# Experimental Analysis and Simulation of a Porous Absorbing Layer for Noise Barriers
## 2 Materials and Methods
### Material Modeling  
#### 2.3.1 Horoshenkhov-Swift Model  
For porous concrete, the Horoshenkov and Swift [20] model for granular porous materials was applied by taking into account porosity, \(\phi\), airflow resistivity, \(\sigma\), tortuosity, \(\alpha_{\infty}\), and the standard deviation of the pore size, and \(\sigma_{p}\), as material parameters. The first three have been calculated empirically [34; 35; 36; 37] and by using an inverse technique approach (see below, in Table 2, experimental and adjusted values). The last parameter, \(\sigma_{p}\), however, was only determined theoretically. These authors suggest to calculate the volumetric density and compressibility module of the porous material using the following equations:  
\[\rho=\frac{\alpha_{\infty}}{\phi}\bigg{(}\rho_{0}-\frac{j\phi\sigma}{\omega \alpha_{\infty}}\widetilde{F}(\omega)\bigg{)}, \tag{8}\]  
\[C=\frac{\phi}{\gamma P_{0}}\left(\gamma-\frac{\rho_{0}(\gamma-1)}{\rho_{0}-j \frac{\sigma\phi}{\omega\alpha_{\infty}N_{pr}}\widetilde{F}\big{(}N_{pr}\omega \big{)}}\right), \tag{9}\]  
in which \(\omega\) corresponds to the angular frequency, \(\gamma\) to the ratio of specific heats, \(P_{0}\) to the atmospheric pressure and \(N_{pr}\) is the Prandtl number, and \(\widetilde{F}(\omega)\) is the viscosity correction function, that can be presented in the form of a Pade approximation as:  
\[\widetilde{F}(\omega)=\frac{1+a_{1}\epsilon+a_{2}\epsilon^{2}}{1+b_{1} \epsilon}, \tag{10}\]  
where \(a_{1}~{}=\theta_{1}/\theta_{2}\), \(a_{2}=\theta_{1}\) and \(b_{1}=a_{1}\). Assuming that the geometry of the pores is circular, in a simplified way, comes the following form factors: \(\theta_{1}=\frac{4}{3}\mathrm{e}^{4\xi}-1\), \(\theta_{2}=\frac{\mathrm{e}^{3\xi/2}}{\sqrt{2}}\), where \(\xi=\big{(}\sigma_{p}~{}\mathrm{ln}(2)\big{)}^{2}\) and \(\epsilon=\sqrt{\mathrm{i}\omega\rho_{0}\alpha_{\infty}/(\sigma\phi)}\).  
In order to theoretically estimate the macroscopic parameters and evaluate a new sound absorption curve of the porous material, an inverse technique based on a genetic algorithm was also applied. This methodology is based on the difference between the theoretical sound absorption coefficient results, obtained with the Horoshenkov- Swift model, and the experimental sound absorption coefficient data. Each macroscopic parameter was given an initial estimation (based on the experimental findings) as well as lower and upper boundary limits in order to be used with this inversion procedure. A variation limit of 10% was imposed. This procedure makes sure the values remain within acceptable bounds. The objective function (11) that follows is then minimized:  
\[Fu=\sum_{i=1}^{nf}\lvert\mathrm{a}_{s}^{a}-\mathrm{a}_{s}^{e}\rvert^{2}, \tag{11}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & **Thickness** & **Tortuosity** & **Porosity** & **Airflow** & **Average Pore Size** \\  & **[m]** & **\(\alpha_{\infty}\)** [-]** & **\(\Phi\)** [-]** & **Resisity** & **Standard Deviation** \\ \hline \multicolumn{6}{c}{Experimental data} \\ \hline AE 2.93 & 0.08 & 2.51 & 0.46 & 4507.88 & 0.25 \\ AE 3.71 & 0.08 & 2.21 & 0.45 & 5044.94 & 0.25 \\ AE 5.18 & 0.08 & 2.37 & 0.46 & 4668.45 & 0.25 \\ \hline \multicolumn{6}{c}{Adjusted data} \\ \hline AE 2.93 & 0.08 & 2.26 (\(-\)9.9\%) & 0.45 (\(-\)2.1\%) & 4720.11 (+4.7\%) & 0.26 (+4\%) \\ AE 3.71 & 0.08 & 2.14 (\(-\)3.2\%) & 0.45 (0.0\%) & 5231.79 (+3.7\%) & 0.27 (+8\%) \\ AE 5.18 & 0.08 & 2.14 (\(-\)9.7\%) & 0.50 (+8.7\%) & 4990.16 (+6.9\%) & 0.27 (+8\%) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experimental and corrected macroscopic parameters results.  
In this equation, _nf_ is the number of discrete frequencies analyzed, \(\alpha_{s}^{a}\) is the absorption coefficient results determined by the Horoshenkov- Swift model and \(\alpha_{s}^{c}\) corresponds to the experimental absorption coefficient data.  
Figure 8 shows a flowchart of the inversion methodology used to determine the macroscopic parameters.