## 3 Numerical Model of Induction Heating a Rotating Steel Cylinder
Full calculation domain with meshing is presented in Figure 5:3 The flowchart of these calculations is shown in Figure 5:4  
The meshing of volume elements was checked automaticaly, resulting in 54.95% non-evaluated elements, 22.4% excellent quality elements, 16.58% good quality elements, 5.34% average quality elements, and only 0.74% poor quality elements. In the initial step of the simulation, for time T = 0 and angle \(\alpha=0\), the distribution of the electromagnetic field in the inductor-cylinder system and the distribution of the active power induced in the cylinder mantle are calculated. These calculations take into account the nonlinear dependence of the magnetic induction **B** on the electromagnetic field intensity **H** (changing the \(\mu_{\mathrm{r}}\) value). This requires multiple iterations of calculations of the induction distribution, B, and of the volumetric active power p\({}_{\mathrm{v}}\) (block [1] in Figure 8), until the convergence conditions are fulfilled. Examples of the results are shown in Figure 9a,b. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \
\begin{tabular}{c} **Symbol of** \\ **Calculations Area** \\ \end{tabular} & \(\mu_{\mathrm{r}}\) **(–)** & \(\gamma\) **(S)** & \(\lambda\) **(W/m K)** & \(\rho\)c **(J/m\({}^{3}\) K)** \\ \hline \(\Omega\Omega\) & 1 & 0 & - & - \\ \hline \(\Omega\Omega\) & 1000 & 0 & - & - \\ \hline \(\Omega\Omega\) & 1 & \(\alpha\) & - & - \\ \hline \multirow{4}{*}{\(\Omega\Omega\)} &
\begin{tabular}{c} According to B = f(H) characteristics \\ built in Flux software database \\ (FLU\_STEEL\_1010\_XC10) by Cedrat \\ \end{tabular} & \(2.94\times 10^{6}\) & \(47\) & \(4.5\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical properties of modelled areas. Figure 7: The calculation domain of the inductors—rotating cylinder system: volume regions (**a**), meshing of volume regions (**b**). For the calculated active power, the temperature distribution in the cylinder area covered by the electromagnetic influence of the inductor is then determined, taking into account the nonlinear relationship of thermal conductivity \(\lambda\) and temperature T. This feedback is represented by block [2] in Figure 8. Due to the fact that, in the modeled process, the surface temperature of the cylinder does not exceed the Curie temperature T\({}_{\text{cur}}\), it is possible, without making a significant error, to uncouple the electromagnetic and temperature fields, so that after calculating the temperature distribution T there is no need to recalculate the active power p\({}_{\text{v}}\) in a given calculation step. This significantly accelerates the calculations. After calculating the temperature distribution on the surface of the cylinder, the transition to the next calculated time interval takes place t = t + \(\Delta t\), i.e., the cylinder is moved by an angle \(\alpha=\alpha+10^{\circ}\). Taking into account the thermal history from the previous interval step, the redistribution of the active power on the surface of the cylinder  
Figure 8: Flowchart of the process of calculating the induction heating of a rotating steel cylinder. Figure 9: Distribution of magnetic induction (**a**) and active power (**b**) on the surface of the cylinder. is calculated again. Of course, the value of the time step \(\Delta\)t depends on the modeled rotational speed of the cylinder. The presented calculation cycle is performed until the rotating cylinder reaches a quasi-steady thermal state. An example of the temperature field distribution on the surface of the rotating cylinder is shown in Figure 10a. Figure 10b shows temperature oscillations along the circumference of the cylinder in a thermally steady state. Based on the calculations, it can be concluded that in a quasi-steady thermal state the temperature fluctuations on the surface of the cylinder along its circumference caused by its rotation are negligible (\(\Delta\)T \(\leq\) 0.2 \({}^{\circ}\)C).