# Design of a Noise Mitigation System Using Lightweight Graded Micro-Porous Material
## 2 Establish Predictive Methods to Calculate Flow Properties of Polydisperse Particles
### Micro-Porous Hollow Bubble (\(\upmu\)-PHB) Composite Material
#### 2.2.3. Calculation of Flow Properties of \(\upmu\)PHB Composite Materials  
In this section, the flow properties of \(\upmu\)PHB composite materials are calculated using two methods--the PSD model-based inverse method and the multivariable-fit inverse method based on G-JCAL/JCAPL. Figure 10. Absorption coefficient versus frequency response of 25 mm thick micro-porous material made of three groups of particle sizes. Figure 9. Microstructure of medium size \(\upmu\)PHB composite material. PSD Model-Based Inverse Method  
The PSD model inverse method [31] was used to calculate the flow properties of uPHB composite materials of all three particle-size groups. However, this method yielded inconsistent flow properties, with values that seemed unrealistic for all three size groups. For example, the method estimated porosity values of 0.219, 0.559, and 0.921 for small, medium, and large particle-size uPHB composite materials. The estimated porosities clearly showed an under-estimation for small and an over-estimation for medium and large particle-size material, with a possibility of taking into account both open porosity and porosity due to micro-holes or cracks in the microbubbles. Similarly, the estimated median pore size (\(\overline{r}\)) was not consistent, with values of 100 um, 90 um, and 90 um for small, medium, and large particle-size uPHB composite materials. From this estimation, we see that the small particle-size uPHB composite material showed a larger median pore size relative to the medium and large particle-size systems, which is highly unlikely. Consequently, based on these inconsistencies, we determined that the PSD model inverse method [31] is unreliable for uPHB composite materials. Multivariable-Fit Inverse Method Based on G-JCAL/JCAPL  
Subsequently, the alternative multivariable-fit inverse method was used for all three size group uPHB composite materials. In this approach, the primary parameter, cementation exponent (m), was established, considering the particles to be nearly spherical or ellipsoid due to surface irregularities, resulting in m > 1.5 [50], unlike monodisperse spherical particles (m \(\approx\) 1.5) [45; 46]. Similarly, other parameters were calculated by curve fitting the experimental absorption coefficient versus frequency (represented by solid lines of the respective three particle-size groups in Figure 10) of the 25 mm thick specimen, thereby calculating the flow properties. This method of curve fitting was chosen as there are no reliable methods as of now to calculate the flow properties of this new material system, apart from expensive direct measurements. The parameters calculated characterize the pore properties of the material and are listed in Table 4.  
The flow properties for small-sized uPHB composite materials were calculated using the G-JCAL equations, while the G-JCAPL equations were used for medium and large particle-size uPHB composite materials and are listed in Table 5. Among the flow properties listed, those with the most significant impact on acoustic response include bonded material porosity, static air flow resistivity, and tortuosity. From Table 5, we see an increase in the porosity of the uPHB composite materials with larger hollow microbubble sizes. While spherical particles typically exhibit a porosity close to 0.40, in this class of materials, due to near-spherical particles, surface roughness, and microstructure complexity, the porosity exceeds 0.40. Additionally, smaller particle sizes are more spherical, with porosities approaching 0.40. The static air flow resistivity decreases with larger microbubble sizes, correlating with the expected increase in pore size. This observed trend is consistent. Similarly, tortuosity increases as microbubble size decreases, as smaller pore sizes create more convoluted paths.