## 5 Conclusions  
In this study, we introduced a design method for expanders for hydrogen liquefaction plants. Blade optimization was conducted to increase the expander efficiency. The capacity of the developed expander for a hydrogen liquefaction system is 0.5 tons of hydrogen liquefied per day. Based on the cycle analysis results of the hydrogen liquefaction system, the meridional and three-dimensional design of the expander were conducted. To improve the efficiency of the expander, the rotor blade meridional shape and blade \(\beta\) angle were optimized. Through the optimization design, the blade height was increased between the blade leading edge and the trailing edge, and the distribution of the blade \(\beta\) angle was rearranged smoothly. Finally, the optimized expander's efficiency increased by 1.98% compared to that of the original model. To investigate the reason for this efficiency improvement, a flow analysis was conducted. The entropy distribution on the suction blade surface was smaller in the optimized model than that of the original model. Through the relative velocity distribution, it was found that the optimized model had a smoother main passage flow and smaller tip leakage flow than the original model. Due to the passage area of the optimized model, which is larger than the original model, the effect of the tip leakage flow on the main passage flow could be reduced. It was confirmed that the optimization of the blade \(\beta\) angle, rotor hub, and shroud meridional contour had an effect on improving the efficiency of the cryogenic expander. The efficiency optimization method in this paper can be used as a reference in designing high-efficiency cryogenic expanders in the future.  
Conceptualization, J.S.; methodology, H.L.; software, J.P.; validation, M.P., D.L.; formal analysis, B.C.; investigation, J.B.; writing--original draft preparation, S.K., Y.L. and B.K.; writing--review and editing, H.L. and A.A.; supervision, J.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the following projects: the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry, and Energy (MOTIE) of the Republic of Korea (Project No.: 20188550000200); the Korea Ministry of Land, Infrastructure, and Transportation and the Korean Agency for Infrastructure Technology Advancement (Project No.: KAIA21 HITP-B151613-03); and a National Research Council of Science and Technology (NST) grant funded by the Ministry of Science, ICT and Future Planning, Korea (Grant No. KIMM-NK231D, KIMM-NK231G).  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Expansion** & \multicolumn{2}{c}{**Efficiency**} & \multicolumn{2}{c}{**\(\Delta\)Efficiency**} & \multicolumn{2}{c}{**Outlet Temperature [K]**} & \multicolumn{2}{c}{**\(\Delta\)Temperature**} \\ \cline{2-7}
**Ratio** & **Original** & **Optimized** & **[\%] *** & **Original** & **Optimized** & **[\%] * \\ \hline
1.74 & 0.750 & 0.764 & 1.9 & 41.306 & 41.167 & 0.34 \\ \hline
1.67 & 0.752 & 0.767 & 1.94 & 41.813 & 41.674 & 0.33 \\ \hline
1.59 (DP) & 0.754 & 0.769 & 1.98 & 42.415 & 42.281 & 0.32 \\ \hline
1.49 & 0.758 & 0.768