**2. Cellular foams: modeling of thermal properties**

Due to their intensive use as insulating materials, thermal properties of cellular foams have been investigated quite intensively [18], and a number of models was reported, first of these date back to the 1930s of the last century [19]. Here we summarize only some of the models adopted.

Placido et al. [19] developed a predictive model which considers that heat transfer takes place by both conductions through solid skeleton and included gas and by radiation across the whole layer. The radiation is attenuated by material microstructures, via scattering and absorption phenomena. As for the contribution of the free convection the heat transfer, it is generally considered negligible due to the very small pore size so that the Raleigh number is much less than the critical value [20, 21].

The one-dimensional thermal conduction occurring in a continuum layer is regulated by the Fourier law:

$$q\_c = -\lambda \frac{\text{d}T}{\text{d}x} \tag{1}$$

where *qc* is the flux of the thermal conduction (W/m<sup>2</sup> ), *λ* is the thermal conductivity (W/(m K)),*T* is the temperature (K), and *x* is the thickness (m). The thermal conductivity in foams results as the sum of bulk thermal conductivity and gas one (λ*<sup>g</sup>* Þ. The radiated effect can be modeled by a diffusive equation as follows:

$$q\_r = -\lambda\_r \frac{\text{d}T}{\text{d}x} \tag{2}$$

where *qr* is the flux of the thermal radiation (W/m<sup>2</sup> ) and *λ<sup>r</sup>* is the radiative conductivity (W/(m K)) as follows:

$$
\lambda\_r = \mathbf{16} \,\sigma\_{\text{Bolz}} \frac{T\_m^3}{\mathbf{3} \, K\_r} \tag{3}
$$

where σBolz is the constant of Stefan-Boltzmann,*Tm* is mean temperature, and *Kr* is the Rosseland coefficient.

As per the conservation law, the final heat flux and the final conductivity will be

$$q\_{\text{tot}} = q\_c + q\_r \tag{4}$$

$$
\lambda\_{\rm tot} = \lambda + \lambda\_{\rm g} + \lambda\_r \tag{5}
$$

For the radiative part, Cunsolo et al. [22] analyzed several analytical methodologies, concluding that those procedures provide results that are less reliable than numerical ones. This difference is basically caused by porosity modeling, while cell size distribution may not affect final outcomes.

Mendes et al. [23] predict the effective thermal conductivity by means of finite volume methods, taking into account both regular cells and real ones.

ratio of published patents/patent applications to the published documents is about 25. It is important to evidence that when the term "acoustic" is added to the previous search, the total drops by about 90%, and most importantly, the above quoted ratio increases up to 87, despite the fact that cellular foams, particularly those with open cells, are widely employed as acoustic insulators [4]. Clearly, despite the technological and market importance of these materials as acoustic insulators, the acoustic aspects and performances of these materials have sparingly

*Results of Scopus search using combination of the indicated keywords. Totals of documents/patens published in 1970–2019 period are also reported (search conducted in December 2019, the lines are used only as eye-guide).*

Among the cellular foam materials, polyurethane foams are widely employed, especially in the building sector [5], due to their quite low thermal conductivity (0.022–0.028 W/(m K)), which makes them one of the materials of choice in the applications that require effective and lightweight insulation. In the recent years, circular economy has become a priority in EU countries. Consistently, recycling of waste into industrial products has become an important issue [6]. As far as foam materials are concerned, whereas elastomers or similar waste residues could be effectively recycled in a polyurethane foaming process or even bio-based binders to form efficient acoustics absorbers [7, 8], reutilization of glass and ceramic waste generally employs high-energy-demanding production process [9–11], often leading to foam materials to be employed as insulators. An important class of materials, difficult to be recycled, is represented by composites such as fiberreinforced thermoset polymers, used in a wide range of industrial application, from automotive to industrial, transportation and naval sectors, etc. [12, 13]. Due to landfill restrictions, thermal (energetic recovery, recovery of fibers/ chemicals), mechanical (filler materials), and chemical (solvolysis) routes are underdevelopment, yet their industrial applications still represent a burden, both economic and technical [13]. Recently, we have developed a novel process to recover glass and fiberglass waste via low-temperature foaming process using natural alginate-based foaming agent as a novel route leading to

sustainable insulating materials, with interesting acoustic absorption properties [14, 15]. An interesting aspect of this research was linked to the applicability and the limitation of literature acoustic models when used to describe the behavior to this novel cellular type of materials. Specifically, the focus is made on the

been addressed in the scientific literature.

**Figure 1.**

**24**

*Figure adapted from [15].*

*Foams - Emerging Technologies*

Klett et al. [24] demonstrated the paramount effect of rigid skeleton compared to included gas.

Öchsner et al. [18] provided a very comprehensive analysis of thermal property simulation and prediction of porous media, highlighting the difference in approaches both from analytical and numerical point of view. Furthermore, they explain how there are three main theoretical approaches:


Many models were built on cells micro–macro structures like Pande et al. [25] focusing on the heat flow propagation direction [26] or on cell distribution [27].

The Monte Carlo approach is used several time in order to compute temperature profiles where there are randomly shaped borders with varying boundary conditions [28].

Yüksel [29] presented a comprehensive review of measurement methods for the determination of thermal conductivity of materials highlighting how for porous ones only heat flow meter and guarded hot plate are useful to this aim.
