# Induction Heating of a Fluidized Pebble Bed: Numerical and Experimental Analysis
## 2 Numerical Model
### Two-Phase Electrothermal Model: Coupled Electromagnetic and Thermal Analysis
#### 2.2.1 Geometry Modeling and Meshing  
DEM software [24] was used to obtain the arrangement of the particles in the randomly-packed monodispersed bed. Table 1 summarizes the input parameters for the DEM calculations. In this simulation, the particles randomly fall under the action of gravity from the upper inlet to the grating of the heating zone, as illustrated in Figure 2. Each particle position was determined for every time step (5 \(\times\) 10\({}^{-7}\) s) using linear spring models. Specifically, the hysteretic linear spring model and linear spring coulomb limit model were applied to model the normal force and the tangential force, respectively [24]. In addition, the linear spring rolling limit is used as rolling resistance model neglecting adhesive force. Figure 4: Cross-section of case 1 mesh (\(d\) = 30 mm) with 15 layers for the sphere subdomain and for its boundary layer inside the fluid subdomain. Figure 3: Cross-section of the computational domain for case 1: pebble bed with \(d\) = 30 mm; subdomains: 1-fluid; 2-solid; 3-tube; 4-insulation. Figure 2: Simulation procedure of the filling process with \(d\) = 30 mm spherical particles into the Al\({}_{2}\)O\({}_{3}\) container using Rocky multibody DEM software [24]. This simulation procedure ends after reaching convergence at \(1\times 10^{-9}\) Joule for the pebble bed's kinetic energy. The obtained sphere center coordinates \(O_{center}\) were then extracted and passed to the ANSYS SpaceClaim geometry model to first generate the geometry using a custom script. The geometry was then transferred to the FLUENT solver for thermal initialization and further simulation work, as shown in Figure 5.  
The resulting three-dimensional domain comprised of cylindrical and spherical subdomains. The hollow cylindrical subdomains represent the insulation and the outer ceramic tube whereas the spherical subdomains represent the solid spheres as shown in Figure 3. The application of Equations (5)-(12) for the spherical subdomain removes the need to model the induction coil thus eliminates the computations associated with the solution of the Maxwell Equations (1) and (2) over the entire simulation domain. The induced heat sources in each sphere (Joule effect) was calculated using Equation (12) as the volumetric power input to each cell inside every spherical subdomain. The calculation of induced volumetric power input was based on the coordinates of every cell centroid in the mesh. Thus, the accuracy of the solution for Equation (12) and for the total electric power input by using the volume integral from Equation (13) depends on the mesh resolution. Therefore, a grid refinement study was conducted on the one hand to ensure a convergence \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Magnitude** \\ \hline particle diameter \(d\) & case 1: 30 mm; case 2: 25 mm; case 3: 20 mm \\ particle number \(n\) & case 1: 61; case 2: 91; case 3: 160 \\ time step \(t\) & \(5\times 10^{-7}\) s \\ Young modulus & \(210\times 10^{9}\) Pa \\ [PERSON]’s ratio & 0.30 \\ restitution coefficient & 0.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Input parameter used for the DEM model. Figure 5: Coupled electromagnetic and thermal analysis with feedback of the temperature field for update of electromagnetic material properties. of the solution for the induced power and on the other hand to obtain a minimum number of mesh elements necessary for a time-efficient and accurate solution process. The results from the grid refinement study, presented in Appendix A, show convergence attained for more than 55 Mio. elements. We applied 57 Mio. mesh elements using 30 inflation layers for time-efficient numerical calculation process of the simulation study from Section 4.2.