## 2 Acoustic Performance Evaluation Indicators for an Expandable Duct Silencing System
### Transmission Loss of Expansion Mufflers Based on the Acoustic Finite Element Method  
The COMSOL Multiphysics 6.2 software's acoustic module can evaluate the acoustic performance of expansion chamber duct silencers in as realistic a scenario as possible by utilizing the Acoustic Finite Element Method (AFEM) to model and simulate acoustic problems. It can also visualize the distribution of sound fields and multiple physical fields. Using an estimated sound pressure function, the AFEM approach first discretizes the muffler's acoustic model into a finite number of non-overlapping linked tiny pieces. Approximate solutions for all elements make up the solution in the whole computational domain, given suitable boundary conditions. Establishing a geometric model of the air duct muffler, defining material parameters, adding boundary conditions, creating grids, setting solver settings, and post-processing the results are all steps in the simulation calculation process that COMSOL Multiphysics uses to finish the air duct muffler. The material used is a porous elastic material with a flow resistance of 1500 Pa-s/m\({}^{3}\). Table 1 displays the structural characteristics of the expansion room air conditioning duct muffler. As shown in Figure 5a, create a three-dimensional model of the expansion chamber duct muffler. The incident sound wave at the inlet and the transmitted sound wave at the outlet are approximately defined as plane wave radiation, with the direction of the sound wave being perpendicular to the cross-section of the air duct. \begin{table}
\begin{tabular}{c c} \hline \hline
**Structural Parameters** & **Value** \\ \hline Cross-sectional area of air duct cavity & 286 cm\({}^{2}\) \\ Dilated chamber cross-sectional area & 656 cm\({}^{2}\) \\ Expansion ratio \(m\) & 2.29 \\ Chamber length \(L\) & 25 cm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structural parameters of expansion chamber air duct mufflers. Figure 4: Transmission loss curves for a single expansion chamber muffler with varying \(L\) values. Figure 3: Transmission loss curves for a single expansion chamber muffler with varying \(m\) values. As illustrated in Figure 5b, remove the hollow portion of the air duct, designate its substance as air, and divide the grid. Determining the right element types and grid sizes throughout the grid partitioning process is essential to increasing the efficiency of the solution. In general, simulation findings are more accurate the finer the network quality but this comes at a higher computing cost. As a result, a free tetrahedral mesh was selected as the mesh element type and an empirical formula was used to determine the mesh element's maximum value, or \(M_{max}\). The number of Degrees of Freedom (DoFs) of the finite element model are 24,824 and the number of elements is 40,708. \[M_{max}=\frac{c}{5f_{max}} \tag{12}\]  
In the formula, \(c\) is the speed of sound, its unit is m/s, and \(f_{max}\) is the maximum solution frequency, its unit is Hz. Designate the surfaces as hard sound field boundaries, except the entrance and outflow. In line with reality, the rigid sound field border prevents sound waves from transmitting or absorbing any sound energy. The expansion chamber duct muffler's inlet and outlet ends have the same acoustic impedance in an ideal gas environment. Thus, the noise reduction amount may be used to compute the expansion chamber duct muffler's transmission loss. The one-dimensional theoretical calculation data and the acoustic finite element simulation values for the expansion chamber duct muffler's transmission loss in the frequency range of 0 Hz to 1500 Hz are displayed in Figure 6.  
Figure 7 illustrates how the theoretical calculation value of the expansion chamber duct muffler's transmission loss is extremely compatible with the findings of the simulation calculations in the frequency range of 0-975 Hz.