## 3 Results and Discussion
A property of materials susceptible to microwaves is the increasing rate of conversion of the microwave into heat when the temperature increases. Therefore, the local overheating will cause better absorbance and conversion of the microwaves into heat. The result will be the occurrence of the unwanted phenomenon called thermal runaway. The Barium Ferrite W-type materials had a similar behavior in the microwave field. The samples \(P_{1}\), \(P_{2}\) and \(P_{3}\) were heated in a microwave field in order to establish to what extent the material defects lead to thermal runaway and to quantify this phenomenon in terms of temperature distribution and the damage suffered by the Barium Ferrite W-type samples. Figure 10 shows the temperature evolution in time for the samples heated in the microwave field. According to Figure 10, thermal runaway occurred only on \(P_{1}\) and the phenomenon started from the very beginning with a peak temperature of 758 \({}^{\circ}\)C. The fast but brief increase in the temperature indicated that high polarization of the edges of defects led to small arc discharges inside the sample. That was the first step of the thermal runaway phenomenon, which occurred without damaging the sample. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Phenomenon** & **Local Overheating** & \multicolumn{2}{c}{**Thermal Runaway**} & \multicolumn{2}{c}{**Arc Discharge**} \\ \hline \multirow{2}{*}{**Sample**} & \(\mathbf{t}\) & \(\mathbf{T_{ss}}\) & \(\mathbf{t}\) & \(\mathbf{T_{ss}}\) & \(\mathbf{t}\) & \(\mathbf{T_{ss}}\) \\  & **(s)** & (\({}^{\circ}\)**C)** & **(s)** & (\({}^{\circ}\)**C)** & **(s)** & **(\({}^{\circ}\)**C)** \\ \hline \(P_{1}\) & 125 & 758 & 400 & 935 & 425 & 1010 \\ \(P_{2}\) & 260 & 486 & 380 & 788 & 425 & 934 \\ \(P_{3}\) & 360 Based on the temperature recorded on the sample surface, a determination of the hottest point can be performed using thermal conduction equations without additional internal heat sources. The temperature of the hottest point was \(T_{SS\_P1}\) = 1462.66 \({}^{\circ}\)C. Similarly, for \(P_{2}\) and \(P_{3}\), the hottest points were determined to be \(T_{SS\_P2}\) = 799.84 \({}^{\circ}\)C and \(T_{SS\_P3}\) = 599.78 \({}^{\circ}\)C, respectively. The thermal transient was computed using the heat diffusion equation, presented in Equation (7). The simulation of the thermal transient have been performed for maximum 15 s.  
\[\frac{\partial T}{\partial t}=\alpha\cdot\left(\frac{1}{r}\cdot\frac{\partial }{\partial r}\cdot\left(r\cdot\frac{\partial T}{\partial r}\right)+\frac{ \partial^{2}T}{\partial z^{2}}\right) \tag{7}\]  
where \(\alpha\) = \(k/\rho c_{p}\) (m\({}^{2}\)/s) is the thermal diffusion, \(r\) (m) is the radial coordinate, meaning the distance to the cylinder axis, and \(z\) (m) is the axial coordinate, meaning the distance along the cylinder height. The graphical representation is presented in Figure 11.  
Figure 11: Transient temperatures in Barium Ferrite W-type samples. Figure 10: Temperatures recorded on the surface of the samples. According to Figure 11, the sample with the highest thermal load is \(P_{1}\), where the heat dissipation is low in the first 5 s and becomes high when the heat reaches the surface of the sample. The temperature decrease is explained by the exposure of the external surface to the ambient temperature. On the other hand, the samples \(P_{2}\) and \(P_{3}\) present similar behavior in terms of the thermal transient. The heat transfer is more efficient and the temperature distributions are uniform. The case of \(P_{3}\) seems to be the most stable, in terms of thermal transfer, with low temperatures and rapid cooling. Figure 12a,b shows the top and side views of the compacted sample at 100 MPa after microwave processing. Regarding the extent to which the samples were damaged by the thermal runaway phenomenon, the sample \(P_{1}\) was the most affected due to the initiation of arc discharge.