## 2 Numerical Model and Calculation Scheme
### Model Validation  
Grid independence verification is an important step in numerical simulations to ensure that the results are not significantly affected by grid resolution. In this study, a grid independence analysis was conducted before experimental verification. A 10-layer grid was employed in the solid domain, whereas in the fluid domain, the grid near the wall was refined to ensure that the thickness of the first layer was less than 1 and the y+ value of the first three layers was less than 5. The time-step for the calculation was set to 0.001 s.  
A grid independence analysis was performed by varying the number of grids and observing their effect on the outlet fuel temperature and cross-sectional average velocity. Figure 5 illustrates the distribution of these parameters for six different grid numbers. The results showed that when the number of grid nodes reached 70,000, there was no significant change in fuel temperature and velocity at the outlet, regardless of a further increase or decrease in the number of grids. This indicated that the results reached a grid-independent state. Based on these findings, it can be concluded that the numerical simulations are sufficiently accurate and reliable, and that the chosen grid size of 70,000 nodes is appropriate for subsequent analysis and experimental verification. To validate the accuracy of the flow and heat transfer calculation model, a common approach is to compare the calculated outer wall temperature with corresponding experimental measurements. The relative deviation between the calculated value (\(T_{w,ext}\)) and the experimental value (\(T_{w,exp}\)) can be calculated using Equation (24). \[\delta_{w}=\frac{T_{w,ext}-T_{w,exp}}{T_{w,exp}}\times 100\% \tag{24}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Working Pressure (MPa)** & **Outlet Fuel Temperature (K)** & **Required Heat Flux (W/m2)** \\ \hline \multirow{3}{*}{5} & 373.15 & 20,855 \\  & 473.15 & 55,735 \\  & 573.15 & 96,793 \\  & 673.15 & 145,355 \\ \hline \multirow{3}{*}{3} & 373.15 & 20,895 \\  & 473.15 & 55,930 \\ \cline{1-1}  & 573.15 & 97,695 \\ \cline{1-1}  & 673.15 & 152,760 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Required heat flux for different outlet fuel temperatures. Figure 5: Grid independence verification. Measurement errors play a significant role in data processing and analysis. These errors can arise from both direct and indirect measurements, with different factors contributing to their overall magnitudes. The measurement error is expressed in Equation (25). \[\sigma=\sqrt{\sum_{i=1}^{n}(a_{i}\sigma_{i})^{2}} \tag{25}\]  
For direct measurements, the errors are primarily influenced by the accuracy of the sensor and the data acquisition system. However, indirect measurements, such as heat flux calculations, involve additional sources of error, including sensor accuracy, data acquisition system accuracy, and the calculation method itself. To quantify the measurement errors, Table 4 summarizes the error expressions associated with the different types of measurements, which are adopted from Equation (25). The table presents the specific error expressions for each measurement type, considering the relevant factors that contribute to the overall measurement uncertainty. These error expressions enable a comprehensive understanding of measurement errors and their potential impact on the experimental results. Based on the analysis and comparison of the calculated and experimental values, the flow and heat transfer calculation models used in this study were validated. The calculated results accurately predicted the changing trend of the outer wall temperature, and there was a good fit between the calculated and experimental values, with an overall relative error of less than 5% (Figure 8). It is worth noting that a relatively large error is observed at the inlet and outlet of the cooling channel, which is located at X-axis < 50 and X-axis > 750 mm.