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A new empirical model for the acoustic properties of loose granular media
Applied Acoustics 64 (2003) 415–432 www.elsevier.com/locate/apacoust
A new empirical model for the acoustic properties of loose granular media N.N. Voroninaa, K.V. Horoshenkovb,* a Institute of Building Physics, 21 Lokomotivnyi Pr., Moscow 127238, Russia School of Engineering, Design and Technology, University of Bradford, Bradford BD7 1DP, UK
b
Received 3 December 2001; received in revised form 24 September 2002; accepted 2 October 2002
Abstract This paper examines physical parameters of loose granular mixes and their empirical relations to the acoustic performance of these mixes. In this work 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 this classification is a useful characteristic for rapid evaluation of the acoustic performance of loose granular mixes. The characteristic impedance and propagation constant have been measured for a representative selection of grain mixes and used to develop a new empirical model. This model relates the above acoustic characteristics to the characteristic particle dimension, porosity, tortuosity and specific density of the grain base, which are routinely measurable parameters. A very good agreement with the experimental data is illustrated in the frequency range of 250–4000 Hz for materials with the grain base of 0.4–3.5 mm and specific densities between 200 and 1200 kg/m3. Unlike many theoretical models for the prediction of the acoustic properties of porous media, the proposed expressions do not involve any special functions of complex argument, empirical shape factors or sophisticated characteristics of porous structure. These are practical enough to be of interest to acoustic and noise control engineers and material manufacturers. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Sound propagation; Loose granulates; Porous media; Acoustic absorption
1. Introduction In recent years the use of acoustic porous materials for indoor and outdoor applications has been stimulated by legislation in the area of environmental noise and public health. Absorbers are now routinely used to improve the acoustic performance * Corresponding author. Tel.: +44-1274-233-877; fax: +44-1274-233-888. E-mail address: [email protected] (K.V. Horoshenkov). 0003-682X/03/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0003-682X(02)00105-6
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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 alternative 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 materials. In this previous work four models [3–6] 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 characteristic 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 particularly helpful to obtain a clear insight into the physical mechanisms of the acoustic absorption in the porous media. In this respect, the availability of independently measured 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 2 discusses the physical parameters, which are required for modelling the acoustic properties of granular media. Section 3 provides the methodology for the experimental investigation, which was conducted on a representative selection of loose granular mixes. Section 4 presents the new empirical expression for the structural characteristic, which then is used in Section 5 to predict the acoustic characteristic impedance and propagation constant. Section 6 presents 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
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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 of compaction 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 dimension 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 porosity, H ¼ 1 m =g , both of which are routinely measurable characteristics. Here m 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 performance of the loose particle mix. The characteristic dimension of the particles can be found provided that the number of thepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi particles in a unit volume, Vg , of the granular mix is known, in which case D ¼ Vg =0:5233. This method yields a 2–5% 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 [9–13]. It has been shown that the thermal dissipation effects in commercial porous materials are typically small (e.g. [14]). The viscous effects are 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 ¼
D0 c 4 10
ð1Þ
where is the dynamic viscosity of air, 0 is the equilibrium density of air and c is the sound speed in air. The parameter can 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 < < 2 and to small grain mixes for 4 1. Recent results for porous materials with high porosity [15] suggest that the acoustic properties of granular materials can be affected by the value g =0 . The acoustic attenuation tends to increase with the reduced value of the grain density.
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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 viscous 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 oscillating 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 frequency range [17], where additional dissipation can take place due to the thermal non-equilibrium in micro-pores. These two effects can be accounted for phenomenologically by introducing a dimensionless parameter g M¼ 3 ; ð2Þ 10 0 where the factor of 103 is a dimensionless normalisation factor. The measured values of the density, porosity, characteristic particle dimension and the dimensionless parameters and M are provided in Table 1 for 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 predicted from the approximate expression [16] q2 ’ 1 þ ð1 HÞð2HÞ1
ð3Þ
However, 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
Vermiculite, g ¼ 1200 kg/m3, M ¼ 0:975 1 385 0.68 2 370 0.69 3 420 0.65 Rubber crumb, g ¼ 1050 kg/m3, M ¼ 0:925 4 590 0.44 5 520 0.54 Perlite, g ¼ 200kg/m3, M ¼ 0:163 6 80 0.60 7 44 0.78 Nitrile foam granulate, g ¼ 165 kg/m3, M ¼ 0:358 8 15 0.91
Characteristic particle size, D (mm)
Parameter
1.4 0.5 0.4
3.1 1.1 0.89
3.5 1.6
7.1 3.6
2.2 0.5
4.9 1.1
1.2
2.29
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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 Exp. (14)
Vermiculite 1 7610 2 56800 3 107000 Rubber crumb 4 3190 5 9800 Perlite 6 4280 7 32800 Nitrile foam granulate 8 1163
Tortuosity, q
Experiment
Exp. (3)
Exp. (4)
Experiment
7580 61000 135000
1.11 1.11 1.13
1.72 1.78 –
1.58 (1.51) 1.63 (1.76) 1.48 (1.45)
2800 13600
1.27 1.21
1.11 1.29
1.13 (1.19) 1.26 (1.23)
– –
1.50 1.08
1.48 –
– –
2800
1.03
1.32
1.31 (–)
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, which are 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 vermiculite 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 higherfrequency limit [9] Wacr ’ q=H;
ð4Þ
and 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 < 500 Hz, so that the tortuosity can be deduced from expression (4) in which Wacr is a routinely measurable constant. Table 2 provides the deduced values of the tortuosity for large grain mixes 1, 4, 5 and 6 and for medium grain mix 2. Generally, there is a good agreement between the values of the turtuosity, which were measured with the three experimental methods (see Table 2). For small grain mixes ( 4 1) the proposed experimental method for measuring tortuosity is likely to fail, because of the relatively high value of the critical frequency (fcr > 4000 Hz), above which the reliable impedance data are usually unavailable.
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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 250–4000 Hz using the standard 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 properties 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 ¼ Wa þ iWi , and propagation constant, ¼ þ i , in the porous samples can be determined from the expressions Ws ¼ Wcothð dÞ and W2s ¼ Wcothð 2dÞ. Here Wa , Wi , are real and imaginary parts of the characteristic impedance, and and are real and imaginary parts of the propagation constant, respectively, and d is 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 laboratories 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 structural characteristic Q. The structural characteristics Q has been introduced in previous work [5,15] and is included in the expression for the real part of the characteristic impedance, Wa ¼ 1 þ Q; f < fcr
ð5Þ
Fig. 1 shows 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 pffiffiffiderived previously for rigid frame porous media [5,15], where Q / ð1 HÞ= HD k , k ¼ 2 f=c being the wave number in air, f is the frequency and c is the sound speed in air. From the comparison of the
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Fig. 1. The experimentally measured and empirically predicted values of the structural characteristic as a function of frequency.
data for granular materials with different values of the parameter M (see Table 1) an empirical expression for the structural characteristic has been deduced and is proposed in the following form Q¼
0:2ð1 HÞð1 þ HÞ2 pffiffiffiffiffiffiffiffiffi : H kD
ð6Þ
The above expression has been used to calculate the predicted values of Q, which are shown in Fig. 1 and compared with the experimental data. Since Q ¼ Wa 1, the above expression can be used to determine the transition frequency fcr, which is calculated from 200ð1 HÞ2 ð1 þ HÞ4 ð7Þ
0 D 2 ðq HÞ2 using the values of the parameters provided in Tables 1 and 2 and the relation Qcr ¼ Hq 1. The critical value Qcr corresponds to the frequency at which the behaviour of the real part of the characteristic impedance W becomes frequency-independent. For frequencies f 5 fcr the real part of the characteristic impedance can be predicted by expression (4). For frequencies f < fcr , expressions (5) and (6) are sufficiently accurate for a majority of practical applications. fcr ¼
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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 the propagation constant. The expressions for these properties have been originally proposed in [5] for materials with low porosity. These expressions have been modified to fit the experimental data for granular materials and are provided below ¼
kQH 1þA
¼ k½1 þ QHð1 þ BÞ
Wi ¼
QH : 1þC
ð8Þ ð9Þ ð10Þ
Here A, B and C are coefficients which depend upon the porosity, structural characteristic and dimensionless parameters M and . 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 pffiffiffiffi A is proportional to 1=ð1 þ QÞ and that the coefficient B is proportional to 1= Q [see Fig. 2(a) and (b)]. The results also suggest that for a given value of Q the coefficient A increases and coefficient B decreases with the increasing value of M. The results for the investigated granular mixes show that for a fixed value of Q the coefficients A and C are proportional to ð1 HÞ and the coefficient B is proportional to 1=ð1 þ HÞ. These dependencies provide the basis to derive the interpolated expressions for the three coefficients, which are provided below ð1 HÞM 1þQ 1 B ¼ pffiffiffiffi Qð1 þ HÞð1 þ Q 2 MÞ A¼
ð11Þ ð12Þ
and 1H C ¼ pffiffiffiffi : Q
ð13Þ
The solid lines in Fig. 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 dimensionless 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 ¼ 0 clim ðWa þ Wi Þ as f !0
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Fig. 2. Experimental and empirically predicted values of the coefficients A, B and C.
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Fig. 2. (continued)
r¼
400ð1 HÞ2 ð1 þ HÞ5 HD 2
ð14Þ
The values of the flow resistivity which are predicted from the above expression are shown in Table 2, where they are compared to the measured data. Good agreement 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. 3–8 show the experimental and predicted values of the characteristic impedance and propagation constant for the eight materials used in the experiments. The agreement 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 predict 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 characteristic impedance is relatively independent of frequency and can be predicted accurately from expression (4). For these mixes the imaginary part of the characteristic impedance is relatively small, Wi < 1, throughout the considered frequency range. The attenuation constant for these materials is limited and the phase velocity is relatively high.
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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. 6. The real (a) and imaginary (b) parts of the propagation constant for vermiculite.
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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 (see Figs. 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 preliminary 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 parameters. 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 nonrigid frame in porous materials with low density and for the absorption in the particles micro-pores. It has been shown that the tortuosity can be predicted or deduced from experimental 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 compaction state of a loose granular mix. The characteristics which the model predicts are the real and imaginary parts of the normalised characteristic impedance and propagation constant. These characteristics are required to predict the efficiency of acoustic porous absorbers and sound propagation over porous soils.
References [1] Swift, MJ. The physical properties of porous recycled materials. PhD thesis, University of Bradford, December 2000. [2] Horoshenkov K, Swift MJ. Acoustic properties of consolidated granular mixes. Appl Acoust 2001; 62(6):665–90. [3] Miki Y. Acoustic properties of porous materials—generalization of empirical models. J Acoust Soc Jap (E) 1990;11:25–8. [4] Wilson K. Simple, relaxational models for the acoustic properties of porous media. Appl Acoust 1997;50:171–88.
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[5] Voronina N. An empirical model for rigid-frame porous materials with low porosity. Appl Acoust 1999;58:295–304. [6] Horoshenkov KV, Attenborough K, Chandler-Wilde SN. Pade´ approximants for the acoustical properties of rigid frame porous media with pore size distribution. J Acoust Soc Am 1998;104:1198– 209. [7] Zou RP, Yu AB. Evaluation of the packing characteristics of mono-sized non-spherical particles. Powder Tech 1996;88:71–9. [8] Bourbie T, Zinszner B. Hydraulic and acoustic properties as a function of porosity in fountainebleau sandstone. J Geophys Res 1985;90(B13):11524–32. [9] Attenborough K. Acoustical characteristics of rigid fibrous absorbents and granular materials. J Acoust Soc Am 1983;73(3):785–99. [10] Rayleigh S. Theory of sound. 2nd ed. 1976. Vol. 2. [11] Zwikker C, Kosten CW. Sound absorbing materials. Amsterdam: Elsevier; 1949. Chapter 2. [12] Biot MA. Theory of elastic wave propagation in a fluid saturated porous solid. Parts I and II. J Acoust Soc Am 1956;28:168–91. [13] Stinson MR. The propagation of plane sound waves in narrow and wide circular tubes, and generalisation to uniform tubes of arbitrary cross-sectional shape. J Acoust Soc Am 1991;89:550–8. [14] Brennan MJ, To MM. Acoustic properties of rigid-frame porous materials—an engineering perspective. Appl Acoust 2001;62(7):793–811. [15] Voronina N. An empirical model for rigid frame porous materials with high porosity. Appl Acoust 1997;51(2):181–98. [16] Umnova O, Attenborough K, Li KM. Cell model calculations of dynamic drag parameters in packings of spheres. J Acoust Soc Am 2000;107(6):3113–9. [17] Boutin C, Royer P, Auriault JL. Acoustic absorption of porous surfacing with dual porosity. Int J Solids Structures 1998;35(34-35):4709–37. [18] Bies DA, Hansen CH. Flow resistance information for acoustical design. Appl Acoust 1980;13:357– 91. [19] Allard J-F, et al. Evaluation of tortuosity in acoustic porous materials saturated by air. Rev Sci Instrum 1994;65:754–5. [20] Leclaire P, Kelders L, Lauriks W, Glorieux C, Thoen J. Determination of the viscous characteristic length in air-filled porous materials by ultrasonic attenuation measurements. J Acoust Soc Am 1996; 99(4):1944–8.
A new empirical model for the acoustic properties of loose granular media N.N. Voroninaa, K.V. Horoshenkovb,* a Institute of Building Physics, 21 Lokomotivnyi Pr., Moscow 127238, Russia School of Engineering, Design and Technology, University of Bradford, Bradford BD7 1DP, UK
b
Received 3 December 2001; received in revised form 24 September 2002; accepted 2 October 2002
Abstract This paper examines physical parameters of loose granular mixes and their empirical relations to the acoustic performance of these mixes. In this work 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 this classification is a useful characteristic for rapid evaluation of the acoustic performance of loose granular mixes. The characteristic impedance and propagation constant have been measured for a representative selection of grain mixes and used to develop a new empirical model. This model relates the above acoustic characteristics to the characteristic particle dimension, porosity, tortuosity and specific density of the grain base, which are routinely measurable parameters. A very good agreement with the experimental data is illustrated in the frequency range of 250–4000 Hz for materials with the grain base of 0.4–3.5 mm and specific densities between 200 and 1200 kg/m3. Unlike many theoretical models for the prediction of the acoustic properties of porous media, the proposed expressions do not involve any special functions of complex argument, empirical shape factors or sophisticated characteristics of porous structure. These are practical enough to be of interest to acoustic and noise control engineers and material manufacturers. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Sound propagation; Loose granulates; Porous media; Acoustic absorption
1. Introduction In recent years the use of acoustic porous materials for indoor and outdoor applications has been stimulated by legislation in the area of environmental noise and public health. Absorbers are now routinely used to improve the acoustic performance * Corresponding author. Tel.: +44-1274-233-877; fax: +44-1274-233-888. E-mail address: [email protected] (K.V. Horoshenkov). 0003-682X/03/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0003-682X(02)00105-6
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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 alternative 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 materials. In this previous work four models [3–6] 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 characteristic 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 particularly helpful to obtain a clear insight into the physical mechanisms of the acoustic absorption in the porous media. In this respect, the availability of independently measured 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 2 discusses the physical parameters, which are required for modelling the acoustic properties of granular media. Section 3 provides the methodology for the experimental investigation, which was conducted on a representative selection of loose granular mixes. Section 4 presents the new empirical expression for the structural characteristic, which then is used in Section 5 to predict the acoustic characteristic impedance and propagation constant. Section 6 presents 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
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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 of compaction 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 dimension 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 porosity, H ¼ 1 m =g , both of which are routinely measurable characteristics. Here m 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 performance of the loose particle mix. The characteristic dimension of the particles can be found provided that the number of thepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi particles in a unit volume, Vg , of the granular mix is known, in which case D ¼ Vg =0:5233. This method yields a 2–5% 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 [9–13]. It has been shown that the thermal dissipation effects in commercial porous materials are typically small (e.g. [14]). The viscous effects are 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 ¼
D0 c 4 10
ð1Þ
where is the dynamic viscosity of air, 0 is the equilibrium density of air and c is the sound speed in air. The parameter can 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 < < 2 and to small grain mixes for 4 1. Recent results for porous materials with high porosity [15] suggest that the acoustic properties of granular materials can be affected by the value g =0 . The acoustic attenuation tends to increase with the reduced value of the grain density.
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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 viscous 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 oscillating 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 frequency range [17], where additional dissipation can take place due to the thermal non-equilibrium in micro-pores. These two effects can be accounted for phenomenologically by introducing a dimensionless parameter g M¼ 3 ; ð2Þ 10 0 where the factor of 103 is a dimensionless normalisation factor. The measured values of the density, porosity, characteristic particle dimension and the dimensionless parameters and M are provided in Table 1 for 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 predicted from the approximate expression [16] q2 ’ 1 þ ð1 HÞð2HÞ1
ð3Þ
However, 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
Vermiculite, g ¼ 1200 kg/m3, M ¼ 0:975 1 385 0.68 2 370 0.69 3 420 0.65 Rubber crumb, g ¼ 1050 kg/m3, M ¼ 0:925 4 590 0.44 5 520 0.54 Perlite, g ¼ 200kg/m3, M ¼ 0:163 6 80 0.60 7 44 0.78 Nitrile foam granulate, g ¼ 165 kg/m3, M ¼ 0:358 8 15 0.91
Characteristic particle size, D (mm)
Parameter
1.4 0.5 0.4
3.1 1.1 0.89
3.5 1.6
7.1 3.6
2.2 0.5
4.9 1.1
1.2
2.29
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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 Exp. (14)
Vermiculite 1 7610 2 56800 3 107000 Rubber crumb 4 3190 5 9800 Perlite 6 4280 7 32800 Nitrile foam granulate 8 1163
Tortuosity, q
Experiment
Exp. (3)
Exp. (4)
Experiment
7580 61000 135000
1.11 1.11 1.13
1.72 1.78 –
1.58 (1.51) 1.63 (1.76) 1.48 (1.45)
2800 13600
1.27 1.21
1.11 1.29
1.13 (1.19) 1.26 (1.23)
– –
1.50 1.08
1.48 –
– –
2800
1.03
1.32
1.31 (–)
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, which are 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 vermiculite 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 higherfrequency limit [9] Wacr ’ q=H;
ð4Þ
and 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 < 500 Hz, so that the tortuosity can be deduced from expression (4) in which Wacr is a routinely measurable constant. Table 2 provides the deduced values of the tortuosity for large grain mixes 1, 4, 5 and 6 and for medium grain mix 2. Generally, there is a good agreement between the values of the turtuosity, which were measured with the three experimental methods (see Table 2). For small grain mixes ( 4 1) the proposed experimental method for measuring tortuosity is likely to fail, because of the relatively high value of the critical frequency (fcr > 4000 Hz), above which the reliable impedance data are usually unavailable.
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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 250–4000 Hz using the standard 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 properties 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 ¼ Wa þ iWi , and propagation constant, ¼ þ i , in the porous samples can be determined from the expressions Ws ¼ Wcothð dÞ and W2s ¼ Wcothð 2dÞ. Here Wa , Wi , are real and imaginary parts of the characteristic impedance, and and are real and imaginary parts of the propagation constant, respectively, and d is 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 laboratories 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 structural characteristic Q. The structural characteristics Q has been introduced in previous work [5,15] and is included in the expression for the real part of the characteristic impedance, Wa ¼ 1 þ Q; f < fcr
ð5Þ
Fig. 1 shows 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 pffiffiffiderived previously for rigid frame porous media [5,15], where Q / ð1 HÞ= HD k , k ¼ 2 f=c being the wave number in air, f is the frequency and c is the sound speed in air. From the comparison of the
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Fig. 1. The experimentally measured and empirically predicted values of the structural characteristic as a function of frequency.
data for granular materials with different values of the parameter M (see Table 1) an empirical expression for the structural characteristic has been deduced and is proposed in the following form Q¼
0:2ð1 HÞð1 þ HÞ2 pffiffiffiffiffiffiffiffiffi : H kD
ð6Þ
The above expression has been used to calculate the predicted values of Q, which are shown in Fig. 1 and compared with the experimental data. Since Q ¼ Wa 1, the above expression can be used to determine the transition frequency fcr, which is calculated from 200ð1 HÞ2 ð1 þ HÞ4 ð7Þ
0 D 2 ðq HÞ2 using the values of the parameters provided in Tables 1 and 2 and the relation Qcr ¼ Hq 1. The critical value Qcr corresponds to the frequency at which the behaviour of the real part of the characteristic impedance W becomes frequency-independent. For frequencies f 5 fcr the real part of the characteristic impedance can be predicted by expression (4). For frequencies f < fcr , expressions (5) and (6) are sufficiently accurate for a majority of practical applications. fcr ¼
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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 the propagation constant. The expressions for these properties have been originally proposed in [5] for materials with low porosity. These expressions have been modified to fit the experimental data for granular materials and are provided below ¼
kQH 1þA
¼ k½1 þ QHð1 þ BÞ
Wi ¼
QH : 1þC
ð8Þ ð9Þ ð10Þ
Here A, B and C are coefficients which depend upon the porosity, structural characteristic and dimensionless parameters M and . 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 pffiffiffiffi A is proportional to 1=ð1 þ QÞ and that the coefficient B is proportional to 1= Q [see Fig. 2(a) and (b)]. The results also suggest that for a given value of Q the coefficient A increases and coefficient B decreases with the increasing value of M. The results for the investigated granular mixes show that for a fixed value of Q the coefficients A and C are proportional to ð1 HÞ and the coefficient B is proportional to 1=ð1 þ HÞ. These dependencies provide the basis to derive the interpolated expressions for the three coefficients, which are provided below ð1 HÞM 1þQ 1 B ¼ pffiffiffiffi Qð1 þ HÞð1 þ Q 2 MÞ A¼
ð11Þ ð12Þ
and 1H C ¼ pffiffiffiffi : Q
ð13Þ
The solid lines in Fig. 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 dimensionless 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 ¼ 0 clim ðWa þ Wi Þ as f !0
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Fig. 2. Experimental and empirically predicted values of the coefficients A, B and C.
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Fig. 2. (continued)
r¼
400ð1 HÞ2 ð1 þ HÞ5 HD 2
ð14Þ
The values of the flow resistivity which are predicted from the above expression are shown in Table 2, where they are compared to the measured data. Good agreement 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. 3–8 show the experimental and predicted values of the characteristic impedance and propagation constant for the eight materials used in the experiments. The agreement 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 predict 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 characteristic impedance is relatively independent of frequency and can be predicted accurately from expression (4). For these mixes the imaginary part of the characteristic impedance is relatively small, Wi < 1, throughout the considered frequency range. The attenuation constant for these materials is limited and the phase velocity is relatively high.
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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. 6. The real (a) and imaginary (b) parts of the propagation constant for vermiculite.
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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 (see Figs. 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 preliminary 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 parameters. 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 nonrigid frame in porous materials with low density and for the absorption in the particles micro-pores. It has been shown that the tortuosity can be predicted or deduced from experimental 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 compaction state of a loose granular mix. The characteristics which the model predicts are the real and imaginary parts of the normalised characteristic impedance and propagation constant. These characteristics are required to predict the efficiency of acoustic porous absorbers and sound propagation over porous soils.
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