**3. Cellular foams: modeling of acoustic properties**

As indicated above, the acoustic behavior of cellular foams, which is the focus of this work, received less attention compared to thermal applications. The acoustic efficiency of the foams can be easily understood when the principles of sound absorption are considered [30, 31]. The incident sound energy (*Ei*) interacts whit the material according to Eq. (6) where *Er*, *Ea*, and *Et* represent, respectively, energy that is reflected, absorbed, and transmitted:

$$E\_i = E\_r + E\_a + E\_t \tag{6}$$

Sound absorption coefficient can be calculated if the surface impedance (*Zs*) is

*<sup>α</sup>* <sup>¼</sup> <sup>1</sup> � *Zs* � *<sup>ρ</sup>*0*c*<sup>0</sup>

where the term *ρ*0*c*0, respectively, density and speed of sound, represents the

The aspects of energy dissipation are associated mainly with viscoelastic phenomena that occur within the rigid material and in the interface between it and the fluid in motion. For this reason, the material can be considered biphasic and can be effectively modeled with the Biot theory [34–36]. Therefore, foams can be defined by an equivalent fluid that features an equivalent density and equivalent bulk

<sup>Δ</sup>*<sup>p</sup>* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>ρ</sup>*�

ibility modulus and *p* corresponds to the acoustic pressure (the tilde symbol (�) specifies that the associated variable is a complex value and related to the frequency). To assess the Biot model with respect to the investigated material, knowledge of macroscopic parameters such as porosity, flow resistance, etc. is necessary. Thus, the knowledge of these parameters allows acoustic behavior to be modeled. These models may use parameters that are based on the definitions from the theory or experimental measurements [37]. Several techniques are, in fact, used to determine the experimental macroscopic parameters, as discussed in Ref. [37]; however,

these measurements are not trivial and often time-consuming.

eq *K*� eq

Consequently, a large number of impedance predictive models for obtaining the sound absorption coefficient have been published [16, 30, 38–40], which can be grossly divided into two groups: empirical and theoretical. The empirical methods were initially developed by applying regression methods to large sets of experimental measurements which clearly links them to the specific material considered [39]. Theoretical methods are based on the physics of the sound propagation in the materials (phenomenological methods) and, desirably, include relationships between microstructure and macroscopic properties [41, 42]. This, in fact, has been the case for polyurethane foams, which, however, typically present a regular and

 

*Zs* þ *ρ*0*c*<sup>0</sup>

 

2

eq are the equivalent density and the equivalent bulk compress-

*p* ¼ 0 (9)

(8)

known according to Eq. (8):

modulus according to the following equation:

*Scheme of sound energy interaction with a sold: energy conservation [32].*

*Thermal and Acoustic Numerical Simulation of Foams for Constructions*

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

impedance of the air.

**Figure 2.**

where *ρ*�

**27**

eq and *K*�

Clearly, as illustrated in **Figure 2**, in in order to minimize transmitted energy, both absorption and reflection must be maximized for our material.

Sound absorption occurs in porous materials essentially via three mechanisms, i.e., (i) interaction of air molecules which vibrate and interact with the pore walls; (ii) air compression and expansion in the pores induced by the entering sound wave, resulting in sound energy transformation into heat; and (iii) vibration and resonance of pore walls [16, 32, 33]. Clearly, all the three mechanisms involve the air located within the pores and its motion that lead to transformation and dissipation of the original sound energy.

From a material point of view, sound absorption coefficient (*α*) is used to quantify the efficiency of the porous sound absorption materials, which can be measured by impedance tube or reverberation chamber by considering its definition in terms of energy:

$$a = 1 - \frac{E\_r + E\_t}{E\_i} = \frac{E\_a}{E\_i} \tag{7}$$

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

### **Figure 2.**

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

Öchsner et al. [18] provided a very comprehensive analysis of thermal property

consideration the microstructure of the system and the influence of the

ii. Resistor approach, where the bulk and gaseous phases are considered like parallel parts which are assumed to be thermal resistors to flux propagation

iii. Phase averaging, where the effective thermal conductivity is obtained by

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

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.

As indicated above, the acoustic behavior of cellular foams, which is the focus of this work, received less attention compared to thermal applications. The acoustic efficiency of the foams can be easily understood when the principles of sound absorption are considered [30, 31]. The incident sound energy (*Ei*) interacts whit the material according to Eq. (6) where *Er*, *Ea*, and *Et* represent, respectively,

Clearly, as illustrated in **Figure 2**, in in order to minimize transmitted energy,

Sound absorption occurs in porous materials essentially via three mechanisms, i.e., (i) interaction of air molecules which vibrate and interact with the pore walls; (ii) air compression and expansion in the pores induced by the entering sound wave, resulting in sound energy transformation into heat; and (iii) vibration and resonance of pore walls [16, 32, 33]. Clearly, all the three mechanisms involve the air located within the pores and its motion that lead to transformation and

From a material point of view, sound absorption coefficient (*α*) is used to quantify the efficiency of the porous sound absorption materials, which can be measured by impedance tube or reverberation chamber by considering its

> *<sup>α</sup>* <sup>¼</sup> <sup>1</sup> � *Er* <sup>þ</sup> *Et Ei*

<sup>¼</sup> *Ea Ei*

(7)

both absorption and reflection must be maximized for our material.

*Ei* ¼ *Er* þ *Ea* þ *Et* (6)

approaches both from analytical and numerical point of view. Furthermore, they

i. Field approach, where the Laplace equation is solved taking into

simulation and prediction of porous media, highlighting the difference in

structural elements in the linear propagation of the flux

profiles where there are randomly shaped borders with varying boundary

explain how there are three main theoretical approaches:

averaging the constituting phases

**3. Cellular foams: modeling of acoustic properties**

energy that is reflected, absorbed, and transmitted:

dissipation of the original sound energy.

definition in terms of energy:

**26**

to included gas.

*Foams - Emerging Technologies*

conditions [28].

*Scheme of sound energy interaction with a sold: energy conservation [32].*

Sound absorption coefficient can be calculated if the surface impedance (*Zs*) is known according to Eq. (8):

$$a = \mathbf{1} - \left| \frac{\mathbf{Z}\_s - \rho\_0 \mathbf{c}\_0}{\mathbf{Z}\_s + \rho\_0 \mathbf{c}\_0} \right|^2 \tag{8}$$

where the term *ρ*0*c*0, respectively, density and speed of sound, represents the impedance of the air.

The aspects of energy dissipation are associated mainly with viscoelastic phenomena that occur within the rigid material and in the interface between it and the fluid in motion. For this reason, the material can be considered biphasic and can be effectively modeled with the Biot theory [34–36]. Therefore, foams can be defined by an equivalent fluid that features an equivalent density and equivalent bulk modulus according to the following equation:

$$
\Delta p + \alpha^2 \frac{\rho\_{\text{eq}}^{\sim}}{K\_{\text{eq}}^{\sim}} p = 0 \tag{9}
$$

where *ρ*� eq and *K*� eq are the equivalent density and the equivalent bulk compressibility modulus and *p* corresponds to the acoustic pressure (the tilde symbol (�) specifies that the associated variable is a complex value and related to the frequency). To assess the Biot model with respect to the investigated material, knowledge of macroscopic parameters such as porosity, flow resistance, etc. is necessary. Thus, the knowledge of these parameters allows acoustic behavior to be modeled. These models may use parameters that are based on the definitions from the theory or experimental measurements [37]. Several techniques are, in fact, used to determine the experimental macroscopic parameters, as discussed in Ref. [37]; however, these measurements are not trivial and often time-consuming.

Consequently, a large number of impedance predictive models for obtaining the sound absorption coefficient have been published [16, 30, 38–40], which can be grossly divided into two groups: empirical and theoretical. The empirical methods were initially developed by applying regression methods to large sets of experimental measurements which clearly links them to the specific material considered [39]. Theoretical methods are based on the physics of the sound propagation in the materials (phenomenological methods) and, desirably, include relationships between microstructure and macroscopic properties [41, 42]. This, in fact, has been the case for polyurethane foams, which, however, typically present a regular and

well-defined structure, as exemplified in **Figure 3**. For such materials, the tetrakaidecahedra unit cells, Kelvin cell, which represents packing of equal-sized objects together to fill space with minimal surface area, can effectively be applied to describe the foam microstructure [41, 43, 44]. This bottom-up approach therefore seems limited to specific materials with a very precise and defined morphology of the cells.

Composite-made foams as those here investigated present a complex structure where the alginate matrix is loaded with the glass-containing powder and thus represents a new class of sustainable materials. Given their composite microstructure, it is of strong interest to assess whether traditional numerical acoustic procedures can be effectively used for describing and forecasting their properties.

In the following paragraphs, the acoustic procedures employed in the present chapter are described. The use of the chosen analytical models is justified by the fact that there are the most used and simple predicting equations found in literature. Thus, an attempt to understand if they could work also with complex foams is important. For detailed description of the experimental details, we refer the reader to our recent publication [15]. The results of this procedures performed on the innovative cellular foams here employed are described in the next section.

From an analytical point of view, the JCA model parameter, i.e., flow resistivity (*σ*), porosity (*ϕ*Þ, tortuosity (*α*∞), and characteristic lengths (*Λ*, *Λ*'), can be determined using Eqs (10)–(14) [49]. These equations are derived by considering some general assumptions such as periodicity of the microstructures, interconnected pore structures, and small Knudsen number values (ratio of the gas molecular size to a

*<sup>ϕ</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>ρ</sup><sup>m</sup>*

*Λ* ¼ *r*<sup>0</sup>

*Λ*' ¼ *r*<sup>0</sup>

where *ρ<sup>m</sup>* is the overall density of the sample and *ρ<sup>b</sup>* is the bulk material

where *η* is the air viscosity, *b* is the square root of area per fiber, and *d* is the

The acoustic model parameters were calculated form Eqs. (10)–(14) using experimental data (SEM measurements, etc.) and then employed as input values for the transfer matrix method (TMM) [51] calculation of the acoustic absorption

As discussed above, the measurement of the above quoted fluid-phase parameters may be difficult to be obtained and time-consuming. The inverse identification

For air flow resistivity, the most used model is Tarnow's one [50]:

*<sup>σ</sup>* <sup>¼</sup> <sup>4</sup>*πη*

*b*2

*ρb*

*ϕ*ð Þ 2 � *ϕ*

*ϕ*

*α*<sup>∞</sup> ¼ 2 � *ϕ* (11)

ð Þ �0*:*64 ln ð Þ� *<sup>d</sup>* <sup>0</sup>*:*<sup>737</sup> <sup>þ</sup> *<sup>d</sup>* (14)

2 1ð Þ � *<sup>ϕ</sup>* (12)

<sup>1</sup> � *<sup>ϕ</sup>* (13)

(10)

**3.1 Acoustic numerical simulation procedure**

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

*Thermal and Acoustic Numerical Simulation of Foams for Constructions*

*3.1.1 Acoustic model parameters*

characteristic pore size):

composing the open cell.

coefficient.

**29**

volume concentration of cylinders.

*3.1.2 Acoustic indirect method*

Thus, a general approach typically employs a semi-phenomenological model.

For the frequency domain, Johnson et al. [45] proposed a model with arbitrary cell shape of the porous material, while Champoux and Allard [46] derived a model that includes the thermal effects inside the porous medium.

At present, the Johnson-Champoux-Allard model appears as the most effective and reliable model to predict frequency behavior throughout the audible range [39]. This model depends on the following parameters:


The JCA model has been further modified by researchers with the aim of improving some aspects: Pannetton [47] considered aspects related to the limp frame, and further modification was incorporated in the Kino's models [48]. To our knowledge, successful application of these models is essentially limited to "simple" foams, such as polyurethane ones, as above quoted.

Foams consist mainly of air interrupted by a very thin solid matrix that constitutes the air cells, which leads to broadband sound absorption properties.

### **Figure 3.**

*SEM pictures of two samples of polyurethane (PU) foam and the relationship with the derived model tetrakaidecahedron unit cell (Kelvin cell) of the PU structure (adapted from [44]). Notice the continuity of the cell microstructure across the figure that includes two samples.*

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

Composite-made foams as those here investigated present a complex structure where the alginate matrix is loaded with the glass-containing powder and thus represents a new class of sustainable materials. Given their composite microstructure, it is of strong interest to assess whether traditional numerical acoustic procedures can be effectively used for describing and forecasting their properties.

## **3.1 Acoustic numerical simulation procedure**

In the following paragraphs, the acoustic procedures employed in the present chapter are described. The use of the chosen analytical models is justified by the fact that there are the most used and simple predicting equations found in literature. Thus, an attempt to understand if they could work also with complex foams is important. For detailed description of the experimental details, we refer the reader to our recent publication [15]. The results of this procedures performed on the innovative cellular foams here employed are described in the next section.

### *3.1.1 Acoustic model parameters*

well-defined structure, as exemplified in **Figure 3**. For such materials, the tetrakaidecahedra unit cells, Kelvin cell, which represents packing of equal-sized objects together to fill space with minimal surface area, can effectively be applied to describe the foam microstructure [41, 43, 44]. This bottom-up approach therefore seems limited to specific materials with a very precise and defined morphology

Thus, a general approach typically employs a semi-phenomenological model. For the frequency domain, Johnson et al. [45] proposed a model with arbitrary cell shape of the porous material, while Champoux and Allard [46] derived a model

At present, the Johnson-Champoux-Allard model appears as the most effective and reliable model to predict frequency behavior throughout the audible range [39].

The JCA model has been further modified by researchers with the aim of improving some aspects: Pannetton [47] considered aspects related to the limp frame, and further modification was incorporated in the Kino's models [48]. To our knowledge, successful application of these models is essentially limited to "simple"

Foams consist mainly of air interrupted by a very thin solid matrix that constitutes the air cells, which leads to broadband sound absorption properties.

*SEM pictures of two samples of polyurethane (PU) foam and the relationship with the derived model tetrakaidecahedron unit cell (Kelvin cell) of the PU structure (adapted from [44]). Notice the continuity of the*

*cell microstructure across the figure that includes two samples.*

that includes the thermal effects inside the porous medium.

)

This model depends on the following parameters:

• Viscous characteristic length *Λ* (μm)

• Thermal characteristic length *Λ*' (μm)

foams, such as polyurethane ones, as above quoted.

• Flow resistivity *σ* (N s m<sup>4</sup>

• Porosity *ϕ* (–)

**Figure 3.**

**28**

• Tortuosity *α*<sup>∞</sup> (–)

of the cells.

*Foams - Emerging Technologies*

From an analytical point of view, the JCA model parameter, i.e., flow resistivity (*σ*), porosity (*ϕ*Þ, tortuosity (*α*∞), and characteristic lengths (*Λ*, *Λ*'), can be determined using Eqs (10)–(14) [49]. These equations are derived by considering some general assumptions such as periodicity of the microstructures, interconnected pore structures, and small Knudsen number values (ratio of the gas molecular size to a characteristic pore size):

$$\phi = 1 - \frac{\rho\_m}{\rho\_b} \tag{10}$$

$$a\_{\approx} = 2 - \phi \tag{11}$$

$$A = r\_0 \frac{\phi(2-\phi)}{2(1-\phi)}\tag{12}$$

$$A' = r\_0 \frac{\phi}{1 - \phi} \tag{13}$$

where *ρ<sup>m</sup>* is the overall density of the sample and *ρ<sup>b</sup>* is the bulk material composing the open cell.

For air flow resistivity, the most used model is Tarnow's one [50]:

$$\sigma = \frac{4\pi\eta}{b^2(-0.64\ln(d) - 0.737 + d)}\tag{14}$$

where *η* is the air viscosity, *b* is the square root of area per fiber, and *d* is the volume concentration of cylinders.

The acoustic model parameters were calculated form Eqs. (10)–(14) using experimental data (SEM measurements, etc.) and then employed as input values for the transfer matrix method (TMM) [51] calculation of the acoustic absorption coefficient.

### *3.1.2 Acoustic indirect method*

As discussed above, the measurement of the above quoted fluid-phase parameters may be difficult to be obtained and time-consuming. The inverse identification

where *Z0* = *ρ0c0* represents the characteristic impedance of the fluid calculated

*zs* <sup>¼</sup> det D½ � <sup>1</sup> det D2�

*α θ*ð Þ¼ <sup>1</sup> � *<sup>R</sup>*<sup>2</sup>

D1 and D2 matrices are obtained from a complete matrix D (combination of transfer matrix of each layer, coupling matrices, and proper boundary conditions)

Generally speaking, cellular foams being porous materials find a large variety of applications, irrespectively of their nature, wherever a lightweight porous material is needed [3], applications as thermal and acoustic insulators being perhaps those most important [54]. Ceramic or glass foam synthesis is traditionally carried out by three routes: (i) replica technique, (ii) use of sacrificial template, and (iii) use of direct foaming agents [2, 55]. There is a common strategy for the first two routes of preparing a precursor of the porous structures at a low temperature. This can be achieved either by impregnation of a "spongelike" material or by using sacrificial particles incorporated in the precursor network. In a subsequent heating step, the sacrificial material is removed, leaving the porous cellular microstructure. In principle this allows to design a specific porous network in the low-temperature synthesis step, which then creates a specific skeleton leading to the porous network during the calcination step. Accordingly, porous structures, ranging from microporous and/or mesoporous to macroporous, could be synthesized [56]. Concerning the third route, the foaming agent is added to the starting mixture. Upon calcination, this agent decomposes generating gas bubbles in the melted material, thus creating the porous structure upon cooling [2, 55]. Typical industrially employed foaming agents are carbonates, particularly in the production of ceramic- and glass-based

As stated in the introduction section, there is an increasing attention to the sustainability of material production, and effectively, acoustical sustainable materials, either natural or made from recycled materials, are quite often a valid alternative to traditional synthetic materials [58, 59]. However, the reutilization of glass and ceramic waste generally employs high-energy-demanding production process [9–11], which clearly impacts the sustainability of this route. Furthermore, the use of sacrificial reagents for the synthesis as stated above, clearly contradicts the principles of sustainable chemistry, whereas there is an increasing need for sustain-

We have recently reported synthesis of open-cell foams based on room temperature co-gelling of alginates with glass waste as a viable and sustainable process for production of glass-based cellular. Alginates biopolymers have been used mostly

and *α θ*ð Þ the sound-absorbing coefficient calculated for any *θ* angle.

**4. Cellular foam from recycled waste: synthesis, microstructure**

(17)

(18)

my multiplying the density *ρ<sup>0</sup>* and speed of sound *c0*. *θ* is the incidence angle assumed to be equal to zero, for sound-absorbing coefficient measured at normal incidence. *Zs* is the surface impedance of a layer of the package considered

*Thermal and Acoustic Numerical Simulation of Foams for Constructions*

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

calculated as follows:

**and material properties**

and

foams [57].

**31**

ability of both processes and products [60].

### **Figure 4.** *Optimization TMM functional scheme.*

methods fit the acoustical experimental data obtained in a standing wave tube to, i.e., acoustic absorption coefficient as a function of frequency, to calculate the fluid-phase parameters.

Accordingly, the five parameters related to the fluid phase were determined by applying an inversion procedure algorithm described in [52, 53] to experimental laboratory acoustic measurements.

The fitting of the experimental data is based on a nonlinear best-fit approach implemented in the ICT\_MAA software (http://www.materiacustica.it/mat\_ Software\_ICT.html).

### *3.1.3 Acoustic TMM numerical simulation*

TMM was used to implement a Johnson-Champoux-Allard model. The general scheme of the TM method is depicted in **Figure 4** where the matrix approach allows introduction of dedicated models according the needed and contemporarily solved.

Eq. (15) reports the general analytical expression for TMM which is normally considered as a two-dimensional problem which considers the impact of a flat acoustic wave on the surface of a structure consistent of two or more layers:

$$\mathbf{V(S\_1) = [T]} \text{ V(S\_2)}\tag{15}$$

The vector V(S1) contains the variables that define the acoustic indicators (pressure, stresses, velocity, etc.) applied to the surface S1, whereas the vector V (S2) contains the same variables for the surface S2. The matrix T is a function of the physical and mechanical parameters associated with each specific layer.

Accordingly, the transfer matrix [T] models the transmission of sound waves through the layered structure. The dimension of the matrix is a function of the type of the layer, i.e., solid, fluid, poroelastic, or viscoelastic.

Assuming hard-wall boundary condition, i.e., the layered structure being immersed in a semi-infinite fluid on both sides, the complex reflection coefficient can be defined as follows:

$$R = \frac{Z\_s \cos \theta - Z\_0}{Z\_s \cos \theta + Z\_0} \tag{16}$$

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

where *Z0* = *ρ0c0* represents the characteristic impedance of the fluid calculated my multiplying the density *ρ<sup>0</sup>* and speed of sound *c0*. *θ* is the incidence angle assumed to be equal to zero, for sound-absorbing coefficient measured at normal incidence. *Zs* is the surface impedance of a layer of the package considered calculated as follows:

$$z\_s = \frac{\det[\mathbf{D}\_1]}{\det[\mathbf{D}\_2]} \tag{17}$$

and

methods fit the acoustical experimental data obtained in a standing wave tube to, i.e., acoustic absorption coefficient as a function of frequency, to calculate the

Accordingly, the five parameters related to the fluid phase were determined by applying an inversion procedure algorithm described in [52, 53] to experimental

The fitting of the experimental data is based on a nonlinear best-fit approach implemented in the ICT\_MAA software (http://www.materiacustica.it/mat\_

TMM was used to implement a Johnson-Champoux-Allard model. The general scheme of the TM method is depicted in **Figure 4** where the matrix approach allows introduction of dedicated models according the needed and contemporarily solved. Eq. (15) reports the general analytical expression for TMM which is normally considered as a two-dimensional problem which considers the impact of a flat acoustic wave on the surface of a structure consistent of two or more layers:

The vector V(S1) contains the variables that define the acoustic indicators (pressure, stresses, velocity, etc.) applied to the surface S1, whereas the vector V (S2) contains the same variables for the surface S2. The matrix T is a function of the

Accordingly, the transfer matrix [T] models the transmission of sound waves through the layered structure. The dimension of the matrix is a function of the type

Assuming hard-wall boundary condition, i.e., the layered structure being immersed in a semi-infinite fluid on both sides, the complex reflection coefficient

> *<sup>R</sup>* <sup>¼</sup> *Zs* cos *<sup>θ</sup>* � *<sup>Z</sup>*<sup>0</sup> *Zs* cos *θ* þ *Z*<sup>0</sup>

physical and mechanical parameters associated with each specific layer.

of the layer, i.e., solid, fluid, poroelastic, or viscoelastic.

V Sð Þ¼ <sup>1</sup> ½ � T V Sð Þ<sup>2</sup> (15)

(16)

fluid-phase parameters.

*Optimization TMM functional scheme.*

*Foams - Emerging Technologies*

**Figure 4.**

Software\_ICT.html).

can be defined as follows:

**30**

laboratory acoustic measurements.

*3.1.3 Acoustic TMM numerical simulation*

$$a(\theta) = \mathbf{1} - |\mathbf{R}^2|\tag{18}$$

D1 and D2 matrices are obtained from a complete matrix D (combination of transfer matrix of each layer, coupling matrices, and proper boundary conditions) and *α θ*ð Þ the sound-absorbing coefficient calculated for any *θ* angle.
