## 3 Numerical Model  
A numerical model is implemented to simulate the pressure loss in a system composed of two dissipative splitter silencers, with a length of 0.9 m, embedded in a duct (see Figure 1). The average pressure at the inlet and outlet of the system is analyzed by using the finite element method (FEM). The stationary turbulent flow \(k\)-\(\epsilon\) model is used for the numerical calculations. This is a two-equation turbulence model that provides the computation of the turbulent kinetic energy, \(k\), defined as the kinetic energy associated with eddies in turbulent flow and the dissipation rate, \(\epsilon\), which is a parameter that is commonly used in the characterization of turbulent flow to determine the amount of energy that is lost due to viscous effects [23]. The turbulent kinetic energy and its dissipation rate are calculated from transport equations solved simultaneously with those governing the mean flow behavior [24]. This model has also the advantages of being computationally efficient and widely validated [25]. Two different profile shapes are simulated in order to study the influence of the profile shape of the silencer in the behavior of the flow and, consequently, in the pressure drop. In this work, the semihexadecagonal shape is used as an approximation of the semicircular shape described in the ISO standard. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Thickness of Splitter** & \(\mathbf{a_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{b_{1}}\) & \(\mathbf{b_{2}}\) \\ \hline
100 mm & 0.235 & 0.017 & \(-\)2.78 & \(-\)2.70 \\
200 mm & 0.255 & 0.015 & \(-\)2.82 & \(-\)2.91 \\
300 mm & 0.294 & 0.0167 & \(-\)2.83 & \(-\)2.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the coefficients for the calculation of the pressure loss coefficient (Equation (5)) in VDI 2081-1. It is also important to note that, due to the geometrical symmetry of the system, and with the purpose of saving computational power and cost, symmetric boundary conditions are considered to simulate only one quarter of the system. Figure 2 shows the geometries of the implemented models, both with rectangular and semihexadecagonal profiles. The considered lengths of the duct downstream and upstream are chosen according to the ISO 14163, in which it is specified that the minimum distances from the silencers for measuring the pressure downstream and upstream are, respectively, 0.5\(d\) and 6\(d\) (\(d\) being the thickness of the splitter). The velocity at which the flow approaches the silencers has also a significant influence on the pressure loss, as high velocities may cause turbulence that increases the drop in the pressure associated with the presence of the splitters and their influence in the behavior of the flow inside the duct. This effect is also studied by considering four different input face velocities, \(U_{0}\), in the models: 5 m/s, 10 m/s, 15 m/s and 20 m/s. The splitter silencers used in the experimental stage of the work (see Section 4) are made of stainless steel, so the contours of these structures are considered rigid walls for the flow in the numerical simulations. The resulting models consist of 172,953 elements in the case of rectangular profiles and 68,407 in the case of semihexadecagonal profiles. The significant difference between the number of elements generated in both meshes, even though they are set by following the same criteria, is associated with the resolution of the mesh required in the vicinity of the profiles to solve the proposed problem. On the one hand, a rectangular profile implies a drastic change in the cross section of the duct, as it is a straight wall that is perpendicular to the propagation of the flow. On the other hand, a semihexadecagonal profile leads to a gradual change that creates a smooth transition from the section of the empty duct to the section in which the silencer is present. Figure 1: A 2D scheme of the studied system, composed of two dissipative splitter silencers embedded in a duct.