## 2 Hybrid Model
### Reciprocating Compressor Sub-Model  
Figure 1 schematically depicts a basic line of reciprocating compressor working unit with spring-loaded suction and discharge valves. The connecting rod converts the rotary motion of crankshaft to the linear movement of piston. In the modelling process, the piston end face, cylinder wall and cylinder cover enclose a varying control volume regulated by the increments of crank-angle. Evaluation of thermodynamic properties is performed by taking advantage of a 0D quasi-steady approach through solving the equations of first thermodynamic law and mass conservation. The equation of mass conservation in the control volume of Figure 1 is given by [8]:  
\[\frac{dm_{cv}}{dt}=\frac{dm_{s}}{dt}-\frac{dm_{d}}{dt} \tag{1}\]  
Mass flow rate through valves can be determined by the following equations which have been successfully and widely used as the standard approach for compressors [8; 20]:  
Figure 1: Schematic of reciprocating compressor sub-model. \[ \begin{split}\frac{dm_{s}}{dt}&=\left\{\begin{array}{ll} \varphi_{s}A_{s}\sqrt{2(p_{s}-p_{cv})\rho_{s}}&\text{for }p_{s}>p_{cv},y_{s}>0\\ -q_{s}A_{s}\sqrt{2(p_{cv}-p_{s})\rho_{cv}}&\text{for }p_{s}<p_{cv},y_{s}>0 \\ \frac{dm_{d}}{dt}&=\left\{\begin{array}{ll}\varphi_{d}A_{d}\sqrt{2(p_{cv}-p _{d})\rho_{cv}}&\text{for }p_{cv}>p_{d},y_{d}>0\\ -\varphi_{d}A_{d}\sqrt{2(p_{d}-p_{cv})\rho_{d}}&\text{for }p_{cv}<p_{d},y_{d}>0 \end{array}\right.\end{split} \right.\end{split} \tag{2}\]  
Without considering the potential and kinetic energy reasonably, energy balance equation over the control volume based on the first law of thermodynamics is governed as follow:  
\[\frac{dQ_{cv}}{dt}+\frac{dm_{s}}{dt}h_{s}=\frac{dm_{d}}{dt}h_{d}+\frac{d}{dt}(me )_{cv}+p_{cv}\frac{dV_{cv}}{dt} \tag{3}\]  
With the equation:  
\[\frac{d(me)_{cv}}{dt}=m_{cv}\frac{de_{cv}}{dt}+e_{cv}\frac{dm_{cv}}{dt} \tag{4}\]  
Equation (3) is re-written as follow:  
\[\frac{de_{cv}}{dt}=\frac{1}{m_{cv}}\bigg{[}\frac{dQ_{cv}}{dt}-p_{cv}\frac{dV_{ cv}}{dt}+\frac{dm_{s}}{dt}h_{s}-\frac{dm_{d}}{dt}h_{d}-e_{cv}\frac{dm_{cv}}{dt} \bigg{]} \tag{5}\]  
in which heat exchange rate is given by the well-known formulation:  
\[\frac{dQ_{cv}}{dt}=\alpha A_{\text{wall}}(T_{\text{wall}}-T_{cv}) \tag{6}\]  
To determine the convective heat transfer coefficient \(\alpha\), the Woschni correlation is used in this paper. It is developed and frequently used for heat transfer studies in IC engines. Also, it has been widely used in reciprocating compressors since, by neglecting the combustion source term, the common peak of the convective coefficient close to a crank angle of 180\({}^{\circ}\) is shifted to the end of the compression step as it is expected for reciprocating compressors [8; 21]. The equation of heat transfer coefficient is defined as:  
\[\alpha=3.26p_{cv}^{0.8}T_{cv}^{-0.546}D_{cv}^{-0.2}u^{0.8} \tag{7}\]  
In the above equation, \(u\) equals to \(6.18u_{p}\) for suction and discharge phases and \(2.28u_{p}\) for compression and expansion phases. The instantaneous in-cylinder working volume from top dead center is:  
\[V_{cv}=V_{cl}+\frac{\pi D_{p}}{4}r_{1}\bigg{[}1-\cos(\omega t)+\frac{r_{2}}{r _{1}}\bigg{(}1-\sqrt{1-\left(r_{1}/r_{2}\right)^{2}\sin^{2}(\omega t)}\bigg{)} \bigg{]} \tag{8}\]  
Reciprocating compressor valves are automatic, i.e., their motion is determined by pressure difference between the cylinder and suction/discharge ambient. Main hypotheses for valve dynamic calculation are: valve is a one-degree-of-freedom system, valve displacement is restricted by limiter and valve plate is rigid.