## 3 Results and Discussion
### FEM Modelling of HF Induction Heating of the Nozzle during Extrusion of a Thermoplastic
As a result of solving the thermal part of the parametric task, a frequency range that satisfies the requirements of high-speed heating was determined. The corresponding task with a variable parameter, the frequency \(f\) of the current \(I\), was numerically solved for \(t=4\) s, the frequency range from 10 to 160 kHz with a step of 10 kHz. The simulation results showed that the heating rate rises with increasing frequency, but at the same time, an increase of inequality in heating over the nozzle surface along the Z axis is observed. Using of the modified inductor shape results in the increase of the heating uniformity in \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & \(\rho\), **kg\(\cdot\)m\({}^{-3}\)** & \(\sigma\), Sm/m** & \(\mu\) & \(\varepsilon\) & \(\mathbf{C_{p}}\), **J\(\cdot\)kg\(\cdot\)1**-\(\cdot\)**K\({}^{-1}\)** & \(\mathbf{\lambda}\), W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\)** \\ \hline Steel AISI420 & 7650 & 1.02\(\cdot\)10\({}^{7}\) & 1500 & 1 & 482 & 25 \\ \hline Copper & 8700 & 5.998\(\cdot\)10\({}^{7}\) & 1 & 1 & 385 & 400 \\ \hline Supermalloy 79 NM & 8600 & 0.001 & 20,000 & 4.5 & 1369 & 1.2 \\ \hline Polyamide PA 6 & 1050 & 0.001 & 1 & 4 & 2000 & 0.33 \\ \hline Aluminium alloy & 2800 & 3.774\(\cdot\)10\({}^{7}\) & 1 & 1 & 920 & 130 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial model data. the active zone of the nozzle along the Z axis from \(\pm\)35 \({}^{\circ}\)C to \(\pm\)13 \({}^{\circ}\)C. By the fourth second of operation of the modified inductor (Figure 5) at the selected frequency \(f\) = 115 kHz and a current of 15 A, the nozzle heats up to 250 \({}^{\circ}\)C.  
Figure 6 shows the experimental and numerically simulated curves of the nozzle cooling when system is operating in its open-loop mode at a set power level of 5% (in % of the maximum power consumed by the inductor) after the nozzle heating at \(V_{ext}\) = 0 mm/s to the stabilization temperature (calculated 306 \({}^{\circ}\)C, observed 287 \({}^{\circ}\)C) and the beginning of polymer extrusion at \(V_{ext}\) = 10 and 20 mm/s. The deviation of the simulation results from the observed temperature readings ranged from 5% to 8%. Under heating conditions similar to Figure 5, the influence of the extrusion speed on the control gain and the time constant of the plant was numerically studied. The simulation results showed that an increase in \(V_{ext}\) leads to a decrease in the nozzle stabilization temperature. Modeling based on the data presented in Figure 7 shows that in order to maintain the steady-state nozzle temperature of 306 \({}^{\circ}\)C while extrusion speed \(V_{ext}\) increases from 0 to 100 mm/s, inductor circuit current should be increased 1.45 times, which results in the rise of the power consumed by the inductor by the factor of 2.1. When increasing \(V_{ext}\) from 0 to 40 mm/s, inductor current should be increased by 1.19 times with the corresponding increase of the power consumed by the inductor by the factor of 1.42. A  
Figure 5: Nozzle temperature distribution with a modified inductor for \(f\) = 115 kHz, \(t\) = 4 s, in the nozzle section \(r\) = 1.1 mm. Figure 6: Experimental and numerically simulated curves of nozzle cooling during operation at a set source power after the beginning of extrusion. decrease of the plant gain with an increase of the extrusion speed is also accompanied by a decrease of the plant time constant (by a factor of 3 with an increase in \(V_{ext}\) from 0 to 100 mm/s and by a factor of 2 with an increase in \(V_{ext}\) from 0 to 40 mm/s). With an increase of the extrusion speed, the ratio of the speed of change of the plant gain to the speed of change of the plant time constant remains and is approximately equal to 0.7, which is important when evaluating the effect of the disturbance signal (removal of heat from the nozzle during extrusion of the polymer) on the quality of control in the closed-loop TACS of the nozzle.