# Parametric Simulations on Leakage and Performance of a Miniature Free-Piston Generator (MFPG)
## 3 Modelling of the MFPG System
### Mass Leakage Model  
The leakage process is modeled by a simplified scenario, in which the combustion chamber is regarded as a control volume and the mass flow rate of the in-cylinder gas through the cylinder-piston gap is considered as an orifice flow [62]:  
\[\frac{dm_{leak}}{dt}=\mu_{cell}A_{leak}\Psi_{lsub,lsup}\sqrt{p_{\mathrm{H}}\rho _{1}} \tag{5}\]  
\[\Psi_{lsub,lsup}=\left\{\begin{array}{cc}\sqrt{\frac{2\gamma}{\gamma-1}\left[ \left(\frac{p_{\mathrm{H}}}{p_{\mathrm{I}}}\right)^{\frac{2}{\gamma}}-\left( \frac{p_{\mathrm{H}}}{p_{\mathrm{I}}}\right)^{\frac{\gamma+1}{\gamma}}\right]} &\frac{p_{\mathrm{H}}}{p_{\mathrm{I}}}>\left(\frac{2}{\gamma+1}\right)^{\frac {\gamma}{\gamma-1}}\\ \left(\frac{2}{\gamma+1}\right)^{\frac{1}{\gamma-1}}\sqrt{\frac{2\gamma}{ \gamma+1}}&\frac{p_{\mathrm{H}}}{p_{\mathrm{I}}}\leq\left(\frac{2}{\gamma+1} \right)^{\frac{\gamma}{\gamma-1}}\end{array}\right. \tag{6}\]  
where \(m_{leak}\) is mass of the leaked gas, kg; \(\mu_{cell}\) is discharge coefficient of the leakage passage; \(A_{leak}\) is leakage area of the piston-cylinder unit, m\({}^{2}\); \(\Psi_{lsub}\), \(\Psi_{lsup}\) are subsonic and supersonic flow functions of the in-cylinder gas leakage; \(p_{\mathrm{I}}\), and \(p_{\mathrm{II}}\) are upstream and downstream pressures of the throttling orifice, Pa; \(\rho_{1}\) and \(\rho_{\mathrm{II}}\) are upstream and downstream densities of the throttling orifice, kg/m\({}^{3}\); \(\gamma\) is the adiabatic exponent.  
Figure 4: Heat transfer model vs. the pressure and temperature of in-cylinder gas. (**a**) In-cylinder gas pressure vs. piston displacement; (**b**) In-cylinder gas pressure vs. expanded piston displacement.  
The simulation of the leakage characteristics of the MFPG prototype is shown in Figure 5 along with that of two other micro heat engines, micro conventional crank shaft engines and micro rotary engines. The three types of micro engines are of the same displacement of 2.5 cc. The leakage characteristics of the micro conventional crank shaft engines and micro rotary engines are collected from the literature [38; 39], while that of the MFPG is calculated by the proposed model. The gap widths are set to be 20, 15 and 10 \(\upmu\)m, respectively. The discharge coefficient \(\mu_{all}\) is assumed to be 0.06 to account for the sealing effect of the lubricant oil in the piston-cylinder gap, as suggested by [PERSON] et al. [37].  
The revolution of the other two micro engines is around 6000-14,000 rpm, while the equivalent engine revolution of the MFPG prototype varies from 600 to 2000 rpm. In comparison with these two micro engines, the lower operation frequency of the MFPG would increase the mass leakage to some extent. However, the much lower compression ratio, compared with the micro conventional crank shaft engines, would help reduce the mass leak of the MFPG significantly. Thus, it is reasonable for the calculated leakage of the MFPG, with the suggested \(\mu_{all}\) 0.06, to be smaller than that of the micro conventional crank shaft engines. Likewise, with the same displacement and with an identical compression ratio, micro rotary engines leak much more severely than the reciprocating engines [39]. Thus, with the suggested \(\mu_{all}\) 0.06, the calculated leakage of the MFPG is smaller than that of the micro rotary engines, which also makes sense. Therefore, the proposed leakage model along with the selected leakage discharge coefficient \(\mu_{all}\) is considered to be a reasonable estimation of mass leakage characteristics of the proposed MFPG prototype.