# Induction Heating of a Fluidized Pebble Bed: Numerical and Experimental Analysis
## 4 Results and Discussion
### Identification of Efficient High-Temperature Materials  
The final parameter study aims to identify suitable material solutions, which ensure high efficiencies and power densities under high-temperature operation beyond \(T_{C}\). Such process temperatures are required for industrial high-temperature applications using EFH and for power plant processes with PH extension. For the second use case, air outlet temperatures up to 1050 \({}^{\circ}\)C are needed, for instance, to reduce the capital expenditures of a Brayton battery system [10; 12]. The approach is similar to the one from Section 4.4. The magnetic permeability \(\mu\) is again independent from bed temperature and is varied from non-magnetic property at \(\mu=1\) to consider the mentioned applications at \(T_{Solid}>T_{C}\) to the strong ferromagnetic property at \(\mu=10^{2}\) for low temperature applications at \(T_{Solid}<T_{C}\). In addition, the electrical conductivity, which is temperature independent in this analysis, is varied in a wide range from \(10^{2}\)\(\Omega^{-1}\)m\({}^{-1}\) to \(10^{8}\)\(\Omega\)\({}^{-1}\)m\({}^{-1}\) to consider electrical semiconductors and conductors as well. For practical reasons, the frequency is maintained constant throughout the process. For industrial applications the frequency is usually below 10 kHz to minimize ohmic losses in the power converter. Therefore, we assume f = 5 kHz and an electrical current of 800 A. Furthermore, geometrical conditions for such an application in the Brayton battery on the megawatt scale are summarized in Table 5.  
Figure 14 illustrates results for the PtH efficiency as contour lines in relation to both varied material properties where the optimal values for \(x_{\delta,\rho\prime\prime}\) from Section 4.4 are additionally illustrated. Contour lines for specific PtH efficiency show the optimal range for both material properties in terms of energy efficiency. Hence, to achieve a high efficiency of \(\eta_{PtH,max}>0.95\) for low temperature applications (\(T_{Solid}<T_{C}\)) the magnetic permeability might be in the range \(10<\mu<100\) for ferromagnetic conductors with an electrical con  
Figure 13: Results for PtH efficiency and volumetric power density with respect to the varied diameter \(d\) = [0.005:0.05] m and \(\delta\) = [1:\(10^{-4}\)] m. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Magnitude** \\ \hline Particle diameter & \(d\) = 0.12 m \\ Number of particles & \(n\) = 4688 \\ Outer packed bed diameter & \(D_{PB}\) = 1.5 m \\ Packed bed height & \(H_{PB}\) = 4.0 m \\ Inductor and wire diameter & \(D_{i}\) = 1.875 m; \(s_{i}\) = 2.5 cm \\ Inductor length & \(l_{i}\) = 5.0 m \\ Number of inductor turns & \(N_{i}\) = 144 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Geometry parameter for application on the megawatt scale. ductivity in the range \(2.0\times 10^{6}<\sigma<15.0\times 10^{6}\)\(\Omega^{-1}m^{-1}\) for steel. For high-temperature applications at \(T_{Solid}>T_{C}\), this contour plot reveals the required electrical conductivity for non-magnetic materials at \(\mu=1\) in between \(40,000<\sigma<190,000\)\(\Omega^{-1}m^{-1}\). A similar contour plot of the induced volumetric power density from Figure 15 additionally reveals values above \(5.5\) MW/m\({}^{3}\) for \(\hat{q}_{ind,PB}\) in a similar \(\sigma\) range at \(\mu=1\). Since the electrical conductivity of steel is normally higher (\(\sigma_{CS}>1.0\cdot 10^{6}\)\(\Omega^{-1}m^{-1}\) for carbon steel), semi-conductive materials are required that are correspondingly high-temperature resistant. Considering these both criteria, electrically conductive ceramics are suitable material options within the identified \(\sigma\)-range.