# Parametric Simulations on Leakage and Performance of a Miniature Free-Piston Generator (MFPG)
## 5 Power Flow and Energy Distribution
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Symbol** & \(\boldsymbol{p_{sc}}\)1  & \(\boldsymbol{x_{ign}}\)1  & \(\boldsymbol{t_{cd}}\)1  & \(\boldsymbol{m_{p}}\)1  \\ \hline Top dead center/m & TDC & \(\uparrow\) & \(\uparrow\) & \(\downarrow\) & \(\downarrow\) \\ Bottom dead center/m & BDC & \(\uparrow\) & \(\uparrow\) & \(\downarrow\) & \(\uparrow\) \\ Piston stroke/m & \(S\) & \(\uparrow\) & \(\downarrow\)\(\uparrow\)\(\downarrow\) & \(\downarrow\) & \(\uparrow\) \\ Compression ratio & \(\varepsilon_{c}\) & \(\uparrow\) & \(\downarrow\) & \(\downarrow\) & \(\uparrow\) \\ Effective compression ratio & \(\varepsilon_{ec}\) & \(\downarrow\) & \(\downarrow\) & \(\uparrow\) & \(\uparrow\) \\ Max piston velocity/m/s & \(\upsilon_{\max}\) & \(\uparrow\) & \(\downarrow\)\(\uparrow\)\(\downarrow\) & \(\downarrow\) & \(\downarrow\) \\ Piston frequency/Hz & \(f_{p}\) & \(\downarrow\) & \(\downarrow\) & \(\downarrow\) & \(\downarrow\) \\ Mass of in cylinder gas at & \(m_{gacc}\) & \(\uparrow\) & \(\downarrow\)\(\uparrow\) & \(\rightarrow\) & \(\rightarrow\) \\ SOC/kg & \(p_{\max}\) & \(\uparrow\) & \(\downarrow\)\(\uparrow\)\(\downarrow\) & \(\downarrow\) & \(\uparrow\) \\ Peak pressure of in-cylinder & & & & & \\ gas/Pa & \(p_{\max}\) & \(\uparrow\) & \(\downarrow\)\(\uparrow\)\(\downarrow\) & \(\downarrow\) & \(\uparrow\) \\ Leakage loss through & & & & & \\ piston-cylinder gap/J & & & & & \\ Heat Loss across cylinder & & & & & \\ wall/J & \(Q_{w}\) & \(\uparrow\)\(\downarrow\) & \(\uparrow\)\(\downarrow\) & \(\downarrow\) & \(\uparrow\) \\ Indicated thermal efficiency 2 & \(\eta_{j,c}\) & \(\uparrow\) & \(\downarrow\)\(\uparrow\)\(\downarrow\) & \(\downarrow\) & \(\downarrow\) \\ Effective thermal efficiency 2 & \(\eta_{j,c}\) & \(\uparrow\)\(\downarrow\) & \(\uparrow\) & \(\downarrow\) & \(\downarrow\) \\ Optimum value & & & 1.15 & 3 & 3.5 & 156 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Influences of parameters on the MFPG performance. In the expansion stroke, the in-cylinder gas starts to work on the piston and pushes the piston towards the BDC, so the value of \(P_{p}\) is positive. After overcoming the frictional power \(P_{f}\) and the electromagnetic power \(P_{ele}\), the residual power is stored in the piston in the form of \(P_{a}\). As the piston moves further, the in-cylinder pressure drops and the gas power \(P_{p}\) becomes insufficient to generate more \(P_{ele}\). Then, the piston slows down to release the stored \(P_{a}\) power and becomes another source of \(P_{ele}\). Thus, the value of \(P_{a}\) is negative. When the piston moves from EPO back to BDC, the \(P_{p}\) is further reduced. The \(P_{a}\) is responsible for overcoming the frictional power \(P_{f}\) and supplying the electric power generation at the same time. The power flow of the entire working cycle of the MFPG system indicates that the energy conversion of the system is a complex dynamic balance of different powers. The two ultimate power states in the process are \(P_{ele}\) and \(P_{p}\). When they are not balanced, their residual power is stored temporarily in the form of \(P_{a}\). It is assumed that the major energy loss of the MFPG system includes the short-cut loss \(Q_{scut}\) caused by scavenging, heat transfer loss \(Q_{w}\) across the cylinder wall, fuel leakage loss \(Q_{lcp}\) through the piston-cylinder gap, the friction loss \(W_{f}\) between the moving/stationary units and mechanical loss \(W_{pump}\) for boosting the scavenging pressure. Based on this assumption, the energy distribution per cycle for this example is predicted and listed in Table 3.  
Figure 11: Simulations of the transient powers of the forces imposed on the piston in a complete cycle. The optimal parameters result in a compression ratio of 5.6. The friction loss takes up 17.9% out of the indicated work.