## 3 Numerical Model of Induction Heating a Rotating Steel Cylinder  
Designing an efficient drying system, in which a wet tape surrounding a rotating drying cylinder is heated by a high-frequency electromagnetic field, requires careful analysis of various phenomena. First, the mechanism of energy exchange between the electromagnetic field source, the inductor, and the cylinder mantle has to be modelled. The resulting model of the thermal field directly influencing the drying tape should then be determined. Several characteristic features of the induction heated drying setup have to be taken into consideration, including the specificity of the electromagnetic field resulting from the type of inductor and its spatial location against the cylinder, the rotating movement of the cylinder, and the mutual dependence of the electromagnetic and thermal fields [31; 32; 33]. All these phenomena can be described analytically by well-known mathematical formalities [34], but a fully functional model of the proposed drying system can be developed only by computer simulations [8; 35; 36; 37]. An illustrative diagram of our numerical model of an induction-heated steel roller is shown in Figure 6.  
Figure 5: Example of measuring the surface temperature of a rotating cylinder using a thermal imaging camera. (**a**) Two-dimensional temperature map of the surface of the rotating cylinder processed by an IR camera. (**b**) One-dimensional temperature distribution along the generatrix of cylinder x. In the modeled system, the following simplifying assumptions were made:  
* The anisotropy of the material properties of the cylinder area \(\Omega\)4 was omitted. It was assumed that during calculation of the power P generated in the cylinder the cylinder does not move. * In the winding area \(\Omega\)3, a harmonic current of 17A flows in the entire volume of the winding (this is the so-called Litz wire). In this area, no active power is released. * Zero conductivity of the ferromagnetic core \(\Omega\)2 is assumed, and the anisotropy of the material properties of this area is omitted. * For the area of the air gap and the environment of the heating system \(\Omega\)1, a compromise is assumed between the size of the division into finite elements and the accuracy of the disappearance of the magnetic field at the edge of the area. * In the induction heated rotating steel cylinder, the conduction current is much greater than the displacement current--i.e., \(\gamma\gg\omega\epsilon_{0}\epsilon_{r}\). * The induction of eddy currents in the cylinder due to its rotational speed is omitted. With the above assumptions, the combined magnetic vector potential for the separated areas of the inductor-cylinder system takes the following forms:  
Figure 6: Areas modeled in the inductor-steel cylinder system with boundary conditions for the electromagnetic field. The figure is not to scale. \(\Omega\)1—area of the air gap and the environment of the heating system; \(\Omega\)2—area of the ferromagnetic core; \(\Omega\)3—winding area; \(\Omega\)4—area of the cylinder, \(\gamma\)—electric conductivity of the \(\Omega\)4 area; \(\omega\)—pulsation of the electromagnetic field, \(\epsilon_{0}\)—vacuum permittivity; \(\epsilon_{r}\)—relative permittivity; \(\mu_{0}\)—magnetic permeability of free space; \(\mu_{r}\)—relative magnetic permeability, \(\alpha\)—heat transfer coefficient, \(\alpha_{r}\)—radiation heat transfer, \(\alpha_{c}\)—convection heat transfer, \(\lambda\)—conduction, T—temperature.