## 4 Results
### Experimental Validation
After comparing the temperature between the CFD simulations and experiments, the velocity distribution was examined in the CFD simulations. The flow region after the injection pipe was a concern because the CCF flow estimation inferred the flow based on the temperature sensors installed in this region. Figure 21 shows the velocity distribution after the water injection under a low flow rate (39.52 GPM). Figure 21 shows the velocity distribution of the cross-section at different locations along the flow direction under the flow rate after 0.5 s of injection. The pipe centerline is marked with a dotted dashed line (--), and the two sensor locations are marked with blue points and black arrows indicating the flow direction. Each examined cross-section is marked with a dashed line (-), and for each velocity distribution, the average main flow rate is marked with a dotted line (\(\cdots\)). The selected cross-sections roughly cover the most important locations in the flow pipe, including the region before the injection pipe, the two exact sensor locations, and some other randomly chosen locations. Before the injection pipe, the velocity profile has a U-shaped curve that is standard in pipe flow. It has a higher local velocity near the centerline and the highest gradient toward zero near the wall. The injected water disturbed the U-shaped velocity profile, as shown in Figure 21.  
After passing the injection pipe, the flow was affected by the physical structure of the pipe, and the flow distribution shows that the upper flow rate sharply decreased and then later recovered. Then, the flow rate passed by the two sensor locations sequentially. The transit time was determined by the flow rate along the path from the first sensor to the second sensor (marked in light blue in Figure 19), but it was hard to examine how the  
Figure 21: Velocity distribution under a low flow rate after water injection. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Flow Rate** & **TC1** & **TC2** \\ \hline Low (\(\sim\)40 GPM, Figure 18) & 0.173 & 0.248 \\ Medium (\(\sim\)133 GPM, Figure 19) & 0.0853 & 0.202 \\ High (\(\sim\)170 GPM, Figure 20) & 0.128 & 0.268 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Root mean squared error between experimental data and CFD simulations. Figure 20: Temperature comparison under a high flow rate (\(\sim\)170 GPM). flow rate changed along this path. Therefore, a more detailed flow-rate distribution along this specific light-blue path is shown in Figure 22. The blue scattered dots represent the flow rate discretely collected along the two sensor paths from the CFD simulations for the lowest flow rate. The red dashed line represents the reference average flow rate mentioned previously. The x-axis only covers the horizontal range between the two sensors. It shows that when the flow reached the first sensor, the flow rate was lower since it had not fully recovered from the disturbance caused by being blocked by the injection pipe. As the flow traveled forward, it recovered and returned to the normal flow rate, which is roughly equal to the reference flow rate in this case. It was quite a different story when the main flow rate was high. Figures 23 and 24 show the same flow-rate distribution plots when the main flow rate was high (178.9 GPM). By comparing the different flow-rate cases, it was found that both suffered from being blocked by the injection pipe. However, the low-flow-rate case exhibited around a 14% decrement, while the high-flow-rate exhibited around a 10% decrement. One possible reason might be that the high flow rate recovers more quickly than the low flow rate. When comparing Figure 22 with Figure 24, another issue was observed where the high-flow-rate case did not return to the average flow rate even after passing the second sensor.