# Modeling and Experimental Verification of Induction Heating of Thin Molybdenum Sheets
## 3 Design of Electro-Thermal Simulation Model of Induction Heating
Equations used within the calculations are listed below:  
\[\rho C_{P}\mathbf{u}*\
abla T+\
abla*\mathbf{q}=Q_{e}+Q_{P}+Q_{vd} \tag{1}\]  
\[\rho C_{P}\mathbf{u}*\
abla T+\
abla*\mathbf{q}=Q_{e}+Q_{td} \tag{2}\]  
\[\mathbf{q}=-k\
abla T \tag{3}\]  
where \"\(\rho\)\" is the material density, \"\(C_{p}\)\" is the specific heat capacity, \"\(T\)\" is the temperature, \"\(q\)\" is the conductive heat flux, \"\(\mathbf{u}\)\" is the velocity vector, \"\(Q_{e}\)\" is the external heat source, \"\(Q_{P}\)\" is heat created by pressure, \"\(Q_{vd}\)\" is the heat source from viscous dissipation, \"\(Q_{td}\)\" is the thermoelastic damping, and \"\(k\)\" is the thermal conductivity. \[-\mathbf{n}*\mathbf{q}=0 \tag{4}\]  
\[Q=Q_{0} \tag{5}\]  
where \"\(\mathbf{n}\)\" is the normal vector on the boundary, \"\(\mathbf{q}\)\" is the conductive heat flux, \"\(Q\)\" is the heat source, and \"\(Q_{0}\)\" is the actual value of the mean value of the electromagnetic losses within the domain given by electromagnetic simulation. Because the simulation's performance (accuracy compared to experimental results) is highly dependent on the mesh complexity and its setting, attention must be paid to the proper definition of the element size. This information for the proposed model of induction heating is listed in Table 5. The mesh for this simulation model is created as Free Tetrahedral for the whole geometry. Time dependency simulation is set for 10 [s] of total time, while the time step is 0.1 [s]. The 0.1 [s] step is time, for which the results are saved, but the real computational time step is given by simulation needs (for each iteration, the convergence settings are different). Parametric Simulation for the Determination of the Optimal Operating Conditions of the Induction Heating  
For the first simulation experiment, we have focused on the parametric change of the distance between the sheet and heating element, while the induction coil geometry is unchanged. The setting for the simulation is as follows:  
* The current flowing through the coil is 150 A. * The thickness of the ferrite core is 10 mm. * The operating frequency of the applied voltage generating EM field is 25 kHz. * Distance (gap) between sheet and heating element is 1:1:10 [mm], i.e., initial distance is 1 mm, step is 1 mm, and the final distance is 10 mm. In Figures 5 and 6, it can be seen that the volumetric loss density, which causes heating of the molybdenum sheet, is dependent on the distance. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Maximum** & **Minimum** & **Maximum** & **Curvature** & **Resolution of** \\
**Element Size [m]** & **Element Size [m]** & **Element Grow [-]** & **Factor [-]** & **Narrow Regions** \\ \hline
8.75 \(\times\) 10\({}^{-3}\) & 3.75 \(\times\) 10\({}^{6}\) & 1.35 & 0.3 & 0.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Mesh settings. The rule that the close induction coil to the sheet is, the higher heating power will be generated is not valid because of the induction heating's physical principles, which is influenced by the system's operation frequency and resonance, which occurs for a given distance [21; 22; 23]. Figure 5 shows that the peak power is generated at the 3 mm of the gap, while below and above this value, volumetric loss decreases. For a better interpretation of the results, the distribution of these losses is shown in Figure 6.  
Figure 5: The dependency of the electromagnetic losses of molybdenum sheet on the distance between the coil and the sheet. Figure 6: Distribution of the volumetric loss density within the molybdenum sheet at different distances between heating element and sheet. The second simulation experiment is focused on the coil geometry's parametric change, while the distance between the sheet and the heating element was constant (3 mm).