## 3 Results and Discussion  
### FEM Modelling of HF Induction Heating of the Nozzle during Extrusion of a Thermoplastic  
In order to determine the plant (heating nozzle, multilayer inductor coil, heatsink) parameters, a FEM mathematical model connecting the thermal and electromagnetic processes of inductive heating of the nozzle during extrusion of the thermoplastic material was developed. The numerical implementation of the formed mathematical model was carried out using the applied software package ComsolMultiphysics. The modelled geometry is shown in Figure 4 and contains the following elements: inductor (Figure 4a), magnetic flux concentrator made of supermalloy 79 NM tape (Figure 4b), an aluminum cylindrical heatsink (Figure 4c), a nozzle (Figure 4d), thermoplastic material (polyamide PA 6) (Figure 4e). The design of the plant made it possible to consider the model in a two-dimensional axisymmetric formulation and use the cylindrical coordinate system r0z. At any specific time, a sinusoidal current \(I\) of a given value flows in the winding of the inductor. To describe electromagnetic processes, a system of differential equa  
Figure 4: Modelled geometry: (**a**) inductor; (**b**) magnetic flux concentrator; (**c**) a heatsink; (**d**) a nozzle; (**e**) thermoplastic material. Figure 3: Functional diagram of the HF generator and measuring circuit. tions with respect to the magnetic vector potential [57], which has a single component \(A_{\phi}:A=(0,A_{\phi},0)\), was used, where the magnetic flux density \(\mathbf{B}=\
abla\times\mathbf{A}\). In order for \(\mathbf{A}\) to be completely determined when solving problems in an alternating electromagnetic field, the Lorentz gauge is applied (1):  
\[\
abla\mathbf{A}+\mu_{0}\mu\varepsilon_{0}\varepsilon\frac{\partial\varphi_{e}}{ \partial t}=0. \tag{1}\]  
For the domains (d) and (c), occupied with nozzle and heatsink (2):  
\[j\omega\sigma\mathbf{A}+\
abla\times\left(\mu_{0}^{-1}\mu^{-1}\
abla\times\mathbf{A} \right)=0; \tag{2}\]  
including \(i\)-th turn of inductor winding (a) (3):  
\[ \begin{array}{cc}j\omega\sigma\mathbf{A}+\
abla\times\left(\mu_{0}^{-1}\mu^{-1} \
abla\times\mathbf{A}\right)=\frac{\sigma U_{i}}{2\pi\tau},&U_{i}=\frac{l+\int_{ S_{i}}j\omega\sigma A_{\phi}dS_{i}}{\sigma\int_{S_{i}}\frac{1}{2\pi\tau}dS_{i}},\\ I=\int_{S_{i}}J_{\phi}dS,&J_{\phi}=-j\omega\sigma A_{\phi}+\frac{\sigma U_{i} }{2\pi\tau};\end{array}  \tag{3}\]  
occupied by the magnetic flux concentrator (b) in a similar manner to (3) when \(I=0\). For the outer air and thermoplastic material (e) (4):  
\[\
abla\times\left(\mu_{0}^{-1}\mu^{-1}\
abla\times\mathbf{A}\right)=0, \tag{4}\]  
where \(\sigma\) is the electrical conductivity of the material; \(\omega\) is the angular frequency; \(j\) is the imaginary unit; \(\varepsilon_{0}\) and \(\mu_{0}\) are electric and magnetic constants; \(\varepsilon\) and \(\mu\) are relative dielectric and magnetic permeability of the medium; \(\varphi_{\mathrm{e}}\) is the electric scalar potential; \(J_{\phi}\) is the current density in the inductor; \(J\) is the current density induced in the nozzle volume; \(r\) is the radial coordinate; \(S_{i}\) is the cross-sectional area of the \(i\)-th turn of the inductor winding; \(U_{i}\) is the potential difference applied to the \(i\)-th turn of the inductor winding; \(I\) is the electric current (setpoint) flowing in the turns of the inductor winding. At the boundaries of the computational domain (\(\partial 100\times 100\) mm) the condition of magnetic isolation is accepted: \(\mathbf{A}=0\). On the axis \(0z\) of symmetry, the following condition is accepted: \(\frac{\partial A}{\partial n}=0\), where \(\mathbf{n}\) is the normal vector on the boundary; \(t\) is the time. At boundary (\(N4\)), the polymer is fed into the nozzle.