# Design of a Noise Mitigation System Using Lightweight Graded Micro-Porous Material
## 2 Establish Predictive Methods to Calculate Flow Properties of Polydisperse Particles
### Polydisperse Smooth-Surface Glass Bead Composite Material
#### 2.1.3 Introduced Method: G-JCAL/JCAPL to Calculate Flow Properties  
This method combines the existing works in the literature to calculate the flow properties of JCAL/JCAPL acoustic models. The flow properties required for the respective acoustic models are listed in Table 1.  
The three primary flow properties--i.e., porosity, static air flow resistivity, and tortuosity--require either direct measurements (expensive) or models to calculate them. In this work, the porosity of the material is measured using the water saturation method [44]; then, we have used a permeability model called the Revil-Glover-Pezard-Zamora (RGPZ) model (initially unpublished discussion from [PERSON], [PERSON], [PERSON], and [PERSON]) developed by [PERSON], [PERSON], and [PERSON] [45] and [PERSON] and [PERSON] [46], which is applicable to the widely varying types of microstructures exhibited by the porous media using the particle diameter (d), porosity (\(\varphi\)), cementation exponent (m) (a parameter that measures the complexity of a material's pore structure and the degree of cementation and consolidation), and parameter constant (a), which depends on the topology of the pore space. For monodisperse spherical solid beads, m \(\approx\) 1.5, a = 8/3 [45; 47; 48]. The flow properties are sensitive to the cementation exponent (m). From the RGPZ model, the permeability (k\({}_{0}\)) of the porous medium assumes that flow occurs in all of the porosity and is given by  
\[\mathrm{k}_{0}=\frac{d^{2}\varphi^{3m}}{4 am^{2}} \tag{1}\]  
where k\({}_{0}\) = permeability, \(\varphi\) = porosity, \(d\) = particle diameter, \(m\) = cementation exponent, and \(a\) = parameter constant (topology of pore space). Once the permeability is known, the static air flow resistivity can be calculated using  
\[\sigma=\frac{\eta}{k_{0}} \tag{2}\]  
where \(\sigma\) = static air flow resistivity and Dynamic Viscosity of air, and \(\eta\) = 1.81 \(\times\) 10\({}^{-5}\) N.s.m\({}^{-2}\). The tortuosity of the porous media [45] is given by  
\[\alpha_{\infty}=\varphi^{1-m} \tag{3}\]  
where \(\alpha_{\infty}\) = tortuosity. Once the primary flow properties are known, the secondary flow properties and viscous (\(\wedge\)) and thermal (\(\wedge^{\prime}\)) characteristic lengths were determined using Equations (4) and (5), respectively, with pore shape factors c = 1 [27] and c\({}^{\prime}\) = 0.67, since \(\Lambda^{\prime}\)\(\approx\) 1.5 \(\Lambda\)[37] for the material with general non-uniform porous structures (\(\Lambda\
eq\Lambda^{\prime}\)), i.e., for the case of solid beads (assumption: non-uniform circular pores) with circular pores. \[\wedge=\frac{1}{c}\sqrt{\frac{8\eta\alpha_{\infty}}{\sigma\varphi}}\qquad\qquad 0.3 \leq c\leq 3.3 \tag{4}\]  
\[\wedge^{\prime}=\frac{1}{c^{\prime}}\sqrt{\frac{8\eta\alpha_{\infty}}{\sigma \varphi}}\qquad\qquad 0.3\leq c^{\prime}\leq 3.3 \tag{5}\]  
where \(\wedge=\) viscous characteristic length, \(\wedge^{\prime}=\) thermal characteristic length and c and c\({}^{\prime}=\) pore shape factors. We have verified the pore shape factor values (c = 1.04 and c\({}^{\prime}\) = 0.69) by inverse calculation using the flow properties determined in reference [37] by applying them to Equations (4) and (5), respectively. Additional secondary flow properties, the static thermal permeability (k\({}^{\prime}_{0}\)) for JCAL, static viscous tortuosity (\(\alpha_{0}\)), and the static thermal tortuosity (\(\alpha^{\prime}_{0}\)) required for the JCAPL model, were calculated in the MATELYS AlphaCell simulator using Equations (6), (7) and (8), respectively, with parameter b = 34 [30] for granular material with monodisperse solid beads that are assumed to have circular pores, and b\({}^{\prime}\) is not established as per the literature; hence, we have assumed it to be 1 as the first approximation. \[\mathrm{k}^{\prime}_{0}=\frac{\varphi}{8}(\Lambda^{\prime})^{2} \tag{6}\]  
where k\({}^{\prime}_{0}\) = thermal permeability.