## 2 Hybrid Model
### Numerical Procedure
However, the time interval \(\Delta t\) will be determined by the criterion of Courant-Friedrichs-Lewy applied to each pipe:  
\[\Delta t\leq\frac{\Delta x}{a+|u|} \tag{20}\]  
In addition, the input complex amplitudes \(p_{FW_{i}}^{*(k)}\) at the \(k\)-th period are achieved through transforming the period time history of pressure component \(p_{FW_{i}}^{(k)}\) by means of FFT algorithm. This transformation is primarily deficient in that the needed values must be equally spaced as an array of \(2^{n}\) (\(n\) is an integer), and the first and the last values must be located separately at the beginning and the end of the compressor period. Therefore, time discretization must be performed with a constant value \(\Delta t=t_{0}/2^{n}\) (\(t_{0}\) indicates compressor period) and be compatible with Equation (20). It is noteworthy that the equivalent duct is quite short. The common time discretization will highly damage the global computational cost. Therefore, two different time-steps \(\Delta t\) and \(\delta t\) are imposed separately to the pipe system and ESDs. This simple approach is much economical for computational cost to handle with a very fine spatial mesh size in particular ducts. After the time-steps are determined, the conservation equations in pipes are solved by employing the LW2 numerical scheme [18]. The flow properties at pipe-ends are updated by using the Trapezoidal Method of Characteristics (TMOC) developed by authors [17]. Although the TMOC formulation is developed for ideal gas, it can be applied to real gas model through simply calling the NIST REFPROP database for gas properties [25]. Without increasing complexity and computational effort, the influence of friction and heat transfer is better evaluated on the characteristic lines and the path-line. The boundary conditions applied, except for the one contiguous to the acoustic FEM characterized element, follow the [PERSON]'s quasi-steady physical models [15]. In the ESDs loop, the compressor sub-model is solved by using the standard 4 th order Runge-Kutta method. Temperature and density are the two independent thermodynamic properties that are enough to pick out other thermodynamic properties. These computations regulated by respective time-steps run several times for the whole compressor period. Then, the pipe system loop and the ESDs loop finish. Subsequently, the main loop calculates the values \(\beta_{i}^{(k+1)}\) that will be imposed, in the next period at each corresponding time-step, as the boundary conditions of the pipes connected to the FEM characterized configuration. The steps are as follows: (1) calculation of time-domain pressure component in the whole \(k\)-th period from the results of unsteady flow computation:  
\[p_{FW_{i}}^{(k)}=p_{ref}\Big{\{}\Big{[}1+\Big{(}a_{ref}/a_{A_{i}}^{(k)}\Big{)} \lambda_{i}^{(k)}\Big{]}/2\Big{\}}^{2\gamma/(\gamma-1)} \tag{21}\]  
(2) calculation of \(p_{FW_{i}}^{*(k)}\) from \(p_{FW_{i}}^{(k)}\) by using FFT algorithm; (3) calculation of \(p_{BW_{i}}^{*(k+1)}\) from \(p_{BW_{i}}^{*(k+1)}\) by using the inversion of FFT algorithm; (5) calculation of \(\beta_{i}^{(k+1)}\) from \(p_{BW_{i}}^{(k+1)}\) based on the equation as follows:  
\[\beta_{i}^{(k+1)}=\Big{(}a_{A_{i}}^{(k+1)}/a_{ref}\Big{)}\bigg{[}2\Big{(}p_{ BW_{i}}^{(k+1)}/p_{ref}\Big{)}^{(\gamma-1)/2\gamma}-1\bigg{]} \tag{22}\]  
Also, the main loop identifies whether or not the numerical procedure has reached the convergence condition of thermodynamic process and flow properties at the monitor points. If the procedure ending has not been achieved, a new calculation period is carried out. This iterative process works until the procedure ending is obtained.