# Energy Consumption Characteristics for Design Parameters of Permanent Magnet-Based Al Billet Heater
## 3 Numerical Analysis and Verification
### Properties and Boundary Conditions of Billet Heating  
Regarding the physical properties of the analysis model for the billet heating characteristics of the induction heating system, the same physical properties and boundary conditions were applied to the real designed system. The physical properties of each component of the heating system, that is, the billet (aluminum), permanent magnet (NdFeB), magnetic housing, and magnetic adapter, are listed in Table 1. For the permanent magnet (NdFeB), graphs of various shapes of physical properties according to the grade are available, and the analysis was performed using the BH curve of NdFeB N35 shown in Figure 3. The effects of temperature on the exterior parts were considered by applying the electric properties (electrical resistivity, relative permeability) and thermal properties (thermal conductivity, specific heat, and volumetric heat capacity), which vary with temperature. Figure 2: Geometric diagram of permanent magnet-based billet induction heating. The model configuration and boundary conditions of the rotary induction heating system are summarized in Table 2. First, the geometric condition was applied for the permanent magnet configuration for system heating. As for the configuration of permanent magnets of a rotary induction heating system that has been actually placed, four magnets were arranged in the tangential direction in a structure such that the magnets face one another. The clearance between the permanent magnet and billet is 2.5 mm, the diameter of the billet is 60 mm, and the length of the billet is 75 mm. The magnet has a size of 65 mm, an internal diameter of 65 mm, an angle of 45\({}^{\circ}\), and a height of 15.43 mm. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Description** & **20 °C** & **100 °C** & **300 °C** & **500 °C** \\ \hline \multirow{4}{*}{Al} & Electrical Resistivity [\(\Omega\cdot\)m] & \(2.65\times 10^{-8}\) & \(3.56\times 10^{-8}\) & \(6.61\times 10^{-8}\) & \(1.23\times 10^{-7}\) \\ \cline{2-6}  & Volumetric Heat Capacity []/m\({}^{3}\cdot\)K] & \(2.40\times 10^{6}\) & \(2.50\times 10^{6}\) & \(2.65\times 10^{6}\) & \(2.90\times 10^{6}\) \\ \cline{2-6}  & Density [kg/m\({}^{3}\)] & 2700 & 2681 & 2640 & 2590 \\ \cline{2-6}  & Thermal Conductivity [W/m\(\cdot\)K] & 236 & 228 & 220 & 210 \\ \hline \multirow{4}{*}{SUS} & Electrical Resistivity [\(\Omega\cdot\)m] & \(7.20\times 10^{-7}\) & \(7.69\times 10^{-7}\) & \(9.00\times 10^{-7}\) & \(1.05\times 10^{-6}\) \\ \cline{2-6}  & Volumetric Heat Capacity []/m\({}^{3}\cdot\)K] & \(3.68\times 10^{6}\) & \(3.90\times 10^{6}\) & \(4.09\times 10^{6}\) & \(4.29\times 10^{6}\) \\ \cline{2-6}
304 & Density [kg/m\({}^{3}\)] & 7960 & 7880 & 7790 & 7710 \\ \cline{2-6}  & Thermal Conductivity [W/m\(\cdot\)K] & 13.4 & 15.35 & 17.3 & 21.3 \\ \hline \multirow{4}{*}{NdFeB} & Flux Density [T] & 1.2 & 1.1 & - & - \\ \cline{2-6}  & Electrical Resistivity [\(\Omega\cdot\)m] & \(1.5\times 10^{-6}\) & \(1.52\times 10^{-6}\) & - & - \\ \cline{2-6}  & Volumetric Heat Capacity []/m\({}^{3}\cdot\)K] & \(3.42\times 10^{6}\) & - & - & - \\ \cline{2-6}  & Density [kg/m\({}^{3}\)] & 7600 & - & - & - \\ \cline{2-6}  & Thermal Conductivity [W/m\(\cdot\)K] & 6.61 & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical properties of billet heating design parts [40,41]. Figure 3: Demagnetization curves as a function of temperature for NdFeB N35. For inducting heating, the billet is rotated by the DC motor with a speed of up to 4000 RPM, considering the operation conditions of the motor. The time taken by the rotor to reach the rotational speed of 4000 RPM was set to 30 s. Furthermore, the sink temperature condition was applied using the infinite box function to the air surrounding the heating system. The infinite box assumption was applied to reflect the infinite realm of air, and it was set to indicate the heat transfer because of the temperature difference between the heating system and air.