# 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
\[\alpha_{0}=\alpha_{\infty}\bigg{(}1+\frac{2\mathrm{k}_{0}\alpha_{\infty}}{ \varphi\mathrm{b}\Lambda^{2}}\bigg{)} \tag{7}\]  
where \(\alpha_{0}\) = static viscous tortuosity, b = pore parameter. \[\alpha^{\prime}_{0}=1+\frac{2\mathrm{k}^{\prime}_{0}}{\varphi{\Lambda^{\prime }}^{2}\mathrm{b}^{\prime}} \tag{8}\]  
where \(\alpha^{\prime}_{0}\) = static thermal tortuosity, b\({}^{\prime}\) = pore parameter. This method of combining [PERSON] et al. 's models [45; 46] for primary flow properties and JCAL/JCAPL equations for secondary flow properties is called G-JCAL/JCAPL. The flow properties calculated from this method were validated for shaped monodisperse glass solid bead specimens [33], wherein the associated parameters (m, a, c, c\({}^{\prime}\), b, b\({}^{\prime}\)) were established, but for any other new granular material system (polydisperse), these parameters need to be established. In this work, a multivariable-fit inverse method is applied to the equations mentioned above of the G-JCAL/JCAPL method for a polydisperse particle-size material system to calculate the flow properties of the JCAL/JCAPL acoustic models. 1. Multivariable-Fit Inverse Method  
A multivariable-fit inverse method was applied to the G-JCAPL method, Equations (1)-(8), to calculate the parameters (m--cementation exponent, a--parameter constant, c and c\({}^{\prime}\)--pore shape factors, b--granular material parameter, and b\({}^{\prime}\) (assumed to 1 as a first approximation)) by curve fitting the present experimental absorption coefficient versus frequency response of a 25 mm thick specimen (see solid red line in Figure 4), thereby calculating the flow properties. The flow properties are listed in the third column of Table 2. These flow properties were used to predict absorption coefficient response in MATELYS AlphaCell using the JCAPL acoustic model and were validated with predictions from the PSD model method (solid pink line) and the experiment (solid red line) for the 25 mm thick specimen shown as black broken lines in \[\mathrm{k}_{0}=\frac{d^{2}\varphi^{3m}}{4 am^{2}} \tag{1}\]0 The predictions agreed well with both the PSD model predictions [35] and the experiment [35]. Also, from Table 2 we see a very small difference in common flow properties between the two different flow property calculation methods. The same independently calculated flow properties were used to predict transmission loss responses in MATELYS AlphaCell using the JCAPL model and compared with the independently measured transmission loss versus frequency using a four-microphone setup according to the ASTM 2611-09 standard [43] for 25 mm specimens as shown in \[\mathrm{k}_{0}=\frac{d^{2}\varphi^{3m}}{4 am^{2}} \tag{1}\]1 The predictions from the flow properties of the multivariable-fit inverse method (broken black line) agreed well with the predictions from the PSD model method (solid pink line) [35] and experimental response (solid red line) [35] for 25 mm thick specimens. This validates that the flow properties calculated from the multivariable-fit inverse method using a 25 mm absorption coefficient response alone can be used to predict the transmission loss response as well. Although there are small differences in flow properties between the methods, the multivariable-fit inverse method is validated with the existing PSD model inverse method through accurate acoustic predictions. Furthermore, it is important to check whether the established flow properties are unique i.e., independent of the specimen geometry for a specific particle-size system. 2. Validation of Acoustic Predictions For a Different Specim Thickness  
To check the uniqueness of the flow properties calculated from the multivariable-fit inverse method listed in the third column of Table 2, it was applied to a 44 mm thick specimen and predictions were made in MATELYS AlphaCell.