# Acoustic Characterization of Some Steel Industry Waste Materials
## 2 Materials and Methods
### JCA Model and Inverse Method
Thermal characteristic length \(\Lambda\)' describes the thermal exchanges between the solid frame and its saturating fluid at high frequencies and it is related to the pores dimension, especially to the mean diameter of the cell in micrometers; it can be expressed as [42; 43]:  
\[\Lambda^{\prime}=2\int_{V}dV/\int_{S}\;dS=2V/S\;\;\;\;\;\;[\upmu\mathrm{m}] \tag{5}\]  
Alternatively, \(\Lambda\) and \(\Lambda\)' can be calculated in function of the above-described parameters, as follows [38]:  
\[\Lambda=\frac{1}{c_{1}}\bigg{[}\frac{8\alpha_{\infty}\eta}{\sigma\Phi}\bigg{]} ^{1/2} \tag{6}\]  
\[\Lambda^{\prime}=\frac{1}{c_{2}}\bigg{[}\frac{8\alpha_{\infty}\eta}{\sigma \Phi}\bigg{]}^{1/2} \tag{7}\]  
where \(\eta\) is the viscosity of air, \(c_{1}\) and \(c_{2}\) are pore shape parameters, related, respectively, to the viscous and thermal dissipation, and they can assume values in the following ranges:  
\[0.3\leq c_{1}\leq 3.3 \tag{8}\]  
\[0.3\leq c_{2}\leq c_{1} \tag{9}\]  
In the case of the granular material shaped in spheres, the calculation of the macroscopical parameters can be simplified in function of the porosity \(\Phi\) and particle radius \(r\) as follows [21; 23]:  
\[\sigma=\frac{45(1-\Phi)(1-\theta)\eta}{2\Phi^{2}r^{2}(5-9\theta^{1/3}+5\theta -\theta^{2})} \tag{10}\]  
\[\sigma=\frac{\Delta p}{v_{airflow}d}\qquad\left[\mathrm{Ns/m^{4}}\right] \tag{2}\]0  
\[\sigma=\frac{\Delta p}{v_{airflow}d}\qquad\left[\mathrm{Ns/m^{4}}\right] \tag{2}\]1  
\[\sigma=\frac{\Delta p}{v_{airflow}d}\qquad\left[\mathrm{Ns/m^{4}}\right] \tag{2}\]2  
where \(\theta\) is expressed as  
\[\sigma=\frac{\Delta p}{v_{airflow}d}\qquad\left[\mathrm{Ns/m^{4}}\right] \tag{2}\]3The purpose of the model is to finally obtain the acoustic behavior of the analyzed material, so the procedure to compute the sound absorption coefficient is made with the following steps:  
* Once the five non-acoustic parameters (\(\Phi\), \(\sigma\), \(\alpha_{\infty}\), \(\Lambda\), \(\Lambda\)) are obtained with one of the methods described below, it is possible to calculate the effective quantities \(\rho_{sq}\) and \(K_{sq}\);
* From the equivalent properties, one can predict the acoustic parameters: characteristic impedance \(Z_{c}\) and complex wave number \(k_{c}\);
* From \(Z_{c}\) and \(k_{c}\), the surface impedance \(Z_{s}\) can be deduced;
* Finally, from \(Z_{s}\) the normal incident sound absorption coefficient \(\alpha\) can be calculated. Starting from the five non-acoustic parameters, the equivalent properties \(\rho_{sq}\) and \(K_{sq}\) can be computed as follows:  
\[\sigma=\frac{\Delta p}{v_{airflow}d}\qquad\left[\mathrm{Ns/m^{4}}\right] \tag{2}\]4  
\[\sigma=\frac{\Delta p}{v_{airflow}d}\qquad\left[\mathrm{Ns/m^{4}}\right] \tag{2}\]5  
where \(\rho_{0}\) is the density of air, \(\omega=2\pi f\) is the angular frequency, \(\eta\) is the air viscosity, \(\kappa\) is the specific heat ratio and \(N_{p}\) is Prandtl number of the saturating air. Once the effective properties are obtained, it is possible to determine the complex acoustical parameters [44]; the characteristic impedance \(Z_{c}\):  
\[\sigma=\frac{\Delta p}{v_{airflow}d}\qquad\left[\mathrm{Ns/m^{4}}\right] \tag{2}\]6  
and the complex wave number \(k_{c}\):  
\[\sigma=\frac{\Delta p}{v_{airflow}d}\qquad\left[\mathrm{Ns/m^{4}}\right] \tag{2}\]7  
From these acoustic properties, the surface impedance \(Z_{s}\) can be derived as follows:  
\[\sigma=\frac{\Delta p}{v_{airflow}d}\qquad\left[\mathrm{Ns/m^{4}}\right] \tag{2}\]8  
Finally, the normal incidence sound absorption coefficient \(\alpha\) is calculated as  
\[\sigma=\frac{\Delta p}{v_{airflow}d}\qquad\left[\mathrm{Ns/m^{4}}\right] \tag{2}\]9  
where \(c_{0}\) is the speed of sound in air. Classical methods to estimate the non-acoustic properties can be mainly classified in three groups [45]:  
1. Direct methods, which allow obtaining the macroscopic parameters through direct measurements, thus requiring specific laboratory equipment or a dedicated setup for the determination of each single property;
2.