## 3 Numerical Model of Induction Heating a Rotating Steel Cylinder
* For the cylinder area \(\Omega\)4: \[\
abla^{2}\mathbf{\Delta}-\mathrm{j}\omega\
u\gamma\mathbf{\Delta}=0\] (1)
* For magnetic shunt areas \(\Omega\)2 and free space \(\Omega\)1: \[\
abla^{2}\mathbf{\Delta}=0\] (2)
* For the inductor area \(\Omega\)3: \[\
abla^{2}\mathbf{\Delta}-\mathrm{j}\omega\
u\gamma\mathbf{\Delta}=-\mathrm{j} \mathrm{J}_{\mathrm{w}}\] (3) where Jw--the forced current density in the inductor winding; \(\omega=2\pi\mathrm{f}\)--supply current pulsation (\(\mathrm{f}\)--frequency); \(\mu=\mu_{0}\)\(\mu_{\mathrm{r}}\)--magnetic permeability of the medium (\(\mu_{0}=4\mu\cdot 10^{-7}\) H\(\mu_{\mathrm{r}}\)--relative magnetic permeability of the medium); \(\gamma\)--conductivity of the medium; Jw--force current. Using the magnetic vector potential A, the volumetric power density \(\mathrm{p}_{\mathrm{v}}\) in the charge can be calculated according to the following formula:  
\[\mathrm{P}_{\mathrm{v}}=\gamma\omega^{2}\mathbf{A}\cdot\mathbf{A}^{*} \tag{4}\]  
where \(\mathbf{A}^{*}\) is the conjugate number of the vector potential \(\mathbf{A}\). To obtain an unambiguous solution of the power density distribution on the cylinder surface, boundary conditions should be formulated at the edge of the calculation area \(\Omega\)1, where by assumption the magnetic field disappears. Therefore,  
\[\mathbf{A}=0 \tag{5}\]  
Condition (5) can also be applied to the plane of symmetry of the modelled system, since the currents in the inductor coil have the opposite conducting direction (the magnetic field force lines eliminate each other). In addition, at the intersection of the areas of the cylinder \(\Omega\)4 (a conductive area), air \(\Omega\)1 (a non-conductive area), and the core of the inductor \(\Omega\)1 (the non-conductive area of the ferromagnets), the normal components of magnetic induction are equal, which gives the following boundary condition for the vector potential:  
\[\mathbf{A}_{\mathrm{i}}\times\mathbf{n}_{\mathrm{ij}}=\mathbf{A}_{\mathrm{j}} \times\mathbf{n}_{\mathrm{ij}} \tag{6}\]  
where i and j are area indicators \(\Omega_{\mathrm{i}}\), and \(\mathbf{n}_{\mathrm{ij}}\) is the normal vector for the contact area surfaces \(\Omega_{\mathrm{i}}\) and \(\Omega_{\mathrm{j}}\). The temperature field of the cylinder area (\(\Omega\)4) in the analyzed system is described by the Fourier-Kirchhoff law:  
\[\
abla^{2}\vartheta-\frac{c\rho}{\lambda}\cdot\frac{\partial\vartheta}{ \partial\mathrm{t}}=-\frac{\mathrm{P}_{\mathrm{v}}}{\lambda} \tag{7}\]  
where: T--cylinder area temperature (\({}^{\circ}\)C); \(c\)--specific heat of the cylinder material (J/kgK); \(\rho\)--resistivity of the cylinder material (\(\Omega\)); \(\lambda\)--specific thermal conductivity of the cylinder material (W/mK); and t--time (s). The unambiguous solution of temperature field distribution is ensured by the Neuman condition at the area between the surface of the cylinder \(\Omega\)4 and the environment \(\Omega\)1:  
Figure 5:0  
where: T\({}_{0}\)--ambient temperature; \(\alpha=\alpha_{\mathrm{conv}}+\alpha_{\mathrm{rad}}\)--the total heat transfer coefficient with the environment consisting of convection (\(\alpha_{\mathrm{conv}}\)) and radiation (\(\alpha_{\mathrm{rad}}\)) heat transfer. On the edges of the \(\Omega\)1 area, the following initial condition was assumed:  
Figure 5:1  
The physical properties of modelled areas are given in Figure 5:2The total heat transfer coefficient between cylinder surface and environment is determined as nonlinear equation: 200 (W/m\({}^{2}\) K), emissivity as 0.8. Another crucial aspect of the modelling method is the inclusion of the rotation of the heated cylinder. This movement was imitated by the superposition of 36 positions of rotation (every 10\({}^{\circ}\)).