**7. Conclusions**

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,

*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].*

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

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

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

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

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

however, has no physical meaning [38, 52].

confirms the reliability of Eq. (19).

science and those used in acoustics.

parameter scarcely affected the goodness of fit.

unity [38].

**40**

**Figure 9.**

*Foams - Emerging Technologies*

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 doping powder loading an unoriented cellular sample morphology.

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 experimental performance.

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 experimental data using TMM calculation and inverse parameter determination.

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 compared to the samples with more regular and oriented morphology.

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 lead physically unreliable values for the other parameters.

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

physically reliable results related to the real microstructure. Thus, a direct link between the materials science property and acoustics has been established.

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