In the current context of sustainable development, engineers and scientists are developing materials having a minimum impact on the environment; they are called green or eco-friendly materials. Some of these materials have shown excellent sound absorption properties compared to conventional materials such as glass and rock wools, with a minimum impact on the environment. In fact, while glass and rock wools are excellent sound absorbers, their recycling is still difficult and their processing is highly energy-consuming (Desarnaulds et al., 2005). An alternative to these wools are fibrous materials made from recycled fibers–this paper focuses on this type of materials. Such materials already replaced successfully glass and rock wools: cotton absorbers made from recycled clothing (shoddies) (Langley et al., 2000) and cellulose insulation made from recycled newspapers and even vegetal wools (Piégay et al., 2021) are a few examples. Reviews of sustainable materials and their applications to noise control can be found elsewhere (Desarnaulds et al., 2005; Langley et al., 2000; Asdrubali, 2006).

The modeling of sound propagation and sound dissipation in traditional porous materials is well known. Reviews can be found elsewhere for fibrous materials (Manning and Panneton, 2013; Lei et al., 2018; Tran et al., 2024a) and general porous media (Allard and Atalla, 2009). For eco-friendly acoustic fibrous materials made from recycled materials, many research works have been published in the past 10 years. Most of these works focus on the characterization of their acoustic properties (ex.: sound absorption coefficient) and on the comparison with traditional glass or mineral wools (Manning and Panneton, 2013; Lorenzana et al., 2002; Lee and Joo, 2003; Kosuge et al., 2005; D’Alessandro and Pispola, 2005; Del Rey et al., 2011; Maderuelo-Sanz et al., 2012). These works show the potential of using fibers from different post-consumer or post-industrial wastes (end-of-life tires, plastic bottles, used carpets, metal shavings, fabrics … ) to fabricate good fibrous sound absorbers. In these works, most of the recycled fibrous materials are nonwoven assemblies, where the fibers are blended and mechanically bonded (ex.: needle punched), thermally bonded or resin bonded. In general, few details are given to link their acoustic performance to their fabrication process, their microstructure or intrinsic parameters with a view to optimizing their sound absorption properties.

Porous materials are frequently acoustically modeled using an equivalent fluid method, assuming the frame is acoustically rigid. Common models for this approach include the 6-parameter Johnson-Lafarge (JL) model and the 5-parameter Johnson-Champoux-Allard (JCA) model. The parameters for these models are static airflow resistivity, open porosity, tortuosity, viscous and thermal characteristic lengths, and static thermal permeability. In the past, numerous theoretical and numerical studies have been conducted to connect these parameters for fiber bundles to their microstructure and acoustic properties, with reviews available elsewhere. (Luu et al., 2017; Pompoli and Bonfiglio, 2020; Tran et al., 2024a) However, such research on acoustic materials made from waste textiles or natural fibers remains limited.

Manning and Panneton (2013) measured the JL parameters in various manufactured samples of shoddies made from waste textiles. They developed empirical formulas that relate the JL parameters to the bulk densities of shoddies, which range from 30 to 180 kg/m³ (or porosity from 86% to 98%), with an average fiber diameter of about 22 μ m. (Manning and Panneton, 2013) Santoni et al. introduced an effective fluid dynamic fiber radius for sound insulation materials made from natural hemp fibers. This diameter, derived from an analytical expression of airflow resistivity and its measurement, ranges between 18 and 27 μ m for the manufactured samples. Except for the viscous characteristic length, which is determined by inversion on an acoustic measurement, existing expressions were used to link the other JCA parameters to this effective fiber radius and bulk density. They applied a homothety law to recalculate all JCA parameters across different compression ranges, covering bulk densities from 50 to 150 kg/m³ (or porosity from 88% to 96%). (Santoni et al., 2019).

For felts made from recycled fibers, Tran et al. used a multiscale numerical approach to link the microstructural features (fiber radius distributions and orientation) to the JL parameters. From their results, they updated the semi-analytical relations of Luu et al. (valid for a single fiber diameter) by including the fiber diameter distribution. Contrary to the latter two studies, this time the relations are based only on microstructural features and not on the bulk density. Also, their relations are validated for fiber diameters varying between 0 and 60 μ m following a Gamma function distribution, and porosity from 65% to 99%. (Tran et al., 2024b). In a subsequent paper, Tran et al. utilized these relations as a basis for optimization, employing an iterative differential evolution algorithm to enhance sound absorption. (Tran et al., 2024a).

All the previous relations on sustainable materials were experimentally validated against measured JL or JCA parameters and their ability to predict the normal incidence sound absorption coefficient obtained using an impedance tube. A common aspect of these studies is that the average diameter of the fibers is less than 30 μ m. Moreover, these studies did not address the optimization of the material for sound absorption by a direct method (i.e., not iterative). In this study, the sound absorption coefficient of nonwoven blends made from a Recycled Nylon Fibers (RNF) derived from carpet waste is investigated. Contrary to the previous studies, the diameter of the fibers is much lager and range between 50 and 90 μ m. Consequently, the main purpose of this work is to propose a robust semi-empirical model to predict and optimize the sound absorption of a RNF blend with only one physical parameter, the bulk density. A general direct optimization approach inspired by the non-dimensional design charts for fibrous sound absorbers by Mechel (Mechel, 1988) is proposed. While this direct optimization approach is developed for the studied RNF nonwoven, it could be extended to any other types of porous materials.

In the following, the acoustic model used to describe the porous materials is first presented. Second, experimental characterizations of several RNF assemblies of different densities, ranging from 30 to 180 kg/m³;, are performed. Empirical relationships between each material property (airflow resistivity, open porosity, tortuosity, viscous and thermal characteristic lengths) and the bulk density are developed. Third, the empirical relationships are used in the acoustic model to derive the semi-empirical model for the RNF blend. This semi-empirical model depends only on the bulk density and thickness of the blend, and the frequency. Fourth, design charts for the RNF sound absorbers are obtained from the semi-empirical model to define optimal sound absorption configurations. To facilitate optimization, a straightforward theoretical procedure is finally developed and validated.

The RNF blend is seen as a porous medium made of a solid phase (the fibers) and a fluid phase (here air), see Figure 1. The fluid phase forms a complex network of interconnected pores in which an acoustic wave can propagate and dissipate by thermal and viscous losses. Typically, the fluid phase network is characterized by five macroscopic parameters. Besides the bulk density ( ρ B ), these parameters are the open porosity ( ϕ ), the static airflow resistivity ( σ ), the tortuosity ( α ∞ ), the viscous characteristic length ( Λ ), and the thermal characteristic length ( Λ ′ ). These parameters are homogeneous at the macroscopic scale H defined in Figure 1. This scale is much larger than the diameter of the fibers, and much smaller than the wavelength λ of the acoustical wave. The macroscopic parameters are used to populate an acoustical propagation model based on an equivalent fluid approach (Panneton, 2007). Following this approach, the elastic deformation of the solid phase is neglected and only sound pressure waves propagate in the fluid network. The sound pressure p in this equivalent fluid is governed by the homogeneous Helmholtz equation Δ p + 2 π f 2 ρ e q K e q p = 0 (1) where f is the frequency in Hertz, Δ is the Laplacian operator, and ρ e q and K e q are the equivalent dynamic density and the equivalent dynamic bulk modulus of the equivalent fluid. In Equation 1, both equivalent properties are frequency-dependent and complex-valued to account for the viscous and thermal losses, respectively. Many models exist to predict these two equivalent properties. A review is given elsewhere. (Allard and Atalla, 2009) In engineering applications dealing with sound-absorbing porous materials, the Johnson-Champoux-Allard (JCA) model is largely used. This model is well adapted to most open-cell porous materials, without being limited to specific high porosity fibrous materials with a given fiber size distribution (ex.: mineral wool based model by Delany and Bazley (1970)). Moreover, it depends on measurable parameters only.

In the present work, the acoustic response of the RNF blend may differ from the classical behavior of simple fiber assemblies, more particularly at higher compaction levels (i.e., higher densities, lower porosities, and higher tortuosities). Consequently, it is preferred to use the general 5-parameter JCA model instead of a specific model. In this model, the equivalent dynamic density and bulk modulus are given by (Equations 5 of Allard and Atalla (2009)) ρ e q f = ρ 0 α ∞ ϕ 1 − j σ ϕ 2 π f ρ 0 α ∞ 1 + j 8 π f ρ 0 α ∞ 2 η σ 2 ϕ 2 Λ 2 (2) and K e q f = γ P 0 ϕ γ − γ − 1 1 − j 4 η π f ρ 0 Pr Λ ′ 2 1 + j π f ρ 0 Pr Λ ′ 2 8 η 1 / 2 − 1 (3) where j 2 = − 1 and ρ 0 , γ , Pr and η are the density, specific heat ratio, Prandtl number, and dynamic viscosity of the saturating fluid, respectively. Note that all the properties defined in this work are in MKS units. For soft fibrous materials (limp materials), it is usually preferred to take into account for the added mass of the fibers by using the corrected dynamic density of the equivalent fluid (Panneton, 2007) ρ e q ′ = ρ e q ρ B + ϕ ρ 0 − ρ 0 2 ρ e q + ρ B − ρ 0 2 − ϕ . (4) Once these two dynamic properties are known, the sound absorption coefficient of a layer of thickness L backed by a hard wall is given by α = 1 − Z s − ρ 0 c 0 Z s + ρ 0 c 0 2 (5) where c ₀ is the speed of sound in the fluid, and Z s is the acoustic surface impedance Z s = ρ e q K e q   coth j 2 π f L ρ e q K e q . (6)

The empirical relationships between the macroscopic parameters of the RNF blend and its bulk density developed in the previous section are valid for the studied range of RNF fiber blends. These empirical relationships (Equations 8, 10, 14–16) can be substituted into the JCA equivalent fluid model (Equations 2–4) to form a semi-empirical model. Doing so, the sound absorption coefficient, Equation 5, now depends on the frequency, the thickness, and bulk density of the NCF blend only: α ( ρ B , L , f ) .

To validate this semi-empirical model, three RNF blends of 30, 120 and 180 kg/m³; are assembled and tested in the acoustical impedance tube described above. Validations are made for 50-mm thick samples. Note that densities 30 and 180 kg/m³; are at the bounds of the range on which the empirical relationships were built. This will test the robustness of the model at its low and high limits. The sound absorption measurements are compared to the predictions obtained with the developed semi-empirical model in Figure 8. Three measurements are done for each density. Good correlations are obtained between the measurements and the predictions of the developed semi-empirical model.

From a noise control point of view, it is always desirable to have a simple solution that is tailored to a given problem. A design engineer would like to know the thickness to use for a given material to reach maximum sound absorption at a given frequency when it is backed by a hard wall. Alternatively, he could want to know the material to use for a fixed thickness to reach maximum absorption at a given frequency. To answer these questions, the engineer needs to test (or simulate when possible) different thicknesses, hoping to find the desired optimal solution. This process is lengthy and may yield no optimal results.

Another optimization process would be to adjust the 5 material parameters to reach a given objective. Such an approach is not realistic because the solution found could not make sense physically. In fact, all the parameters of the material are related to its microstructure. Apart from using a microstructural model and sophisticated simulation tools (Perrot et al., 2008; Tran et al., 2024b), one needs to perform a systematic study on the relationship between sound absorption and the material properties. In the past, some optimization studies on fibrous absorbers have been done. Shoshani and Yakubov (2000) made a numerical investigation to obtain optimal parameters for a sound-absorbing fiber web in terms of thickness, porosity, density and coupling parameter (function of resistivity). Their analysis did not take into account for the fact that the material parameters are linked together in a blend (e.g.,: porosity does not vary independently from resistivity, nor density). While their study gives interesting insights, it cannot be systematically applied in a real situation. More recently, Yang et al. (2011) investigated experimentally the effects of the bulk density on the sound-absorbing behavior of a fiber assembly at different compaction levels using a similar setup as the one used in Figure 3. They noticed that there exists an optimum density at which the sound absorption passes through a maximum at a so-called critical frequency. Nevertheless, they did not make a systematic optimization study to link the critical frequency and optimum density to the material properties and thickness. However, such a systematic study had already been made by Mechel (1988) to obtain design charts for fibrous sound absorbers. Based on a one-parameter equivalent fluid empirical model (here the parameter was resistivity), he observed that the sound absorption coefficient is governed by two non-dimensional parameters: k 0 L and σ L / Z 0 , where k 0 = 2 π f / c 0 is the free field wavenumber in air. From this observation, he built design charts at different angles of incidence of the acoustic wave. The design charts can be used to find optimal values of thickness and resistivity to reach a maximum absorption at a given frequency. While his charts are of practical interest, they were built from a one-parameter empirical model valid for a given set of fibrous materials.

For the RNF blend under study, the developed semi-empirical model may be used to find the optimal parameters that will maximize the sound absorption of the blend when backed by a hard wall. This is possible since the only material parameter of the model is the bulk density. Once the optimal bulk density is found, the fiber blend is fully determined; there is no need to directly know what the 5 macroscopic parameters are. Consequently, the optimization process could be posed by one of the following expressions: max     ρ B ∈ 30,180     α ρ B , f 0 , L max     L ∈ 0 , ∞      α ρ B , f 0 , L max ρ B ∈ 30,180 , L ∈ 0 , ∞     α ρ B , f 0 , L (17) where f 0 is the frequency at which the absorption must reach a maximum. The difficulty in using directly Equation 17 lies in the frequency behavior of the absorption that can show several maxima. Another approach is inspired by the work of Mechel (1988) It consists of computing the normal incidence sound absorption coefficient α ( ρ B , L , f ) from the developed semi-empirical model, and plotting absorption charts in function of the non-dimensional parameters F = k 0 L and R = σ L Z 0 . Here, since the relationship is known between resistivity and bulk density, the second parameter is replaced by σ ( ρ B ) L Z 0 , where σ ρ B is given by Equation 10. Such charts are given in Figure 9 for three different thicknesses: 25, 50, and 100 mm. The charts are similar to the normal incidence absorption chart obtained by Mechel. However, contrary to Mechel, a single chart is not sufficient to represent the studied materials. In fact, the absorption chart also depends on the thickness, and not only on R and F. This is explained by the fact the studied material cannot be described by the resistivity only, as in the empirical model by Mechel. This supports the choice of using the more complex and general five-parameter JCA model to build our semi-empirical model.

For each chart in Figure 9, one notes that more than one maximum exists. The first maximum occurs at the so-called critical frequency f ₁ For this frequency, one can clearly determine the optimal non-dimensional parameters R and F. One can note that the optimal value for R is slightly influenced by the thickness. For the three thicknesses, its optimal values are 1.47, 1.57 and 1.57, respectively. For a thickness of 50 mm and Z ₀ = 414 Nsm^-3, this yields an optimal resistivity of 12 986 Nsm^-4 or, using Equation 10, an optimal bulk density of 116 kg/m³;. As for the non-dimensional parameter F, one can note that its optimal value increases with the thickness. For 25, 50, and 100 mm, the optimal values of F are 1.00, 1.12 and 1.27, respectively. For a thickness of 50 mm and c ₀ = 343 m/s, this yields an optimal frequency of 1,224 Hz. As demonstrated, the charts in Figure 9 can be used to quickly identify a configuration yielding maximum absorption. However, if the thickness has to be fixed to a value not given in Figure 9, one needs to interpolate.

Another, more straightforward, approach is to identify relationships between the bulk density, the thickness and the critical frequency at which the first maximum of sound absorption occurs. To proceed this way, one can obtain the sound absorption coefficient for a range of admissible bulk densities and thicknesses on which the empirical relations were built in the previous section. Here, the sound absorption coefficient α ( ρ B , L , f ) is computed for the following admissible combinations: ρ B = 30 : 2 : 180 kg/m³;, L = 10 : 10 : 250 mm, and k 0 L = 0 : 0.01 : 2.5 . For each thickness, the optimal bulk density ρ B opt is identified together with the optimal non-dimensional parameter k e L opt at which the first maximum of sound absorption occurs. Here, k e is the complex wave number in the material. These optimal absorption behaviors are plotted in Figure 10 in function of k ₀ L and k e L . One notes that for all thicknesses, the maximum absorption occurs around the non-dimensional parameter k 0 L opt ≈ 1.30 or k e L opt ≈ 2.04 . The ratio between these two optimal parameters, r p → a = k 0 L opt k e L opt ≈ 0.62 , can be viewed as a projection factor from the porous material wavenumber domain to the air wavenumber domain.

The identified optimal value k e L opt ≈ 2.04 may also be found theoretically. In fact, for a sound absorbing material of thickness L backed by a hard wall, the maximum absorption is mainly reached when the maximum viscous dissipation is reached. This maximum dissipation occurs approximately when the squared RMS acoustic velocity in the material is maximum. Under a normal incidence plane wave, if the origin of the x-axis is at the hard wall, the squared RMS acoustic velocity in the material is given by v RMS 2 = 4 A 2 Z e 2 L ∫ 0 L ⁡ sin k e x 2 d x (18) where A is the amplitude of the pressure wave in the material, Z e = ρ e K e and k e = 2 π f ρ e K e are the characteristic impedance and wave number in the material. The maximum of Equation 18 occurs at optimal non-dimensional parameter k e L = k e L opt ≈ 2.2 , which is about the one found from the simulations. This value is an approximation and depends also on the thermal losses and material properties.

From the previous simulations and discussions, it is clear that a relationship exists between ρ B opt and k e L opt . Moreover, for a given thickness, k e L opt yields the critical frequency at which the first maximum peak occurs: f 1 p = c 0 2 π k e L opt L ≈ 112 L (19) with c 0 = 343 m/s at standard atmospheric conditions. The previous frequency is in the porous material wavenumber domain. To obtain its value in the air wavenumber domain, the following conversion is needed: f 1 a = r p → a f 1 p (20) with r p → a ≈ 0.62 . From the simulation results, the relationships between these parameters are plotted in Figure 11 for the calculated thicknesses. Also the following power law is obtained for the optimal bulk density: ρ B opt = 27.958 L 0.437 (21)

In the following, Equations 19–21 will be used to find the optimal RNF blend for different sound absorber design objectives.

In this paper, a semi-empirical model was derived to predict the normal incidence sound absorption coefficient of a Recycled Nylon Fiber (RNF) blend. The semi-empirical model is based on the general five-parameter Johnson-Champoux-Allard (JCA) for porous materials in which the five parameters (i.e., open porosity, airflow resistivity, tortuosity, viscous and thermal characteristic lengths) are replaced by characterized empirical relationships expressed in terms of the bulk density of the blend only. From this semi-empirical model, sound absorption charts were built to help design optimized RNF absorbers. A more systematic design procedure based on equations was also worked out. The procedure allows for a quick determination of the optimal parameters of a mixture to achieve maximum absorption. Three design problems were addressed and validated by experiments 1) to determine the optimal thickness and bulk density to reach maximum absorption at a given frequency, 2) to find the optimal density for a given thickness, and 3) to find the optimal thickness for a given bulk density. While the optimization procedure was developed for RNF absorbers, a similar procedure could be developed for any other fibrous-like sound absorbers. More specifically, the proposed method could be applied to cotton felts which have smaller fiber diameters and are often used in the automotive industry. This could be a simpler way to optimize these felts for sound absorption compared to a complex multi-scale analysis as done elsewhere (Tran et al., 2024b).