# Sustainable Materials from Waste Paper: Thermal and Acoustical Characterization
## 3 Test Procedure
### Acoustic Tests  
The normal incidence sound absorption coefficients \(\alpha_{0}\) were measured by a BSWA SW 260 impedance tube, according to the ISO 10534-2 [62], following the transfer-function method. The measurements were performed at 18 \(\pm\) 2 \({}^{\circ}\)C and 40 \(\pm\) 2% RH. Two different cylindrical samples of 30 mm and 60 mm diameter were tested for each type of mix (Figure 2). The measurements were performed on both faces of each specimen. A 20 W loudspeaker was placed at the end of the tube in order to produce an acoustic plane wave on the front surface of the experimented sample set on the opposite side of the tube. Two \(\frac{1}{4}\)\" microphones on the tube wall measured the pressure at two fixed positions, allowing us to calculate the transfer function between the measured signals. The normal incidence reflection factor and the normal incidence sound absorption curve were estimated based on the complex transfer function between the measured signals and the distance between the two microphones and the exposed surface of the sample. The frequency range from 125 to 5000 Hz was taken into account, considering two different measurement configurations. The 60 mm tube was used for the low and medium frequency range (125 \(\div\) 2500 Hz), while the 30 mm tube was adopted for high frequency range (800 \(\div\) 5000 Hz). \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Mix Type**} & **Bulk Density** & **Bulk** & **Thermal** & **Thermal** & **Volumetric Heat** & **Specific Heat** \\  & \(\rho_{bulk}\) & **Porosity \(\varepsilon\) (-)** & **Conductivity** & **Diffusivity** & **Capacity** & **Capacity** \\  & **(kg\(\cdot\)m\({}^{-3}\))** & **Porosity \(\varepsilon\) (-)** & **\(\lambda\) (W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\))** & **\(\alpha\) (10\({}^{-7}\cdot\)m\({}^{2}\cdot\)s\({}^{-1}\))** & **\(\rho c\) (10\({}^{5}\)J\(\cdot\)m\({}^{-3}\cdot\)K\({}^{-1}\))** & **c (J\(\cdot\)K\({}^{-1}\cdot\)Kg\({}^{-1}\))** \\ \hline \multirow{2}{*}{SP} & 200 & 0.868 & 0.071 & 3.62 & 1.96 & 980 \\  & (7.1) & (0.003) & (0.0038) & (0.0179) & (0.0433) & (37.7) \\ \hline \multirow{2}{*}{CG} & 318 & 0.784 & 0.085 & 1.45 & 5.85 & 1839 \\  & (5.4) & (0.002) & (0.0045) & (0.0242) & (0.5677) & (139.9) \\ \hline \multirow{2}{*}{FB} & 348 & 0.680 & 0.093 & 2.89 & 3.20 & 919 \\  & (12.9) & (0.002) & (0.0065) & (0.0278) & (0.0678) & (35.8) \\ \hline \end{tabular}
\end{table} 
Table 3: Physical and thermal properties (The values in the brackets are the standard deviation). The measurement procedure was controlled by means of a graphic user interface developed in MATLAB(r) (2018, Mathworks, Natick, MA, USA) and using a sine sweep to feed the loudspeaker. The measured sound absorption coefficients were then compared with the values estimated by the semi-phenomenological prediction model developed by [PERSON] (later on referred as JCA) [63; 64] to describe the complex density of an acoustical porous material with a motionless skeleton and an arbitrary pore shape. More refined models were not used at this stage to limit the number of estimated parameters. According to the model, the absorption coefficients of the tested materials were obtained after determining the characteristic impedance \(Z_{c}\) and the wavenumber \(k\) expressed as a function of the effective (or dynamic) bulk density \(\rho_{e}\) and bulk modulus \(k_{e}\) as follows:  
\[Z_{C}=(k_{e}\,\rho_{e})^{0.5} \tag{3}\]  
\[k=(\rho_{e}/k_{e})^{0.5} \tag{4}\]  
These two parameters are correlated with the physical characteristics of porosity \(\varepsilon\), tortuosity \(\tau\), viscous characteristic length \(\Lambda\), thermal characteristic length \(\Lambda^{\prime}\), air flow resistivity \(\sigma\) (i.e., the air flow resistance per unit thickness) and bulk density. The two characteristic lengths are defined as the effects of viscosity and thermal dissipative forces produced inside the porous structure.