## 4 Results and Discussion
### Effect of Mass Flow Rate on Heat and Mass Transfers in the Curved Cooling Channel  
In the curved channel, the centrifugal force and secondary flow were directly proportional to the inlet mass flow rate of fuel. The change of the inlet mass flow rate may induce different heat and mass transfer characteristics. Therefore, in this section, the variations of fuel conversion and heat transfer coefficient distributions with the inlet mass flow rate, \(m\), are analyzed to study the secondary flow effect on the heat and mass transfers of hydrocarbon fuels. The three different mass flow rates were set to be 1.341 g/s, 1.788 g/s and 2.235 g/s, respectively. The relevant parameters are listed in Table 1.  
Furthermore, results in Section 4.1 indicate that the distributions of fuel temperature and conversion in the cross section of the cooling channel present non-uniformity, which significantly affects fuel heat-absorbing capacity. Therefore, the non-uniformity coefficient is introduced to study the influence of inlet mass flow on the heat and mass transfer of hydrocarbon fuels in a curved cooling channel. Therefore, the non-uniformity coefficient is introduced to study the influence of inlet mass flow rate. The non-uniformity coefficients of fuel temperature, \(R_{T}\), and coefficients of fuel conversion, \(R_{Z}\), are defined as follows [41]:  
\[R_{T}=\frac{T_{\text{max}}-T_{\text{min}}}{T_{mean}}\times 100\%, \tag{15}\]  
\[R_{Z}=\frac{Z_{\text{max}}-Z_{\text{min}}}{Z_{mean}}\times 100\%, \tag{16}\]  
where \(T_{\text{max}}\), \(T_{\text{min}}\) and \(T_{mean}\) are the maximum, minimum, and mass-weighted average temperatures in the cross section of the cooling channels. \(Z_{\text{max}}\), \(Z_{\text{min}}\) and \(Z_{mean}\) are the maximum, minimum, and mass-weighted average conversions in the cross sections of the cooling channels.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & \(m\) **(g/s)** & \(T_{in}\) **(K)** & \(P_{in}\) **(MPa)** & \(\mathbf{Re_{in}}\) & \(q^{\prime\prime}\) **(W/m2)** \\ \hline
1 & 1.341 & 700 & 3 & \(6.0\times 10^{4}\) & \(5.0\times 10^{6}\) \\
2 & 1.788 & 700 & 3 & \(8.0\times 10^{4}\) & \(6.67\times 10^{6}\) \\
3 & 2.235 & 700 & 3 & \(1.0\times 10^{5}\) & \(8.33\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Boundary conditions in different mass flow rate cases.  
Figure 10: Variation of heat transfer coefficient along the flow direction.  
With the increase in inlet mass flow rate, the centrifugal force became larger. The average velocity of secondary flow in the cross-sectional planes increased with the inlet mass flow rate, as shown in Figure 11. The heat and components in the near-wall region were easily transported to the main flow. As shown in Figures 12 and 13, the non-uniformity coefficients of fuel temperature and conversion decreased when the inlet mass flow rate increased. Therefore, with the increasing inlet mass flow rate, the fuel heat-absorbing capacity increased, and the heat transfer coefficient increased, as shown in Figure 14. When the inlet mass flow rate changed from 1.341 g/s to 2.235 g/s, the heat transfer coefficient at \(x/D_{h}=80\) increased by 30.5%. The increasing inlet mass flow rate enhanced the heat and mass transfers and improved the utilization level of the fuel heat-absorbing capacity in the curved cooling channel.  
Figure 11: Secondary flow velocity with different mass flow rates along the flow direction.  
Figure 12: Distribution of \(R_{T}\) with different mass flow rates along the flow direction.