## 3 Simulation and Evaluation Method  
In Reference [15], a transient cooling model of revers Brayton refrigerator was built, which used the method of dual non-steady time steps to connect dynamic characteristics of the turboexpander and cryogenic heat exchanger. Similar to the reverse Brayton refrigerator, the normal temperature part of the cryogenic turbo-based system operates under relatively stable boundary conditions, and the dynamic performance of the system is mainly affected by the low temperature part. In a cryogenic turbo-based system, with related component model inserted in the transient model, the dynamic performance of the system could be simulated and predicted. As a critical component of the turbo-based refrigerator, matching performances of the turboexpander with a blower were researched and the mathematical model was built in References [4; 15]. The working power and rotating speed are two equilibrium parameters between the expander and blower. The relations between the expander and blower could be presented as the following [4; 15]:  
\[p_{B}T_{0E}=M\frac{p_{0E}}{(u_{1}/c_{s})^{3}}\Big{[}1-\varepsilon_{E}{}^{(1-k )/k}\Big{]}^{0.5}\eta_{E}T_{0B}, \tag{1}\]  
\[M=\frac{\pi A_{\min}n_{d}D_{1E}{}^{3}}{169.7(1+\beta)\mu Q_{d}D_{2B}{}^{2}} \Bigg{[}\frac{n_{z}-1}{Z_{0E}}\bigg{(}\frac{2}{n_{z}+1}\bigg{)}^{\frac{n_{z}+ 1}{(n_{z}-1)}}\Bigg{]}^{0.5}, \tag{2}\]  
\[\frac{u_{1}}{c_{s}}=\frac{\pi D_{1E}n}{60\sqrt{2h_{Es}}}=\frac{\pi\sqrt{(k-1) /(kR)}}{60\sqrt{2\Big{[}1-\varepsilon_{E}{}^{(1-k)/k}\Big{]}}}\frac{nD_{1E}} {\sqrt{T_{0E}}}. \tag{3}\]  
As shown in the above equations, \(M\) is nearly constant for a certain refrigerator, and \(T_{0B}\) is certain at a specific working condition. The essential of calculating matching characteristics is to obtain relations among \(p_{B}\), \(p_{0E}\), \(\varepsilon_{E}\), \(T_{0E}\) and \(n\). The key factor is to obtain the expansion characteristics, which reflect the mutual interactions of \(\eta_{E}\), \(\varepsilon_{E}\), and \(u_{1}/c_{s}\) in performance curves [23]. When the expansion work is recycled by a motor, there are constant rotating speed together with constant brake work. As shown in Equation (3), \(u_{1}/c_{s}\) is just related to \(\varepsilon_{E}\), \(T_{0E}\) at this time. Also, the constant brake means that the expansion work is certain. The expansion work is as below:  
\[W_{E}=\frac{k}{k-1}RT_{0E}\Big{[}1-\varepsilon_{E}{}^{(1-k)/k}\Big{]}q_{mE} \eta_{E}, \tag{4}\]  
Therefore, with a motor to recycle the output energy, the essential of the matching characteristics is also to obtain the performance curve. ANSYS CFX was performed to simulate the expansion performance, and the Shear Stress Transport model was applied to resolve the turbulence term [15]. The working fluid was the real air. The inlet boundary conditions were the mass flow rate and static temperature, and the outlet one was the static pressure. The temperature contours from CFX simulations of the expander under the design point are presented in Figure 3. The gas was first expanded in the nozzles and then further expanded and cooled in the expansion wheel. In the diffuser, the temperature increased a bit, which was due to the diffusion effect. As shown in the temperature contours, the variations of temperature in the expansion process were uniform. According to the analysis and the CFX simulation, the unified turboexpander dynamic model could be established, as shown in Figure 4.  
The flow chart of the simulation method is shown in Figure 5. With a brake of energy consumption, the simulation method is the same as the one in Reference [15]. In the dynamic cooling process, the total computation time is \(t_{t}\), and the \(1^{\text{st}}\) time step is \(\Delta t\). In the \(1^{\text{st}}\) time step, \(\Delta\tau\) is divided as the \(2^{\text{nd}}\) time step. In \(\Delta t\), inlet temperature of the heat exchanger at the cold side is assumed to be equal to the expander outlet temperature, and they are considered as constant.