# Induction Heating of a Fluidized Pebble Bed: Numerical and Experimental Analysis
## 4 Results and Discussion
### Numerical Results and Validation  
For the simulation study, the particle diameter \(d\) and the flow velocity \(w_{f}\) were varied analogously to the experimental investigation from previous section. These parameters are summarized in Table 4 together with measured values for frequency \(f\) and inductor's peak current \(l_{i}\). Both values are influenced by the system's inductance that varies with changing packed bed geometry and its temperature dependent material properties \(\sigma\) and \(\mu\). Therefore, these electrical inputs were also reported for calculating the electromagnetic field solution. Figure 9 shows the comparison of simulation and experimental results for the total induced power and the outlet temperature. The curve characteristics of both comparison criteria show a matching progression with respect to the Reynolds number. The characteristic of \(P_{ind,tot}\) shows a maximum at moderate \(Re\) numbers. This maximum is caused by temperature dependent electrical material properties \(\mu\) and \(\sigma\), which vary with flow conditions inside the pebble bed. Accordingly, low \(Re\) numbers result in high solid temperatures near the Curie point at 768 \({}^{\circ}\)C and high \(Re\) numbers lead to an improved convective heat transport along with a decrease in solids temperature to 200 \({}^{\circ}\)C. In this temperature range, there is a significant variation in the electrical material properties as illustrated in Figure 10 influencing the heat generation inside the solid. This figure shows a maximum of the function \(\mu/\sigma\), which causes the maximum in the induced power of Figure 9. The discussion in the following section points this effect further out. The quantitative comparison between experimental and simulation results in Figure 9 shows a good agreement for the induced power \(P_{ind,tot}\) of the pebble bed and the resulting averaged outlet temperature \(T_{f,out}\). The maximum deviation is below 15% for Reynolds numbers above 2000. For smaller Re numbers, however, the solid temperature approaches the Curie point at \(T_{C}\) = 768 \({}^{\circ}\)C of the used carbon steel where the magnetic permeability \(\mu\) changes non-linearly as illustrated in Figure 10. The uncertainty of this \(\mu(T_{S})\) func \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Air Inlet**} & \multicolumn{1}{c}{**0.62 m/s**} & \multicolumn{1}{c}{**1.27 m/s**} & \multicolumn{1}{c}{**1.87 m/s**} & \multicolumn{1}{c}{**1.46 m/s**} & \multicolumn{1}{c}{**3.05 m/s**} \\ \hline \multirow{4}{*}{\(\Delta\)**C**} & \(d\) = 20 mm & 347.89 & 328.05 & 323.14 & 323.16 & 311.62 \\ \cline{2-7}  & \(d\) = 25 mm & 341.54 & 331.33 & 331.32 & 326.21 & 322.85 \\ \cline{2-7}  & \(d\) = 30 mm & 341.05 & 333.98 & 328.74 & 327.04 & 323.53 \\ \hline \multirow{4}{*}{\(\Delta\)**C**} & \(d\) = 20 mm & 20.68 & 21.23 & 21.36 & 21.38 & 21.56 \\ \cline{2-7}  & \(d\) = 25 mm & 20.84 & 21.30 & 21.41 & 21.57 & 21.57 \\ \cline{1-1} \cline{2-7}  & \(d\) = 30 mm & 21.16 & 21.65 & 21.97 & 22.01 & 22.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Input parameter for simulation study: frequency and electric current measured at coil terminals for the electromagnetic field solution, and inlet velocity boundary condition for coupled thermal analysis. Figure 10: Relative magnetic permeability \(\mu\) and electrical resistivity \(1/\sigma\) as a function of solid temperature \(T_{Solid}\) for used steel UNS G10060 [5; 25; 29]. tion is the major reason for resulting deviations in the electromagnetic field solution and consequently in the obtained temperature field solution. Moreover, the assumption of a homogenous external magnetic field as well as negligible radiation effects at high solid temperatures are additional causes for the deviations from the experimental data.