## 5 Results and Discussion
### Numerical Results  
To verify the reliability of the numerical analysis, experimental values and simulation values were compared for NPR (the permanent magnets arrangement of the highest temperature distribution) and experiment 3 (the arrangement of the lowest temperature distribution) in Table 7. Only an error of maximum 1.08% was shown, proving the validity of the numerical analysis results.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & **D1 (cm)** & **D2 (cm)** & **TC 1 (\({}^{\circ}\)C)** & **TC 2 (\({}^{\circ}\)C)** & **TC 3 (\({}^{\circ}\)C)** & **TC 4 (\({}^{\circ}\)C)** \\ \hline \multirow{3}{*}{NN} & Predict & \multirow{3}{*}{18} & \multirow{3}{*}{21.06} & 35.16 & 35.19 & 35.19 & 35.08 \\  & Experiment & & & 34.63 & 34.68 & 34.69 & 34.57 \\ \cline{2-6}  & \multicolumn{2}{c}{_TC’s Relative error (Predict vs. Exp.)_} & \multicolumn{2}{c}{_1.5\%_} & \multicolumn{2}{c}{_1.45\%_} & \multicolumn{2}{c}{_1.42\%_} \\ \hline \multirow{3}{*}{NPR} & Predict & \multirow{3}{*}{17.78} & \multirow{3}{*}{18.80} & 35.51 & 35.35 & 35.25 & 35.09 \\  & Experiment & & & 35.09 & 35.19 & 35.09 & 34.99 \\ \cline{1-1} \cline{2-6}  & \multicolumn{2}{c}{_TC’s Relative error (Predict vs. Exp.)_} & \multicolumn{2}{c}{_1.18\%_} & \multicolumn{2}{c}{_0.45\%_} & \multicolumn{2}{c}{_0.45\%_} & \multicolumn{2}{c}{_0.28\%_} \\ \hline \end{tabular}
\end{table}
Table 6: Optimization results for the two regression models (NN and NPR).  
The velocity contour was used to visualize the behavior of the ferrofluid in the Pyrex glass tube. Numerical analysis was performed for all experiments. Figure 8 shows two experiments (experiment 3 and NPR) among them. In experiment 3 (see Figure 8a), most of the fluid behavior is located at the bottom of the glass tube. There were cells only near permanent magnets, and heat diffusion of the fluid was concentrated near there. The high-temperature fluid in the lower part could not reach the upper part of the tube. Therefore, the overall temperature distribution was low. In the case of NPR (see Figure 8b), the fluid behavior was different from experiment 3. The fluid concentration around the permanent magnets was similar to experiment 3, but there was a difference in that ferrofluid circulated over a wide range in the tube. An eddy was actively generated by the external magnetic field. It means that the high-temperature fluid at the bottom affected the upper part of the tube. That is, the energy exchange occurred most actively, and the temperature distribution was the highest. Through numerical analysis, the variation of the fluid behavior was confirmed according to the arrangement of permanent magnets.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & **TC 1 (\({}^{\circ}\)C)** & **TC 2 (\({}^{\circ}\)C)** & **TC 3 (\({}^{\circ}\)C)** & **TC 4 (\({}^{\circ}\)C)** \\ \hline \multirow{4}{*}{Experiment 3} & Numerical analysis & 33.88 & 33.86 & 33.84 & 33.83 \\  & Experiment & 34.03 & 34.05 & 33.99 & 34.01 \\ \cline{2-6}  & _TC’s Relative error_ & _0.44\%_ & _0.56\%_ & _0.44\%_ & _0.53\%_ \\ \hline \multirow{4}{*}{NPR} & Numerical analysis & 34.84 & 34.81 & 34.80 & 34.78 \\  & Experiment & 35.09 & 35.19 & 35.09 & 34.99 \\ \cline{2-6}  & _TC’s Relative error_ & _0.71\%_ & _1.08\%_ & _0.83\%_ & _0.60\%_ \\ \cline{1-1}  & _(Numerical. vs. Exp.)_ & _0.71\%_ & _1.08\%_ & _0.83\%_ & _0.60\%_ \\ \hline \hline \end{tabular}
\end{table}
Table 7: Validations of numerical analysis for two experiments (Experiment 3 and NPR).  
Figure 8: Velocity contour for two experiments (**a**) Experiment 3, (**b**) NPR.