**5.2 Acoustic indirect method**

As discussed in the preceding section, the modeling of the acoustic properties of porous materials requires to determine physical parameters of the porous solid, namely, airflow resistivity, open porosity, tortuosity, and viscous and thermal characteristic lengths [84]. In the recent years, an inversion method can be applied which consists in a best-fit procedure of the experimental acoustic data to provide all these parameters as the output has become a popular methodology [52]. Such an approach could successfully be applied to a number of different types of porous materials [38]. This is exemplified in **Figure 7** which reports the comparison between the measured and calculated trends for a free inversion of the rock wool sample.

The picture reported in **Figure 7** clearly suggests the effectiveness of this procedure since the modeled data visibly better fit the experimental data compared to the analytical model reported in **Figure 6**. As discussed in Section 4, the final goal of the modeling procedures is to acquire a predictive capacity and, most importantly, the capability to properly correlate the microstructure of the investigated material with its sound-absorbing capacity [42]. This clearly would open new horizons for the material development by trying to develop correlations between the synthesis conditions and material properties [44]. In this respect, it important to recall that the inversion procedure involves a best fit of an experimental curve using a number of parameters, 5 for the JCA model, which can increase up to 8, according to the model considered [30, 85].

The inversion procedure algorithm was therefore applied to the experimental acoustic measurements using three different approaches: in the first one, no restriction has been applied to the inverse procedure. In the second one, restrictions were applied to the values obtained from the modified analytical model. The limitations were applied in terms of upper and lower limits of the flow resistivity (*σ*) within which the inverse procedure can fit. In the third one, the thermal characteristic length (*Λ*') value was imposed based on the experimental data (pore radius in **Table 2**) in the inverse procedure. The choice of these restrictions is motivated by

### **Figure 7.**

*Comparison of modeled and measured values for rock conditions and wool using parameters obtained from the free inversion procedure. Figure adapted from [15].*

Eq. (19), which is obtained by modifying the formulation of Archie for the tortuosity [83], is therefore proposed as a partial modification of Eq. (11). Eq. (19) is able to provide a reliable fit, up to 2500 Hz, as shown in **Figure 6**, because this

*Sound absorption coefficient as a function of frequency: analytical model (calculated with TMM) vs experimental values obtained for rock wool, samples A, B, and C and sample A using modified tortuosity*

*<sup>α</sup>*mod,<sup>∞</sup> <sup>¼</sup> <sup>1</sup>

The exponent of the open porosity in Eq. (19) is calculated by a curve-fitting procedure of all measured results. The value of the tortuosity calculated using Eq. (19) is included in **Table 2** for material A. As shown in **Figure 6** (modified model), using the modified tortuosity parameter (*α*mod,∞) as input, the TMM simu-

Accordingly, an important finding of this part of this study is the demonstration of the necessity of adapting the analytical calculation of the parameters for the JCA

*<sup>ϕ</sup>*<sup>12</sup>*:*<sup>72</sup> (19)

model depends only on the open porosity:

*Eq. (19). Figure adapted from [15].*

*Foams - Emerging Technologies*

**Figure 6.**

**36**

lation nicely fits the experimental data.

the fact that these parameters are those usually experimentally measured in, respectively, acoustic and material science studies.

**Figure 8** compares the experimental sound-absorbing coefficient and the complex impedance for the three materials with the calculated, ones using TMM, vs frequency. A quite good agreement between the fitted and experimental curves is found, unrestricted fitting giving the best result for sample A.

To properly assess the goodness of fit, an attempt was performed using standard

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ *xi* � *μ* 2

(20)

deviation calculated as reported in Eq. (20):

**Material Flow**

*Table adapted from [15].*

*a*

**Table 3.**

I. Fitting with no restriction

**resistivity (***σ***) ((N s) m**�**<sup>4</sup>**

**)**

*DOI: http://dx.doi.org/10.5772/intechopen.91727*

III. Fitting using experimental pore dimension

*value of three experiments in the range 200–3000 Hz.*

**Porosity (***ϕ***) (–)**

*Thermal and Acoustic Numerical Simulation of Foams for Constructions*

II. Fitting using restricted method (based on analytical model)

**Tortuosity (***α***∞) (–)**

A 17,744 0.87 6.78 91 194 0.0142

A 59,676 0.81 4.34 53 53 0.0251

A 59,181 0.82 2.88 29 29 0.0323

*Standard deviation of the calculated* α *from the experimental values assuming that the latter represent the average*

**Viscous characteristic length (***Λ***) (μm)**

**Thermal characteristic length (***Λ***') (μm)**

**Standard deviation<sup>a</sup> (***σ***)**

regular structures compared to those irregular.

**6. Comparison of the results**

original paper [15].

**39**

*σ* ¼

*Parameters obtained from inverse procedure using different fitting approaches.*

1 *N* X*<sup>N</sup> i*¼1

The calculated values and standard deviation of the calculated *α* from the experimental values assuming that the latter represent the average value, being an average of three measurements in the range 200–3000 Hz, are reported in **Table 3**. The values of the calculated deviation shows (i) minor effects of the restriction on the parameters on the goodness of fit upon variation of the fitting procedure, (ii) unrestricted fit is slightly better than those restricted, and (iii) samples A and B are much better fitted than sample C (compare full data in Ref. [15]). As for the latter aspect, this could be related to the irregular pore morphology sample C. Accordingly, the fitting procedure, based on an idealized structure, fits better

Let us now discuss the four sets of parameter derived by the above described procedures (**Tables 2** and **3**), which were used to calculate the TMM-based forecast of the sound adsorption capability and compared with the experimental data (**Figure 8**). At first, we observe that by either restricting the inverse fitting procedure or using the parameters calculated in the modified analytical model or imposing the measured *Λ*' value, the calculated TMM profile fits slightly worse the experimental

However, it is important to consider that the derived *ϕ*, *α*, *Λ* and *Λ*' parameters

should be properly related to the real microstructure of foam materials, which should help to discriminate the proper fitting model among of the four considered. For this purpose, **Figure 9** compares the obtained acoustic parameters for the four sets expressed in terms of relative percentages. For the scope of this paper, we limit the analysis to sample A, being sufficient to provide relevant considerations and insights. For a full comparison of the three samples, we refer the reader to our

data compared to those obtained by the unrestricted inverse method.

r

### **Figure 8.**

*Comparison of modeled and measured values for sample A using parameters obtained from the inversion procedure, using free inversion, restricted analytical model values and imposing the measured* Λ*' value: absorbing coefficient vs frequency and complex impedance. Figure adapted from [15].*

*Thermal and Acoustic Numerical Simulation of Foams for Constructions DOI: http://dx.doi.org/10.5772/intechopen.91727*


*Table adapted from [15].*

*a Standard deviation of the calculated* α *from the experimental values assuming that the latter represent the average value of three experiments in the range 200–3000 Hz.*

### **Table 3.**

the fact that these parameters are those usually experimentally measured in,

*Comparison of modeled and measured values for sample A using parameters obtained from the inversion procedure, using free inversion, restricted analytical model values and imposing the measured* Λ*' value:*

*absorbing coefficient vs frequency and complex impedance. Figure adapted from [15].*

**Figure 8** compares the experimental sound-absorbing coefficient and the complex impedance for the three materials with the calculated, ones using TMM, vs frequency. A quite good agreement between the fitted and experimental curves is

respectively, acoustic and material science studies.

*Foams - Emerging Technologies*

**Figure 8.**

**38**

found, unrestricted fitting giving the best result for sample A.

*Parameters obtained from inverse procedure using different fitting approaches.*

To properly assess the goodness of fit, an attempt was performed using standard deviation calculated as reported in Eq. (20):

$$\sigma = \sqrt{\frac{1}{N} \sum\_{i=1}^{N} (\mathbf{x}\_i - \boldsymbol{\mu})^2} \tag{20}$$

The calculated values and standard deviation of the calculated *α* from the experimental values assuming that the latter represent the average value, being an average of three measurements in the range 200–3000 Hz, are reported in **Table 3**.

The values of the calculated deviation shows (i) minor effects of the restriction on the parameters on the goodness of fit upon variation of the fitting procedure, (ii) unrestricted fit is slightly better than those restricted, and (iii) samples A and B are much better fitted than sample C (compare full data in Ref. [15]). As for the latter aspect, this could be related to the irregular pore morphology sample C. Accordingly, the fitting procedure, based on an idealized structure, fits better regular structures compared to those irregular.
