# Influence of Selected Model Parameters on the Electromagnetic Levitation Melting Efficiency
## 3 Results
### The Forces  
The capability of levitation inside the inductor is a crucial matter in this case, and it is expressed in the difference between the \(F_{z}\) and gravitational force \(Q_{B}\) acting on the batch shown in Figure 6. For the tested variant, the following remarks might be drawn:  
* For each current, the frequency maximum of \(F_{z}\) was achieved for \(\alpha=\frac{311}{32}\) and \(z_{B}=0.035\) m;
* The maximum of \(F_{z}\) was obtained for \(f=30\) kHz;
* An increase in \(f\) causes an increase in \(F_{z}\) (in the range \(f\) from 5 to 30 kHz);
* An increase in \(\alpha\) causes an decrease in \(F_{z}\) (in the range \(\alpha\) from \(\frac{611}{32}\) to \(\frac{311}{32}\));
* Every simulation performed for \(f=5\) kHz resulted in impossibility of batch levitation. The absolute condition for achieving levitation is that the electromagnetic buoyancy force \(F_{Z}\) is in the the balance with gravitational force \(Q_{B}\). From a theoretical point of view, for each inductor, there should be two points where this balance occurs [21]. In our experiment, both points were observed for a frequency of 30 kHz and an angle of \(\frac{511}{32}\). Figure 5: The axes and forces applied to the batch. \(F_{z}\) is ascending force, \(Q_{B}\) is gravity force, and \(F_{y}\) and \(F_{x}\) are horizontal force applied to the batch. If the batch is knocked out above the top equilibrium point, the electromagnetic buoyancy force \(F_{Z}\) becomes lower than gravity force \(Q_{B}\); thus, the batch will start to descend. However, if the batch passes below this equilibrium point, the \(F_{Z}\) overcome \(Q_{B}\), so the batch will be pushed back towards the top equilibrium. Therefore, the top equilibrium point should be considered as stable, which means that the batch should levitate in it during a melting process [3]. On the other hand, a different situation can be observed for a bottom equilibrium point. If the batch goes below this equilibrium point, the \(Q_{B}\) will be higher than \(F_{Z}\), so it will pull the batch further down. Above the bottom equilibrium point, the batch is pushed up towards the top equilibrium point. This is why the batch will be pushed back towards the top equilibrium point. This is why the batch will be pushed back towards the top equilibrium point. Figure 6: Difference between vertical components \((F_{z})\) of the force and gravity \((Q_{B})\) applied to the batch in N for different values of the frequency \((f)\), different positions of the batch \((z_{B})\) and different values of the angle \((\alpha)\). Positive values, where levitation occurs, are marked by a dark blue color. this bottom point should be considered an unstable one. Both equilibrium points (bottom and top) are shown in Figure 7d. The distance between those is correlated with \(\alpha\), so it increases while \(\alpha\) grows. Figure 7: Cont. The following observations were made:  
* For \(\alpha=\frac{5\Pi}{32}\) (Figure 7d), the value of \(F_{z}\) resembles a square function with a negative slope. The batch is levitating near the top wire of the inductor. * For more narrow angles from \(\frac{5\Pi}{32}\) (Figure 7a-c), \(F_{z}\) rises with \(z_{B}\) as long as the position is below the \(z_{B}=0.35\) m. This is a result of the fact that the wires are close to the batch, which increase \(F_{z}\), pushing the batch up. The top wire with opposite current slightly counteracts it, and the batch achieves a stable position above the top wire. * For \(\frac{6\Pi}{32}\) angle (Figure 7e), \(F_{z}\) significantly decreases while \(z_{B}\) rises. This is due to the fact that the top wire is too far away to have efficient impact on the batch. In this case the batch is levitating. The values of the horizontal components \((F_{x},F_{y})\) of the force applied to the batch are negligibly small in comparison the to \(F_{z}\); therefore their presentation was omitted in this paper.