## 4 Result and Discussion
### Simulation of Energy Recovery  
To recycle the energy, the expansion work could be absorbed by a motor or a compressor. The common advantage of the two modes is that the output energy is recycled, and the energy efficiency is raised. While, the dynamic characteristics of the turboexpander and refrigerator are quite different under the two operation modes. As a motor is employed, the rotating speed of turboexpander is fixed, which means the expansion work is also fixed. With a compressor applied, the expansion work is absorbed by the coupled compressor, which raises the pressure of gas before flowing into the expander [14]. The rotating speed and other coupling parameters might vary during the cooling process, which was presented in detail in Reference [14]. As a result, the energy and economic efficiencies under two modes were also different. The energy and economy efficiency of the energy recycle method with a motor will be studied in detail in the present work. As shown in Figure 8, the dynamic characteristics of the refrigerator under different rotating speeds are presented, both of which operated at the expansion work of 800 W. As shown in Figure 8a,b, the cooling curves are much smoother than those of the refrigerator with certain brake pressure, which means the cooling process is more stable. With fixed expansion work and rotating speed, the variations of expansion parameters might be more controllable, resulting in the slow change of the refrigerator. That is, the more constrains, the less uncertainty. It also can be seen that, with a lower rotating speed, the temperature declines faster first and then slower. As shown in Figure 8c, the heat exchanger effectiveness was nearly the same at the beginning of the cooling process, and increased with the rotating speed later. Although a higher rotating speed was related to higher effectiveness, the refrigerator has poor performance. This shows that the expander efficiency played a more important role in this refrigerator. As shown in Figure 8d, the ultimate temperature of the refrigerator decreases with the increase of rotating speed. Figure 8: Dynamic cooling characteristics with a motor: (**a**) Refrigerator cooling characteristics; (**b**) cooling characteristics at the beginning; (**c**) effectiveness of the heat exchanger; (**d**) ultimate refrigerating temperature. The same expansion work represents the same energy recycled. Whereas, the refrigerating temperature and cooling capacity were different. At this expansion work, the greater rotating speed, which is corresponding to higher effectiveness of heat exchanger, results in relatively weaker cooling performance finally. The changing trend is a comprehensive reflect of dynamic characteristics of both the heat exchanger and turboexpander. As shown in Figure 9, the expansion performances are illustrated. Under rotating speed of 170 krpm, the expansion ratio is much higher at the beginning, and it is corresponding to a lower expansion efficiency. With the cooling of the refrigerator, the expansion ratio decreases first and then increases. The expansion efficiency increases until to the optimal value, and then decreases. But during the most period, the expansion efficiency is relatively high. That's why at this speed, the refrigerator reveals a better cooling performance. As the rotating speed decreases, the variation tendencies of the expansion ratio and efficiency begin to change obviously. The starting pressure of expander becomes smaller and the turning point gradually disappears. The expansion ratio increases and efficiency declines, which leads to less satisfactory performance. Although the refrigerator may achieve good performance under lower rotating speed, the starting pressure is quite high and changes dramatically in the beginning. Those were unfavorable to the stability and reliability of the cryogenic refrigerator. Both the cooling effect and operational performance should be considered in the actual application. The required compression work of the front-end compressor was also analyzed.