An empirical model for rigid-frame porous materials with low porosity

Applied Acoustics 58 (1999) 295±304 www.elsevier.com/locate/apacoust

An empirical model for rigid-frame porous materials with low porosity N. Voronina* Research Institute of Building Physics, Moscow, Russia Received 29 April 1998; received in revised form 4 November 1998; accepted 11 November 1998

Abstract Experimental data obtained for porous ceramic with low porosity have been analyzed to improve empirical formulas of characteristic impedance and propagation constant derived for highly porous materials in a previous paper (Voronina, N. An empirical model for rigid frame porous materials with high porosity. Appl. Acoust. 1997;51(2):181±98). The new empirical model has been compared with Attenborough's theory using experimental and calculated results obtained by Champoux and Stinson (Champoux, Y., Stinson. M.R., Measurement of the characteristic impedance and propagation constant of materials having high ¯ow resistivity. Journal Acoust. Soc. Am. 1991;90(4):2182±91) for a sample of ®lter ceramic QF-130. Good agreement has been noted between data measured and calculated by theoretical and empirical models. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Experimental acoustic parameters; Empirical formulas

Nomenclature co D f H k Q W

sound velocity of air pore diameter frequency porosity wave number structural characteristic characteristic impedance

* Tel.: +7-482-4076. 0003-682X/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S000 3-682X(98)0007 6-0

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Wa Wj 

0 

real part of W imaginary part of W attenuation coecient phase constant dynamic viscosity coecient density of air ¯ow resistivity

1. Introduction It is known that the acoustical properties of porous media are completely summarized if both the characteristic impedance, W, and the propagation constant, (l/m) can be speci®ed. It is indisputable that a theory is very important to describe a sound propagation through materials. However, from a practical point of view an empirical method may be also capable of predicting acoustic parameters on the basis of physical values characterizing a medium structure. In a previous paper [1] an experimental study has been carried out on rigid-frame highly porous materials including a foam-gypsum, a foam-slag-concrete and a foamceramic. The structural characteristic, Q introduced as an acoustic parameter for a quantitative estimation of energy losses in porous media has been de®ned as s 1 ÿ H 200 Qˆ HD k0 C0

…1†

where, H is porosity; D is a pore diameter (m);  ˆ 1:85  10ÿ5 is dynamic viscosity coecient (Pa s); o is air density (kg/m3); CO is sound velocity in air (m/s); k ˆ 2f=c0 is wave number (1/m); f is frequency (Hz). From the comparative analysis of experimental results empirical formulas for acoustic parameters have been derived in terms of the structural characteristic. Assuming B ˆ 0 in Eq. (6) presented in Ref. [1] they can be written as W ˆ Wa ÿ iWi ˆ 1 ‡ Q ÿ i

ˆ  ‡ i ˆ

Q p 1 ‡ Q= …1 ‡ Q†2

kQ p ‡ ik …1 ‡ Q† 1 ‡ 0:5Q= …1 ‡ Q†2

…2†

…3†

where Wa , Wi are the real and the imaginary parts of W; ; are the attenuation coecient and the phase constant.

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Thus, if porosity and the pore diameter are known then the structural characteristic and acoustic parameters W and may be calculated by Eqs. (1±3). However, this study has been specialized to highly porous materials when H > 0:75. Now, it is necessary to explore the range of application of the empirical model for rigid-frame materials having low porosity. Nine ceramic samples (0:26 < H < 0:5) and two samples of a foam-gypsum (H ˆ 0:72) have been investigated. Numerical values of H and D for these materials are presented in Table 1. The characteristic impedances and the propagation constants have been measured by means of standard methods [2]. Experimental results are shown in Figs. 1 and 2.

Table 1 Physical parameters of porous materials Sample

Porosity H

Pore diameter D  106 (m)

Structural character Q4000 at 4000 Hz

Ceramic 1 2 3 4 5 6 7 8 9

0.26 0.3 0.33 0.33 0.33 0.4 0.4 0.5 0.5

110 84 93 71 200 98 46 180 73

4.6 5.3 4.4 5.7 2.1 3.4 7.3 1.4 3.4

Foam-gypsum 10 11

0.72 0.72

80 45

1.4 2.6

Table 2 Acoustic parameters W and calculated by models 1 and 2 for a sample of foam-slag-concrete with H ˆ 0:9 and D ˆ 100  10ÿ6 m f (Hz)

Model 1

W ˆ W ÿ iWi 250 500 1000 2000 4000

2.56ÿi 2.1ÿi 1.78ÿi 1.55ÿi 1.39ÿi

ˆ  ‡ i …1=m† 250 500 1000 2000 4000

6.1+i 11.6 8.9+i 19.2 12.8+i 32.5 18.4+i 56.7 26.6+i 102

0.67 0.49 0.35 0.25 0.18

Model 2 2.48ÿi 2.04ÿi 1.74ÿi 1.52ÿi 1.37ÿi

0.69 0.46 0.32 0.22 0.16

6.3+i 11 9+i 18.2 12.8+i 31 18+i 54.5 25+i 100

Experimental 2.67ÿi 2.1ÿi 1.8ÿi 1.52ÿi 1.42ÿi

0.65 0.45 0.34 0.17 0.16

5.6+i 12 11+i 20 13+i 34 16+i 60 26+i 111

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2. Empirical formulas for acoustic parameters The results obtained show clearly that the acoustic parameters have marked dependence on porosity. From the comparison of the data in Fig. 1 for test samples having equal pore diameters it has p been found that the structural characteristic Q ˆ Wa ÿ 1 changes as …1 ÿ H†= H, In this connection the previous Eq. (1) has been corrected and presented in the form s 1 ÿ H 200 Q ˆ p HD k0 c0

…4†

The numerical values of Q4000 calculated by Eq. (4) at f ˆ 4000 Hz are shown in Table 1 for all test samples. They can be used in determing the structural characteristic at frequency of interest assuming that Q is proportional fÿ0:5p.  It should be noted that with increase of porosity when the value of H is close to H, there is a small di€erence between Eq. (1) and Eq. (4). It may be taken into account in predicting the structural characteristic for highly porous materials. Fig. 2(a) Illustrates that the imaginary part of the characteristic impedance depends on porosity also. The experimental points obtained for samples with different porosity lie between two functions Wi ˆ Q and Wi ˆ 0:5Q plotted by dashed lines. One can see that the value of Wi tends to Q at low porosity but it is always smaller than the structural characteristic. Therefore, the imaginary part of W can be written as Wi ˆ Q=…1 ‡ a†

…5†

where a is a coecient that depends on H and Q.

Fig. 1. Structural characteristic as a function of frequency for porous materials listed in Table 1: (1) sample 7; (2) sample (4); (3) sample 1; (4) sample 9; (5) sample 11.

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Fig. 2. Acoustic parameters as a function of structural characteristic: (a) the imaginary part of characteristic impedance; (b) the phase constant for porous materials (Table 1) having porosity: (1) H ˆ 0:26; (2) H ˆ 0:3; (3) H ˆ 0:33; (4) H ˆ 0:4; (5) H ˆ 0:5; (6) H ˆ 0:72.

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The experimental values of  are shown in Fig. 3(a). From the comparison of the results ®rst at Q=constant and after that at H=constant the empirical expression has been obtained as p …6† a ˆ H2 ‰1 ‡ 10 Q…1 ÿ H†=H…1 ‡ Q2 †Š The data in Fig. 2(b) have been analysed in a similar way. It has been found that the p relation of …=k† varies as HQ. Taking into account the observed inequality …=k† < HQ the empirical formula for the attenuation coecient has been written as p  ˆ kQ H=…1 ‡ b†

…7†

where b is a coecient that depends on H and Q.

Fig. 3. Experimental values of (a) coecient ; (b) coecient b vs structural characteristic for porous materials having porosity: (1) H ˆ 0:26; (2) H ˆ 0:3; (3) H ˆ 0:33; (4) H ˆ 0:4; (5) }H ˆ 0:5; (6) H ˆ 0:72.

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The experimental data provided for the coecient b and presented in Fig. 3(b) show that the smaller the porosity, the greater the quantity b. The ®nal analytical formula has been derived as p b ˆ …1 ÿ H2 †=H Q

…8†

The results obtained for the phase constant [Fig. 2(c)] gave a clear indication of the validity of … =k† ÿ 1 ˆ HQ. Thus, making use of Eqs. (5±8), the empirical formulas for acoustic parameters have been established as W ˆ W ÿ iWi ˆ 1 ‡ Q ÿ i

1‡

H2 ‰1

Q p ‡ 10 Q…1 ÿ H†=H…1 ‡ Q2 †Š

p kQ H

ˆ  ‡ i ˆ p ‡ ik…1 ‡ HQ† 1 ‡ …1 ÿ H†2 =H Q

…9†

…10†

The acoustic parameters calculated by Eqs. (9) and (10) using (4) are plotted by solid curves in Figs. 1 and 2. One can see that there is a good agreement between calculated and measured data. Now, one needs to examine the ®t of the new model for materials with high porosity. In Table 2 the results calculated by Eqs. (2) and (3) (model 1) and by Eq. (9) and Eq. (10) (model 2) are presented for a sample of a foam-slag-concrete having H ˆ 0:9 and D ˆ 10ÿ4 m. It Is clear that both models yield similar values of W and

. Therefore, the new empirical formulas Eqs. (9) and (10) can be used to predict acoustical behaviour of highly porous materials also. It is important to note that the empirical equation for air ¯ow resistivity, (=o Co †, mÿ1 may be de®ned in terms of porosity and the structural characteristic by Eqs. (9) and (10) as   p  H 2 H‡ ˆ lim …Wa  ‡ Wi † ˆ kQ 0 c0 k!0 1 ‡ H2

…11†

In a case when porosity and air ¯ow resistivity of a porous material are known the structural characteristic may be determined from Eq. (11) in the form s …1 ‡ H2 † p p Qˆ 0 c0 k H…1 ‡ H ‡ H2 †

…12†

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3. Comparison between theoretical and empirical models In the work [3] by Champoux and Stinson the characteristic impedance and the propagation constants have been measured for porous ceramic QF-130 and predicted by applying the theory of Attenborough [4] using experimental values of ¯ow resistivity, porosity and tortuosity listed in Table 3. The numerical value of 0.52 for the pore shape factor (Table 3) has been selected by authors to provide the best overall ®t of the theory to the measured data. The acoustic parameters W, and are shown in Fig. 4. Table 3 Physical parameters of porous ceramic QF ÿ 130 Porosity 0.432

Flow resistivity (cgs-rayl/cm)

Tortuosity

Shape factor

43.6

1.64

0.52

Fig. 4. Acoustic parameters as a function of a frequency: (a) the real part of characteristic impedance; (b) the imaginary part of characteristic impedance.

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Fig. 4. (c) the attenuation coecient; (d) the phase constant measured and calculated by (solid curves) the theory of Attenborough; (dashed curves) the empirical model for a porous ceramic QF ÿ 130 having physical parameters in Table 3.

Now, it is interesting, to compare the theory and the experimental results with the empirical model derived here. Taking into account the dimension of (=0 C0 ), cmÿ1 and k, cmÿ1 in Eq. (12) the qualities  ˆ 43:6 cgs-rayl cmÿ1 and H ˆ 0:432 have been used to determine the structural characteristic for ceramic QF-130 in frequency range 100±1000 Hz. The acoustic parameters computed by the empirical model using the numerical values of Q are plotted by dashed curves in Fig. 4. As seen from Fig. 4(a,b) a satisfactory agreement among the theoretical, empirical and experimental data has been obtained for the characteristic impedance, although there some discrepancies at low frequencies. It should be noted that the propagation constant calculated by Eq. (10) [Fig. 4(c,d)] is con®rmed experimentally and is close to that predicted by the theory. Thus, the comparison of the results has shown that there is a quantitative agreement between the theoretical and empirical models.

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4. Conclusion Characteristic impedances and propagation constants have been measured for low-porosity rigid-frame materials. Comparative analysis of the results has permitted the acoustic parameters to be represented by empirical functions depending explicitly on porosity and a pore diameter that characterised medium structure. The empirical model has been compared to experimental data and predictions obtained by Champoux and Stinson [3]. It has been found that the predictions of the empirical formulas are con®rmed experimentally and have a quantitative agreement with those based on the theory. References [1] Voronina N. An empirical model for rigid frame porous materials with high porosity. Appl. Acoust. 1997;51(2):181±98. [2] Voronina N. Physic-mathematical model of a porous absorbing material. Tr./NIISF, Building and acoustic means of noise protection, 1988. p. 44±53. [3] Champoux Y, Stinson MR. Measurement of the characteristic impedance and propagation constant of materials having high ¯ow resistivity. Journal Acoust. Soc. Am. 1991;90(4):2182±91. [4] Attenborough K. On the acoustic slow wave in air-®lled granular media. Journal Acoust. Soc. Am. 1987;81(1):93±102.