# Influence of Selected Model Parameters on the Electromagnetic Levitation Melting Efficiency
## 4 Discussion  
The presented experiment concerned multi-variant calculations for the levitation melting model of a certain class of the inductors. In the experiment, three parameters have been changed: half apex angle, current frequency, and the vertical position of the batch. Subsequently, the forces applied to the batch, the power inducted in the batch, and the power supplied to the inductor were controlled. Therefore observations were taken for the following:  
* The maximum efficiency where levitation occurs is for \(f=10\) kHz, \(\alpha=\frac{2\Pi}{32}\) and \(z_{B}=0.03\) m. This variant was selected as most promising one for future optimisation. * Vertical component (\(F_{2}\)) of the force applied to the batch achieves the maximum for \(f=30\) kHz, \(\alpha=\frac{3\Pi}{32}\) and \(z_{B}=0.035\) m.
* The highest efficiency was reached for \(f=10\) kHz, \(\alpha=\frac{2\Pi}{32}\), and \(z_{B}=0.015\) m, even though the batch in this position did not levitate. * The power induced in the batch (\(P_{B}\)) is maximum for \(f=35\) kHz, \(\alpha=\frac{2\Pi}{32}\), and \(z_{B}=0.01\) m; however, the batch did not levitate. * The maximum of total power supplied to the system (\(P_{T}\)) was obtained for \(f=35\) kHz, \(\alpha=\frac{6\Pi}{32}\), and \(z_{B}=0.04\) m. However, levitation did not occur. * The melting of the batch for the selected variant was confirmed (for the increased inductor current). Moreover, all results obtained during this experiment provided valuable information for subsequent stage of research, which is the optimization of the melting efficiency:  
* The efficiency of the process (power ratio) will be used as the cost function in future optimization. * According to results obtained for all considered variants, no local extrema were found in the solution space. Therefore, the classical local-search algorithms should be sufficient for conducting the future optimization. * Furthermore, the hard condition of the batch levitation allows rough constraints to be applied to model parameters, reducing the size of the solution space. It was found that \(f\) must be greater than \(5\) kHz, and if it is equal or lower than
* \(15\) kHz, the \(\alpha\) should be less than \(\frac{6\pi}{32}\). * \(10\) kHz, the \(\alpha\) should be less than \(\frac{5\pi}{32}\) and the \(z_{B}\) should be greater than \(0.02\) m.
* \(25\) kHz, the \(z_{B}\) should be greater than \(0.01\) m.  
Future research will be conducted in a few ways. Firstly, the model will be confirmed in physical measurement. Furthermore, the simulation will be expanded for the batch position where stable equilibrium point occurs. Furthermore the optimisation of performance with parameters with the greatest potential will be commenced, and finally, the optimized physical inductor for metal melting process will be constructed. Conceptualization, R.P. and B.N. ; methodology, B.N. ; software, B.N. ; formal analysis, R.P. ; investigation, L.M. ; resources, B.N. ; writing--original draft preparation, B.N. ; writing--review and editing, L.M. and R.P. ; supervision, R.P. All authors have read and agreed to the published version of the manuscript. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Current [A]** & \(P_{B}\) [W] & \(t_{m}\) for Radiation [s] & \(t_{m}\) for Radiation and Convection [s] \\ \hline
200 & 0.6073 & 464 & did not occurs \\
300 & 1.3665 & 149 & 239 \\
500 & 3.7958 & 48 & 54 \\ \hline \hline \end{tabular} \(t_{m}\) is a time to melt the batch.  
\end{table} 
Table 2: Results of the simulation of the batch heating for the different current. **Funding:** This research was funded by Silesian University of Technology grant number BKM-776/RM4/2020; 11/040/BKM20/0019. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.