## 3 Results and Discussion
### Case 1: Geometries with Multi Ports  
A double-acting reciprocating air compressor is firstly studied, which is equipped with two suction valves and two discharge valves. The suction chamber is geometrically identical with the discharge one and both have three boundaries, as shown in Figure 5. Two of them fit together with the valves and the third one with the pipeline. Geometric properties and thermodynamic conditions of this examined compressor are listed in Table 1. The commercial code VirtualLab is used to perform the acoustic FEM simulations in the scope of frequencies between 1 Hz and 1501 Hz with a step of 10 Hz. The thermodynamic conditions reported in Table 1 are considered. The fluid domain is discretized with the unstructured, tetrahedral elements. The cell size is about 20 times smaller than the minimum wavelength considered [13], which can behave a well balance between accuracy and computational time. For each chamber, three simulations are required. Each simulation is carried out by imposing particle velocity on one boundary and anechoic termination on the others. By processing numerical results, the transmission and reflection coefficients of each chamber are calculated corresponded to each predefined frequency. For the sake of clarity, only the operations of discharge chamber are shown in Figure 6 when the particle velocity is imposed on the boundary of head-end valve. Moreover, the acoustic coefficients obtained can be extensively applied for different thermodynamic conditions by expressing them as a function of the wavenumber which corresponds to each frequency. This simplest approach needs no further acoustic FEM simulations of the chamber. In addition, the acoustic coefficients are interpolated to obtain the acoustic operations versus the compressor harmonics. It is worth pointing out that the compressor model with in-pipe 1D gas dynamics has been successfully applied to very simple compression system. A comparison analysis is performed between \begin{table}
\begin{tabular}{c c c} \hline \hline
**Specification** & **Value** & **Unit** \\ \hline Rotational speed & 950 & rpm \\ Cylinder diameter & 105 & mm \\ Stroke & 110 & mm \\ Diameter of the pipeline & 50 & mm \\ Length of suction pipe & 0.47 & m \\ Length of discharge pipe & 0.65 & m \\ Suction temperature & 304.15 & K \\ Suction pressure & 0.802 & MPa \\ Discharge pressure & 2.1 & MPa \\ Discharge temperature & 367.15 & K \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main specifications of the reciprocating compressor. Figure 5: The examined chamber: (**a**) fluid domain and (**b**) finite element mesh. it and the developed hybrid model with acoustic FEM characterization. Figure 7 shows the comparison of pressure profile and its amplitude-frequency characteristics at the suction and discharge pipe-ends connected to the chamber. Generally, the two models give similar results when an acoustic response is not dominated by chamber. In particular, the attention is focused on the influence of complex geometries on compressor simulation. It can be seen that the model with 1D gas dynamics poorly involves the acoustic response of chamber as it simply considers the chamber as a volume cavity. Conversely, the hybrid model predicts clearly marked acoustic response of the chamber due to the acoustic FEM characterization. Figure 8 depicts the in-cylinder pressure variation between the two models. The predictions are in good agreement due to the low response of compressor cycle to the pressure pulsation inside pipelines. However, by comparing in detail the thermodynamic cycle at the suction and discharge phase, it is noticed that in-cylinder pressure oscillation trend is directly linked to the pulsating pressure profiles inside the pipeline system. Figure 6: Amplitude of the transmission and reflection coefficients of discharge chamber with particle velocity imposed on the boundary of head-end valve.