# 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
#### 2.1.2 [PERSON]'s Pore-Size Distribution Model Inverse Method  
The inverse method curve fits the present experimental reflection coefficient (real and imaginary parts) versus frequency (250 Hz to 1600 Hz) data for polydisperse solid beads using the Nelder-Mead function minimization technique [38]. All the equations required for predictions and minimization functions were carefully compiled in MATLAB and provided to us by Dr. [PERSON] [31; 42]. The calculated flow properties are listed in the second column [35] of Table 2. The predictions for 25 mm thick specimens (solid pink color) were carried out in MATLAB's AlphaCell by the PSD model using the flow properties listed in the second column of Table 2 and are validated with experimental results (solid red color) as shown in Figure 4. The predictions agree well with the experiment. These predictions have been reported in our previous works [35] and re-reported here.  
The same independently calculated flow properties from the PSD model inverse method listed in the second column of Table 2 were used to predict transmission loss responses in MATELYS AlphaCell using the PSD model and compared with the independently measured transmission loss versus frequency using a four-microphone setup (see Figure 5) according to the ASTM 2611-09 standard [43] for 25 mm specimens. The transmission loss is measured by means of the 'two load' transfer function method, by acquiring the sound pressure in four fixed microphone impedance tube setups: two consecutive data acquisitions are carried out for a sample by changing the end conditions of the tube (an anechoic-absorptive end and a free end). The test again is limited to normal incidence, can encounter sample edge effects on measurements, and is not suitable for large-scale field testing, etc. Figure 6 shows transmission loss versus frequency response, wherein the predictions (solid pink line) agreed well with the experiments (solid red line) for 25 mm  
Figure 4: Absorption coefficient versus frequency of two test specimens, 25 mm and 44 mm thickness by impedance test [35]. Also shown are the predictions.  
Figure 3: Two-microphone normal incidence impedance tube setup.  
coefficient predictions (solid pink line) agreed well with the experiment (solid blue line) (Figure 4). Similarly, the same flow properties were used to make the transmission loss predictions in MATELYS AlphaCell and compare them with the independently measured transmission loss versus frequency as shown in Figure 6. The predictions (solid pink line) agreed well up to a 1200 Hz frequency with the experiment (solid blue line), but the predictions deviated from the experiment beyond 1200 Hz for a 44 mm thick specimen. Overall, the agreement is reasonable. These results have been reported in our previous works [35] and re-reported here.  
This validates that the flow properties calculated from the 25 mm reflection coefficient (real and imaginary parts) response alone can be used to predict both the absorption coefficient and the transmission loss response for a smooth-surface polydisperse spherical bead material system of any thickness. Thus, from these assessments, we confirm that [PERSON]'s PSD model-based inverse method [31] can be applied to a smooth-surface polydisperse spherical bead material system, and the flow properties are unique for a specific particle-size material system when the ratio of the minimum specimen dimension to mean particle diameter is about 10 or more. In this case, the ratio is 12.5. This is demonstrated for the first time.