# Parametric Simulations on Leakage and Performance of a Miniature Free-Piston Generator (MFPG)
## 4 Parametric Study on Leakage and Performance
### Effect of the Combustion Duration  
Combustion duration \(t_{cd}\) is an important parameter reflecting the combustion process. The \(t_{cd}\) is varied from 2.5-4 ms in the simulation while all the other parameters are kept the same. As implied by Figure 9a,b, the curve of the in-cylinder gas during compression overlaps each other and the mass leak \(m_{lcp}\) is nearly flat, meaning that the variation of \(t_{cd}\) has no significant effect on the fuel loss. On the other hand, the mass of in-cylinder gas at exhaust port opening \(m_{gepo}\) decreases with the extending of \(t_{cd}\), which means the mass leakage during expansion \(m_{lcp}\) increases. Although the prolongation of \(t_{cd}\) lowers the \(p_{\text{max}}\), as shown in Figure 9c, which helps to reduce the \(m_{lcp}\), the increased expansion time \(t_{exp}\) is mainly responsible for the increase in \(m_{lcp}\).  
With a shorter \(t_{cd}\), a system releases the heat in a shorter time, which is closer to the ideal Otto cycle. The \(p_{\text{max}}\) is therefore enhanced, as presented in Figure 9c. The TDC increases, as shown in Figure 9d, as a result of elevated in-cylinder gas pressure and the effective compression ratio \(\varepsilon_{cc}\) hence decreases slightly. The BDC also increases due to the enhanced \(p_{\text{max}}\). By taking the difference between the positions of TDC and BDC, the piston stroke \(S\) shows an increasing trend and so does the compression ratio \(\varepsilon_{c}\). The increased piston peak velocity \(v_{\text{max}}\) and piston frequency \(f_{p}\) are also the results of enhanced \(p_{\text{max}}\).  
Figure 9e shows that the indicated thermal efficiencies, under the three hypotheses, all increase monotonously with the reducing of combustion duration \(t_{cd}\), mainly due to the aforementioned higher degree of isochoric combustion. There are few effects of \(t_{cd}\) on the three major consumption work, \(W_{consm}\), \(W_{pump}\), and \(W_{f}\), so the effective thermal efficiencies \(\eta_{c,a}\), \(\eta_{c,b}\) and \(\eta_{c,c}\) show nearly the same trends with their corresponding indicated thermal efficiencies. The \(\Delta\eta_{i,a,b}\), which reflects the effect of leakage loss, decreases insignificantly with respect to the increase in \(t_{cd}\) as implied by Figure 9f. The result is consistent with the aforementioned conclusion that \(t_{cd}\) has no significant effect on \(m_{gsoc}\). The variation of \(t_{cd}\) brings in two effects on heat loss. As \(t_{cd}\) increases, the maximum combustion temperature \(T_{\text{max}}\) decreases, which is beneficial for heat loss reduction. On the other hand, time consumed by expansion \(t_{exp}\) increases, which extends the heat transfer time and intensifies the heat loss. The \(\Delta\eta_{i,b,c}\), which reflects the effect of the heat loss, decreases slightly with  
Figure 9: The influence of combustion duration on leakage and system performance. (**a**) Mass variations of in-cylinder gas in function of stroke; (**b**) Mass leakage; (**c**) Pressure variations of in-cylinder gas in function of stroke; (**d**) Piston dynamic characteristics; (**e**) Thermal efficiencies; (**f**) Indicated thermal efficiency drops. Figure (**a**,**c**,**d**) apply the same legend displayed in figure (**c**). The blue, green, red and black solid lines stand for the combustion durations of 2.5, 3.0, 3.5 and 4.0 ms, respectively. The symbol \(f_{p}\) in figure (**d**) stands for piston frequency, Hz.  
respect to the prolongation of \(t_{cd}\), indicating that the influence of \(T\) on the heat loss is dominant in these cases.