# Green Shipping--Multifunctional Marine Scrubbers for Emission Control: Silencing Effect
## 2 Materials and Methods
### The Case Studies: Experimental Tests and Numerical Analyses
#### 2.2.2 FEM Simulations of Transmission Loss  
The numerical simulations were performed using the software Actran VI [29]. The geometries were discretized with the Actran mesh generator. First, the surfaces of the studied geometries were meshed, then the volume mesh was generated starting from the meshed surface to discretize the interior volume. The guidelines [29; 30; 31] suggest using a minimum of 8-10 linear elements per wavelength: in this case, the base size is 20 mm and 10 mm with a deviation of 0.1 mm for the simple expansion chamber and the model-scale scrubber, respectively. A tetrahedral mesh was used due to its high flexibility with complex geometries (e.g., scrubber with perforated plates and pipes). A mesh sensitivity study was not performed since the mesh size was calculated on the basis of the maximum wavelength to be considered, so the number of elements is sufficient to adequately reconstruct it. Considering 10 elements per wavelength, as in this case, the error on the pressure estimation remains less than 3% [31]. In Table 2, the mesh parameters ensuring the mesh quality [29; 32] are reported. Figure 8: Details of the bushing used to connect the expansion chamber and the impedance tube. Figure 9 shows the discretized geometries. Two configurations are used to analyse the geometries: the simple one is analysed as it is (Figure 9(a1,b1)), and the other one considers the discontinuity introduced by the connection (bushing in Figure 8) with the impedance tube (Figure 9(a2,b2)). The bushing discontinuity was modelled by considering a sudden change in diameter from 35 mm (prototypes pipe diameter) to 45 mm (impedance tube diameter). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Max** & **Mean** & **Min** \\ \hline Jacobian & 1 & 1 & 1 \\ Aspect Ratio & 6.42 & 1.76 & 1.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mesh quality parameters. Figure 9: Discretized expansion chamber (**a1**) without and (**a2**) with the pipe discontinuity and model-scale scrubber (**b1**) without and (**b2**) with the pipe discontinuity. The TL of the model-scale scrubber was evaluated with and without the presence of water at the bottom (Figure 6). Perforated elements were modelled inside of the software environment considering their viscous dissipation. This was modelled by imposing a transfer admittance \(A\) derived from the fluid characteristics and perforation geometry as follows:  
\[A=\frac{1}{Z_{p}}\ \, \tag{10}\]  
\(Z_{p}\) is the transfer impedance of the perforated element expressed as follows:  
\[Z_{p}=\frac{Z}{\sigma}\ \, \tag{11}\]  
where \(Z\) is the transfer impedance for a single hole expressed by [PERSON]'s formulation, and \(\sigma\) is the porosity parameter reported in the following equations [33] according to a formulation that is valid for triangular meshes:  
\[Z= \tag{12}\]  
\[\sigma= \frac{2\pi a^{2}}{\sqrt{3}d^{2}}\, \tag{13}\]  
where \(\mu\) is the dynamic viscosity of the fluid, \(L\) is the thickness of the perforated shell, \(a\) is the hole radius, \(d\) is the hole spacing, \(\omega\) is the angular frequency, \(\rho\) is the fluid density, and \(k_{s}\) the shear wavenumber, which can be expressed as follows:  
\[k_{s}=\sqrt{\frac{\omega\rho}{\mu}}\ \, \tag{14}\]  
As discussed in Section 2.1.2, at the inlet, the duct modes were used to model the incident wave, imposing a plane wave propagation in a frequency range 0-1700 Hz and 0-2300 Hz for the expansion chamber and the model-scale scrubber, respectively. The upper limits were selected according to the ISO standard guidelines reported in Section 2.2 [21]. In order to avoid reflection, free mode propagation was set in the direction opposite to the excitation. At the outlet, the anechoic condition was modelled using the duct modes and by setting free mode propagation. Environmental conditions and fluid characteristics were set in the same way as they were during the experimental tests (Section 2.2.1, Table 1).