# Influence of Selected Model Parameters on the Electromagnetic Levitation Melting Efficiency
## 3 Results
### Selected Variant  
To achieve a high economic result of the melting process, the highest possible power efficiency should be obtained, but only if the batch is in the upper equilibrium point. Due to fact that the parameters are isolated points, the highest that levitated was selected. Figure 10: The efficiency for different values of the frequency \((f)\), different positions of the batch \((z_{B})\), and different values of the angle \((\alpha)\). The darker color corresponds to higher value. Therefore, as the starting point for future optimisation, the variant \(f=10\) kHz, \(\alpha=\frac{2\Pi 1}{32}\), and \(z_{B}=0.03\) m was selected as the most promising one. For the selected variant, the stability analysis of the horizontal batch position was also performed. For the different \((x,y)\), the calculation was conducted and is presented in Table 1. The horizontal forces (\(F_{x}\) and \(F_{y}\)) act in opposing direction to the horizontal displacement like in a pendulum. As a finial analysis, it must be verified if, for a selected variant, power efficiency is sufficient to melt the batch. The calculations were conducted according to Equations (10)-(12) for the initial temperature of 20 \({}^{\circ}\)C. The energy needed for the batch melting was obtained from (10). \[E_{M}=E_{H}+E_{L}, \tag{10}\]  
where:  
* \(E_{H}\) is the energy needed to increase the temperature to melting temperature,
* \(E_{L}\) is the latent heat energy. The calculations of \(E_{H}\) and \(E_{L}\) are based on (11) and (12). \[E_{H}=C\cdot m\cdot\Delta T, \tag{11}\]  
where:  
* \(C\) is the specific heat capacity,
* \(m\) is the weight of the batch,
* \(\Delta T\) is the difference between starting temperature and melting temperature. \[E_{L}=L_{H}\cdot m, \tag{12}\]  
where:  
* \(L_{H}\) is the latent heat. Heat losses during melting occur due to two main reasons: radiation and convection. Therefore, the calculations were divided into two parts. In the first, only radiation causing heat losses were taken under consideration, because in actual applications, melting of the batch was performed in a vacuum [29]. In the second part, these calculations were complemented by heat losses caused by convection. It has to be emphasized, that such calculations are very simplified. In reality, every alternation of the temperature modifies the batch properties and implies a requisite of recalculation of power inducted in the batch. This makes an iterative approach to EM-T calculations necessary (according to Figure 1). The equation for the energy of heating and radiation cooling the batch is shown in (13). \[E_{B}(t)=E_{I}(t)-E_{R}(t), \tag{13}\]  
where:  
* \(E_{B}(t)\) is the energy in the batch in the time domain,
* \(E_{I}(t)\) is the energy inducted in the batch by electromagnetic field in the time domain,
* \(E_{R}(t)\) is the energy lost by radiation in the time domain. \begin{table}
\begin{tabular}{c c c c} \hline \(x\) **[m]** & \(y\) **[m]** & \(F_{x}\) **[N]** & \(F_{y}\) **[N]** \\ \hline
0 & 0 & \(1.92\cdot 10^{-5}\) & \(-1.03\cdot 10^{-5}\) \\
0.003 & 0 & \(-1.55\cdot 10^{-5}\) & \(-1.92\cdot 10^{-5}\) \\ \(-0.003\) & 0 & \(2.93\cdot 10^{-5}\) & \(2.26\cdot 10^{-5}\) \\
0 & 0.003 & \(-1.48\cdot 10^{-5}\) & \(-1.36\cdot 10^{-5}\) \\
0 & \(-0.003\) & \(-2.48\cdot 10^{-5}\) & \(2.26\cdot 10^{-5}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation results for various horizontal positions of the batch. The power inducted on the batch was calculated during simulation. Therefore, the energy to heat the batch could be calculated according to (14). \[E_{I}(t)=t\cdot P_{B}, \tag{14}\]  
where:  
* \(P_{B}\) is the power inducted in the batch,
* \(t\) is the time. The energy lost to radiation is calculated based on (15).