaginary parts of the characteristic impedance, and and

are real and imaginary parts of the propagation constant, respectively, and dis the

layer thickness. The experimental data for the above characteristics were applied to

develop and propose new empirical expressions for the acoustic properties of porous

granulates. The reproducibility of the experimental data between the two labora-

tories was within 10%.

4. Structural characteristic for granular materials

Experimental results for the characteristic impedance and propagation constant

have been used to determine the effects of the porosity and grain size on the struc-

tural characteristic Q. The structural characteristics Q has been introduced in pre-

vious work [5,15] and is included in the expression for the real part of the

characteristic impedance,

Wa 1 Q; f< fcr 5Fig. 1shows the experimentally determined frequency dependence of the structural

characteristic Q for three granular mixes. The results confirm that the functional

behaviour of this characteristic is similar to that derived previously for rigid frameporous media [5,15], where Q / 1 H = HD

ffiffiffik

p , k 2f=c being the wave number

in air,fis the frequency andc is the sound speed in air. From the comparison of the



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data for granular materials with different values of the parameter M(seeTable 1) an

empirical expression for the structural characteristic has been deduced and is pro-

posed in the following form

Q 0:2 1 H 1 H 2

HffiffiffiffiffiffiffiffiffikD

p : 6

The above expression has been used to calculate the predicted values ofQ, which

are shown inFig. 1and compared with the experimental data.Since Q Wa 1, the above expression can be used to determine the transition

frequency fcr, which is calculated from

fcr200 1 H 2 1 H 4

0D 2 q H 2 7

using the values of the parameters provided in Tables 1 and 2 and the relation

Qcr qH 1. The critical value Qcr corresponds to the frequency at which the beha-viour of the real part of the characteristic impedance Wbecomes frequency-inde-

pendent.

For frequencies f5fcr the real part of the characteristic impedance can be pre-dicted by expression (4). For frequencies f< fcr, expressions (5) and (6) are suffi-

ciently accurate for a majority of practical applications.

Fig. 1. The experimentally measured and empirically predicted values of the structural characteristic as a

function of frequency.



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5. Empirical expressions for the acoustic properties

Expression (6) for the structural characteristic can also be used to predict the

imaginary part of the characteristic impedance and real and imaginary parts of thepropagation constant. The expressions for these properties have been originally

proposed in[5] for materials with low porosity. These expressions have been mod-

ified to fit the experimental data for granular materials and are provided below

kQH1 A 8

k1 QH1 B 9

Wi

QH

1 C:

10

Here A, B and C are coefficients which depend upon the porosity, structural

characteristic and dimensionless parameters Mand .

Fig. 2(a)(c) shows the experimentally determined values of the coefficients A, B

and C as functions of the structural characteristic Q for materials 1, 4 and 6. The

results suggest that the coefficient A is proportional to 1=1 Q and that the coef-ficientB is proportional to 1=

ffiffiffiffiQ

p [seeFig. 2(a) and (b)]. The results also suggest that

for a given value ofQ the coefficient A increases and coefficientB decreases with the

increasing value ofM. The results for the investigated granular mixes show that for

a fixed value of Q the coefficients A and C are proportional to1 H and thecoefficient B is proportional to 1=1 H.These dependencies provide the basis to derive the interpolated expressions for the

three coefficients, which are provided below

A 1 H M1 Q 11

B 1ffiffiffiffiQ

p 1 H 1 Q 2M 12

and

C 1 HffiffiffiffiQ

p : 13

The solid lines inFig. 2(a)(c)show the calculated values of the coefficients A, B

and C for materials 1, 4 and 6 as functions of the structural characteristic Q.

Expressions (11)(13) provide a close fit to the experimental results throughout the

considered range of values of the structural characteristic, porosity and dimension-

less parameter M.

Empirical expressions (6)(11) can also be used to predict the flow resistivity of

granular mixes, which is determined from the low frequency limit r 0clim

f!0Wa Wi as



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Fig. 2. Experimental and empirically predicted values of the coefficientsA,B and C.



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r 4001 H21 H5

HD 2 14

The values of the flow resistivity which are predicted from the above expression

are shown inTable 2,where they are compared to the measured data. Good agree-

ment between the measured and predicted data is observed for materials 1, 2, 4 and

8. Expression (12) is less accurate in the case of materials 3 and 5, which can be

attributed to the high sensitivity of the flow resistivity to the material compaction.

Figs. 38show the experimental and predicted values of the characteristic impedance

and propagation constant for the eight materials used in the experiments. The agree-ment between the experimental results and the proposed empirical model for all the

eight granular mixes is good throughout the considered frequency range. The model

does not involve any special functions of complex argument and is easy to implement.

The classification according to the value of the parameter is very useful to pre-

dict the behaviour of the real part of the characteristic impedance of a granular mix.

The results show that for large grain mixes with 5 2 (materials 1, 4, 5, 6 and 8 in

Table 1) the transition frequency fcr is low. In these cases the real part of its char-

acteristic impedance is relatively independent of frequency and can be predicted

accurately from expression (4). For these mixes the imaginary part of the char-

acteristic impedance is relatively small, Wi


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Fig. 3. The real (a) and imaginary (b) parts of the normalised characteristic impedance of vermiculite.



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Fig. 4. The real (a) and imaginary (b) parts of the normalised characteristic impedance of loose rubber crumb.



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Fig. 5. The real (a) and imaginary (b) parts of the normalised characteristic impedance of perlite and

nitrile granulate.



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Fig. 7. The real (a) and imaginary (b) parts of the propagation constant for loose rubber crumb.



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Fig. 8. The real (a) and imaginary (b) parts of the propagation constant for perlite and nitrile granulate.



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The transition frequency of small (materials 3 and 7) and medium grain materials

(material 2) is relatively high and is outside the considered frequency range (see

Figs. 3 and 5). The behaviour of the real part of the characteristic impedance for

these materials is frequency dependent and can be predicted closely by expression(5). The attenuation coefficient for materials 2, 3 and 7 is relatively large and the

phase velocity is relatively small (seeFigs. 6 and 8). These properties are desirable in

the design of efficient acoustic absorbers. In this respect, the classification according

to the value of the parameter makes easier the correct choice of the optimal grain

mix.

6. Conclusions

An experimental investigation has been carried out to determine the characteristic

impedance and propagation constant in a representative selection of loose granular

materials. A new classification of granular media has been proposed which is related

to the characteristic particle dimension and the specific density of the grain base. It

has been shown that the proposed classification is a useful characteristic for pre-

liminary estimation of the acoustic performance of loose granular mixes.

The experimental results have been used to develop a new empirical model, which

can predict reliably the acoustic performance of loose granular mixes. The model

requires knowledge of the characteristic particle dimension, porosity, tortuosity and

the specific density of the grain base, most of which are routinely measurable para-meters. The introduction of the specific density of the grain base is important and

can empirically account for the effects of friction between the elements of the non-

rigid frame in porous materials with low density and for the absorption in the par-

ticles micro-pores.

It has been shown that the tortuosity can be predicted or deduced from experi-

mental data. Unlike many other models, the proposed model does not require the

knowledge of the flow resistivity which values can be very sensitive to the compac-

tion state of a loose granular mix. The characteristics which the model predicts are

the real and imaginary parts of the normalised characteristic impedance and propa-

gation constant. These characteristics are required to predict the efficiency ofacoustic porous absorbers and sound propagation over porous soils.

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A New Empirical Model