## 2 Hybrid Model
### Numerical Procedure  
As stated before, the hybrid model consists of the compressor, gas dynamic and acoustic FEM characterization sub-models. The former two sub-models work in the time domain, whereas the last  
Figure 2: Schematic acoustic description of a multi-port configuration. one is in frequency domain. Reciprocal interaction among sub-models is determined by the coupling algorithm, as shown in Figure 3. Moreover, in order to guarantee the implementation of this method, the compressor suction/discharge valve channels are approximately treated as equivalent short ducts (ESDs). The numerical procedure is inspired by the points that the in-pipe pressure perturbation in the way of 1D unsteady flow may be decomposed as a forward pressure component and a backward pressure component [24]:  
\[p_{FW}=p_{ref}\Big{\{}\Big{[}1+\Big{(}a_{ref}/a_{A}\Big{)}\lambda\Big{]}/2 \Big{\}}^{2\gamma/(\gamma-1)},\quad p_{BW}=p_{ref}\Big{\{}\Big{[}1+\Big{(}a_{ ref}/a_{A}\Big{)}\beta\Big{]}/2\Big{\}}^{2\gamma/(\gamma-1)} \tag{18}\]  
where \(\lambda\) and \(\beta\) are given by:  
\[\lambda=\frac{a}{a_{ref}}+\frac{\gamma-1}{2}\frac{u}{a_{ref}},\quad\beta= \frac{a}{a_{ref}}-\frac{\gamma-1}{2}\frac{u}{a_{ref}}; \tag{19}\]  
and a multi-port complex fluid domain could be well described by the transmission and reflection coefficients through acoustic FEM characterization. Mutual transformation between time domain and frequency domain is obtained by using Fast Fourier Transform (FFT) and its inverse. The flowchart of the hybrid approach is shown in Figure 4. The acoustic FEM simulations of complex geometries are first performed based on the approach described in Section 2.3. The obtained acoustic coefficients are then introduced in main procedure. The time steps are determined based on FFT and CFL condition. The flow computation for main pipes are carried out based on the approach described in Section 2.2 for one whole cycle, and then the compressor thermodynamic calculation (Section 2.1) and flow calculation in ESDs are also performed for a whole cycle. Subsequently, the hybrid coupling procedure described in Section 2.4 is carried out. The obtained acoustic boundary will used in next cycle. As coupling of acoustic characterization needs time-domain variables \(p_{FW_{i}}\) for a necessarily whole period, the numerical procedure follows a period iterative fashion. The procedure includes three loops totally. The pipe system loop is performed to solve the governing equations in the main pipe and the boundary conditions. Subsequently, the ESDs loop is performed with solution of compressor thermodynamics and the unsteady flow in ESDs. The above two loops are included inside a main loop which regulates the well-running of execution. Also, the main loop calculates the values \(\beta_{i}^{(k+1)}\) with \(k+1\) indicating the period-iterative times, which will be imposed in the next period as boundary conditions at the pipe-end connected to the FEM characterized configuration. Figure 3: Schematic description of the compressor hybrid model. Before implementing the loop calculation, it is firstly to perform acoustic FEM simulations of the complex geometries at the given mean thermodynamic conditions in the range of frequencies interested in. The results are processed to determine the transmission and reflection coefficients indexed on wave number concerning to each simulation frequency. The coefficient values will be then interpolated based on multiples of the compressor fundamental frequency for loop calculation. In addition, in order to extend application of the acoustic representation to different thermodynamic conditions, a simple correction that keeps the wave number the same is employed instead of further FEM calculation [24]. The time-step of loop calculation must be properly set. It is common sense that the setup of spatial mesh size \(\Delta x\) is carried out by users with a suitable trade-off between accuracy and computational  
Figure 4: Flowchart of the hybrid model programme layout. cost.