# Parametric Simulations on Leakage and Performance of a Miniature Free-Piston Generator (MFPG)
## 4 Parametric Study on Leakage and Performance
### Effect of the Piston Mass  
Piston mass \(m_{p}\) is another important factor affecting the natural frequency of the piston\(f_{p}\). The piston mass of the physical MFPG prototype is 156 g, so the adjusting range of \(m_{p}\) is set to be 100-184 g with the other parameters unchanged. According to Figure 10a, the mass of the in-cylinder gas at EPC \(m_{gpc}\) is almost the same in the four cases and so is the gas mass at SOC \(m_{gpc}\), which means that the variation of \(m_{p}\) has no significant effect on \(m_{lcp}\) as directly presented in Figure 10b. This is mainly attributed to the insignificant differences of the in-cylinder gas pressure during combustion, as shown in Figure 10c, and nearly the same compression time \(t_{cp}\) as implied by Figure 10d. During the expansion, the peak velocity \(v_{\text{max}}\) of the heavier piston is obviously lower than that of the light ones, which results in the extended time of expansion \(t_{exp}\) and leads to the enlarged mass leakage \(m_{lexp}\) in this range. Figure 10: The influence of piston mass 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 characeristics; (**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 piston mass of 100, 128, 156 and 184 mm, respectively. The symbol \(f_{p}\) in figure (**d**) stands for piston frequency, Hz. Figure 10c indicates that \(m_{p}\) has little effect on the in-cylinder gas pressure during the compression. However, \(p_{\text{max}}\) increases when a heavier piston is applied, which is mainly attributed to the increased effective compression ratio \(\epsilon_{ec}\), since the mass of the in-cylinder gas at SOC \(m_{gsoc}\) is almost the same and hardly affects the \(p_{\text{max}}\). The heavier the piston, the closer the TDC with respect to the cylinder head, so the \(\epsilon_{ec}\) increases accordingly and promotes the \(p_{\text{max}}\). In addition, the inertia effect becomes more pronounced with the increase in \(m_{p}\), the piston, therefore, responds slowly when expansion. Thus, the degree of isovolumetric combustion is hence increased and further enhances the \(p_{\text{max}}\). It is notable that the heavier the piston, the faster the in-cylinder gas pressure decreases in the expansion stroke, which is due to the more severe leakage of \(m_{lexp}\) with increase in the piston mass. As the piston mass increases, the piston stops farther with a larger value of BDC, as is indicated in Figure 10d. This is because the heavier piston leads to higher \(p_{\text{max}}\), which produces more work and needs a longer distance to offset. For the heavier pistons, the longer BDC implies increased acceleration length in the next cycle and therefore the higher speed at SOC. The piston kinetic energy is, therefore, greater and the piston is closer to the cylinder head, resulting in a decrease in TDC. Hence, the piston stroke \(S\) and compression ratio \(\epsilon_{c}\) both increase accordingly. In contrast, when the pistons expand back towards the BDC, all the pistons start at almost the same position, so the advantage of acceleration distance for the heavier pistons disappears and results in lower \(v_{\text{max}}\). It is known that the natural frequency is inversely proportional to the square root of the mass, thus the piston frequency \(f_{p}\) decreases as the piston mass increases. The effective compression ratio \(\epsilon_{ec}\) and the degree of isochoric combustion both increase with an increase in the piston mass, so the \(\eta_{i,a}\) increases, as presented in Figure 10e. The \(\eta_{i,b}\) keeps the same trend with \(\eta_{i,a}\) because the fuel leakage indicator \(m_{lcp}\) is little affected by the variation of \(m_{p}\).