## 6 Conclusions  
In this paper, the mathematical model of the actuator and the CFD model of the compressor cylinder and the valve plate are established. The relationship between different backflow clearance and the increase of cylinder pressure is studied, and puts forward an optimized design of actuator parameters considering backflow clearance, which can increase the life and performance of the stepless capacity control system of reciprocating compressor and lay a foundation for the overall optimization of the system. (1) Firstly, the mathematical model of the actuator was established. Then, the parameters of the experimental test were taken as an example for numerical analysis. Combined with the experimental data, the rebound coefficient was determined to be \(-0.075\). On this basis, the motion simulation curve of the actuator was modified. Compared with the experimental motion curve, the error is small and the ejection time difference is 4 ms, which verifies the correctness of the model. Finally, the relationship between impact velocity, spring stiffness and hydraulic pressure is analyzed. In the ejection process, the impact velocity is positively correlated with the hydraulic force, and negatively correlated with the spring stiffness. In the withdrawal process, the impact velocity has nothing to do with the hydraulic force, but has a positive correlation with the spring stiffness. (2) The CFD model of the compressor cylinder and the valve plate was built based on the experimental platform of the capacity control system. By analyzing the data, it can be seen that the relationship between backflow clearance and the maximum gas resultant force and the corresponding angle increases exponentially. When the backflow clearance is 1.25 mm, the pressure increases by 26.1%, and the compressor can still run safely and stably. When the backflow clearance is 1.5 mm, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & **Design 1** & **Design 2** & \begin{tabular}{c} **Decrease** \\ **Percentage** \\ \end{tabular} & **Design 3** & \
\begin{tabular}{c} **Decrease** \\ **Percentage** \\ \end{tabular} & **Design 4** &
\begin{tabular}{c} **Decrease** \\ **Percentage** \\ \end{tabular} \\ \hline Spring stiffness (N/m) & 18,425 & 11,588.82 & 37.10\% & 11,575.84 & 37.17\% & 11,606.55 & 37.01\% \\ Hydraulic force (N) & 190.9 & 87.44 & 54.20\% & 94.25 & 50.63\% & 101.82 & 46.66\% \\ Ejection impact velocity (m/s) & 0.836 & 0.093 & 88.83\% & 0.17 & 79.48\% & 0.23 & 72.83\% \\ Withdrawal impact velocity (m/s) & 0.752 & 0.354 & 52.91\% & 0.42 & 44.22\% & 0.48 & 35.55\% \\ Backflow clearance (mm) & 0 & 0.00125 & — & 0.00065 & — & \(2.00\times 10^{-5}\) & — \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The comparison of different designs. Figure 14: Pareto optimal solutions of hydraulic force. the pressure increases by 43.72%, and the compressor can no longer run stably for a long time. Therefore, the 1.25 mm backflow clearance is taken as the maximum backflow clearance. At this point, the maximum gas resultant force of the valve plate is 69.565 N.  
(3) With spring stiffness and hydraulic force as design variables, impact velocity and backflow clearance as objective functions, multi-objective optimization was conducted based on NSGA-II. Except for the backflow clearance, all the other parameters after optimization decreased significantly, with the spring stiffness decreased by about 37%, the hydraulic force decreased by about 50.5%, the ejection impact velocity decreased by about 80.38%, and the withdrawal impact velocity decreased by about 44.23%. Finally, according to the influence of backflow clearance on the running state of the compressor, a set of design parameters of the actuator was selected from the Pareto optimal solution set. (4) In this paper, we consider the effect of gas resultant force of the valve plate on the design of the actuator. There are many factors that affect the gas resultant force of the valve plate, such as backflow clearance, leakage, valve form and so on.