# Energy Consumption Characteristics for Design Parameters of Permanent Magnet-Based Al Billet Heater
## 5 Conclusions  
The electromagnetic-thermal characteristics of a billet heater using permanent magnets were numerically analyzed in this study. To this end, the correlation between the experiment and analysis results was examined, and the reliability of the analysis method was obtained by verifying the analysis method. The electromagnetic-thermal analysis was performed using the analysis method that was proven to be reliable. The electromagnetic-thermal analysis was performed by selecting design parameters such as the polarity direction, the number of permanent magnets, and eccentric distance. Then, the heating characteristics were examined through the prediction of temperature distribution according to changes in the design parameters and analysis of the results. The results of the analysis model of the billet heater using permanent magnets until reaching 450 \({}^{\circ}\)C were compared with the experiment result to achieve reliability within an error rate of 3%. The analysis results using this model confirmed that the flux penetration and concentration area inside the billet changed by the polarity direction of the permanent magnet, and the heating efficiency varied significantly according to the change in the polarity direction. In particular, even though the flux model of the normal direction had the same surface adjacent to the billet, the heating value inside the billet decreased as the number of magnets increased. For the flux model of the tangential direction, the heating performance gradually increased with the number of magnets; however, the inflection point was formed in a system of 12 permanent magnets. The cause of this result was found to be the fact that the magnetic flux formed inside the billet due to the arrangement of magnets had a significant effect on the heating efficiency. In the four-magnet model, the normal model for the flux direction improved the heating time by approximately 73% and the power loss by approximately 5% compared to the tangential mode. Furthermore, the examination of the clearance between the billet and permanent magnet confirmed that a smaller clearance was associated with higher efficiency, and a nonlinear characteristic was verified. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **Magnetic Flux** & **Current** & **Power loss Density** \\  & & **Density** & **Density** & **[W/m\({}^{3}\)]** \\ \cline{3-5}  & & **[T]** & **[\(\times\)10\({}^{6}\), A/m\({}^{2}\)]** & **[W/m\({}^{3}\)]** \\ \hline \hline \multirow{2}{*}{Case01} & Min & \(136\times 10^{-8}\) & \(-265\) & \(828\times 10^{-9}\) \\ \cline{2-5}  & Max & 1.456 & 265 & \(186\times 10^{7}\) \\ \hline \multirow{2}{*}{Case02} & Min & \(734\times 10^{-9}\) & \(-266\) & \(394\times 10^{-9}\) \\ \cline{2-5}  & Max & 1.439 & 266 & \(188\times 10^{7}\) \\ \hline \multirow{2}{*}{Case03} & Min & \(178\times 10^{-8}\) & \(-273\) & \(564\times 10^{-10}\) \\ \cline{2-5}  & Max & 1.862 & 273 & \(193\times 10^{7}\) \\ \hline \multirow{2}{*}{Case04} & Min & \(184\times 10^{-8}\) & \(-288\) & \(313\times 10^{-8}\) \\ \cline{2-5}  & Max & 2.378 & 288 & \(220\times 10^{7}\) \\ \hline \multirow{2}{*}{Case05} & Min & \(641\times 10^{-9}\) & \(-354\) & \(633\times 10^{-8}\) \\ \cline{2-5}  & Max & 2.961 & 354 & \(333\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Analysis results of normal direction model. Regarding the effect of eccentricity, the trend of flux linkage was changed by the change in the air gap flux density. Furthermore, the heating value increased as the trend of the loss of current density and eddy current inside the billet changed due to the increase or decrease in the clearance between the billet and permanent magnet, and the heating efficiency increased on average. That is, it was confirmed that the power loss increased by about 1.79% in the four permanent magnets to the tangential model, and increased by about 10.51% in the 12 permanent magnets to the tangential model when an eccentricity of 2 mm was applied at a clearance of 2.5 mm.