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A comparison of impedance models for the inverse estimation of the non-acoustical parameters of granular absorbers
Applied Acoustics 104 (2016) 119–126
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A comparison of impedance models for the inverse estimation of the non-acoustical parameters of granular absorbers Pedro Cobo ⇑, Francisco Simón Institute of Physical and Information Technologies (ITEFI), CSIC, Serrano 144, 28006 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 29 June 2015 Received in revised form 30 October 2015 Accepted 10 November 2015 Available online 28 November 2015 Keywords: Sound absorption Impedance models Simulated Annealing Random beads packings
a b s t r a c t An impedance model relates acoustic properties of porous materials with non-acoustical parameters. Although such parameters can be measured, specific equipment is required for each of them, so that numerical methods have been proposed for estimating their value from a more manageable measurement of the normal incidence absorption coefficient in an impedance tube. This inverse procedure requires both an impedance model and an inversion technique. This paper compares four impedance models, Miki, Hamet–Berengier, Johnson–Allard–Champoux and Champoux–Stinson, when Simulated Annealing is used for the inverse estimation of the non-acoustical parameters of three granular materials, consisting of packings of small spherical glass beads of different diameters. Some of these parameters have also been measured, so that they can be compared with these estimated by the proposed inverse method. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Sound absorbing materials are profusely used in Acoustics for increasing the transmission loss across multilayer walls, decreasing the reverberation in enclosures, or dissipating the noise produced by diverse sources [1]. Most of sound absorbing materials are porous-type, which consist of a solid (rigid or flexible) skeleton and air in between. Sound absorption in porous materials is produced by the particle velocity and thermal gradients between the solid and gas phases of the material. Based on their microstructural configuration, porous absorbing materials can be classified as cellular (foams), fibrous (mineral wools) or granular (aquarium gravel, rubber crumbs) [2]. The most used sound absorbing materials are synthetic foams and fibrous wools since they offer an excellent acoustical performance at a moderate cost. Nowadays, however, there is a trend toward recycled materials, which are of the granular type, and may come from the waste generated in other production plants [3]. A clear example of this trend is the use as absorbing material of rubber crumbs coming from the recycling of out-of-use tyres [4]. The acoustical performance of an absorbing material can be predicted through an impedance model that relates acoustical variables, such as the complex characteristics impedance, Zs, and the complex wave number, ks, with non-acoustical parameters, such ⇑ Corresponding author. Tel.: +34 91 5618806; fax: +34 91 4117856. E-mail address: [email protected] (P. Cobo). http://dx.doi.org/10.1016/j.apacoust.2015.11.006 0003-682X/Ó 2015 Elsevier Ltd. All rights reserved.
as air flow resistivity, r, porosity, /, or tortuosity, q2. A variety of empirical, phenomenological, and microstructural impedance models have been proposed [5,6]. Empirical models provide regression equations for the acoustical variables based on experimental data. The Delany–Bazley [7] and Miki [8] models, fruitfully used in modeling cellular and fibrous materials, as well as the Voronina–Horoshenkov [9] model, recommended in the modeling of loose granular materials, are of the empirical type. The phenomenological approach models the material as a compressive fluid where viscous, due to particle velocity gradients, and thermal, due to thermal gradients, dissipation occurs. The Hamet–Berengier model [10], successfully used in the propagation of sound through porous pavements, is of the phenomenological type. In recent years, substantial effort has been spent in the development of microstructural models that describe the propagation of sound waves across the porous material. The Johnson–Allard–Champoux [6] and the Champoux–Stinson [11] models are of the microstructural type. In general, the more sophisticated the model is, the more non-acoustical parameters are required. In an ordinary use of a sound absorbing model (direct modeling), Fig. 1, the non-acoustical parameters are first measured and then entered into the impedance model equations to obtain its sound absorption coefficient as a function of frequency. However, some of the parameters are difficult to measure, and specific equipment is required to measure each of the non-acoustical parameters. Some authors have proposed a multiscale approach to compute these parameters from specific finite-element analyses
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Fig. 1. Direct and inverse modeling in absorbing materials. The curve at the right is the normal incidence absorption coefficient.
based on realistic representations of the actual microstructure of the porous material [12]. Alternatively, they can be inversely estimated from the measurement of the sound absorption coefficient at normal incidence [13,14]. As illustrated in Fig. 1, the normal incidence absorption coefficient, as routinely measured in an impedance tube using the transfer function method [15], is used to estimate the non-acoustical parameters. This inverse approach requires the assumption of an impedance model and an inversion technique. The objective of the inversion technique is to obtain the non-acoustical parameters that minimize the difference between the measured and modeled absorption curves. Genetic [13], Simulated Annealing [14], and non-lineal best fitting algorithms [13] have been used to estimate the non-acoustical parameters of porous materials and microperforated panels [15]. The main aim of this paper is to carry out a comparison of impedance models in the inverse estimation of the non-acoustical parameters of a granular material consisting of random packings of identical small glass beads. Simulated Annealing (SA in the following) will be assumed as the inversion technique due to the excellent performance in previous inverse problems [14,16]. The three-parameters Miki model, the phenomenological Hamet– Berengier model (HB in the following), and the microstructural Jo hnson–Allard–Champoux (JAC in the following) and Champoux– Stinson (CS in the following) models will be compared. All these models share the three following parameters: air flow resistivity, r, porosity, /, and tortuosity, q2. JAC and CS are five-parameters models. JAC uses, besides the above mentioned three parameters, the viscous, K, and thermal, K0 , characteristics lengths. CS uses instead the viscous, sp, and thermal, sk, shape factors. Non-acoustical parameters of loosely packed small spheres have been measured by Allard et al. [17]. Umnova et al. [18,19] theoretically predicted these parameters using a cell model with an adjustable cell radius which allows for hydrodynamic interactions between the spheres. Gasser et al. [20] predicted the acoustical properties of regular packing of small hollow beads by means of homogenization and FEM computations. Zielinski [12] applied also FEM techniques to study the influence of the regular sphere packings in the computation of the non-acoustical parameters. Therefore, there exist abundant results in the literature to which the results provided by the inverse estimation proposed in this paper can be compared with. The paper is organized as follows. The Miki, HB, JAC and CS impedance models are reviewed first in Section 2. The inverse estimation of the non-acoustical parameters by Simulated Annealing is examined then in Section 3. Section 4 describes in more detail the granular absorber, the measurement of some of its non-acoustical parameters and absorption coefficient, and the estimation of such non-acoustical parameters by SA. Finally, the main conclusions of this study are outlined in Section 5.
2. Impedance models Assuming a time dependence exp(ixt), the input acoustic impedance to an absorbing layer of thickness d backed by a rigid wall is [1]
Z i ðxÞ ¼ Z s coth½iks d;
ð1Þ
where Zs is the complex characteristic acoustic impedance and ks the complex wave number of the absorbing material. Once the input acoustic impedance, Zi, is known, the normal incidence reflection, R, and absorption, a0, coefficients can be calculated from
R¼
Zi Z0 ; Zi þ Z0
ð2Þ
a0 ¼ 1 jRj ; 2
ð3Þ
where Z0 is the air characteristic impedance. In the following, the equations for the complex characteristic acoustic impedance and the complex wave number are given for each impedance model. The common non-acoustic parameters for the four models are r the air flow resistivity, / the porosity, and q2 the tortuosity (notice that a1 is used also by some authors for tortuosity). 2.1. The empirical model of Miki The empirical equations for the Miki model are
0:632 0:632 Z s ¼ /q 1 þ 0:07 rfe þ 0:107i rfe 0:618 0:618 ; þ 0:160i rfe ks ¼ xc0q 1 þ 0:109 rfe
ð4Þ
where
re ¼
/ r: q2
ð5Þ
2.2. The phenomenological model of Hamet and Berengier The HB model is characterized by the equations
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qg ðxÞK g ðxÞ / qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ks ¼ x qg ðxÞ=K g ðxÞ
Zs ¼
where
qg ðxÞ ¼ q0 q2 1 þ K g ðxÞ ¼ cP0 1 þ
ifl f
ð6Þ
ðc 1Þ 1 fifh
!1
ð7Þ
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and
fl ¼ fh ¼
121
2.5. Comparison of the four models
/r 2pq0 q2
ð8Þ
r
2pq0 NPr
being q0 the air density, c the specific heats ratio of air, P0 the ambient atmospheric pressure, and NPr the Prandtl number. 2.3. The microstructural model of Johnson, Allard and Champoux The JAC model is based on the equations
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qg ðxÞK g ðxÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ks ¼ x qg ðxÞ=K g ðxÞ Z s ¼ /1
ð9Þ
with
qg ðxÞ ¼ K g ð xÞ ¼
q0 q2 /
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#
" 1þ
r/ 4q4 gq0 x 1þi ; 2 r2 K2 /2 iq q0 x
cP 0 =/
1 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K02 N Pr q0 x 8g c ðc 1Þ 1 i K02 N q x 1 þ i 16g Pr
ð10Þ
ð11Þ
0
being g the dynamic viscosity of air, and K, K0 the viscous and thermal characteristic lengths, respectively. 2.4. The microstructural model of Champoux and Stinson In the case of the CS model, the equations are
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qg ðxÞK g ðxÞ / qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ks ¼ x qg ðxÞ=K g ðxÞ
Zs ¼
ð12Þ
with
qg ðxÞ ¼ q0 q2 i K g ð xÞ ¼
r/Fðkp Þ ; x
ð13Þ
3. Inverse estimation of the non-acoustical parameters The problem of finding the set of non-acoustical parameters that minimizes the difference between the measured and theoretical absorption curves can be regarded actually as a global optimization problem, being the non-acoustical parameters the design variables, with the objective function defined as
Fðam Þ ¼
cP0 Kk 1 þ 2ðc1ÞTð K
; Þ
ð14Þ
X jaMeas ðf n Þ aThe;m ðf n Þj;
ð20Þ
n
where fn is a discrete frequency within the range of optimization,
k
aMeas ðf n Þ is the measured absorption curve of the sample, and aThe;m ðf n Þ is the theoretical absorption curve provided by the partic-
where
Fðkp Þ ¼
It would be an interesting exercise to compare the absorption curves provided by the four models for the same values of the common non-acoustical parameters. For example, let us choose the values for (r, /, q2) measured by Allard et al. [17] for glass beads of mean diameter D = 1.46 mm, namely, (r, /, q2) = (12.21 kN s m4, 0.4, 1.37). Fig. 2 shows the corresponding absorption curves for a layer thickness d = 5 cm. For the case of JAC model (K, K0 ) = (90 lm, 320 lm), and for the CS model (sp, sk) = (1.3, 0.9) in accordance with usual values in porous materials [10]. As it can be seen in Fig. 2, all the models give different curves for the same set of values, meaning that only one should fit right with the actual measured absorption curve. Usually, each model works better with the type of material its development is based on; but all of them should work appropriately with almost any kind of (granular) porous material. In this sense, an inverse method is quite convenient as it provides the values of the intrinsic properties that will give the right absorption curves for a given model (the one used in the inverse method). This way, once the non-acoustical parameters are obtained for a given model, they can be used for design or engineering porpoises. The curves provided by the Miki and HB models differ each other (the HB curve is shifted toward higher frequencies). Therefore, it can be anticipated that both models will afford different values of the non-acoustical parameters when the inverse procedure will be applied. The JAC and CS models also yield different curves, each other and compared with the Miki and HB curves, and this is likely due to the other two parameters to be fitted. Furthermore, since there is more chance to match the measured curves with a five-parameters than a three-parameters model, a better performance of the JAC and CS models could be expected.
pffiffiffiffiffiffi pffiffiffiffiffiffi 1 kp iTðkp iÞ pffiffiffiffi ; T ðkp iÞ 4 ffi 1 2 k pffiffiffi i
ð15Þ
ular impedance model for the combination of non-acoustical parameters (ðrm ; /m ; q2m ; dm Þ for the case of the Miki and HB
p
kp ¼ sp
1=2 8q2 q0 x ; r/
ð16Þ
Kk ¼ kk
pffiffiffiffiffiffiffiffiffiffiffiffi NPr i;
ð17Þ
kk ¼ sk
1=2 8q2 q0 x ; r/
ð18Þ
and
TðnÞ ¼
J 1 ðnÞ ; J 0 ðnÞ
ð19Þ
being J0 and J1 the zero and first Bessel functions, respectively. sp and sk are the shape factors related with the viscous and thermal microstructure, respectively.
Fig. 2. Normal incidence absorption curves of a packing of spheres of diameter 1.46 mm provided by each impedance model for a layer thickness of d = 5 cm.
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models, ðrm ; /m ; q2m ; Km ; K0m ; dm Þ for the case of the JAC model, and ðrm ; /m ; q2m ; spm ; sqm ; dm Þ for the case of the CS model). Notice that the layer thickness, dm, is considered also a parameter to be estimated by this inverse procedure. The design variables are constrained within the range
rm 2 ½1000; 15; 000 N s m4
4. Results
/m 2 ½0:1; 0:6 q2m 2 ½1:2; 1:6
Km 2 ½50; 300 lm K0m 2 ½100; 400 lm spm 2 ½1:2; 1:5 skm 2 ½0:75; 1:2 dm 2 ½4:8; 10:2 cm
ð21Þ
Simulated Annealing (SA), a numerical method inspired in a natural process, called annealing, is specially appropriate to solve the above outlined constrained optimization problem [21]. Let DE be the energy change from one state to another at a temperature T. The Boltzmann law establishes that the probability that the system change from one state to another is [21]
1 DE exp PrfEg ¼ ; CðTÞ KBT
ð22Þ
where C(T) is a normalization factor and KB is the Boltzmann constant. Thus, when the temperature is high, there is a high probability to accept a new state with a higher energy. However, as the temperature decreases states with higher energy are more unlikely, and eventually, when the temperature approaches to zero, only minimum energy state will likely be accepted. In this process, slow cooling is essential, allowing ample time for redistribution of atoms as they lose mobility. SA optimization of functions exploits the analogy with this natural annealing process. The application of the Boltzmann scheme to the function optimization problem is named the Metropolis algorithm [22]. To apply this algorithm to the optimization of a function is necessary to provide the following: (a) an objective function, analogous to the energy in natural annealing; (b) a space of configurations, similar to the states in metals; (c) a random mechanism to change for one configuration to another; (d) a control parameter T, analogous to the temperature in natural annealing, and (e) a cooling schedule that establishes the decreasing velocity of T. In this work, the objective function to be minimized is the sum of the absolute difference between measured and theoretical absorption curves F(am) defined by Eq. (20). The space of configurations consists of combinations of the non-acoustical parameters within the range defined by Eq. (21). To pass from one configuration to another, only one parameter is changed randomly, this ensuring that both configurations are in the vicinity of each other. To implement the Metropolis algorithm, a DF = F(am)new F(am)old factor is defined for a new configuration at temperature T. The new configuration is accepted whether DF < 0 or Pr(T) = exp(DF/T) > /, being / a random number in the interval [0, 1]. Therefore, configurations with lesser values of the objective function are accepted unconditionally, while configurations with larger values of the objective function are accepted only if the corresponding Boltzmann factor is larger than /. This mechanism of choosing always configurations with DF < 0 but sometimes configurations with DF > 0 prevents the objective function from being trapped in a local minimum. The cooling schedule includes the initial temperature, Tini, the final temperature, Tfin, and the cooling velocity geometrical law
T kþ1 ¼ fT k ;
with the cooling factor 0 < f < 1. The algorithm proceeds from Tk to Tk+1 when a number of iterations are attained. The process finalizes either when the final temperature, Tfin, is reached or when a number of changes are done without success in finding a better configuration.
ð23Þ
In order to illustrate the above described inverse estimation method, three loose granular samples of small glass beads have been prepared. The samples consist of random packings of glass beads of diameters D = 1, 2, and 3 mm (see Fig. 3). First, the three samples were subjected to a series of tests to characterize its non-acoustical properties (r, /, q2). The absorption coefficient was measured as well since it is the input parameter of the inverse method. Later, the estimation of the non-acoustical parameters by SA is carried out. Finally the agreement degree between the measured and estimated by SA parameters is analyzed. 4.1. Measurement of the non-acoustical parameters The flow resistivity, r, plays a main role in the acoustic behavior of porous materials. It takes into account the resistance exerted by the porous material against the air flowing through, and is defined as
r¼
DP A ; U d
ð24Þ
where DP is the air pressure difference across the porous layer, U is the velocity of the mean flow of air, A is the cross-sectional area of the layer, and d is the thickness. The flow resistivity is measured according to the standardized ISO method [23] which consists on passing a flow of air through the sample layer of known A and d, so that r is determined using Eq. (24) once U and DP have been measured. The porosity, /, quantifies the amount of air inside the material and can be defined as the ratio of the air volume inside the material, Va, to the total volume taken up by the material, VT. It was measured by the geometrical method proposed by Zielinski [12].
Fig. 3. A sample of the small glass beads of diameter 1 mm.
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4.3. Inverse estimation of the non-acoustical parameters
Table 1 Measured non-acoustical parameters of packing beads of different diameters.
Resistivity, r (kN s m Porosity, / Tortuosity, q2
4
)
D = 1 mm
D = 2 mm
D = 3 mm
14.86 ± 0.44 0.405 ± 0.012 1.485 ± 0.044
6.544 ± 0.074 0.405 ± 0.012 1.454 ± 0.053
2.880 ± 0.063 0.405 ± 0.012 1.548 ± 0.050
The assumed identical beads were randomly poured in a cylinder to fill a certain height, so that the total volume of the layer, VT, could be calculated. Furthermore, the total number of beads filling the cylindrical recipient could be estimated by dividing the total weight by that of a single bead. Since the volume of a single bead is known (pD3/6), the total volume occupied by the packing of beads, and hence the porosity, can be calculated. The tortuosity, q2, is related to the total path the wave has to cover, with respect to a straight path across a material layer. It is obtained by the equation [24]
q2 ¼ /
Rc ; Rf
123
ð25Þ
where Rc and Rf are the electrical resistances of the sample saturated in a fluid and that of the fluid itself, respectively, and / is the porosity. A series of measurements were carried out according to the methods described above. A total of five tests were done for each quantity (r, /, and q2). Table 1 summarizes the results of the measurements of these non-acoustical parameters. Uncertainty values calculated for a 95% confidence level are also provided for the measured values. These results will then be compared with those estimated by SA. Notice that, while porosity does not depend on the particles size, the flow resistivity decreases, and the tortuosity increases, with the beads diameter.
In this Section, the SA algorithm described in Section 3 is used, with Tini = 1, Tfin = 106 and f = 0.95, to minimize the objective function defined in Eq. (20) with the constraints defined by Eq. (21). The algorithm finalizes either when the final temperature is reached or when 50 changes of parameters are done without success in finding a better configuration. To illustrate the functioning of the SA algorithm, the evolution of the temperature and objective function is shown in Fig. 5 for a case. At the beginning, when the temperature is still high, there is a high probability to accept successful configurations with higher values of the objective function. This probability, however, decreases as the temperature cools, and eventually, when the temperature is very low, only configurations with lower values of the objective functions are accepted. This is the way as SA finds the global minimum value of the objective function without be trapped in a local minimum. Now, the SA algorithm is combined with each of the four impedance models to estimate the non-acoustical parameters corresponding to each of the three samples (D = 1, 2 and 3 mm) and two thicknesses (d = 5 and 10 cm). For every sample, the SA algorithm is run five times, so that the average non-acoustical parameters, as well as the standard deviations, can be obtained. Tables 2–4 summarizes the results for the three samples of D = 1, 2 and 3 mm, respectively. For each D, the summarized non-acoustical parameters correspond to the mean of those of
4.2. Measurements of the absorption curves For each packing, two samples of the loose glass beads, with nominal thicknesses d = 5 cm and d = 10 cm, were prepared to be measured in a B&K 4206 impedance tube using the standard ISO10534-2 [15]. The transfer function between two B&K 4187 microphones separated 2 cm was used to calculate the normal incidence sound absorption coefficient of the samples mounted at one end of the tube with inner diameter 29 mm, in the frequency range from 200 Hz to 6500 Hz. Fig. 4 shows the six measured absorption curves.
Fig. 5. Evolution of the control parameter (T) and objective function (F(am)) in a run of the SA algorithm.
Fig. 4. Normal incidence absorption curves of the three packing beads samples for a layer of thickness (a) d = 5 cm, and (b) d = 10 cm.
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Table 2 Non-acoustical parameters for the sample D = 1 mm.
Measured SA + Miki Error (%) SA + HB Error (%) SA + JAC Error (%) SA + CS Error (%)
r (N s m4)
/
q2
14860 ± 440 14950 ± 450 0.6 14950 ± 310 0.6 14140 ± 230 4.8 14260 ± 280 4.0
0.405 ± 0.012 0.415 ± 0.005 2.5 0.368 ± 0.002 9 0.440 ± 0.008 8.6 0.428 ± 0.007 5.7
1.485 ± 0.044 1.477 ± 0.036 0.54 1.560 ± 0.011 5.1 1.454 ± 0.046 2.1 1.492 ± 0.044 0.5
r (N s m4)
/
q2
6544 ± 74 8200 ± 420 25.3 8960 ± 150 36.9 7610 ± 260 16.3 5300 ± 190 19.0
0.405 ± 0.012 0.422 ± 0.07 4.2 0.398 ± 0.003 1.7 0.447 ± 0.006 10.3 0.442 ± 0.007 9.1
1.454 ± 0.053 1.530 ± 0.053 11 1.678 ± 0.025 15.4 1.520 ± 0.042 4.5 1.529 ± 0.026 5.1
r (N s m4)
/
q2
2880 ± 63 3740 ± 420 29.80 7330 ± 440 154.5 2260 ± 210 21.5 2070 ± 160 28.1
0.405 ± 0.012 0.464 ± 0.025 14.5 0.367 ± 0.010 9.3 0.433 ± 0.007 6.9 0.487 ± 0.05 20
1.548 ± 0.050 1.510 ± 0.043 2.4 1.555 ± 0.052 0.45 1.423 ± 0.048 8 1.506 ± 0.035 2.7
K (lm) or sp
K0 (lm) or sk
110 ± 2
240 ± 6
1.30 ± 0.03
1.10 ± 0.01
K (lm) or sp
K0 (lm) or sk
185 ± 3
343 ± 8
1.39 ± 0.03
0.95 ± 0.01
K (lm) or sp
K0 (lm) or sk
275 ± 3
350 ± 2
1.41 ± 0.07
0.99 ± 0.02
Table 3 Non-acoustical parameters for the sample D = 2 mm.
Measured SA + Miki Error (%) SA + HB Error (%) SA + JAC Error (%) SA + CS Error (%)
Table 4 Non-acoustical parameters for the sample D = 3 mm.
Measured SA + Miki Error (%) SA + HB Error (%) SA + JAC Error (%) SA + CS Error (%)
Fig. 6. Measured and SA estimated normal incidence absorption curves for the glass beads of D = 1 mm and for (a) d = 5 cm, and (b) d = 10 cm.
P. Cobo, F. Simón / Applied Acoustics 104 (2016) 119–126
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Fig. 7. Measured and SA estimated normal incidence absorption curves for the glass beads of D = 2 mm and for (a) d = 5 cm, and (b) d = 10 cm.
Fig. 8. Measured and SA estimated normal incidence absorption curves for the glass beads of D = 3 mm and for (a) d = 5 cm, and (b) d = 10 cm.
d = 5 and 10 cm. The values included in the fifth and sixth columns of these Tables are K and K0 , for the SA + JAC case and sp and sk, for the SA + CS case. For each estimated non-acoustical parameter, the relative error, with respect to the measured one, is also included in Tables 2–4. In the three samples, the HB model provides the worse estimation, as indicated by the highest relative errors. The other three models affords good estimations of the (r, /, q2) parameters, being the JAC model the one with the lowest relative errors. The non-acoustical parameters reported in Tables 2–4 are similar to these measured or calculated by others authors. For instance, Allard et al. [17] provided the measured values (r, /, q2, K, K0 ) = (12210 N s m4, 0.4, 1.37, 90 lm, 320 lm) for random packings of glass beads of D = 1.46 mm. Gasser et al. [20] obtained the numerical values (/, q2, K, K0 ) = (0.36, 1.65, 260 lm, 400 lm) for sphere packings of D = 2 mm. Umnova et al. [18] also applied numerical techniques to obtain the values (/, q2, K) = (0.4, 1.37, 90 lm) for sphere packings of D = 1.46 mm. Figs. 6–8 shows the measured and SA estimated for each model corresponding to the samples of D = 1, 2 and 3 mm, respectively. The SA estimated curves are obtained with the non-acoustical parameters summarized in Tables 2–4. Again, the HB model seems to yield the absorption curves that match worse to the measured
ones. The absorption curves provided by the other three models, on the other hand, give curves that fit reasonably well to the measured curves.
5. Conclusions In this paper, a comparison of impedance models in the inverse estimation of the non-acoustical parameters of a granular material consisting of random packings of identical small glass beads has been carried out. The parameters have been inversely estimated from the measurement of the sound absorption coefficient at normal incidence in an impedance tube using the transfer function method. This inverse approach requires the assumption of an impedance model and an inversion technique. The objective of the inversion technique is to obtain the non-acoustical parameters that minimize the difference between the measured and modeled absorption curves. Simulated Annealing has been assumed as the inversion technique due to the excellent performance in previous inverse problems. The three-parameters Miki model, the phenomenological Hamet–Berengier model, and the microstructural J ohnson–Allard–Champoux and Champoux–Stinson models, have
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been chosen for the comparison. All these models share three non-acoustical parameters: air flow resistivity, r, porosity, /, and tortuosity, q2. The Johnson–Allard–Champoux and the Champoux–Stinson are five-parameters models that use, besides the above mentioned three parameters, the viscous, K, and thermal, K0 , characteristics lengths, the first one, and the viscous, sp, and thermal, sk, shape factors, the latter. The comparison of the estimated parameters with those measured in three granular samples demonstrated that the Hamet–Berengier model provides the worse estimation, as indicated by the highest relative errors. The other three models afford good estimations of the parameters, being the Johnson–Allard–Ch ampoux model the one with the lowest relative errors. This result was expected, since the Hamet–Berengier model was proposed for characterizing porous pavements. Furthermore, since there is more chance to match the measured curves with a five-parameters than a three-parameters model, a better performance of the Johnson–Allard–Champoux and Champoux–Stinson models could also be anticipated. Acknowledgements The technical assistance of Eduardo de Andrés and Marco Cortés is kindly acknowledged. References [1] Beranek LL, Vér IL. Noise and vibration control engineering. New York: John Wiley & Sons; 1992. [2] Arenas JP, Crocker MJ. Recent trends in porous sound-absorbing materials. Sound Vib 2010(July):12–7. [3] Asdrubali F, Schiavoni S, Horoshenkov KV. A review of sustainable materials for acoustic applications. Build Acoust 2012;19:283–312. [4] Pfretzschner J, Rodríguez RM. Acoustic properties of rubber crumbs. Polym Test 1999;18:81–2. [5] Attenborough K, Li KM, Horoshenkov K. Predicting outdoor sound. London: Taylor & Francis; 2007.
[6] Allard JF, Atalla N. Propagation of sound in porous media. Chichester: John Wiley & Sons; 2009. [7] Delany ME, Bazley EN. Acoustical properties of fibrous absorbent materials. Appl Acoust 1970;3:105–16. [8] Miki Y. Acoustic properties of porous materials – generalizations of empirical models. J Acoust Soc Jpn 1990;11:25–8. [9] Voronina NN, Horoshenkov KV. Acoustic properties of unconsolidated granular mixes. Appl Acoust 2004;65:673–91. [10] Berengier MC, Stinson MR, Daigle GA, Hamet JF. Porous road pavements: acoustic characterization and propagation effects. J Acoust Soc Am 1997;101:155–62. [11] Champoux Y, Stinson MR. On acoustical models for sound propagation in rigid frame materials and the influence of shape factors. J Acoust Soc Am 1992;92:1120–31. [12] Zielinski TG. Microstructure-based calculations and experimental results for sound absorbing porous layers of randomly packed rigid spherical beads. J Appl Phys 2014;116(034905):1–17. [13] Bonfiglio P, Pompoli F. Inversion problems for determining physical parameters of porous materials: overview and comparison between different methods. Acta Acust United Acust 2013;99:341–51. [14] Cobo P, Moraes E, Simón F. Inverse estimation of the non-acoustical parameters of loose granular absorbers by Simulated Annealing. Build Environ 2015. [15] ISO 10534-2. Acoustics – determination of sound absorption coefficient and impedance in impedance tubes. Part 2: Transfer function method; 1998. [16] Ruiz H, Cobo P, Jacobsen F. Optimization of multiple-layer microperforated panels by simulated annealing. Appl Acoust 2011;72:772–6. [17] Allard JF, Henry M, Tizianel J, Kelders L, Lauriks W. Sound propagation in airsaturated random packings of beads. J Acoust Soc Am 1998;104:2004–7. [18] Umnova O, Attenborough K, Li KM. Cell model calculations of dynamic drag parameters in packing of spheres. J Acoust Soc Am 2000;107:3113–9. [19] Umnova O, Attenborough K, Li KM. A cell model for the acoustical parameters of packing of spheres. Acta Acust United Acust 2001;87:226–35. [20] Gasser S, Paun F, Bréchet Y. Absorptive properties of rigid porous media: application to face centered cubic sphere packing. J Acoust Soc Am 2005;117:2090–9. [21] Press WH, Flannery BP, Teukolski SA, Vetterling WT. Combinatorial minimization: method of simulated annealing. In: Numerical recipes. Cambridge: Cambridge Press; 1986. [22] Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. Equation of state calculations by fast computing machines. J Chem Phys 1953;21:1087–92. [23] ISO 9053. Acoustics – materials for acoustical applications – determination of airflow resistance; 1991. [24] Allard JF, Castagnede B, Henry M, Lauriks W. Evaluation of tortuosity in acoustic porous materials saturated by air. Rev Sci Instrum 1994;65:754–5.
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A comparison of impedance models for the inverse estimation of the non-acoustical parameters of granular absorbers Pedro Cobo ⇑, Francisco Simón Institute of Physical and Information Technologies (ITEFI), CSIC, Serrano 144, 28006 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 29 June 2015 Received in revised form 30 October 2015 Accepted 10 November 2015 Available online 28 November 2015 Keywords: Sound absorption Impedance models Simulated Annealing Random beads packings
a b s t r a c t An impedance model relates acoustic properties of porous materials with non-acoustical parameters. Although such parameters can be measured, specific equipment is required for each of them, so that numerical methods have been proposed for estimating their value from a more manageable measurement of the normal incidence absorption coefficient in an impedance tube. This inverse procedure requires both an impedance model and an inversion technique. This paper compares four impedance models, Miki, Hamet–Berengier, Johnson–Allard–Champoux and Champoux–Stinson, when Simulated Annealing is used for the inverse estimation of the non-acoustical parameters of three granular materials, consisting of packings of small spherical glass beads of different diameters. Some of these parameters have also been measured, so that they can be compared with these estimated by the proposed inverse method. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Sound absorbing materials are profusely used in Acoustics for increasing the transmission loss across multilayer walls, decreasing the reverberation in enclosures, or dissipating the noise produced by diverse sources [1]. Most of sound absorbing materials are porous-type, which consist of a solid (rigid or flexible) skeleton and air in between. Sound absorption in porous materials is produced by the particle velocity and thermal gradients between the solid and gas phases of the material. Based on their microstructural configuration, porous absorbing materials can be classified as cellular (foams), fibrous (mineral wools) or granular (aquarium gravel, rubber crumbs) [2]. The most used sound absorbing materials are synthetic foams and fibrous wools since they offer an excellent acoustical performance at a moderate cost. Nowadays, however, there is a trend toward recycled materials, which are of the granular type, and may come from the waste generated in other production plants [3]. A clear example of this trend is the use as absorbing material of rubber crumbs coming from the recycling of out-of-use tyres [4]. The acoustical performance of an absorbing material can be predicted through an impedance model that relates acoustical variables, such as the complex characteristics impedance, Zs, and the complex wave number, ks, with non-acoustical parameters, such ⇑ Corresponding author. Tel.: +34 91 5618806; fax: +34 91 4117856. E-mail address: [email protected] (P. Cobo). http://dx.doi.org/10.1016/j.apacoust.2015.11.006 0003-682X/Ó 2015 Elsevier Ltd. All rights reserved.
as air flow resistivity, r, porosity, /, or tortuosity, q2. A variety of empirical, phenomenological, and microstructural impedance models have been proposed [5,6]. Empirical models provide regression equations for the acoustical variables based on experimental data. The Delany–Bazley [7] and Miki [8] models, fruitfully used in modeling cellular and fibrous materials, as well as the Voronina–Horoshenkov [9] model, recommended in the modeling of loose granular materials, are of the empirical type. The phenomenological approach models the material as a compressive fluid where viscous, due to particle velocity gradients, and thermal, due to thermal gradients, dissipation occurs. The Hamet–Berengier model [10], successfully used in the propagation of sound through porous pavements, is of the phenomenological type. In recent years, substantial effort has been spent in the development of microstructural models that describe the propagation of sound waves across the porous material. The Johnson–Allard–Champoux [6] and the Champoux–Stinson [11] models are of the microstructural type. In general, the more sophisticated the model is, the more non-acoustical parameters are required. In an ordinary use of a sound absorbing model (direct modeling), Fig. 1, the non-acoustical parameters are first measured and then entered into the impedance model equations to obtain its sound absorption coefficient as a function of frequency. However, some of the parameters are difficult to measure, and specific equipment is required to measure each of the non-acoustical parameters. Some authors have proposed a multiscale approach to compute these parameters from specific finite-element analyses
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Fig. 1. Direct and inverse modeling in absorbing materials. The curve at the right is the normal incidence absorption coefficient.
based on realistic representations of the actual microstructure of the porous material [12]. Alternatively, they can be inversely estimated from the measurement of the sound absorption coefficient at normal incidence [13,14]. As illustrated in Fig. 1, the normal incidence absorption coefficient, as routinely measured in an impedance tube using the transfer function method [15], is used to estimate the non-acoustical parameters. This inverse approach requires the assumption of an impedance model and an inversion technique. The objective of the inversion technique is to obtain the non-acoustical parameters that minimize the difference between the measured and modeled absorption curves. Genetic [13], Simulated Annealing [14], and non-lineal best fitting algorithms [13] have been used to estimate the non-acoustical parameters of porous materials and microperforated panels [15]. The main aim of this paper is to carry out a comparison of impedance models in the inverse estimation of the non-acoustical parameters of a granular material consisting of random packings of identical small glass beads. Simulated Annealing (SA in the following) will be assumed as the inversion technique due to the excellent performance in previous inverse problems [14,16]. The three-parameters Miki model, the phenomenological Hamet– Berengier model (HB in the following), and the microstructural Jo hnson–Allard–Champoux (JAC in the following) and Champoux– Stinson (CS in the following) models will be compared. All these models share the three following parameters: air flow resistivity, r, porosity, /, and tortuosity, q2. JAC and CS are five-parameters models. JAC uses, besides the above mentioned three parameters, the viscous, K, and thermal, K0 , characteristics lengths. CS uses instead the viscous, sp, and thermal, sk, shape factors. Non-acoustical parameters of loosely packed small spheres have been measured by Allard et al. [17]. Umnova et al. [18,19] theoretically predicted these parameters using a cell model with an adjustable cell radius which allows for hydrodynamic interactions between the spheres. Gasser et al. [20] predicted the acoustical properties of regular packing of small hollow beads by means of homogenization and FEM computations. Zielinski [12] applied also FEM techniques to study the influence of the regular sphere packings in the computation of the non-acoustical parameters. Therefore, there exist abundant results in the literature to which the results provided by the inverse estimation proposed in this paper can be compared with. The paper is organized as follows. The Miki, HB, JAC and CS impedance models are reviewed first in Section 2. The inverse estimation of the non-acoustical parameters by Simulated Annealing is examined then in Section 3. Section 4 describes in more detail the granular absorber, the measurement of some of its non-acoustical parameters and absorption coefficient, and the estimation of such non-acoustical parameters by SA. Finally, the main conclusions of this study are outlined in Section 5.
2. Impedance models Assuming a time dependence exp(ixt), the input acoustic impedance to an absorbing layer of thickness d backed by a rigid wall is [1]
Z i ðxÞ ¼ Z s coth½iks d;
ð1Þ
where Zs is the complex characteristic acoustic impedance and ks the complex wave number of the absorbing material. Once the input acoustic impedance, Zi, is known, the normal incidence reflection, R, and absorption, a0, coefficients can be calculated from
R¼
Zi Z0 ; Zi þ Z0
ð2Þ
a0 ¼ 1 jRj ; 2
ð3Þ
where Z0 is the air characteristic impedance. In the following, the equations for the complex characteristic acoustic impedance and the complex wave number are given for each impedance model. The common non-acoustic parameters for the four models are r the air flow resistivity, / the porosity, and q2 the tortuosity (notice that a1 is used also by some authors for tortuosity). 2.1. The empirical model of Miki The empirical equations for the Miki model are
0:632 0:632 Z s ¼ /q 1 þ 0:07 rfe þ 0:107i rfe 0:618 0:618 ; þ 0:160i rfe ks ¼ xc0q 1 þ 0:109 rfe
ð4Þ
where
re ¼
/ r: q2
ð5Þ
2.2. The phenomenological model of Hamet and Berengier The HB model is characterized by the equations
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qg ðxÞK g ðxÞ / qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ks ¼ x qg ðxÞ=K g ðxÞ
Zs ¼
where
qg ðxÞ ¼ q0 q2 1 þ K g ðxÞ ¼ cP0 1 þ
ifl f
ð6Þ
ðc 1Þ 1 fifh
!1
ð7Þ
P. Cobo, F. Simón / Applied Acoustics 104 (2016) 119–126
and
fl ¼ fh ¼
121
2.5. Comparison of the four models
/r 2pq0 q2
ð8Þ
r
2pq0 NPr
being q0 the air density, c the specific heats ratio of air, P0 the ambient atmospheric pressure, and NPr the Prandtl number. 2.3. The microstructural model of Johnson, Allard and Champoux The JAC model is based on the equations
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qg ðxÞK g ðxÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ks ¼ x qg ðxÞ=K g ðxÞ Z s ¼ /1
ð9Þ
with
qg ðxÞ ¼ K g ð xÞ ¼
q0 q2 /
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#
" 1þ
r/ 4q4 gq0 x 1þi ; 2 r2 K2 /2 iq q0 x
cP 0 =/
1 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K02 N Pr q0 x 8g c ðc 1Þ 1 i K02 N q x 1 þ i 16g Pr
ð10Þ
ð11Þ
0
being g the dynamic viscosity of air, and K, K0 the viscous and thermal characteristic lengths, respectively. 2.4. The microstructural model of Champoux and Stinson In the case of the CS model, the equations are
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qg ðxÞK g ðxÞ / qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ks ¼ x qg ðxÞ=K g ðxÞ
Zs ¼
ð12Þ
with
qg ðxÞ ¼ q0 q2 i K g ð xÞ ¼
r/Fðkp Þ ; x
ð13Þ
3. Inverse estimation of the non-acoustical parameters The problem of finding the set of non-acoustical parameters that minimizes the difference between the measured and theoretical absorption curves can be regarded actually as a global optimization problem, being the non-acoustical parameters the design variables, with the objective function defined as
Fðam Þ ¼
cP0 Kk 1 þ 2ðc1ÞTð K
; Þ
ð14Þ
X jaMeas ðf n Þ aThe;m ðf n Þj;
ð20Þ
n
where fn is a discrete frequency within the range of optimization,
k
aMeas ðf n Þ is the measured absorption curve of the sample, and aThe;m ðf n Þ is the theoretical absorption curve provided by the partic-
where
Fðkp Þ ¼
It would be an interesting exercise to compare the absorption curves provided by the four models for the same values of the common non-acoustical parameters. For example, let us choose the values for (r, /, q2) measured by Allard et al. [17] for glass beads of mean diameter D = 1.46 mm, namely, (r, /, q2) = (12.21 kN s m4, 0.4, 1.37). Fig. 2 shows the corresponding absorption curves for a layer thickness d = 5 cm. For the case of JAC model (K, K0 ) = (90 lm, 320 lm), and for the CS model (sp, sk) = (1.3, 0.9) in accordance with usual values in porous materials [10]. As it can be seen in Fig. 2, all the models give different curves for the same set of values, meaning that only one should fit right with the actual measured absorption curve. Usually, each model works better with the type of material its development is based on; but all of them should work appropriately with almost any kind of (granular) porous material. In this sense, an inverse method is quite convenient as it provides the values of the intrinsic properties that will give the right absorption curves for a given model (the one used in the inverse method). This way, once the non-acoustical parameters are obtained for a given model, they can be used for design or engineering porpoises. The curves provided by the Miki and HB models differ each other (the HB curve is shifted toward higher frequencies). Therefore, it can be anticipated that both models will afford different values of the non-acoustical parameters when the inverse procedure will be applied. The JAC and CS models also yield different curves, each other and compared with the Miki and HB curves, and this is likely due to the other two parameters to be fitted. Furthermore, since there is more chance to match the measured curves with a five-parameters than a three-parameters model, a better performance of the JAC and CS models could be expected.
pffiffiffiffiffiffi pffiffiffiffiffiffi 1 kp iTðkp iÞ pffiffiffiffi ; T ðkp iÞ 4 ffi 1 2 k pffiffiffi i
ð15Þ
ular impedance model for the combination of non-acoustical parameters (ðrm ; /m ; q2m ; dm Þ for the case of the Miki and HB
p
kp ¼ sp
1=2 8q2 q0 x ; r/
ð16Þ
Kk ¼ kk
pffiffiffiffiffiffiffiffiffiffiffiffi NPr i;
ð17Þ
kk ¼ sk
1=2 8q2 q0 x ; r/
ð18Þ
and
TðnÞ ¼
J 1 ðnÞ ; J 0 ðnÞ
ð19Þ
being J0 and J1 the zero and first Bessel functions, respectively. sp and sk are the shape factors related with the viscous and thermal microstructure, respectively.
Fig. 2. Normal incidence absorption curves of a packing of spheres of diameter 1.46 mm provided by each impedance model for a layer thickness of d = 5 cm.
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P. Cobo, F. Simón / Applied Acoustics 104 (2016) 119–126
models, ðrm ; /m ; q2m ; Km ; K0m ; dm Þ for the case of the JAC model, and ðrm ; /m ; q2m ; spm ; sqm ; dm Þ for the case of the CS model). Notice that the layer thickness, dm, is considered also a parameter to be estimated by this inverse procedure. The design variables are constrained within the range
rm 2 ½1000; 15; 000 N s m4
4. Results
/m 2 ½0:1; 0:6 q2m 2 ½1:2; 1:6
Km 2 ½50; 300 lm K0m 2 ½100; 400 lm spm 2 ½1:2; 1:5 skm 2 ½0:75; 1:2 dm 2 ½4:8; 10:2 cm
ð21Þ
Simulated Annealing (SA), a numerical method inspired in a natural process, called annealing, is specially appropriate to solve the above outlined constrained optimization problem [21]. Let DE be the energy change from one state to another at a temperature T. The Boltzmann law establishes that the probability that the system change from one state to another is [21]
1 DE exp PrfEg ¼ ; CðTÞ KBT
ð22Þ
where C(T) is a normalization factor and KB is the Boltzmann constant. Thus, when the temperature is high, there is a high probability to accept a new state with a higher energy. However, as the temperature decreases states with higher energy are more unlikely, and eventually, when the temperature approaches to zero, only minimum energy state will likely be accepted. In this process, slow cooling is essential, allowing ample time for redistribution of atoms as they lose mobility. SA optimization of functions exploits the analogy with this natural annealing process. The application of the Boltzmann scheme to the function optimization problem is named the Metropolis algorithm [22]. To apply this algorithm to the optimization of a function is necessary to provide the following: (a) an objective function, analogous to the energy in natural annealing; (b) a space of configurations, similar to the states in metals; (c) a random mechanism to change for one configuration to another; (d) a control parameter T, analogous to the temperature in natural annealing, and (e) a cooling schedule that establishes the decreasing velocity of T. In this work, the objective function to be minimized is the sum of the absolute difference between measured and theoretical absorption curves F(am) defined by Eq. (20). The space of configurations consists of combinations of the non-acoustical parameters within the range defined by Eq. (21). To pass from one configuration to another, only one parameter is changed randomly, this ensuring that both configurations are in the vicinity of each other. To implement the Metropolis algorithm, a DF = F(am)new F(am)old factor is defined for a new configuration at temperature T. The new configuration is accepted whether DF < 0 or Pr(T) = exp(DF/T) > /, being / a random number in the interval [0, 1]. Therefore, configurations with lesser values of the objective function are accepted unconditionally, while configurations with larger values of the objective function are accepted only if the corresponding Boltzmann factor is larger than /. This mechanism of choosing always configurations with DF < 0 but sometimes configurations with DF > 0 prevents the objective function from being trapped in a local minimum. The cooling schedule includes the initial temperature, Tini, the final temperature, Tfin, and the cooling velocity geometrical law
T kþ1 ¼ fT k ;
with the cooling factor 0 < f < 1. The algorithm proceeds from Tk to Tk+1 when a number of iterations are attained. The process finalizes either when the final temperature, Tfin, is reached or when a number of changes are done without success in finding a better configuration.
ð23Þ
In order to illustrate the above described inverse estimation method, three loose granular samples of small glass beads have been prepared. The samples consist of random packings of glass beads of diameters D = 1, 2, and 3 mm (see Fig. 3). First, the three samples were subjected to a series of tests to characterize its non-acoustical properties (r, /, q2). The absorption coefficient was measured as well since it is the input parameter of the inverse method. Later, the estimation of the non-acoustical parameters by SA is carried out. Finally the agreement degree between the measured and estimated by SA parameters is analyzed. 4.1. Measurement of the non-acoustical parameters The flow resistivity, r, plays a main role in the acoustic behavior of porous materials. It takes into account the resistance exerted by the porous material against the air flowing through, and is defined as
r¼
DP A ; U d
ð24Þ
where DP is the air pressure difference across the porous layer, U is the velocity of the mean flow of air, A is the cross-sectional area of the layer, and d is the thickness. The flow resistivity is measured according to the standardized ISO method [23] which consists on passing a flow of air through the sample layer of known A and d, so that r is determined using Eq. (24) once U and DP have been measured. The porosity, /, quantifies the amount of air inside the material and can be defined as the ratio of the air volume inside the material, Va, to the total volume taken up by the material, VT. It was measured by the geometrical method proposed by Zielinski [12].
Fig. 3. A sample of the small glass beads of diameter 1 mm.
P. Cobo, F. Simón / Applied Acoustics 104 (2016) 119–126
4.3. Inverse estimation of the non-acoustical parameters
Table 1 Measured non-acoustical parameters of packing beads of different diameters.
Resistivity, r (kN s m Porosity, / Tortuosity, q2
4
)
D = 1 mm
D = 2 mm
D = 3 mm
14.86 ± 0.44 0.405 ± 0.012 1.485 ± 0.044
6.544 ± 0.074 0.405 ± 0.012 1.454 ± 0.053
2.880 ± 0.063 0.405 ± 0.012 1.548 ± 0.050
The assumed identical beads were randomly poured in a cylinder to fill a certain height, so that the total volume of the layer, VT, could be calculated. Furthermore, the total number of beads filling the cylindrical recipient could be estimated by dividing the total weight by that of a single bead. Since the volume of a single bead is known (pD3/6), the total volume occupied by the packing of beads, and hence the porosity, can be calculated. The tortuosity, q2, is related to the total path the wave has to cover, with respect to a straight path across a material layer. It is obtained by the equation [24]
q2 ¼ /
Rc ; Rf
123
ð25Þ
where Rc and Rf are the electrical resistances of the sample saturated in a fluid and that of the fluid itself, respectively, and / is the porosity. A series of measurements were carried out according to the methods described above. A total of five tests were done for each quantity (r, /, and q2). Table 1 summarizes the results of the measurements of these non-acoustical parameters. Uncertainty values calculated for a 95% confidence level are also provided for the measured values. These results will then be compared with those estimated by SA. Notice that, while porosity does not depend on the particles size, the flow resistivity decreases, and the tortuosity increases, with the beads diameter.
In this Section, the SA algorithm described in Section 3 is used, with Tini = 1, Tfin = 106 and f = 0.95, to minimize the objective function defined in Eq. (20) with the constraints defined by Eq. (21). The algorithm finalizes either when the final temperature is reached or when 50 changes of parameters are done without success in finding a better configuration. To illustrate the functioning of the SA algorithm, the evolution of the temperature and objective function is shown in Fig. 5 for a case. At the beginning, when the temperature is still high, there is a high probability to accept successful configurations with higher values of the objective function. This probability, however, decreases as the temperature cools, and eventually, when the temperature is very low, only configurations with lower values of the objective functions are accepted. This is the way as SA finds the global minimum value of the objective function without be trapped in a local minimum. Now, the SA algorithm is combined with each of the four impedance models to estimate the non-acoustical parameters corresponding to each of the three samples (D = 1, 2 and 3 mm) and two thicknesses (d = 5 and 10 cm). For every sample, the SA algorithm is run five times, so that the average non-acoustical parameters, as well as the standard deviations, can be obtained. Tables 2–4 summarizes the results for the three samples of D = 1, 2 and 3 mm, respectively. For each D, the summarized non-acoustical parameters correspond to the mean of those of
4.2. Measurements of the absorption curves For each packing, two samples of the loose glass beads, with nominal thicknesses d = 5 cm and d = 10 cm, were prepared to be measured in a B&K 4206 impedance tube using the standard ISO10534-2 [15]. The transfer function between two B&K 4187 microphones separated 2 cm was used to calculate the normal incidence sound absorption coefficient of the samples mounted at one end of the tube with inner diameter 29 mm, in the frequency range from 200 Hz to 6500 Hz. Fig. 4 shows the six measured absorption curves.
Fig. 5. Evolution of the control parameter (T) and objective function (F(am)) in a run of the SA algorithm.
Fig. 4. Normal incidence absorption curves of the three packing beads samples for a layer of thickness (a) d = 5 cm, and (b) d = 10 cm.
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Table 2 Non-acoustical parameters for the sample D = 1 mm.
Measured SA + Miki Error (%) SA + HB Error (%) SA + JAC Error (%) SA + CS Error (%)
r (N s m4)
/
q2
14860 ± 440 14950 ± 450 0.6 14950 ± 310 0.6 14140 ± 230 4.8 14260 ± 280 4.0
0.405 ± 0.012 0.415 ± 0.005 2.5 0.368 ± 0.002 9 0.440 ± 0.008 8.6 0.428 ± 0.007 5.7
1.485 ± 0.044 1.477 ± 0.036 0.54 1.560 ± 0.011 5.1 1.454 ± 0.046 2.1 1.492 ± 0.044 0.5
r (N s m4)
/
q2
6544 ± 74 8200 ± 420 25.3 8960 ± 150 36.9 7610 ± 260 16.3 5300 ± 190 19.0
0.405 ± 0.012 0.422 ± 0.07 4.2 0.398 ± 0.003 1.7 0.447 ± 0.006 10.3 0.442 ± 0.007 9.1
1.454 ± 0.053 1.530 ± 0.053 11 1.678 ± 0.025 15.4 1.520 ± 0.042 4.5 1.529 ± 0.026 5.1
r (N s m4)
/
q2
2880 ± 63 3740 ± 420 29.80 7330 ± 440 154.5 2260 ± 210 21.5 2070 ± 160 28.1
0.405 ± 0.012 0.464 ± 0.025 14.5 0.367 ± 0.010 9.3 0.433 ± 0.007 6.9 0.487 ± 0.05 20
1.548 ± 0.050 1.510 ± 0.043 2.4 1.555 ± 0.052 0.45 1.423 ± 0.048 8 1.506 ± 0.035 2.7
K (lm) or sp
K0 (lm) or sk
110 ± 2
240 ± 6
1.30 ± 0.03
1.10 ± 0.01
K (lm) or sp
K0 (lm) or sk
185 ± 3
343 ± 8
1.39 ± 0.03
0.95 ± 0.01
K (lm) or sp
K0 (lm) or sk
275 ± 3
350 ± 2
1.41 ± 0.07
0.99 ± 0.02
Table 3 Non-acoustical parameters for the sample D = 2 mm.
Measured SA + Miki Error (%) SA + HB Error (%) SA + JAC Error (%) SA + CS Error (%)
Table 4 Non-acoustical parameters for the sample D = 3 mm.
Measured SA + Miki Error (%) SA + HB Error (%) SA + JAC Error (%) SA + CS Error (%)
Fig. 6. Measured and SA estimated normal incidence absorption curves for the glass beads of D = 1 mm and for (a) d = 5 cm, and (b) d = 10 cm.
P. Cobo, F. Simón / Applied Acoustics 104 (2016) 119–126
125
Fig. 7. Measured and SA estimated normal incidence absorption curves for the glass beads of D = 2 mm and for (a) d = 5 cm, and (b) d = 10 cm.
Fig. 8. Measured and SA estimated normal incidence absorption curves for the glass beads of D = 3 mm and for (a) d = 5 cm, and (b) d = 10 cm.
d = 5 and 10 cm. The values included in the fifth and sixth columns of these Tables are K and K0 , for the SA + JAC case and sp and sk, for the SA + CS case. For each estimated non-acoustical parameter, the relative error, with respect to the measured one, is also included in Tables 2–4. In the three samples, the HB model provides the worse estimation, as indicated by the highest relative errors. The other three models affords good estimations of the (r, /, q2) parameters, being the JAC model the one with the lowest relative errors. The non-acoustical parameters reported in Tables 2–4 are similar to these measured or calculated by others authors. For instance, Allard et al. [17] provided the measured values (r, /, q2, K, K0 ) = (12210 N s m4, 0.4, 1.37, 90 lm, 320 lm) for random packings of glass beads of D = 1.46 mm. Gasser et al. [20] obtained the numerical values (/, q2, K, K0 ) = (0.36, 1.65, 260 lm, 400 lm) for sphere packings of D = 2 mm. Umnova et al. [18] also applied numerical techniques to obtain the values (/, q2, K) = (0.4, 1.37, 90 lm) for sphere packings of D = 1.46 mm. Figs. 6–8 shows the measured and SA estimated for each model corresponding to the samples of D = 1, 2 and 3 mm, respectively. The SA estimated curves are obtained with the non-acoustical parameters summarized in Tables 2–4. Again, the HB model seems to yield the absorption curves that match worse to the measured
ones. The absorption curves provided by the other three models, on the other hand, give curves that fit reasonably well to the measured curves.
5. Conclusions In this paper, a comparison of impedance models in the inverse estimation of the non-acoustical parameters of a granular material consisting of random packings of identical small glass beads has been carried out. The parameters have been inversely estimated from the measurement of the sound absorption coefficient at normal incidence in an impedance tube using the transfer function method. This inverse approach requires the assumption of an impedance model and an inversion technique. The objective of the inversion technique is to obtain the non-acoustical parameters that minimize the difference between the measured and modeled absorption curves. Simulated Annealing has been assumed as the inversion technique due to the excellent performance in previous inverse problems. The three-parameters Miki model, the phenomenological Hamet–Berengier model, and the microstructural J ohnson–Allard–Champoux and Champoux–Stinson models, have
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been chosen for the comparison. All these models share three non-acoustical parameters: air flow resistivity, r, porosity, /, and tortuosity, q2. The Johnson–Allard–Champoux and the Champoux–Stinson are five-parameters models that use, besides the above mentioned three parameters, the viscous, K, and thermal, K0 , characteristics lengths, the first one, and the viscous, sp, and thermal, sk, shape factors, the latter. The comparison of the estimated parameters with those measured in three granular samples demonstrated that the Hamet–Berengier model provides the worse estimation, as indicated by the highest relative errors. The other three models afford good estimations of the parameters, being the Johnson–Allard–Ch ampoux model the one with the lowest relative errors. This result was expected, since the Hamet–Berengier model was proposed for characterizing porous pavements. Furthermore, since there is more chance to match the measured curves with a five-parameters than a three-parameters model, a better performance of the Johnson–Allard–Champoux and Champoux–Stinson models could also be anticipated. Acknowledgements The technical assistance of Eduardo de Andrés and Marco Cortés is kindly acknowledged. References [1] Beranek LL, Vér IL. Noise and vibration control engineering. New York: John Wiley & Sons; 1992. [2] Arenas JP, Crocker MJ. Recent trends in porous sound-absorbing materials. Sound Vib 2010(July):12–7. [3] Asdrubali F, Schiavoni S, Horoshenkov KV. A review of sustainable materials for acoustic applications. Build Acoust 2012;19:283–312. [4] Pfretzschner J, Rodríguez RM. Acoustic properties of rubber crumbs. Polym Test 1999;18:81–2. [5] Attenborough K, Li KM, Horoshenkov K. Predicting outdoor sound. London: Taylor & Francis; 2007.
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