## 2 Experimental Process
### Design of Experiments  
To analyze the heat transfer performance according to the permanent magnet arrangement, a design of experiments (DOE) was conducted [14]. As design parameters, the height of permanent magnets (D1, D2 of Figure 2c) was set. The center position of the permanent magnet was set to (A), and the upper surface of the heating pad was set to (B) (see Figure 2c). Based on the glass tube, the distance corresponding to the side is D1, and to the right side is D2. The DOE was processed using Ansys Design Xplorer software. Latin hypercube sampling (LHS) was selected as the DOE type, and the parameter sets for the design variables (height of the permanent magnet) were set according to LHS. The advantage of LHS is the permanent magnet arrangement can be randomly determined without overlapping the design points between the parameter sets. Nine experiments were conducted with different permanent magnet arrangements (see Table 3). The resulting temperature for each of the TC wires in each experiment is shown in Table 4. In experiment 4 and 8, it was numerically confirmed that temperature was high. Based on this result, an optimization process was performed.  
In addition, we considered two experimental uncertainties; the random error was the permanent magnet vertical position adjustment (D1, D2), and the systematic error was the measurement error caused by the thermocouple (TC). It was assumed that a random error occurs about 0.25 mm in the D1 and D2 adjustment. The uncertainty of the TC is known to be about 0.013 \({}^{\circ}\)C at 0.006 mV/\({}^{\circ}\)C at 25 \({}^{\circ}\)C and was ignored.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Experiment** & **TC 1 (\({}^{\circ}\)C)** & **TC 2 (\({}^{\circ}\)C)** & **TC 3 (\({}^{\circ}\)C)** & **TC 4 (\({}^{\circ}\)C)** \\ \hline Experiment 1 & 34.5627 & 34.5769 & 34.5188 & 34.4329 \\ Experiment 2 & 34.7507 & 34.7574 & 34.7345 & 34.6549 \\ Experiment 3 & 34.0390 & 34.0583 & 33.9920 & 34.0150 \\ Experiment 4 & 35.0921 & 34.9866 & 34.8003 & 34.9575 \\ Experiment 5 & 35.0339 & 34.8213 & 34.8070 & 34.7308 \\ Experiment 6 & 34.7378 & 34.7169 & 34.7249 & 34.3848 \\ Experiment 7 & 34.9472 & 34.7990 & 34.7724 & 34.5860 \\ Experiment 8 & 35.2011 & 34.1153 & 35.1518 & 34.9218 \\ Experiment 9 & 34.8204 & 34.9160 & 34.8537 & 34.8118 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Temperature result for each TC based on all DOE cases.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Properties** & **Value** \\ \hline Temperature of Air & 292.65 K \\ Initial temperature of the Ferrofluid (T\({}_{i}\)) & 292.65 K \\ High temperature (T\({}_{\mathrm{h}}\)) surface of the Pyrex glass tube & 333.15 K \\ Magnetic flux density of the permanent magnet (M) & 0.325 T \\ Velocity of reciprocating linear motor & 17.5 cm/s \\ \hline \hline \end{tabular}
\end{table}
Table 2: Operating and boundary conditions applied in this study.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Experiment** & **Distance of D1** & **Distance of D2** \\ \hline Experiment 1 & 13.3 \(\pm\) 0.025 cm & 8.0 \(\pm\) 0.025 cm \\ Experiment 2 & 21.3 \(\pm\) 0.025 cm & 13.3 \(\pm\) 0.025 cm \\ Experiment 3 & 5.3 \(\