# Induction Heating of a Fluidized Pebble Bed: Numerical and Experimental Analysis
## 2 Numerical Model
### Two-Phase Electrothermal Model: Coupled Electromagnetic and Thermal Analysis
#### 2.2.2 Coupled Electromagnetic and Thermal Solution  
The solution for electromagnetic field was calculated based on the harmonic analysis described by Equations (5)-(12) and coupled with the temperature field solution according to Figure 5. The solution for the temperature field of the solid and fluid phase was obtained numerically using a commercially available fluid dynamics package FLUENT 21.1. The calculation for the solid phase relies on steady state condition of the heat Equation (14) formulated in spherical coordinate system and extended by the source term from Equation (12). This expression was implemented into the solving process as a custom volumetric heat source using a user defined function. The volumetric power density \(p_{ind}\) is fully converted into heat and replaces the right-hand side of Equation (14) as the Joule heat volumetric power density \(\hat{q}_{ind}\):  
\[\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(\lambda_{s}r^{2}\frac{ \partial T_{s}}{\partial r}\right)+\frac{1}{r^{2}\sin^{2}\theta}\frac{ \partial}{\partial\varphi}\left(\lambda_{s}\frac{\partial T_{s}}{\partial \varphi}\right)+\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\lambda_{s}\frac{\partial T_{s}}{\partial\theta}\right)=p_{ ind}=\hat{q}_{ind} \tag{14}\]  
The fluid phase is calculated by solving the steady state incompressible Navier Stokes Equations using a pressure-based solver. Turbulence is modeled using the \(\kappa\)-\(\omega\)-turbulence model. Gravity effects are disregarded for the fluid phase, since the conceptual investigation is carried out using air flow with high Reynolds numbers. Thermal properties for dry air are taken from the FLUENT material database considering a temperature dependence for the specific heat capacity and for the density. But constant values for viscosity (1.7894 \(\times\) 10\({}^{-5}\) kg m\({}^{-1}\) s\({}^{-1}\)) and thermal conductivity (0.0242 W m\({}^{-1}\) K\({}^{-1}\)) are applied. In addition, radiation is neglected since the investigation is conducted at temperatures significantly below the Curie point of carbon steel (768 \({}^{\circ}\)C).  
To obtain sufficiently accurate simulation results, temperature dependences of the solid material are incorporated into the electromagnetic solver. The temperature dependency of the electrical conductivity \(\sigma\) and the magnetic permeability \(\mu\) were considered via empirical models obtained from [3; 25]. In addition, magnetic field dependence of \(\mu\) was considered based on the external field \(H_{o}\)[5]. Finally, the total induced power density given by Equation (13) and the averaged fluid outlet temperature were set as convergence criteria for the solution process. Thus, the simulation finishes after reaching convergence at \(10^{-5}\).  
The boundary conditions (BCs) are illustrated in Figure 3. The BC for the inlet is an isothermal mass flow at a constant ambient temperature of 293 K. The wall temperature of the outer domain wall directly borders the water-cooled induction coil. Therefore, the wall temperature is set to be isothermal at the 323 K as measured in the setup from Figure 6. Finally, the outlet boundary condition is a simple pressure outflow.