## 4 Results and Discussion of the Initial Cooling Channel
### Steady-State Numerical Simulation Results with Phase-Change Cooling
With further increases in heat flux density or decreases in water flow velocity, there is a high risk of heat transfer deterioration due to boiling dry out. Figure 8: Temperature distribution of the solid at inlet flow rates of (**a**) 0.05, (**b**) 0.1, (**c**) 0.2, and (**d**) 0.5 m/s. (**e**) Temperature distribution along the axis at different inlet flow rates. Figure 9: (**a1–d1**) Fluid phase fraction distribution and (**a2–d2**) temperature distribution when the inlet flow rate is (**a**) 0.5, (**b**) 0.2, (**c**) 0.1, and (**d**) 0.05 m/s, respectively. The average convective heat transfer coefficients at different flow rates were calculated and summarized in Table 6. At an inlet velocity of 0.5 m/s, the fluid is mainly single-phase liquid. The heat transfer relies primarily on single-phase liquid forced convection, with the convection heat transfer coefficient mainly depending on the fluid velocity. At this velocity, the water flow rate is maximized, resulting in good cooling effects with minimal internal boiling heat transfer phenomena. At inlet velocities of 0.2 m/s and 0.1 m/s, the flow is in the saturated nucleate boiling region, relying mainly on boiling heat transfer. The cooling effect on the rudder shaft is relatively good, but the overall convection heat transfer effect decreases due to reduced fluid velocity, making the cooling effect slightly worse than at 0.5 m/s. Under the 0.1 m/s condition, dual-phase forced convection heat transfer regions appear near the outlet, with risks of further deterioration. At an inlet velocity of 0.05 m/s, the gas content increases further, and the fluid undergoes two-phase forced convective heat transfer with slower fluid velocities, resulting in insufficient cooling effects. Due to direct contact with steam in some wall regions, there is a high likelihood of heat transfer deterioration and a sharp increase in wall temperature with increased heat flux density or disturbed water flow velocity. This inlet velocity condition is considered quite dangerous and unsuitable for cooling solutions. Observing the average surface pressure at five cross sections along the fluid flow direction at different velocities as shown in Figure 10, it is noted that the surface pressure decreases along the flow direction, reducing to atmospheric pressure at the outlet (see Table 7). Higher inlet velocities correspond to larger average pressure drops between the inlet and outlet. Considering the above factors, for the initial cooling structure, an inlet water velocity of 0.2 m/s is deemed more reasonable. On the one hand, it provides good cooling effects for the rudder shaft and, while meeting project cooling requirements, can save 60% of the total cooling water compared to an inlet velocity of 0.5 m/s. The pressure drop between the inlet and outlet is smaller, approximately two-thirds of that at 0.5 m/s. On the other hand, the 0.2 m/s condition has a higher average convection heat transfer coefficient (referencing the inlet temperature) compared to 0.1 m/s, with only a slight increase in pressure drop. Additionally, based on boiling mode analysis, the 0.2 m/s condition is less prone to heat transfer deterioration at the channel top, offering better safety. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Flow Rate (m/s)** & 0.05 & 0.1 & 0.2 & 0.5 \\ \hline
**Average convection heat transfer coefficient (W/m\({}^{2}\)-K)** & 7545 & 8510 & 12,973 & 6142 \\ \hline \end{tabular}
\end{table} 
Table 6: Average convection heat transfer coefficient at different flow rates.