# On the Dependence of Acoustic Pore Shape Factors on Porous Asphalt Volumetrics

^{*}

## Abstract

**:**

## 1. Introduction

_{den}and/or L

_{night}).

^{3}on average), and about 54% (16.8 µg/m

^{3}on average) of the total, respectively [39]. PA pavements allow significantly reducing (up to 90%) the concentration of total suspended solids, total Kjeldahl nitrogen, chemical oxygen demand, total metals (e.g., copper, lead, and zinc), nutrients, mineral oils, and other soluble and anthropogenic pollutants (from non-exhaust traffic-related sources such as brake, tire, and clutch) into runoff water [40,41]. For PA durability and clogging (see, e.g., [42]), during the lifetime of porous asphalt pavement, traffic-related particles (e.g., wear particles from both tires and roads due to the tire-road interaction), and natural particles (e.g., sand and dust) obstruct the air voids causing the long-term loss of the sound absorption. By acting on maintenance (e.g., by periodic cleanings), on mixture design (i.e., improving the mix gradation, e.g., using gap-graded thin and very thin overlays of 10–25 mm, [42,43]), and on layer design (e.g., using two-layer asphalt mixtures [44]), the clogging risk can be reduced.

## 2. Background

_{0}′) are used to represent the viscous and the thermal dissipation of energy due to the presence of pores (see Section 4.2 for more details). In summarizing, even with its limitations in terms of frequency, the JCAL model was selected because of its suitability to be applied to highly porous mixes and because of its structure.

_{ρ}and s

_{K}were introduced. These latter are (i) The thermal shape factor and the viscous shape factor, respectively. (ii) They take into account the energy dissipation due to the presence of pores. (iii) Depend on tortuosity, porosity, and resistivity of all the material and on resistivity, length, and cross-sectional area of each pore. The expressions used to define both the shape factors are similar (cf. Equations (5) and (7) in Section 4.1), and the only difference refers to the use, as weights in the summations, of the cross-sectional areas. In particular, s

_{ρ}is more affected by the narrower parts of the pores, while the wider parts are more important for s

_{K}. Note that the use of the single shape factor model (i.e., using the s

_{ρ}parameter only) or the generalized two shape factor model (i.e., using s

_{ρ}and s

_{K}together) depends on the type of materials [51]. The aforementioned model was used by Stinson et al. (1997) [52] for the acoustical characterization of the porous road pavements. Note that in the abovementioned study, 10 cm diameter and 4-cm-thick porous pavement samples were tested, the airflow resistivity, r = 55 kNs/m

^{4}, porosity, Ω = 15%, tortuosity, q

^{2}= 2.5, and the shape factors s

_{p}and s

_{K}were assumed equal to 1. This latter is one of the most important models used to characterize the sound absorption of PA concrete objects, and, for this reason, it was used in the study presented in this paper and, in the following, is called the STIN model. Note that its structure and development interact with well-known milestones, including (1) the flow resistance-focused model after Delany and Bazley (cf. [60,61,62]). (2) Many studies in the literature dealing with the relationship among the three main factors (resistivity, porosity, and tortuosity (cf. [63]). (3) The three-parameter model after Hamet et al. (cf. [52,63,64,65]).

_{0}′, static viscous tortuosity, τ

_{0}, and static thermal tortuosity, τ

_{0}′). On the other hand, pore structure elastic parameters describe the solid phase viscoelastic behavior. In the case of small deformation, Young’s moduli, Poisson’s ratio, and structural damping coefficients can be used as pore structure elastic parameters. Furthermore, Otaru (2020) [53] reported several techniques to enhance the acoustic absorption of porous metals, i.e., (1) Eliminating resonance (vibration) and reducing acoustical energy. (2) Using materials that are able to withstand microstructure manipulation (e.g., mechanical alteration). (3) Using materials that have small interconnected pores. (4) Using materials that allow sound pressure waves to fully penetrate the interior of the microstructure. (5) Using surfaces that are characterized by smaller pores (generally, sound absorption is improved if sound wave impacts smaller pores first). (6) Bearing in mind that characteristics such as structural morphology, pore size, pore openings, and pore volume of the materials depend on filler size and shape, packing density, arrangements, and the applied pressure. (7) Increasing the porous layer thickness that affects (increase) the pores’ non-uniformity, which in turn increases the high-frequency dynamic tortuosity. (8) Increasing the presence of back cavities or air gaps (the inclusion of air gaps allows reducing the thickness while maintaining the absorption potential of the material because a part of the sound energy is converted into heat by the Helmholtz resonance effect). (9) Increasing hole drilling/rolling of metal foams and the patterns in the arrangement of the space fillers (e.g., packing of spheres). Note that some of the techniques listed above are the same as those reported above for PA pavements (e.g., increase the layer thickness), and the others may be used in the case of PA. Finally, Otaru (2020) [66] stated that the quantitative assessment of pore structure-related parameters (e.g., pore volume fraction and open porosity) of porous metallic materials can be carried out by combining high-resolution tomography and 3D advanced image processing (e.g., volume rendering, segmentation, 3D editing, thresholding filtering). Finally, methods based on ultrasonic sound and the Q-delta approach can be used to quantify Λ, Λ′, and q

^{2}(tortuosity) [68].

## 3. Objectives and Tasks

## 4. Methods and Materials

#### 4.1. Impact of Shape Pore Factors on the Acoustic Absorption in the STIN Model

_{K}, thermal shape factor, whereas ρ(ω) depends on the viscous shape factor, s

_{ρ}.

^{2}) and two supplementary energy loss-related parameters s

_{ρ}and s

_{K}. These parameters (a.k.a., viscous and thermal pore shape factors, respectively [52]) were introduced to relate the behavior of real pore shapes to that of circular pores and represent the influence of the cross-sectional shape of the pore (i.e., the deviation from circular) [52,71,72].

_{ρ}is the viscous pore shape factor ratio, and, in the high-frequency limit, is related to the permeability coefficient k

_{0}(i.e., a dimensionless parameter that is constant for a given pore shape, i.e., 2 for circular pores, 3 for slits, 5/3 for triangular equilateral pores, and 1.78 for square pores) as follows [72]:

_{ρ}is related to the airflow resistivity, r, within the pores [73], the dynamic viscosity of the air inside the pores (η), and the hydraulic radius of the pores (r

_{h}, i.e., the ratio between the cross-sectional area and circumference of the pore) through the following expression [72]:

_{ρ}is 0.5 for circular pore materials, about 0.41 for slit-pore materials, and about 0.55 for triangular pore materials [72]; (2) the resistance corresponding to σ is measured by considering the real part of the normal-incidence flow impedance at very low frequency/low Reynolds numbers, cf. [72,74]. The following table (Table 1) contains values of the pore shape factors included in the STIN model for common materials.

_{ρ}and s

_{K}comes from the following expressions [51,52], which represent the microstructural model. The dynamic density (microstructural approach) is:

_{0}is the density of air, ω is the angular frequency (=2πf, where f is the frequency), α

_{∞}is almost equal to the tortuosity (q

^{2}), r is the airflow resistivity of the porous structure of the material, Ω is the porosity of the of air-filled connected pores of the material, and F(λρ) is:

_{ρ}, which is given by the expression:

_{ρ}is the viscous pore shape factor, which is an adjustable parameter that is connected (for real granular structure with complex geometry) to the viscosity dependence inside the material [52]. For the dynamic bulk modulus (microstructural model), we use:

_{0}is the ambient atmospheric pressure, N

_{pr}is the Prandtl number, T( ) is the ratio between Bessel functions of first and zero order, and the parameter λ

_{K}is:

_{K}is the thermal pore shape factor.

_{ρ}(from 0.5 to 4.0) corresponds to the reduction in the maximum sound absorption coefficient and the reduction in the abscissa (peak frequency). Note that an opposite effect is obtained if s

_{K}increases in the same range (from 0.5 to 4.0).

#### 4.2. Impact of Shape Pore Factors on the Acoustic Absorption Based on the JCAL Model

_{0}′). Viscous characteristic length Λ (measured in μm) and thermal characteristic length Λ′ (measured in μm) are two parameters used in the JCAL model [77,79] to take into account the viscous and thermal effects that occur in porous materials filled with fluid. Because of the not simple geometry of the pores in ordinary porous materials, a direct calculation of the two parameters mentioned above is not possible [80]. Hence, the simple case of sound propagation in porous materials with cylindrical pores (i.e., cylindrical tubes having a circular cross-section) is commonly used to derive an approximation of the aforementioned lengths and define important concepts such as tortuosity.

_{∞}, which is, as defined above, almost equal to the tortuosity, q

^{2}), static airflow resistivity (σ), air density (ρ

_{0}), shear viscosity (η; while the volume viscosity is neglected), atmospheric pressure P

_{0}, the specific heat ratio of air (γ = C

_{p}/C

_{v}), the Prandtl number (N

_{pr}), the viscous (Λ) and thermal (Λ′) characteristic lengths, and the static thermal permeability (k

_{0}′). The aforementioned complex parameters can be written as [64]:

_{0}′ was introduced by Lafarge to accurately describe the thermal effect at the low frequencies. In particular, it represents the low-frequency limit of the dynamic thermal permeability (k′), and it describes the thermal exchanges between the frame and saturating fluid as the static viscous permeability (k

_{0}) describes the viscous forces [66]. It can be estimated using the expression:

_{0}, static viscous permeability, is the ratio of η and σ, where η is the dynamic viscosity of air, and σ is the static airflow resistivity. For acoustical materials, the range of values for the static thermal permeability (k

_{0}′) is approximately 10

^{−10}–10

^{−8}m

^{2}.

_{pr}) can be calculated using the expression:

_{0}= 0 °C, and pressure P

_{0}= 1.0132 × 10

^{5}Pa), the air viscosity is η = 1.84 × 10

^{−5}kg/(m·s), and the air thermal conductivity k = 2.6 × 10

^{−2}W/(m·K). For air at T = 18 °C and pressure P

_{0}, the air density ρ

_{0}= 1.213 kg/m

^{3}, the air adiabatic bulk modulus K

_{0}= 1.42 × 10

^{5}Pa, the speed of sound in air is c

_{0}= 342 m/s, the air characteristic acoustic impedance Z

_{0}= 415 Pa·s/m, the air specific heats ratio γ = 1.4, and Pr = 0.71 [77].

^{2}), affect the high-frequency behavior of the complex effective density (ρ) and the complex bulk modulus (K) of the fluid into the pores. Λ only depends on the geometry of the frame and does not depend on the characteristics of the fluid. The viscous effects are located in a very small region close to the walls of the pores. Hence, by neglecting the small contribution of the boundary-layer region, Λ can be defined using the expression:

_{i}(r) is the local microscopic velocity of the fluid inside the pores that occupy a volume (V), while u

_{i}(r

_{w}) is the local microscopic velocity of the fluid at the surface (S), which is the area occupied by the pore walls (w) in the representative elementary volume. Λ is related to the airflow resistivity (r), and, for this reason, it can be estimated using the expression:

_{0}′ on the sound absorption spectrum modeled using the JCAL model. Figure 3a,b show the influence of Λ and Λ′ when they vary between 100 and 1000 μm, while Figure 3c shows the effect of k

_{0}′ varying between 1 × 10

^{−}

^{10}m

^{2}and 1 × 10

^{−}

^{8}m

^{2}. The ranges above were derived from Table 2.

#### 4.3. Pore Shape Factors and Acoustic Absorption

_{K}and Λ) or decreases (s

_{ρ}, Λ′, and k

_{0}′) of the sound absorption coefficient (a

_{0}). At the same time, they affect the position of the maximum (a

_{0,max}, i.e., the point of maximum f, Hz), according to Table 3.

^{2}= 0.97):

_{20}stands for permeability measured at 20 °C.

#### 4.4. Experiments

## 5. Results

_{ρ}, s

_{K}, Λ, Λ′, k

_{0}′) vary for the 10 cases (cores) under consideration, for the 2 considered models (STIN and JCAL), under the hypothesis of having a single layer (1L), or 2 layers (2L (UP) and 2L (LOW)), with tests carried out from above (interface type-pavement, 2L (UP)) or from below (2L (LOW)). Note that:

- These values were obtained as a result of the optimization process. Assuming for t (1L) and Ωc the actual values (with a specific tolerance of ±30%), the resistivity was derived using Equation (17) (r
_{est}, which allows obtaining results better than those obtained using r_{meas}), while the remaining parameters were derived through the optimization; - These optimal values refer to the minimization of errors around the peak. This means that in the minimization process, attention was paid to fitting the values of frequency and absorption around the peak or the peaks. Consequently, this often implied to fit a maximum around 0.7–0.9 for frequencies around 0.8–1.2 kHz;
- 2L simulations always provided results at least comparable to the ones given by 1L-simulations;
- The word “Good” refers to appreciable goodness of fit (peak well simulated), while “Bad” to the opposite situation.

_{ρ}, s

_{K}, Λ, Λ′, k

_{0}′):

- s
_{ρ}, maximum is 5.0, its minimum is 0.5, the average is 3, with a coefficient of variation (ratio of the standard deviation to the mean) of about 49%; - s
_{K}maximum is 5.0, its minimum is 0.5, the average is 1.3, with a coefficient of variation of about 83%; - Λ maximum is 790, its minimum is 5, its average is 316, with a coefficient of variation of about 86%;
- Λ′ maximum is 828, its minimum is 15, its average is 414, with a coefficient of variation of about 63%;
- k
_{0}′ maximum is 1 × 10^{−8}, its minimum is 1 × 10^{−10}, its average is 5 × 10^{−9}, with a coefficient of variation of about 84%.

^{2}was derived:

- STIN, 1L: R
^{2}(Ω) = 0.58; R^{2}(r_{est}__{UP}) = 0.95; R^{2}(a_{0}) = {0.09–0.99}, when the outliers were not discarded; - STIN, 2L(UP)): R
^{2}(Ω) = 0.92; R^{2}(r_{est}__{UP}) = 0.99; - STIN, 2L(LOW)): R
^{2}(Ω) = 0.98; R^{2}(r_{est}__{LOW}) = 0.72; - STIN, 2L: R
^{2}(a_{0}) = {0.49–0.99}; - JCAL, 1L: R
^{2}(Ω) = 0.40; R^{2}(r_{est}__{UP}) = 0.99; R^{2}(a_{0}) = {0.003–0.99}, when the outliers were not discarded; - JCAL, 2L(UP)): R
^{2}(Ω) = 0.91; R^{2}(r_{est}__{UP}) = 0.97; - JCAL, 2L(LOW)): R
^{2}(Ω) = 0.94; R^{2}(r_{est}__{LOW}) = 0.74; - JCAL, 2L: R
^{2}(a_{0}) = {0.80–0.99}.

- For the correlations between identical parameters in different models, thickness, porosity, and resistivity are well correlated to each other (high correlations; R = 0.89–0.94), while tortuosity showed low-to-negligible correlations;
- For the correlations involving the porosity (Ω) derived by the STIN model, Ω is moderately correlated with the resistivity (R = −0.69), is low correlated with thickness (R = −0.39) and tortuosity (R = −0.31). At the same time, the JCAL model allowed deriving a porosity that is low correlated with resistivity (R = −0.55) and thermal characteristic length Λ′ (R = −0.54) and is low correlated with tortuosity (R = 0.20) and viscous characteristic length Λ (R = 0.28). Negligible correlations are observed otherwise (−0.04 ≤ R ≤ 0.28);
- For the correlations involving the resistivity (r
_{est}), the STIN model provides values moderately correlated with the porosity (R = −0.69) and lowly correlated with the tortuosity (R = 0.40). The JCAL model yields values that are moderately correlated with porosity (R = −0.55) and that are low correlated with tortuosity (R = 0.31). Negligible correlations are observed otherwise (−0.01 ≤ R ≤ 0.24); - For the correlations involving tortuosity (q
^{2}), the values obtained through the STIN model have low correlation with porosity (R = −0.31) and resistivity (R = 0.4), while those returned by the JCAL model are low correlated with thickness (R = −0.45), resistivity (R = 0.31), and static thermal permeability k_{0}′ (R = 0.38). Negligible correlations are observed otherwise (−0.17 ≤ R ≤ 0.25); - For the correlations involving pore factors (i.e., s
_{ρ}, s_{K}, Λ, Λ′, k_{0}′), the low correlations are observed: (1) Between viscous characteristic length Λ and thermal characteristic length Λ′ (R = −0.49). (2) Between thermal characteristic length Λ′ and static thermal permeability k_{0}′ (R = −0.42). (3) Negligible correlations were observed between the STIN-related pore factors (R = −0.08). (4) Low-to-negligible correlations are observed between STIN-related shape factors and JCAL-related pore factors (−0.24 ≤ R ≤ 0.15).

- For the correlations between identical parameters in different models, thickness, porosity and resistivity are very high-to-high correlated with each other (R = 0.82–0.97), while tortuosity shows low (R = 0.46) correlations;
- For the correlations involving porosity (Ω), the STIN model, Ω corresponds to values that are moderately correlated with resistivity (R = −0.52) and lowly correlated with the viscous shape factor s
_{ρ}(R = 0.35) and the thermal shape factor s_{K}(R = −0.42). The JCAL model shows porosities that are moderately correlated with the thermal characteristic length Λ′ (R = −0.51) and low correlated with resistivity (R = −0.37). Negligible correlations are obtained otherwise (−0.00 ≤ R ≤ 0.16); - For the correlations involving the resistivity (r
_{est}), for the STIN model, r_{est}results moderately correlated with porosity (R = −0.52) and viscous shape factor s_{ρ}(R = −0.51), and lowly correlated with thickness (R = 0.34). The JCAL model provides values of resistivity low correlated with thickness (R = 0.31), porosity (R = −0.37), viscous characteristic length Λ (R = 0.32). Negligible correlations are observed otherwise (−0.28 ≤ R ≤ 0.21); - For the correlations involving tortuosity (q
^{2}), the STIN model shows a moderate correlation of this parameter with the viscous shape factor s_{ρ}(R = −0.65). The JCAL model showed low correlations between tortuosity and thickness (R = −0.47) and static thermal permeability k_{0}′ (R = 0.39). Negligible correlations were derived otherwise (−0.22 ≤ R ≤ 0.15); - For the correlations involving pore factors (i.e., s
_{ρ}, s_{K}, Λ, Λ′, k_{0}′), low correlations are observed between viscous characteristic length Λ and thermal characteristic length Λ′ (R = −0.48), between thermal characteristic length Λ′ and static thermal permeability k_{0}′ (R = −0.40), and between viscous shape factor s_{ρ}and thermal characteristic length Λ (R = −0.42), while negligible correlations (R = 0.02) are observed between the STIN-related pore factors, and between STIN-related shape factors and JCAL-related pore factors (except for the low correlation, R = −0.42, between viscous shape factor s_{ρ}and thermal characteristic length Λ′).

- For porosity, the STIN model shows an inverse proportionality between porosity and resistivity, as well as thermal shape factors (s
_{K}). The JCAL model shows an inverse proportionality of Ω with the thermal characteristic length Λ′; - For resistivity, the STIN model exhibits its inverse proportionality with viscous shape factors (s
_{ρ}); - For tortuosity, for the STIN model, an inverse proportionality with viscous shape factors (s
_{ρ}) is obtained. At the same time, the JCAL model shows an inverse proportionality with thickness; - For pore factors, the best (inverse) proportionalities are observed between the couples Λ-Λ′ (R = −0.48; JCAL model), Λ′-k
_{0}′ (R = −0.40; JCAL model), and s_{ρ}-Λ (R = −0.47; STIN model-JCAL model).- ○
- For the viscous shape factor (sρ), higher values correspond to lower thickness (R = −0.25), higher porosity (R = 0.35), lower resistivity (R = −0.51), and tortuosity (R = −0.65). When the two cases 2 and 3 are not considered, the absolute value of the Pearson coefficients increases. Estimates take into account the inverse relationship with resistivity (sρ = A × rest−0.5), where A is a calibration factor, and which is consistent with the generalized model for porous materials (cf. [51]). In this case, the Pearson coefficient yields an appreciable value (−0.91);
- ○
- For the thermal shape factor (sK), higher values correspond to higher thickness (R = 0.14), lower porosity (R = −0.42), higher resistivity (R = 0.21), and lower tortuosity (R = −0.17). The fact that the Pearson coefficient for Ω-sK is negative could depend on thermal losses;
- ○
- For the viscous characteristic length (Λ), higher values correspond to higher thickness (R = 0.39), porosity (R = 0.16), and resistivity (R = 0.32), and to lower tortuosity (R = −0.10);
- ○
- For the thermal characteristic length (Λ′), higher values correspond to higher thickness (R = 0.26), lower porosity (R = −0.51), and higher resistivity (R = 0.15) and tortuosity (R = 0.13);
- ○
- For the static thermal permeability (k0′), higher values correspond to lower thickness (R = −0.52), higher porosity (R = 0.11), lower resistivity (R = −0.28), and higher tortuosity (R = 0.39).

_{0,max}), it is better to act on:

- The viscous shape factor (s
_{ρ}), which should be as low as reasonably achievable, and this can be obtained principally reducing resistivity and tortuosity (and, in a less effective way, reducing the thickness and increasing the porosity). At the same time, it is noted that lower values of s_{ρ}correspond to higher points of maximum (frequency of the maximum of the sound absorption spectrum, cf. Figure 2), which could affect its potential to minimize the corresponding spectrum of the particular noise source; - The viscous characteristic length (Λ), which should be increased. This can be obtained by increasing thickness (and, in a less effective way, increasing porosity and resistivity and reducing the tortuosity). Importantly, as mentioned above, for s
_{ρ}, Λ affects the absorption peak in terms of value and frequency. This should be considered in terms of mix design.

- Λ (which refers to the viscous characteristic lengths) exhibits a moderate positive relationship with porosity and resistivity and a moderate negative relationship with tortuosity;
- Λ′ (which refers to the thermal characteristic lengths) yields a moderate negative relationship with porosity and Λ, while it shows a moderate positive relationship with resistivity and tortuosity. Note that the relationship between Λ and Λ′ (Λ′ > Λ) complies with the fact that Λ′ is related to the largest size of the pores while Λ to the smallest ones;
- k
_{0}′ (which refers to the static thermal permeability) yields a moderate negative relationship with r, Λ, and Λ′. It has a moderate positive correlation with porosity and tortuosity.

_{ρ}, s

_{K}, Λ, Λ′, and k

_{0}′) and porosity (Ω; dimensionless, %), airflow resistivity (r; kN × s/m

^{4}), and tortuosity (q

^{2}; dimensionless). Figure 6a,b refer to the STIN model, while Figure 6c–e refer to the JCAL one.

## 6. Conclusions

_{0}) plays an important role during design and acceptance procedures. This parameter depends not only on geometric and volumetric factors (i.e., thickness, porosity, tortuosity, airflow resistivity, and pore shape factors) but also on pore shape factors (related to thermal and viscous effects inside the medium). Unfortunately, there is a lack of available relationships for the prediction of pore shape-related factors in the design and acceptance procedures of PAs.

_{0}values measured on PA samples (i.e., a

_{0}maximum around 0.7–0.9 for frequencies around 0.8–1.2 kHz, fitted using the inverse problem applied considering the best range 600–1600 Hz for the optimization). One-layer (1L) and two-layer (2L) models were used (because of possible clogging phenomena and inhomogeneity).

_{ρ}and s

_{K}(the pore shape factors derived using the STIN model) have the potential for governing a

_{0}spectrum in terms of peak position, when they vary within the values indicated in the literature (i.e., s

_{ρ}= 0.4–3.1, and s

_{K}= 0.3–5.2). The same applies to the JCAL shape pore factors, when they vary within the values indicated in the literature, i.e., Λ = 0.7–2100 μm, Λ′ = 5–1850 μm, and k

_{0}′ = 9 × 10

^{−8}–11 × 10

^{−8}).

- All the shape factors show quite reliable correlations (Pearson coefficients greater than |0.51| and R
^{2}that reached 0.70) with porosity or resistivity. At the same time, they exhibit a high coefficient of variation (i.e., 49–86%), and this calls for further research; - s
_{ρ}and s_{K}(which refer to the viscous and thermal effects inside the narrower and the wider parts of the pores, respectively) can be estimated based on resistivity and tortuosity (equations with R^{2}= 0.66–0.70 were found when the two-layer approach was used to characterize the samples); - Λ (viscous effects) can be estimated based on resistivity and tortuosity (R
^{2}= 0.40–0.68 using the two-layer approach), Λ′ (thermal effects) can be estimated based on porosity and resistivity (R^{2}= 0.50–0.55 using the two-layer approach), and k_{0}′ can be estimated based on resistivity and tortuosity (R^{2}= 0.50–0.56 using the two-layer approach); - PAs with high values of max sound absorption coefficient (e.g., 0.8) can be obtained mainly acting on: (1) The viscous shape factor (s
_{ρ}), which should be as low as reasonably achievable. This can be obtained by reducing resistivity and tortuosity (and, in a less effective way, reducing the thickness and increasing the porosity). (2) The viscous characteristic length (Λ), which should be increased. This can be obtained by increasing thickness; - The most important factors for the acoustic design of dense graded friction courses (DGFCs) are the porosity and the viscous shape factor, while further investigations are needed on the static thermal permeability. In contrast, porous European mixtures (PEMs) and open graded friction courses (OGFCs) mainly depend on tortuosity and resistivity, while further investigations are needed on the thermal shape factor.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Influence of the parameters (

**a**) Λ, (

**b**) Λ′, and (

**c**) k

_{0}′ on the JCAL’s spectrum (one layer).

**Figure 5.**Instruments used to derive (

**a**) the sound absorption coefficient, (

**b**) the porosity, (

**c**) the airflow resistivity, and (

**d**) the hydraulic permeability of the samples used in this study.

**Figure 6.**Relationships involving the shape factors (

**a**) s

_{ρ}, (

**b**) s

_{K}, (

**c**) Λ, (

**d**) Λ′, and (

**e**) k

_{0}′. Note. Log. = logarithmic curve, Exp. = exponential curve.

Ref. | Range of s_{ρ} (dim.less) | Range of s_{K} (dim.less) | Short Notes |
---|---|---|---|

[52] | 1 | 1 | Porous pavement (microstructural model with core samples. d = 10 cm; t = 4 cm; r = 55,000 Ns/m^{4}, Ω = 15%, q^{2} = 2.5). |

[52] | 1.14 | 0.44 | Porous asphalt (microstructural model/Circular pores. r = 55,000 Ns/m^{4}, Ω = 15%, q^{2} = 2.5). |

[52] | 1.14 | 0.88 | Porous asphalt (microstructural model constant cross-section with a modification of the shape along the pore axis. r = 55,000 Ns/m^{4}, Ω = 15%, q^{2} = 2.5). |

[75] | 3.1 | 0.350 | Porous absorber made up of ground tire rubber (GTR), vermiculite and expanded polystyrene (EPS. r = 8865 Ns/m^{4}, Ω = 61.7%, q^{2} = 2.749, D = 408.7 kg/m^{3}, binder concentration 5%, grain size < 2.0 mm). |

[75] | 2.590 | 0.285 | Porous absorber GTR 88% (r = 14,551 Ns/m^{4}, Ω = 53.5%, q^{2} = 2.402, D = 547.6 kg/m^{3}, binder concentration 12% grain size 1.0–3.0 mm). |

[51] | 0.93 | 5.15 | Two-diameter model porous material (i.e., a specially fabricated sample consisting of two porous layers with two different diameters, i.e., about 1.5 mm and about 0.3 mm. r = 68.7 cgs rayls/cm, Ω = 39.2%, q^{2} = 4.06). |

[76] | 1.2–1.34 | 0.83–0.9 | Loose aquarium gravel (r = 4850 ± 626 Ns/m^{4}, Ω = 43.4% ± 0.24%, q^{2} = 1.37 ± 0.17, layer 1 t = 5 ± 0.2 cm and layer 2 t = 10 ± 0.2 cm). |

[72] | 0.408–0.816 (rectangular pores) −0.548–1.095 (triangular pores) | n.a. | Two model porous materials were built. The first one containing rectangular pores (about 0.15 × 0.17 mm; Ω = 0.9, q^{2} = 1.44, and r = 8.71 cgs rayl/cm). The second one containing triangular pores (about 0.34 × 0.37 mm. Ω = 0.3, q^{2} = 1.44, and r = 14.2 cgs rayl/cm). |

_{ρ}= viscous pore shape factor; s

_{K}= thermal pore shape factor; d = diameter; t = thickness; r = airflow resistivity; D = bulk density; Ω = porosity; q

^{2}= tortuosity; n.a. = not available.

Ref. | Range of Λ (μm) | Range of Λ′ (μm) | Range of k_{0}′(m ^{2}) | Short Notes |
---|---|---|---|---|

[89] | 0.73–1.2 | 64–100 | n.a. | Composite materials made of adhesive mortar and scrap tire rubber particles (E = 281–6140 MPa; D = 0.92–1.52 g/cm^{3}; t = 0.5–0.58 cm; Ω = 44–83%; q^{2} = 1.5–3.5, r = 3 × 10^{3}–26 × 10^{5} Ns/m^{4}). |

[90] | 30–182 | 60–400 | n.a. | Fibrous materials (felt, fiberglass, polyester fibers). q^{2} = 1–1.06. |

[90] | 5–450 | 15–690 | n.a. | Cellular materials (cellular rubber, melamine foam, metal foam, plastic foam, poroelastic foam, polymide foam, polylactide and polyethylene glycol foam, polyurethane foam). q^{2} = 1.01–4.45. |

[90] | 5.1–550 | 15.4–830 | n.a. | Granular materials (lead shot, gravel, glass breads, perlite; t = 0.01–0.9 cm; q^{2} = 1.1–3.84). Other q^{2}: Open porous asphalt = 2–3.3. Asphalt = 1.8. Compacted soil = 1.4. Forest floor = 1.1. Soft soil = 1.3. Snow old crusted = 4. Snow new = 1.5–2.7. |

[90] | 49–770 | 131–582 | n.a. | Porous aluminum, porous ceramic, snow. q^{2} = 1.1–3.3. |

[91] | 104–154 7–45 55–69 38–72 | 104–292 112–368 78–142 84–392 | n.a. | Four materials are frequently used in aerospace and building applications for thermal and sound insulation. Material A (Low resistivity plastic foam. t = 5 cm; Ω = 98%; D = 9 kg/m^{3}; r = 1 × 10^{4} Ns/m^{4}; q^{2} = 1–1.07); Material B (High airflow resistivity plastic foam. t = 5 cm; Ω = 99%; D = 5 kg/m^{3}; r = 4 × 10^{4} Ns/m^{4}; q^{2} = 1–2.6); Material C (Low density fibrous materials. t = 1.8 cm; Ω = 99%; D = 5.5 kg/m^{3}; r = 1.5 × 10^{4} Ns/m^{4}; q^{2} = 1–1.05); Material D (High-density fibrous materials. t = 8 cm; Ω = 99%; D = 40 kg/m^{3}; r = 1.3 × 10^{4} Ns/m^{4}; q^{2} = 1–1.15). |

[92] | 420 790 300 370 | 830 260 940 810 | n.a. | 3D-printed specimens. Sample 1: body centered cubic, BCC (t = 6 cm; Ω = 80%; r = 1.3 × 10^{3} Ns/m^{4}; q^{2} = 1.52). Sample 2: BCC (t = 5 cm; Ω = 71%; r = 7 × 10^{3} Ns/m^{4}; q^{2} = 2.55). Sample 3: face centered cubic, FCC (t = 6.5 cm; Ω = 77%; r = 4.3 × 10^{3} Ns/m^{4}; q^{2} = 2). Sample 4: similar to Weaire–Phelan structure, A15 (t = 6.5 cm; Ω = 77%; r = 2.3 × 10^{3} Ns/m^{4}; q^{2} = 2). |

[93] | 245–251 | 418–430 | n.a. | Date palm fibers (t = 2–4 cm; D = 65 kg/m^{3}; r = 0.91 × 10^{3}–1 × 10^{3} Ns/m^{4}; Ω = 93%; q^{2} = 2.9). |

[94] | 120 | 500 | n.a. | Reticulated foams (q^{2} = 1.2; Ω = 98%; r = 4 × 10^{4} Ns/m^{4}). |

[84] | 180–202 132–134 249–273 | 429–610 292–370 650–750 | n.a. | Helium saturated sample measured at ultrasonic frequencies (70–600 kHz). Sample 1: t = 1 cm; Ω = 98%; q^{2} = 1.052. Sample 2: t = 0.9 cm; Ω = 97%; q^{2} = 1.042. Sample 3: t = 0.9 cm; Ω = 97%; q^{2} = 1.054. |

[82] | 199 (with c = 0.56) 147 (with c = 0.94) | 291 (with c′ = 2.6) 287 (with c′ = 2.1) | n.a. | Ultrasonic measurements on air-filled porous samples. Sample 1: t = 0.2–1 cm; Ω = 98%; q^{2} = 1.06. Sample 2: t = 0.2–0.9 cm; Ω = 97%; q^{2} = 1.12. Air characteristics: D = 1.2 kg/m^{3}; η = 1.85 × 10^{−5} kg/m × s; γ = 1.4; N_{pr} = 0.71. |

[79] | 610 (with c = 0.3) 2100 (with c = 0.03) | 1.3 × 10^{−8}–1.7 × 10^{−8} | Measurements of dynamic compressibility of air-filled porous materials at audible frequencies. Foam (r = 6 × 10^{3} Ns/m^{4}; Λ = 0.3 × 10^{−8} m^{2}; k_{0}′ = 1.3 × 10^{−8} m^{2}). Glass wool (r = 2.3 × 10^{3} Ns/m^{4}; Λ = 0.8 × 10^{−8} m^{2}; k_{0}′ = 1.7 × 10^{−8} m^{2}). | |

[95] | 226 37 | 226 121 | n.a. | Rigid open-cell porous materials (partially reticulated foams). Sample 1: Ω = 90%; q^{2} = 7.8; r = 25 × 10^{3} Ns/m^{4}. Sample 2: Ω = 99%; q^{2} = 1.98; r = 65 × 10^{3} Ns/m^{4}. |

[83] | 197–209 19.7–20.3 7–7.2 | n.a. | n.a. | Polyurethane foam (Low r = 2.3 × 10^{3} Ns/m^{4}; Ω = 96%; q^{2} = 1.29). Metal foam (Medium r = 50 × 10^{3} Ns/m^{4}; Ω = 89%; q^{2} = 1.27). Rock wool (High r = 150 × 10^{3} Ns/m^{4}; Ω = 93%; q^{2} = 1). Air characteristics: D = 1.15–1.19 kg/m^{3}; η = 1.83 × 10^{−5}–1.84 × 10^{−5} Ns/m^{2}. |

[87] | n.a. | 340–400 (with c′ = 0.98) 120–145 (with c′ = 1.34) 51–67 (with c′ = 2.84) | 11 × 10^{−10}–206 × 10^{−10} | Polyurethane foam (t = 1.3 cm; r = 2.3 × 10^{3} Ns/m^{4}; Ω = 96%; q^{2} = 1.28; D = 60 kg/m^{3}; k_{0}′ = 120 × 10^{−10}–200 × 10^{−10} m^{2}). Glass wool (t = 1.2 cm; r = 24.3 × 10^{3} Ns/m^{4}; Ω = 98%; q^{2} = 1.01; D = 53 kg/m^{3}; k_{0}′ = 28 × 10^{−10}–34 × 10^{−10} m^{2}). Rock wool (t = 1.1 cm; r = 51.2 × 10^{3} Ns/m^{4}; Ω = 97%; q^{2} = 1.06; D = 183 kg/m^{3}; k_{0}′ = 13 × 10^{−10}–14 × 10^{−10} m^{2}). |

[96] | 76–96 190–222 54–80 | n.a. | n.a. | Industrial method (low-frequency ultrasound) to quickly measure tortuosity and viscous characteristic length. Glass beads (t = 1.3–2.15 cm; q ^{2} = 1.37–1.4). Plastic foam (t = 1.3 cm; r = 3.6 × 10^{3} Ns/m^{4}; q^{2} = 1.06; Ω = 99%). Felt (t = 1.8 cm; r = 26 × 10^{3} Ns/m^{4}; q^{2} = 1; Ω = 98%). |

[97] | 1000 | 1850 | 4.8 × 10^{−8}–9.16 × 10^{−8} | Open-cell aluminum foam (Ω = 92%). |

[98] | 5–200 77–155 | 5–400 207–240 | n.a. | Industrial data related to a wide variety of porous materials (Ω = 70–99%; r = 1.5 × 10^{3}–2 × 10^{5} Ns/m^{4}; q^{2} = 1–3). Polyurethane foam (Ω = 95–97%; r = 14 × 10 ^{3}–17 × 10^{3} Ns/m^{4}; q^{2} = 1.6–2.3). |

[99] | 51–240 | 51–240 | n.a. | Theoretical materials (t = 1.3–4.5 cm; Ω = 55–95%; q^{2} = 1–33.2; r = 12 × 10 ^{3}–1 × 10^{5} Ns/m^{4}. |

[100] | 47–69 | 159–196 | n.a. | Polyurethane foam (D = 66.22 Kg/m^{3}; r = 24.2–57.8 Ns/m^{4}; Ω = 95%; E = 61 kPa; q ^{2} = 3.2–3.6). |

[101] | 155–160 | 310–320 | 2.9 × 10^{−9}–3.1 × 10^{−9} | Foam-formed cellulose materials (D = 37.3–38.18 kg/m^{3}; ΩΩ = 98.3–98.5%; r = 5770–6200 Ns/m ^{4}; q^{2} = 1.007–1.009). |

_{0}′ = static thermal permeability; t = thickness; r = airflow resistivity; D = bulk density; Ω = porosity; q

^{2}= tortuosity; E = Young’s modulus; N

_{pr}= Prandtl number; η = dynamic viscosity; γ = specific heat ratio; n.a. = not available.

Case | Parameters | If | Then | Average First Derivatives |
---|---|---|---|---|

1 | t = 4 cm, Ω = 18%, r = 7200 Ns/m^{4}, q^{2} = 3, s_{ρ} = 0.5–4.0, s_{K} = 2 | s_{ρ}↑ | a_{0,max} ↓f (a _{0,max}) ↓ | −0.0150 Hz^{−1} |

2 | t = 4 cm, Ω = 18%, r = 7200 Ns/m^{4}, q^{2} = 3, s_{ρ} = 3, s_{K} = 0.5–4.0 | s_{K}↑ | a_{0,max} ↑f (a _{0,max}) ↑ | 0.00130 Hz^{−1} |

3 | t = 4 cm, Ω = 18%, r = 7200 Ns/m^{4}, q^{2} = 3, Λ = 100–1000 μm, Λ′ = 600 μm, k_{0}′ = 1 × 10^{−9} m^{2} | Λ↑ | a_{0,max} ↑f (a _{0,max}) ↑ | 0.00040 μmHz^{−1} |

4 | t = 4 cm, Ω = 18%, R = 7200 Ns/m^{4}, q^{2} = 3, Λ = 300 μm, Λ′ = 100–1000 μm, k_{0}′ = 1 × 10^{−9} m^{2} | Λ′↑ | a_{0,max} ↓f (a _{0,max}) ↑ | −0.0007 μmHz^{−1} |

5 | t = 4 cm, Ω = 18%, r = 7200 Ns/m^{4}, q^{2} = 3, Λ = 300 μm, Λ′ = 600 μm, k_{0}′ = 1 × 10^{−10}–1 × 10^{−8} m^{2} | k_{0}′↑ | a_{0,max} ↓↑f (a _{0,max}) ↑ | −0.00002 m^{2} Hz^{−1} |

^{2}= tortuosity; s

_{ρ}= viscous pore shape factor; s

_{K}= thermal pore shape factor; Λ = viscous characteristic length; Λ′ = thermal characteristic length; k

_{0}′ = static thermal permeability; f

_{max}= frequency associated with maximum of the sound absorption spectrum; a

_{0,max}= maximum of the sound absorption coefficient spectrum; ↓ = decrease; ↑ = increment; ↓↑ = constancy.

# | t (cm) | AV (%) | Ωc (%) | a_{0,max}(dim.less) | f (a_{0,max})(Hz) | r_{meas_UP}(Ns/m ^{4}) | r_{meas_LOW}(Ns/m ^{4}) | k_{20}(cm/s) |
---|---|---|---|---|---|---|---|---|

1 | 3.62 | 22.47 | 20.88 | 0.82 | 1182 | 3911 | 3633 | 0.21 |

2 | 6.31 | 18.53 | 16.94 | 0.78 | 1004 | 77,983 | 5333 | 0.02 |

3 | 6.22 | 15.33 | 16.02 | 0.80 | 1278 | 25,837 | 4089 | 0.05 |

4 | 4.36 | 23.67 | 20.81 | 0.85 | 1118 | 3472 | 3798 | 0.25 |

5 | 4.56 | 26.47 | 24.47 | 0.87 | 986 | 1858 | 1871 | 0.44 |

6 | 6.13 | 24.99 | 24.67 | 0.89 | 806 | 2733 | 2242 | 0.38 |

7 | 5.17 | 24.52 | 22.14 | 0.76 | 862 | 1966 | 2068 | 0.34 |

8 | 4.12 | 23.10 | 22.13 | 0.92 | 1068 | 2416 | 2484 | 0.28 |

9 | 3.96 | 24.90 | 20.62 | 0.81 | 1102 | 2750 | 2832 | 0.27 |

10 | 4.63 | 25.65 | 23.76 | 0.84 | 1020 | 2416 | n.a. | 0.28 |

Standards | - | [102] | [102,103] | [104] | - | [105] | [105] | [106] |

_{0,max}= maximum of the sound absorption spectrum measured using the Kund’t tube (Figure 5a); f(a

_{0,max}) = frequency corresponding to a

_{0,max}; r

_{meas_UP}and r

_{meas_LOW}= airflow resistivity measured using the instrument in Figure 5c from the upper, UP, and the lower, LOW, surface of the samples; k

_{20}= permeability at 20 °C measured using the instrument in Figure 5d; dim.less = dimensionless.

**Table 5.**Values of the main parameters and goodness of fit of the models STIN and JCAL (optimization: 600–1600 Hz).

Case | N.of Layers | Model | |||||||

STIN | Simulation Goodness | ||||||||

t(cm) | Ω(%)(dim.less) | r_{est}(kNs/m^{4}) | q^{2}(dim.less) | s_{ρ}(dim.less) | s_{K}(dim.less) | STIN | |||

1 | 1L | 3.62 | 17.88 | 7.22 | 2.92 | 3.23 | 2.18 | Good | |

2L (UP) | 1.75 | 21.76 | 6.15 | 1.52 | 4.26 | 1.03 | Good | ||

2L (LOW) | 1.87 | 20.00 | 5.75 | 7.87 | 2.25 | 0.87 | |||

2 | 1L | 6.31 | 14.45 | 19.31 | 1.24 | 1.62 | 5.49 | Good | |

2L (UP) | 3.26 | 19.94 | 26.69 | 5.07 | 0.50 | 0.50 | Bad | ||

2L (LOW) | 3.05 | 13.94 | 35.84 | 10.00 | 5.50 | 0.50 | |||

3 | 1L | 6.22 | 19.02 | 11.92 | 9.23 | 0.50 | 0.50 | Bad | |

2L (UP) | 3.02 | 19.02 | 10.84 | 4.03 | 0.50 | 0.50 | Bad | ||

2L (LOW) | 3.20 | 13.02 | 20.12 | 10.00 | 5.50 | 0.50 | |||

4 | 1L | 4.36 | 17.81 | 6.20 | 2.21 | 2.95 | 1.75 | Good | |

2L (UP) | 2.43 | 21.26 | 5.08 | 1.51 | 3.77 | 2.18 | Good | ||

2L (LOW) | 1.93 | 20.03 | 4.91 | 6.31 | 2.98 | 2.53 | |||

5 | 1L | 4.56 | 21.47 | 4.27 | 2.57 | 2.90 | 0.52 | Good | |

2L (UP) | 2.67 | 25.22 | 3.72 | 1.79 | 4.68 | 1.15 | Good | ||

2L (LOW) | 1.89 | 25.15 | 3.31 | 7.65 | 1.65 | 0.86 | |||

6 | 1L | 6.13 | 21.67 | 4.73 | 2.21 | 2.30 | 0.72 | Good | |

2L (UP) | 3.07 | 25.88 | 4.08 | 1.26 | 4.43 | 0.88 | Good | ||

2L (LOW) | 3.06 | 25.03 | 3.60 | 5.64 | 0.94 | 1.04 | |||

7 | 1L | 5.17 | 20.39 | 4.40 | 2.13 | 4.25 | 0.60 | Good | |

2L (UP) | 2.06 | 22.23 | 4.16 | 2.69 | 4.20 | 2.76 | Good | ||

2L (LOW) | 3.11 | 21.33 | 4.03 | 2.24 | 2.92 | 2.43 | |||

8 | 1L | 4.12 | 19.13 | 5.88 | 3.04 | 2.24 | 1.68 | Good | |

2L (UP) | 1.91 | 23.51 | 4.98 | 1.45 | 3.52 | 0.71 | Good | ||

2L (LOW) | 2.21 | 21.69 | 4.57 | 7.81 | 1.22 | 0.81 | |||

9 | 1L | 3.96 | 17.63 | 6.08 | 2.75 | 2.85 | 0.50 | Good | |

2L (UP) | 2.12 | 22.24 | 5.46 | 1.54 | 4.77 | 0.60 | Good | ||

2L (LOW) | 1.84 | 20.87 | 4.88 | 8.50 | 2.24 | 0.82 | |||

10 | 1L | 4.63 | 20.76 | 5.75 | 2.27 | 3.04 | 0.99 | Good | |

2L (UP) | 2.70 | 24.67 | 5.08 | 1.39 | 4.57 | 0.71 | Good | ||

2L (LOW) | 1.93 | 22.94 | 4.47 | 7.96 | 2.33 | 2.40 | |||

Standard dev. | |||||||||

1L | 1.0 | 2.2 | 4.7 | 2.2 | 1.0 | 1.5 | |||

2L (UP) | 0.5 | 2.2 | 7.0 | 1.3 | 1.6 | 0.8 | |||

2L (LOW) | 0.6 | 4.1 | 10.6 | 2.3 | 1.6 | 0.8 | |||

Case | N.of Layers | JCAL | Simulation Goodness | ||||||

t(cm) | Ω(%)(dim.less) | r_{est}(kNs/m^{4}) | q^{2}(dim.less) | Λ(μm) | Λ′ (μm) | k_{0}′ (m^{2}) | JCAL | ||

1 | 1L | 3.62 | 18.53 | 5.59 | 3.04 | 301.7 | 564.0 | 9 × 10^{−10} | Good |

2L (UP) | 1.87 | 23.88 | 5.59 | 9.63 | 746.1 | 15.0 | 1 × 10^{−8} | Good | |

2L (LOW) | 1.75 | 17.88 | 5.59 | 10.00 | 5.0 | 718.7 | 1 × 10^{−8} | ||

2 | 1L | 6.31 | 19.94 | 27.57 | 10.00 | 510.3 | 776.2 | 1 × 10^{−10} | Bad |

2L (UP) | 3.46 | 19.35 | 27.57 | 3.64 | 451.9 | 366.1 | 1 × 10^{−10} | Good | |

2L (LOW) | 2.85 | 14.97 | 27.57 | 9.87 | 24.5 | 451.3 | 1 × 10^{−8} | ||

3 | 1L | 6.22 | 19.02 | 10.84 | 7.03 | 789.6 | 787.0 | 1 × 10^{−10} | Bad |

2L (UP) | 2.79 | 19.00 | 15.48 | 3.50 | 787.9 | 15.1 | 1 × 10^{−8} | Bad | |

2L (LOW) | 3.43 | 13.03 | 15.48 | 7.63 | 86.8 | 816.5 | 1 × 10^{−10} | ||

4 | 1L | 4.36 | 18.71 | 4.86 | 1.82 | 176.4 | 827.9 | 4 × 10^{−10} | Good |

2L (UP) | 2.76 | 21.13 | 4.86 | 4.82 | 624.0 | 22.2 | 1 × 10^{−8} | Good | |

2L (LOW) | 1.60 | 20.47 | 4.86 | 7.33 | 6.2 | 677.5 | 1 × 10^{−8} | ||

5 | 1L | 4.56 | 22.43 | 3.30 | 2.52 | 242.8 | 402.2 | 1 × 10^{−9} | Good |

2L (UP) | 3.25 | 25.07 | 3.30 | 4.72 | 522.5 | 25.4 | 1 × 10^{−8} | Good | |

2L (LOW) | 1.31 | 22.45 | 3.30 | 9.27 | 7.7 | 645.7 | 1 × 10^{−8} | ||

6 | 1L | 6.13 | 24.05 | 3.65 | 2.46 | 388.6 | 318.9 | 8 × 10^{−9} | Bad |

2L (UP) | 4.26 | 25.75 | 3.65 | 4.04 | 515.1 | 425.7 | 1 × 10^{−10} | Good | |

2L (LOW) | 1.87 | 23.86 | 3.65 | 5.20 | 6.6 | 409.2 | 2 × 10^{−9} | ||

7 | 1L | 5.17 | 21.65 | 3.98 | 2.27 | 140.7 | 435.9 | 1 × 10^{−8} | Good |

2L (UP) | 3.21 | 24.43 | 3.98 | 5.39 | 349.3 | 32.8 | 5 × 10^{−9} | Good | |

2L (LOW) | 1.96 | 21.57 | 3.98 | 5.65 | 5.9 | 321.1 | 7 × 10^{−9} | ||

8 | 1L | 4.12 | 19.14 | 4.54 | 2.99 | 412.3 | 487.3 | 1 × 10^{−9} | Good |

2L (UP) | 2.76 | 23.65 | 4.54 | 5.67 | 695.4 | 27.4 | 5 × 10^{−9} | Good | |

2L (LOW) | 1.36 | 20.55 | 4.54 | 8.73 | 6.1 | 510.1 | 9 × 10^{−9} | ||

9 | 1L | 3.96 | 18.53 | 4.69 | 2.92 | 309.5 | 486.5 | 9 × 10^{−10} | Good |

2L (UP) | 2.55 | 22.31 | 4.69 | 6.32 | 592.0 | 51.4 | 7 × 10^{−9} | Good | |

2L (LOW) | 1.41 | 19.81 | 4.69 | 8.35 | 6.8 | 436.7 | 8 × 10^{−9} | ||

10 | 1L | 4.63 | 20.76 | 4.48 | 2.06 | 185.5 | 518.6 | 7 × 10^{−10} | Good |

2L (UP) | 2.85 | 25.07 | 4.48 | 5.39 | 578.6 | 423.5 | 1 × 10^{−10} | Good | |

2L (LOW) | 1.78 | 23.18 | 4.48 | 5.59 | 6.92 | 414.5 | 5 × 10^{−9} | ||

Standard dev. | |||||||||

1L | 1.0 | 1.9 | 7.4 | 2.7 | 194.5 | 176.6 | 4 × 10^{−9} | ||

2L (UP) | 0.6 | 2.4 | 7.8 | 1.8 | 134.6 | 183.6 | 4 × 10^{−9} | ||

2L (LOW) | 0.7 | 3.5 | 7.8 | 1.8 | 25.5 | 162.9 | 4 × 10^{−9} |

_{est}= airflow resistivity estimated using the Equation (17); q

^{2}: tortuosity; s

_{ρ}= viscous pore shape factor (STIN model); s

_{K}= thermal pore shape factor (STIN model); Λ = viscous characteristic length (JCAL model); Λ′ = thermal characteristic length (JCAL model); k

_{0}′ = static thermal permeability (JCAL model).

Models | STIN | JCAL | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Parameters | t | Ω | r_{est} | q^{2} | s_{ρ} | s_{K} | t | Ω | r_{est} | q^{2} | Λ | Λ′ | k_{0}′ | |

STIN | t | 1.00 | −0.39 | 0.19 | −0.20 | −0.24 | 0.14 | 0.93 | −0.17 | 0.25 | −0.40 | 0.18 | 0.40 | −0.47 |

Ω | −0.39 | 1.00 | −0.69 | −0.31 | −0.04 | −0.21 | −0.33 | 0.89 | −0.65 | −0.15 | 0.11 | −0.47 | 0.20 | |

r_{est} | 0.19 | −0.69 | 1.00 | 0.40 | 0.05 | −0.01 | 0.19 | −0.64 | 0.94 | 0.29 | 0.01 | 0.22 | −0.15 | |

q^{2} | −0.20 | −0.31 | 0.40 | 1.00 | −0.24 | −0.25 | −0.34 | −0.57 | 0.28 | 0.54 | −0.52 | 0.48 | 0.15 | |

s_{ρ} | −0.24 | −0.04 | 0.05 | −0.24 | 1.00 | −0.08 | −0.04 | 0.05 | −0.14 | 0.03 | 0.02 | −0.24 | 0.11 | |

s_{K} | 0.14 | −0.21 | −0.01 | −0.25 | −0.08 | 1.00 | 0.13 | 0.07 | 0.18 | 0.18 | −0.05 | 0.15 | −0.10 | |

JCAL | t | 1.00 | −0.06 | 0.24 | −0.45 | 0.38 | 0.25 | −0.51 | ||||||

Ω | −0.06 | 1.00 | −0.55 | −0.20 | 0.28 | −0.54 | 0.11 | |||||||

r_{est} | 0.24 | −0.55 | 1.00 | 0.31 | 0.12 | 0.20 | −0.16 | |||||||

q^{2} | −0.45 | −0.20 | 0.31 | 1.00 | −0.17 | 0.17 | 0.38 | |||||||

Λ | 0.38 | 0.28 | 0.12 | −0.17 | 1.00 | −0.49 | −0.13 | |||||||

Λ′ | 0.25 | −0.54 | 0.20 | 0.17 | −0.49 | 1.00 | −0.42 | |||||||

k_{0}′ | −0.51 | 0.11 | −0.16 | 0.38 | −0.13 | −0.42 | 1.00 |

Models | STIN | JCAL | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Parameters | t | Ω | r_{est} | q^{2} | s_{ρ} | s_{K} | t | Ω | r_{est} | q^{2} | Λ | Λ′ | k_{0}′ | |

STIN | t | 1.00 | −0.52 | 0.34 | −0.22 | −0.25 | 0.14 | 0.93 | −0.24 | 0.32 | −0.41 | 0.18 | 0.42 | −0.49 |

Ω | −0.52 | 1.00 | −0.52 | 0.01 | 0.35 | −0.42 | −0.44 | 0.82 | −0.53 | 0.05 | −0.06 | −0.43 | 0.24 | |

r_{est} | 0.34 | −0.52 | 1.00 | 0.05 | −0.51 | 0.21 | 0.34 | −0.41 | 0.97 | 0.06 | 0.33 | 0.18 | −0.36 | |

q^{2} | −0.22 | 0.01 | 0.05 | 1.00 | −0.65 | −0.17 | −0.38 | −0.36 | 0.02 | 0.46 | −0.47 | 0.44 | 0.18 | |

s_{ρ} | −0.25 | 0.35 | −0.51 | −0.65 | 1.00 | 0.02 | −0.03 | 0.54 | −0.52 | −0.16 | 0.17 | −0.42 | 0.14 | |

s_{K} | 0.14 | −0.42 | 0.21 | −0.17 | 0.02 | 1.00 | 0.13 | −0.08 | 0.34 | 0.26 | −0.10 | 0.21 | −0.11 | |

JCAL | t | 1.00 | −0.10 | 0.31 | −0.47 | 0.39 | 0.26 | −0.52 | ||||||

Ω | −0.10 | 1.00 | −0.37 | 0.00 | 0.16 | −0.51 | 0.11 | |||||||

r_{est} | 0.31 | −0.37 | 1.00 | 0.15 | 0.32 | 0.15 | −0.28 | |||||||

q^{2} | −0.47 | 0.00 | 0.15 | 1.00 | −0.10 | 0.13 | 0.39 | |||||||

Λ | 0.39 | 0.16 | 0.32 | −0.10 | 1.00 | −0.48 | −0.14 | |||||||

Λ′ | 0.26 | −0.51 | 0.15 | 0.13 | −0.48 | 1.00 | −0.40 | |||||||

k_{0}′ | −0.52 | 0.11 | −0.28 | 0.39 | −0.14 | −0.40 | 1.00 |

Material Science | Acoustic Inputs | s_{ρ}Viscous Effects (Narrow Sections of the Pores) | s_{K}Thermal Effects (Wider Sections of the Pores) | Λ Viscous Effects | Λ′ Thermal Effects | k_{0}′Thermal Effects | |||
---|---|---|---|---|---|---|---|---|---|

AV, n_{eff} | Ω ↑ | ↑ | ↓ | ↑ | ↓ | ↑ | |||

K_{2}_{0}, r_{est} | r_{est} ↑ | ↓ | ↑ | ↑ | ↑ | ↓ | |||

q^{2} ↑ | ↓ | ↓ | ↓ | ↑ | ↑ | ||||

Partial Equations (Two-layer Approach) | Partial Equations (Two-layer Approach) | ||||||||

s_{ρ} (Ω) | 1L | s_{ρ} = 2.5303 ln (Ω) − 4.8488 | R^{2} = 0.0929 | Λ (q^{2}) | 1L | Λ = 293.19 ln (q^{2}) +8.6725 | R^{2} = 0.6781 | ||

2L | s_{ρ} = 1 × 10^{−4} Ω^{3.2527} | R^{2} = 0.1730 | 2L | Λ = 83,125 q^{2} − 3.902 | R^{2} = 0.3060 | ||||

s_{K} (Ω) | 1L | s_{K} = −10.54 ln (Ω) + 32.464 | R^{2} = 0.7023 | Λ′ (q^{2}) | 1L | Λ′ = 38.608 q^{2} + 417.18 | R^{2} = 0.3372 | ||

2L | s_{K} = −0.0461 Ω + 2.2976 | R^{2} = 0.0153 | 2L | Λ′ = 55.326 q^{2} − 40.416 | R^{2} = 0.2080 | ||||

Λ (Ω) | 1L | Λ = −464.2 ln (Ω) + 1741 | R^{2} = 0.0470 | k_{0}′ (q^{2}) | 1L | k_{0}′ = 4 × 10^{−9} exp − (0.403 q^{2}) | R^{2} = 0.4988 | ||

2L | Λ = 0.0194 exp (0.3737 Ω) | R^{2} = 0.1431 | 2L | k_{0}′ = 1 × 10^{−9} q^{2} + 4 × 10^{−10} | R^{2} = 0.2865 | ||||

Λ′ (Ω) | 1L | Λ′ = 6394.2 exp − (0.122 Ω) | R^{2} = 0.5456 | Global Equations (single-layer approach) | |||||

2L | Λ′ = −776.1 ln (Ω) + 2709.1 | R^{2} = 0.1088 | |||||||

k_{0}′ (Ω) | 1L | k_{0}′ = 1 × 10^{−9} Ω − 2 × 10^{−8} | R^{2} = 0.4593 | ||||||

2L | k_{0}′ = −7 × 10^{−10} Ω + 2 × 10^{−8} | R^{2} = 0.1807 | s_{ρ} (Ω) | s_{ρ} = 0.173 Ω^{−0.8927} | R^{2} = 0.1220 | ||||

s_{ρ} (r_{est}) | 1L | s_{ρ} = −1.49 ln (r_{est}) + 5.4277 | R^{2} = 0.4864 | s_{ρ} (r_{est}) | s_{ρ} = 4.0314 exp − (0.081r_{est}) | R^{2} = 0.3713 | |||

2L | s_{ρ} = 4.0057 exp − (0.085r_{est}) | R^{2} = 0.3864 | s_{ρ} (q^{2}) | s_{ρ} = −1.362 ln (q^{2}) + 4.2939 | R^{2} = 0.4784 | ||||

s_{K} (r_{est}) | 1L | s_{K} = 0.2666 r_{est}^{−0.5288} | R^{2} = 0.6614 | s_{K} (Ω) | s_{K} = −3.89 ln (Ω) + 13.192 | R^{2} = 0.2169 | |||

2L | s_{K} = 2.6293 r_{est}^{−0.535} | R^{2} = 0.2005 | s_{K} (r_{est}) | s_{K} = 0.4716 ln (r_{est}) + 0.52 | R^{2} = 0.0439 | ||||

Λ (r_{est}) | 1L | Λ = 193.03 ln (r_{est}) + 9.3394 | R^{2} = 0.4045 | s_{K} (q^{2}) | s_{K} = −0.316 ln (q^{2}) + 1.6993 | R^{2} = 0.0379 | |||

2L | Λ = 186.15 ln (r_{est}) + 23.343 | R^{2} = 0.1024 | Λ (Ω) | Λ = 18.289 Ω^{−58.628} | R^{2} = 0.0249 | ||||

Λ′ (r_{est}) | 1L | Λ′ = 195.51 ln (r_{est}) + 219.7 | R^{2} = 0.5033 | Λ (r_{est}) | Λ = 188.93 ln (r_{est}) + 17.961 | R^{2} = 0.1547 | |||

2L | Λ′ = 261.7 r_{est}^{−0.339} | R^{2} = 0.0145 | Λ (q^{2}) | Λ = 687.36 exp − (0.316 q^{2}) | R^{2} = 0.1706 | ||||

k_{0}′ (r_{est}) | 1L | k_{0}′ = 2 × 10^{−8} r_{est}^{−1.779} | R^{2} = 0.5654 | Λ′ (Ω) | Λ′ = −1221 ln (Ω) + 4136.4 | R^{2} = 0.2690 | |||

2L | k_{0}′ = 7 × 10^{−9} exp − (0.101r_{est}) | R^{2} = 0.1404 | Λ′ (r_{est}) | Λ′ = 72.572 ln (r_{est}) + 276.29 | R^{2} = 0.0257 | ||||

s_{ρ} (q^{2}) | 1L | s_{ρ} = 4.5752 × 10^{−0.224} q^{2} | R^{2} = 0.7036 | Λ′ (q^{2}) | Λ′ = 12.945 q^{2} + 328.44 | R^{2} = 0.0170 | |||

2L | s_{ρ} = −1.538 ln (q^{2}) + 4.714 | R^{2} = 0.6025 | k_{0}′ (Ω) | k_{0}′ = 3 × 10^{−12} Ω^{2.1138} | R^{2} = 0.0183 | ||||

s_{K} (q^{2}) | 1L | s_{K} = −1.659 ln (q^{2}) + 3.1085 | R^{2} = 0.3002 | k_{0}′ (r_{est}) | k_{0}′ = 5 × 10^{−9} exp − (0.124r_{est}) | R^{2} = 0.2168 | |||

2L | s_{K} = 0.0322 ln (q^{2}) + 1.2268 | R^{2} = 0.0009 | k_{0}′ (q^{2}) | k_{0}′ = 6 × 10^{−10} q^{2} + 2 × 10^{−9} | R^{2} = 0.1538 |

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## Share and Cite

**MDPI and ACS Style**

Praticò, F.G.; Fedele, R.; Briante, P.G.
On the Dependence of Acoustic Pore Shape Factors on Porous Asphalt Volumetrics. *Sustainability* **2021**, *13*, 11541.
https://doi.org/10.3390/su132011541

**AMA Style**

Praticò FG, Fedele R, Briante PG.
On the Dependence of Acoustic Pore Shape Factors on Porous Asphalt Volumetrics. *Sustainability*. 2021; 13(20):11541.
https://doi.org/10.3390/su132011541

**Chicago/Turabian Style**

Praticò, Filippo Giammaria, Rosario Fedele, and Paolo Giovanni Briante.
2021. "On the Dependence of Acoustic Pore Shape Factors on Porous Asphalt Volumetrics" *Sustainability* 13, no. 20: 11541.
https://doi.org/10.3390/su132011541