## 2 Numerical Calculation Model
### The Boundary Condition Setting of the Calculation
#### 2.3.2 Boundary Conditions for Liquid Nitrogen Boiling Simulation  
In addition to the fluid pulsation caused by the pump, the system will also experience vibration due to the boiling of liquid nitrogen. As the heat load generated by the first crystal is approximately 90 W, and the boiling point of liquid nitrogen at atmospheric pressure is 77.36 K, boiling of the coolant may occur. Therefore, conducting an in-depth study of the boiling phenomenon inside the cooling system is essential.  
Firstly, a comprehensive steady-state fluid-thermal two-way interaction analysis is conducted to obtain temperature field data for both solid and fluid under stable conditions. As shown in Table 3, the inlet liquid nitrogen flow velocity is 0.995 m/s, and the temperature is 77.35 K, representing saturated liquid nitrogen under standard conditions. The heat flux density at the beam spot is \(2.25\times 10^{7}\) W/m\({}^{2}\), forming a rectangular shape with dimensions of 2 mm \(\times\) 2 mm. Its geometric center is located on the meridian of the first crystal, 30 mm away from the left end face of the first crystal. Next, the temperature field data at the fluid-solid interface will be extracted for use as boundary conditions in the research of liquid nitrogen boiling. To investigate the boiling of liquid nitrogen, the volume of fluid (VOF) two-phase flow method is employed to simulate the motion of liquid nitrogen and nitrogen gas [15; 16]. The pump operates at a rated power with a flow rate of 3 L/min and an operating pressure at standard atmospheric pressure. The initial temperature of liquid nitrogen inside the flow domain is 77 K.  
The complete incompressible conservation of momentum equation was solved using the SIMPLEC method. Energy and momentum use a second-order upwind scheme, and the equation of volume fraction by the Geo-Reconstruct scheme, and turbulent kinetic energy and turbulent dissipation rate are discretized using a first-order upwind scheme. The pressure uses the PRESTOI scheme. Adaptive time steps are used with a global maximum Courant number (CFL) of 0.5 and convergence residuals are controlled below \(1\times 10^{-5}\). The simulation covers a fluid time of \(0\sim 3\) s, with the first 2 s dedicated to excluding the initial liquid nitrogen inside the pipes and allowing the flow field to reach a stable state under actual operating conditions. Data sampling is performed only on the flow field at 3 s, with a sampling interval of \(5\times 10^{-4}\) s. During the calculation, the minimum time step is \(2\times 10^{-5}\) s, and the maximum time step is \(5.5\times 10^{-4}\) s. With the support of a 48-core 2.7 GHz high-performance workstation, the fluid calculation took 70 h. The relevant physical properties of liquid nitrogen and nitrogen gas are shown in Figure 5[17; 18].  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Type** & **Numerical Value** \\ \hline Average flow velocity \(v_{a}/\text{m}\cdot\text{s}^{-1}\) & 0.995 \\ Outlet pressure/Pa & 101, 325 \\ Frequency \(f\)/Hz & 40, 60, 80, 100 \\ Frequency \(f\)/Hz & 200, 300, 400 \\ Amplitude of pulsation \(A\) & (0.1\%, 0.25\%, 0.5\%, 0.75\%, 1\%)\(\cdot v_{a}\) \\ Computing time & 40\(\cdot\frac{1}{f}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Boundary conditions of pulsation effect.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Type** & **Numerical Value** \\ \hline Velocity of inlet/m \(\cdot\text{s}^{-1}\) & 0.995 \\ Temperature of inlet/K & 77.35 \\ Thermal power/W & 90 \\ The initial temperature of liquid nitrogen/K & 77 \\ Global maximum Courant number & 0.5 \\ Minimum time step/s & \(2\times 10^{-5}\) \\ Maximum time step/s & \(5.5\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: The initial boundary conditions of liquid nitrogen boiling.