# Energy Consumption Characteristics for Design Parameters of Permanent Magnet-Based Al Billet Heater
## 4 Heating Characteristic Effects on Design Parameters
### Effects of Magnet Arrangement Clearance  
As a result of analyzing the characteristics of each design parameter, the maximum for the number of magnets could not be analyzed and it continued to decrease for the centrifugal magnetic field model. Hence, in this section, a model with the circumferential direction of the magnetic field was set, and the optimal conditions were derived by optimizing the distance gap between magnets under the same volume of magnets. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Magnetic Flux Density [T]**} & \multicolumn{2}{c}{**Current Density [\(\times\)10\({}^{6}\), A/m\({}^{2}\)]**} & \multicolumn{2}{c}{**Power Loss Density [W/m\({}^{3}\)]**} \\ \hline \multirow{2}{*}{Case01} & Min & 796 \(\times\) 10\({}^{-9}\) & \(-\)266 & 455 \(\times\) 10\({}^{-9}\) \\ \cline{2-5}  & Max & 1.398 & 266 & 188 \(\times\) 10\({}^{7}\) \\ \hline \multirow{2}{*}{Case02} & Min & 194 \(\times\) 10\({}^{-8}\) & \(-\)211 & 997 \(\times\) 10\({}^{-11}\) \\ \cline{2-5}  & Max & 1.350 & 211 & 119 \(\times\) 10\({}^{7}\) \\ \hline \multirow{2}{*}{Case03} & Min & 605 \(\times\) 10\({}^{-9}\) & \(-\)176 & 825 \(\times\) 10\({}^{-9}\) \\ \cline{2-5}  & Max & 1.258 & 176 & 826 \(\times\) 10\({}^{6}\) \\ \hline \multirow{2}{*}{Case04} & Min & 200 \(\times\) 10\({}^{-8}\) & \(-\)152 & 131 \(\times\) 10\({}^{-9}\) \\ \cline{2-5}  & Max & 1.248 & 152 & 614 \(\times\) 10\({}^{6}\) \\ \hline \multirow{2}{*}{Case05} & Min & 671 \(\times\) 10\({}^{-9}\) & \(-\)134 & 200 \(\times\) 10\({}^{-8}\) \\ \cline{2-5}  & Max & 1.243 & 134 & 479 \(\times\) 10\({}^{6}\) \\ \hline \multirow{2}{*}{Case06} & Min & 574 \(\times\) 10\({}^{-9}\) & 120 & 200 \(\times\) 10\({}^{-9}\) \\ \cline{2-5}  & Max & 1.238 & \(-\)120 & 385 \(\times\) 10\({}^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Analysis results with different clearances. Figure 14: Changes in torque ripple according to eccentricity. Figure 13: Changes in current density according to eccentricity. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Eccentricity**} & \multirow{2}{*}{**0**} & \multirow{2}{*}{**0.5**} & \multirow{2}{*}{**1.0**} & \multirow{2}{*}{**1.5**} & \multirow{2}{*}{**2.0**} \\
**[mm]** & & & & & & \\ \hline Power Loss & NM = 4 & 3.23 & 3.24 & 3.25 & 3.27 & 3.29 \\ \cline{2-6}
[kW] & NM = 12 & 3.49 & 3.50 & 3.57 & 3.70 & 3.90 \\ \hline Torque Ripple & NM = 4 & 0.01 & 0.02 & 0.06 & 0.15 & 0.37 \\ \cline{2-6}
[mN-m] & NM = 12 & 0.08 & 0.13 & 0.21 & 0.70 & 0.91 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Mean heating value and torque ripple in the billet. (NM = Number of magnets). was set to 0.5 mm considering the manufacturing and assembly tolerances, and the inner radius (\(R_{0}\)) of the permanent magnet was set to 30.5 mm. Therefore, the combinations for the analysis of the design parameters, including the angle (\(\theta\)) and outer radius (\(R_{1}\)) of the permanent magnets and excluding the fixed design parameters, are listed in Table 7.  
The results of the electromagnetic-thermal analysis according to the changes in the angle and outer radii of the billet heater using permanent magnets are shown in Figure 16 and Table 8. As the angle of the permanent magnet increased, the surfaces adjacent to the billet increased and the outer radius of the permanent magnet decreased at the same time. In other words, the clearance between the permanent magnets decreased. When the results of Case01 to Case04 were analyzed, the larger the surface adjacent to the billet was, the higher the magnetic flux density, current density, and power loss, and the higher the heating efficiency inside the billet. Consequently, an increase in the torque and a reduction in the heating time can be predicted. With the change in the angle of the permanent magnet, the clearance between the permanent magnets gradually became narrower.