## 3 Results and Discussion
### FEM Modelling of HF Induction Heating of the Nozzle during Extrusion of a Thermoplastic
The non-stationary thermal process in the nozzle area (d), thermoplastic (e), and heatsink (c) is described by the heat transfer Equation (5):  
\[pC_{p}\frac{\partial T}{\partial t}+pC_{p}p\
abla T-\
abla(\lambda\
abla T)=q, \tag{5}\]  
where \(\rho\), \(C_{p}\), and \(\lambda\) denote the density, specific heat capacity, and thermal conductivity of the relevant materials; \(\mathbf{v}\) is the speed equal to zero for all domains except thermoplastic (e). The ratio of extrusion speed \(V_{ext}\) to thermoplastic velocity in the nozzle channel \(V_{f}\) is close to \(V_{ext}=V_{f}(D_{n}/d_{n})^{2}\), where \(D_{n}\) is the diameter of the inner channel of the nozzle equal to \(2\) mm, \(d_{n}\) is the nozzle forming hole diameter is \(0.4\) mm. In this context \(V_{f}\) usually takes the values from \(0.5\)-\(5\) mm per sec and it is determined by the feeder. Heat dissipation during the flow of a viscous liquid is ignored, since it is several orders less than the heat supplied from the heated nozzle [58; 59; 60]. Moreover, it is known from [61] that until the thermoplastic reaches the nozzle output, its melting occurs mainly at the boundary with the channel walls. Then, when assessing the influence of the polymer velocity through the nozzle channel on the temperature of the nozzle, the polymer was considered as a solid body, regardless of its temperature. The conditions of constant consumption (6) were set:  
\[\
abla\cdot\mathbf{v}=0. \tag{6}\]where \(q\) is the specific capacity of the volume heat source. Heat is generated due to the flow of HF currents \(q_{r}\) and magnetic hysteresis losses \(q_{m}\) (7):  
\[q=q_{r}+q_{m},\ q_{r}=\frac{\mathbf{J}^{*}}{2\sigma}=\frac{1}{2}\omega^{2}\sigma \Big{(}A_{\Phi}A_{\Phi}^{*}\Big{)},\ q_{m}=\frac{1}{2}Re(j\omega\mathbf{B}\mathbf{H}^{*}). \tag{7}\]  
On the boundary (\(N\)1) corresponding to the surface of the nozzle, the conditions of radiation and convection (8) were set:  
\[-\frac{\lambda\partial T}{\partial\mathbf{n}}=\varepsilon_{T}\sigma_{sb}\ \Big{(}T^{4}-T_{0}^{4}\Big{)}+k(T-T_{0}), \tag{8}\]  
where \(\sigma_{sb}\) is the Stefan-Boltzmann constant. The surface emissivity of stainless steel (AISI420) was taken equal to \(\varepsilon_{T}=0.2\). The heat transfer coefficient of steel and aluminum to air was taken equal to \(k=20\) W/(m\({}^{2}\cdot\)K). The ambient temperature was taken equal to \(T_{0}\) = 20 \({}^{\circ}\)C on the boundary (\(N\)2) in a similar way to (\(N\)1) without radiation. On the boundary (\(N\)3) conditions of the thermal insulation are the following: \(-\frac{\lambda\partial T}{\partial\mathbf{n}}=0\). On the boundary (\(N\)4) for the polymeric material, the following Equation (9) was set:  
\[pC_{p}v(T-T_{0})=-\frac{\lambda\partial T}{\partial\mathbf{n}}. \tag{9}\]  
On the axis of symmetry: \(\frac{\partial T}{\partial r}=0\). The initial data: \(A_{\Phi}|_{t=0}=0,\ T\ |_{t=0}=T_{0}\). The data for modeling are given in the Table 1.  
Numerical Implementation of the Coupled Model of Induction Heating of the Nozzle during Extrusion of a Thermoplastic  
Here and below in Section 2, for measuring the temperature of a lightweight nozzle during the experiments, we used the previously mentioned UT-325 temperature meter with a K-type thermocouple soldered to the nozzle surface at a height of z = 8.5 mm (half the height of the active zone of the nozzle, i.e., half the distance from the tip to the heat break). During the numerical simulation, an ideal contact probe was used in the same position. In the work of [53], the comparison speed results of the temperature change of a lightweight nozzle and CE were presented. The results of experiments and numerical simulation of the cooling process at different polymer extrusion speeds showed that the temperature change rate of the lightweight nozzle is about six times higher than that of the heated part (hot-end) of the CE and increases with the extrusion speed rising. Due to the high thermal inertia of the heated part of the CE, the sensor is not sensitive to changes in the extrusion speed.