## 1 Introduction  
A new climate regime, POST-2020, was agreed upon at the climate change conference held in Paris in December 2015. In order to implement this new climate system, all countries must strive to achieve their greenhouse gas reduction targets. To this end, many efforts are being made to develop clean energy worldwide. Representatively, there have been studies on the production and use of hydrogen energy. Recently, a full-scale study on the development of core technologies for hydrogen liquefaction plants began in South Korea. A hydrogen liquefaction plant is a closed-loop system, and it consists of a compressor, heat exchanger, cold box, and expander. The expander is one of the core components in a cryogenic system. In the cryogenic expander, the working fluid is expanded under extremely low temperatures, e.g., below 93 K. Therefore, some of the parts, i.e., the rotor, shaft, and labyrinth seal, are operated under cryogenic conditions. In general, the cryogenic expander rotates at a high speed, from tens of thousands to hundreds of thousands of revolutions per minute, so rotordynamic stability is important. The technologies required to develop the cryogenic expander are as follows: aerodynamic design technology with an expander loss model for cryogenic working fluids, seal and insulation technology, thermal stress analysis technology, high-speed oil-free bearing technology, clearance prediction and control technology, secondary flow stabilization technology, and expander power control technology. However, because of the information security of the industry, the introduction of these technologies in the literature or at conferences has been limited. However, it is possible to identify the latest technology trends and technical issues through the publications of a few companies that produce cryogenic expanders [1; 2; 3]. There are several methods to exhaust the output power generated by the expander. First, a compressor is installed on the opposite side of the expander with the same shaft. It can be applied to compress the working fluid at the cryogenic system [4]. This expander-compressor assembly is called a compander. When it is used as a compander, there is an advantage in that the high pressure required in the cryogenic system can be acquired from the compressor. However, there is a disadvantage in that the system operation is complicated. Second, the power generated from the expander drives the brake compressor to exhaust energy [5; 6; 7; 8]. Alternatively, there is a method of converting the power generated by the expander into heat energy using an eddy current breaker. It has an advantage in that the size of the system can be reduced and the control can be made simple. However, it dissipates the energy generated by the expander; therefore, it is not preferable for a large-capacity plant. Third, the expander rotates the generator, and electric power can be acquired from the generator. If the capacity of the expander is large enough, the application of a generator can be considered. As described above, the method of utilizing expander energy can be determined by the characteristics of the cryogenic system. In this research, the brake compressor is connected to the expander so that the speed of the expander can be controlled by adjusting the compressor's throttle valve. [PERSON] et al. conducted blade optimization to reduce the swirl flow and cavitation in the draft tube of a cryogenic liquid turbine [9]. [PERSON] et al. investigated the effect of the blade loading distribution and blade lean angles on the cavitation in the draft tube of a cryogenic expander [10]. However, there has been little research on the optimization of cryogenic expander efficiency. To develop a highly efficient cryogenic system, it is necessary to increase the efficiency of the expander. In this paper, the optimization of an expander rotor was conducted to improve expander efficiency. The expander rotor rotates at tens of thousands of revolutions per minute under cryogenic conditions.