## 5 Optimized Design of the Cooling Channel and Its Cooling Performance
### Transient Simulation Result Analysis  
Addressing the issues of insufficient inlet flow and safety, the inlet condition with a velocity of 0.2 m/s was adopted before \(t=550\) s, and then the cooling water flow rate was increased to 0.3 and 0.5 m/s, respectively, with standard atmospheric boundary conditions \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Inlet Flow** & **Outlet Cross** & **Middle** & **Middle** & **Middle** & **Inlet Cross** & **Average Inlet and Outlet Pressure** \\
**Rate (m/s)** & **Section (Pa)** & **Section (Pa)** & **Section (Pa)** & **Section (Pa)** & **Section (Pa)** & **Drop (Pa)** \\ \hline
0.1 & 0 & 55.4372 & 161.968 & 1326.6 & 3068.06 & 3068.06 \\ \hline
0.2 & 0 & 12.163 & 141.859 & 1730.83 & 3649.56 & 3649.56 \\ \hline \end{tabular}
\end{table} 
Table 10: Average gauge pressure at five sections along the flow direction at different flow velocities. Figure 18: Temperature distribution along the axis at different inlet flow rates. at the outlet. As of the current analysis, the transient calculations for the improved cooling scheme have reached a maximum time of 1250 s. At this time point (\(t=1250\) s), the overall temperature distribution on the surface of the rudder shaft is similar to the steady-state calculations. The highest temperatures are located at the top, reaching approximately 478 and 488 \({}^{\circ}\)C under inlet velocities of 0.5 and 0.3 m/s, respectively, as shown in Figure 19. Additionally, higher temperatures are observed at the junctions between the rudder shaft and the upper and lower flanges, where heat flux density is significant. In other areas, particularly in the lower half of the rudder shaft, temperatures generally remain below 100 \({}^{\circ}\)C. Overall, the surface temperature of the rudder shaft is slightly lower under the 0.5 m/s condition, though the difference is minimal, with more noticeable variations only in the microchannel region. Figure 20b shows the variation in the maximum surface temperature of the rudder shaft over time after the inlet flow rate increases, specifically for \(t>600\) s. It can be observed that, under both inlet velocity conditions, the maximum temperature of the rudder shaft rises rapidly between approximately 600 and 1050 s, with an increase exceeding 120 \({}^{\circ}\)C in both cases. Under the 0.5 m/s inlet condition, the maximum temperature reaches approximately 505 \({}^{\circ}\)C around \(t=1050\) s, while under the 0.3 m/s condition, it reaches about 515 \({}^{\circ}\)C, which is 10 \({}^{\circ}\)C higher. Following this, the maximum temperature of the rudder shaft begins to decrease rapidly, dropping to 478 \({}^{\circ}\)C at \(t=1250\) s under the 0.5 m/s condition, and to 488 \({}^{\circ}\)C, at \(t=1250\) s under the 0.3 m/s condition, with minimal difference between the two. Overall, during the heating phase before \(t=1050\) s, the rate of temperature increase under the 0.5 m/s inlet condition is lower than that under the 0.3 m/s condition. However, during the cooling phase after \(t=1050\) s, the temperature change rates under both conditions become similar. When considering the heat flux density changes depicted in Figure 6a, the heat flux density at the top of the rudder shaft peaks at \(t=1000\) s and then decreases rapidly, while other regions maintain their peak values. In contrast, in the transient calculation results of this round, the maximum temperature of the rudder shaft under both inlet velocity conditions peaks at approximately \(t=1250\) s, showing a lag of about 50 s relative to the heat flux boundary condition, which is consistent with theoretical expectations. It is noteworthy that the maximum temperature achieved with the improved scheme is higher than that of the original scheme. This is because the improved scheme employed an inlet velocity of 0.2 m/s during the initial 550 s, which is only 67% and 40% of the original scheme's velocity, respectively. This led to a faster temperature rise in the early stages, accumulating more thermal energy.