**6. Comparison of the results**

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 data compared to those obtained by the unrestricted inverse method.

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 original paper [15].

characteristic length values, the complex foam cells of our materials feature a parallelepiped interconnected geometry which appears consistent with the similar-

*Thermal and Acoustic Numerical Simulation of Foams for Constructions*

A novel class of sustainable innovative acoustic insulation materials has been described in the present paper. The use of a natural alginate-based gelling agent allows efficient incorporation of waste glass and fiberglass powders. The analysis of the microstructure indicates a strong sensitivity of the pore morphology, on particle dimensions of the doping powder and its amount. The formation of oriented regular cell patterns was attributed to the presence of a large amount of small particles that favors heterogeneous nucleation of ice formation leading to mono-dimensional freezing process. Consistently, using coarse particles produces at comparable dop-

Five different forecasting methods including traditional analytical, a modified analytical with a new proposed equation, and inverse procedures were employed to determine the JCA parameters related to the sound-absorbing properties of foam materials. TMM to assess the reliability of the different procedures in comparison to

The analytical modeling of the JCA parameters, namely, tortuosity, viscous characteristic length, thermal characteristic length, porosity, and flow resistivity showed some limitations of the applicability of the traditional equation, because they are strongly related to fibrous materials rather than foams and a new equation for the determination of the tortuosity was proposed and validated against experi-

The use of the inverse determination of the physical parameters allowed to provide an insight between the materials' properties and acoustic performance: consistent with SEM microstructural analysis indicated comparable foam properties for materials A and B, material C being somewhat different, a situation well consistent with the acoustic performance. As in fact, the sound-absorbing performance depends on cell shape and dimension identified by the thermal lengths. Thus, using the same foaming agent with different doping powders leads to different sound absorption trends: volcano-shaped for materials A and B with glass powder and flat for material C with fiberglass inclusions, as the decline of the sound absorption being less important. The effects of cell orientation impact the acoustic properties as the unoriented cell morphology leads to enhanced sound absorption capacity com-

mental data using TMM calculation and inverse parameter determination.

pared to the samples with more regular and oriented morphology.

lead physically unreliable values for the other parameters.

An important warning arises from the present data which is the fact that unrestricted fitting may lead to a reliable acoustic profile, corresponding to a local minimum that, however, may not have a physical relationship with the materials properties, e.g., pore morphology. As a matter of fact, the performed sensitivity analysis indicated tortuosity as a factor that heavily affects the fit, which may easily

Finally, it has been clearly shown that the "traditional" analytical model for determination of JCA parameters cannot be a priori applied to these novel materials due to their complex structure: modification of the calculation of the tortuosity was necessary, and a new equation for the determination of the tortuosity is proposed that has been assessed; the results of the inverse procedure, using the thermal characteristic length derived from the SEM micrographs as imposed parameter, well agree with the modified analytical model. The use of measured values of thermal characteristic length in the inverse procedure is recommended in order to obtain

ing powder loading an unoriented cellular sample morphology.

ity of the two parameters.

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

the experimental performance.

**41**

**7. Conclusions**

### **Figure 9.**

*Comparison of modeled JCA values for sample A with modified analytical, the inversion techniques: values are expressed as % of the maximum value observed for each material/parameter. Figure adapted from [15].*

A perusal of **Figure 9** immediately reveals that the "free fit" inverse method computes significantly different values for the *α*∞, *Λ* and *Λ*' parameters compared to the other fits. Since the calculated values are unrelated to the physical nature of our materials, this is a clear indication that the fit end with a local minimum which, however, has no physical meaning [38, 52].

To be noticed is that the calculated tortuosity (*α*∞) shows good agreement for both the modified analytical method and the inversion method with the fixed *Λ*' parameter. Since *Λ*' parameter is evaluated from experimental data, this observation confirms the reliability of Eq. (19).

It is worth to remind that conventional materials such as lightweight, fibrous materials (e.g., fiberglass and rock wool) and reticulated foams (e.g., polyurethane and melamine open-cell foams) typically feature porosity and tortuosity very close to unity. In contrast, our and other materials feature tortuosity factors well above unity [38].

The modified analytical methods and fixed *Λ*' value inverse procedure show a good agreement for *ϕ* e *α* parameters, whereas no method accurately estimates the value of the *Λ*' parameter. Our data indicate that this parameter should be measured experimentally using SEM or an equivalent technique to get a reliable result. This observation highlights direct link between the parameters used in the material science and those used in acoustics.

Both the analytical model and the free inverse fitting (**Table 3**) lead to a low value of the *σ* parameter compared to the other methods. This however may be explained by the fact that a sensitivity analysis [52] revealed that variation of this parameter scarcely affected the goodness of fit.

The pore geometry is associated with viscous and thermal characteristic lengths [45], the average size of the foam cells being correlated to the thermal characteristic length (*Λ*'). As for the characteristic viscose length *Λ*, this parameter, albeit linked to pore geometry, can hardly be derived from the microstructural characterization, whereas its influence is important since narrowing the interconnections between the foam cells, blocks the fluid movement and transition, resulting in improved sound absorption characteristics. As for the similarity of the thermal and viscous

characteristic length values, the complex foam cells of our materials feature a parallelepiped interconnected geometry which appears consistent with the similarity of the two parameters.
