# Energy Consumption Characteristics for Design Parameters of Permanent Magnet-Based Al Billet Heater
## 3 Numerical Analysis and Verification
### Comparison between Numerical Analysis and Experiment  
#### 3.3.1 Analysis of the Base Model  
Electromagnetic-thermal analysis was performed for the model of the induction heating system using permanent magnets arranged in the tangential direction, as proposed in the previous section. The finite element model and boundary conditions are shown in Table 2. Furthermore, the magnetic flux density, current density, power loss, and temperature distribution were examined to verify the heating characteristics of the billet, and the results are shown in Figure 4. As shown in Figure 4, the magnetic flux distribution inside the billet is concentrated in the outermost part owing to the high-speed rotation of the billet. This phenomenon can be attributed to the induced electromotive force caused by an electric field and magnetic field. It appears owing to the eddy current, which forms a closed loop in the local area inside the conductor to cancel the internal magnetic field. Consequently, it can be observed that the maximum current and power loss occur at the same position where the magnetic flux density is concentrated due to the current density.  
The temperature distribution of the induction heating system indicates that the surface of the billet where the magnetic flux density is concentrated has the highest temperature, and the part where the magnetic flux density is concentrated in the main heating source. Moreover, it was found that the heating performance may vary with the amount of power loss.  
Regarding the heating performance of the rotary induction heating system, the energy consumption can be derived using the torque and heating time as follows:  
\[P=\frac{NTt}{m\eta} \tag{6}\]  
where, \(m\), \(N\), \(T\), and \(t\) denote the billet weight, rotational speed, torque, and heating time, respectively. Equation (6) shows that the energy consumption is proportional to the rotational speed, torque, and heating time. Therefore, it can be seen that the relationship between the rotation speed and heating time is inversely proportional to the same energy consumption and torque. On the other hand, if the power loss and rotation speed are constant, it can be seen that the torque and heating time are inversely proportional. The energy consumption of the rotary induction heating system was calculated using a power factor of (\(\eta\)) = 0.8, which was derived by the ratio of the motor's active power and apparent  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Condition** & **Value** & **Unit** \\ \hline \multirow{4}{*}{Geometric Condition} & Pole & 4 & EA \\ \cline{2-4}  & Magnetic Direction & Tangential & - \\ \cline{2-4}  & Clearance & 2.5 & mm \\ \hline \multirow{4}{*}{Boundary Condition} & Rotational speed & 4000 & RPM \\ \cline{2-4}  & Sink Temperature & 293 & K \\ \cline{1-1} \cline{2-4}  & Uniform Heat Source & variable & W/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Boundary conditions for numerical analysis model.  
power. It was found that the rotary induction heating system had an energy consumption of 302.57 kWh/ton.