**5.1 Analytical model and acoustic performance**

Experimentally measured acoustic absorption coefficient for the samples A, B, and C are reported in **Figure 6**. Samples A and B show comparable shape of the curves where sample B features better global sound-absorbing properties compared to A: the highest absorption coefficient observed for sample B is 0.998 at 2190 Hz. Sample C features a maximum of absorption at about 2100 Hz followed by a slow decline, at variance with samples A and B where a rapid decline is observed. Clearly, different morphologies of sample C compared to A and B lead to different acoustic properties. For comparison, a reference rock wool sample features a nearly linear increase of the sound absorption coefficient with a maximum value of ca. 0.85 at 2900 Hz.

As highlighted above, the application of the analytical model by calculating the JCA parameters using Eqs. (10)–(14) was one of the important aspects of this study. The question is, are the widely employed state-of-the-art parameter formulations applicable to a novel type of material?

To answer this question, we report, for the sake of conciseness, only the result obtained for sample A, but equivalent results have been obtained for samples B and C [15].

In the first instance, in order to model the sound absorption coefficient, the five parameters were calculated according to Eqs. (10)–(14) using the measured densities and the dimensions of the cells evaluated from the SEM micrographs (**Table 2**). As perusal of the data reported in **Table 2** reveals a close similarity of the calculated parameters notwithstanding the dissimilarity in their nature and morphology. This demonstrates that the analytical model is not suitable for this kind of cellular foam microstructure.

The frequency trends of the sound absorption coefficient, which were calculated using these parameters as input for the TMM procedure, are shown in **Figure 6**.

The results (**Figure 6**) show that the analytical model procedure as implemented using Eqs. (10)–(14) cannot be reliably applied to the complex foam structures, at variance with the rock wool sample which is properly modeled. The observation is in line with the above reported comments on the limits of the applicability of this methodology to fibrous materials [48, 50].

The above presented microstructural data show that the morphology and dimensions of the foam cells depend on the addition of the glass-containing powders. Since the powder is incorporated into the walls of the cells, increasing its amount will result in an extension of the free path for the wave propagating within the material itself. As a consequence, a modification of the tortuosity parameter is expected. For this reason, we consider the tortuosity factor as calculated by Eq. (11), developed for fibrous like materials, to inadequately describe this type of novel material. Notice that a modification of the tortuosity parameter changes the sound absorption leaving the thickness of the material unchanged.


### **Table 2.**

*Analytical model results: flow resistivity, porosity (*ϕÞ*, tortuosity (*α∞*), tortuosity* α*mod*,∞*, and characteristic lengths (*Λ*,* Λ'*).*

The process conditions strongly affect freeze-drying synthesis since directional freezing of the ice particles can be easily achieved leading to novel morphologies such as monoliths [69, 78]. This technique can be widely applied, and also alginatebased gels were produced in an anisotropic form [70]. Ordinary freezing conditions were employed for the synthesis, which suggest that this effect should not be operative in our case. It is well-known that during the crystallization of ice, both solute and suspended particles/gels are segregated from the ice crystals. This may generate an ice-templating effect where the morphology of the material is dictated by the crystallized solvent [79]. A large number of small particles favors heterogeneous nucleation providing a large number of nucleation centers [80, 81]. The large amount of small particles in the glass-containing samples A and B increase the ice front velocity promoting formation of a columnar morphology [82], accounting for the morphology detected by SEM. Sample C contains much less small particles, and the rate of nucleation decreases compared to that of particle growth (ice crystallization). This generates an isotropic pattern of the open cells in sample C. The large pore dimension is in line with the higher particle size of the fiberglass compared to

*Microstructure and properties of the alginate foams: average area and radius of the foam pores, density, and*

**Sample A B C Rock wool**

EC (MPa) 5.2 4.2 3.4 1.0 Standard deviation (MPa) 0.6 0.3 0.1 0.1

Radius mean value (μm) 29 38 75 Porosity 0.85 0.91 0.93

) 0.011 0.019 0.074

) 0.011 0.009 0.033

) 186 201 250 150

Thus, the crystallization conditions and the particle distribution in the starting waste material appear to represent factors capable of directing the microstructure leading to distinct cell morphology and dimension. This is an important aspect as the aim of the study is to find correlation between the microstructure and acoustic

The data reported in **Table 1** show clear trends for the density and the compression modulus which can be correlated with the dimension of the open cells. For a fixed volume, the higher the pore area, the lower the number of cells, which means that the density increases in the sequence samples A, B and C and the opposite occurs for the compression modulus. Data for a rock wool sample are also

In this section the results of the acoustic performance and application of the different procedures to model the acoustic performance of these novel materials are discussed, first using the analytical procedure to calculate the model parameters and

included in **Table 1**, as a standard sample for the acoustic studies.

**5. Cellular foam from recycled waste: acoustic studies**

glass materials [79, 80].

Pore medium area (mm2

*Foams - Emerging Technologies*

Standard deviation (mm<sup>2</sup>

Density (kg/m<sup>3</sup>

*Table adapted from [15].*

*compression modulus.*

**Table 1.**

properties of these materials.

then using the TMM approach.

**34**

### **Figure 6.**

*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 Eq. (19). 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 model depends only on the open porosity:

$$a\_{\text{mod},\text{ss}} = \frac{1}{\phi^{12.72}}\tag{19}$$

model to the specific material analyzed. In this case the predictive value of the tortuosity clearly appears strictly related to the nature of the sample. Accordingly,

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

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

The inversion procedure algorithm was therefore applied to the experimental

*Comparison of modeled and measured values for rock conditions and wool using parameters obtained from the*

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

the "traditional" analytical model will not be considered further.

*Thermal and Acoustic Numerical Simulation of Foams for Constructions*

**5.2 Acoustic indirect method**

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

to the model considered [30, 85].

sample.

**Figure 7.**

**37**

*free inversion procedure. Figure adapted from [15].*

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 simulation nicely fits the experimental data.

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

model to the specific material analyzed. In this case the predictive value of the tortuosity clearly appears strictly related to the nature of the sample. Accordingly, the "traditional" analytical model will not be considered further.
