# Induction Heating of a Fluidized Pebble Bed: Numerical and Experimental Analysis
## 2 Numerical Model
### Particle Model: Electromagnetic Field Solution  
The electromagnetic analysis is described by [PERSON]'s equations evaluated in a frequency domain. We examine the case of a single sphere plunged into a given uniform harmonic magnetic field \(\mathbf{B}\) oriented parallel to the inductor axis, adopting spherical coordinates \((r,\varphi,\theta)\), neglecting displacement currents [3] and assuming linear constitutive relations. [PERSON]'s equations are thus written as  
\[rot\ \mathbf{H}=\sigma\mathbf{E}\ \ \ \text{with}\ div\ \mathbf{B}=0 \tag{1}\]  
\[rot\ \mathbf{E}=-j\omega\mu\mu_{0}\mathbf{H}\ \text{with}\ \ lapl\ \mathbf{E}=0 \tag{2}\]  
The solution is analogous to [PERSON]'s [3] and [PERSON]'s approach [22] with the implementation of Bessel functions to Equation (2), assuming that the amplitude of the field strength \(H_{0}\) outside the sphere is deduced from the inductor length \(l_{i}\), its windings \(N_{i}\), the current \(\ l_{i}\) and the applied factor \(0.4\pi\)[22].  
\[H_{0}=\frac{\ l_{i}\ N_{i}\ 0.4\pi}{\ l_{i}\ K_{i}} \tag{3}\]  
considering [PERSON]'s correction factor for inductors with finite length-to-diameter ratio [23].  
\[K_{i}=1+0.44\ d_{i}/l_{i} \tag{4}\]  
The electric field \(\ \mathbf{E}\) is then calculated in circumferential direction using the unit vector \(\mathbf{i_{\varphi}}\) from Equation (5) inside the sphere  
\[\mathbf{E_{\varphi}}=-C\sin\theta r\ (\psi+1)\mathbf{i_{\varphi}} \tag{5}\]  
where \(C\) is a constant finally given by Equation (11), which has to be determined by the initial and boundary conditions. In this case, the boundary conditions are specifiedas continuous tangential components of \(\mathbf{H}\) and \(\mathbf{E}\) at the boundary of the sphere. This returns a value for \(\psi\) as a function of the sphere radius \(r\) and of the damping coefficient \(k\) from Equation (9)  
\[\psi=\frac{(2\mu-1)-F(k\cdot r)}{(\mu+1)+F(k\cdot r)}\quad\ \forall\ \psi\in\ \mathbb{C} \tag{6}\]  
with \(\ \mu\) being the relative magnetic permeability and \(F(k\cdot r)\) given by the function  
\[F(k\cdot r)=\frac{k\cdot r\ S_{1}^{\prime}(k\cdot r)}{S_{1}(k\cdot r)} \tag{7}\]  
The function \(S_{1}(k\cdot r)\) is related to the J Bessel function using the following equation  
\[S_{1}(k\cdot r)=r^{-1/2}J_{1.5}(k\cdot r) \tag{8}\]  
The constant \(k\) is the damping coefficient of the electromagnetic wave [3], which describes the skin effect in an electrically conducting body:  
\[k=\delta^{-1}+j\delta^{-1} \tag{9}\]  
According to this effect, the current density decreases from the surface of the sphere toward its center. Consequently, the induced power is concentrated in the surface layer which is called the skin depth and calculated by Equation (10)  
\[\delta=\sqrt{\frac{1}{\pi\ f\ \sigma\ \mu\ \mu_{0}}} \tag{10}\]  
with \(f\) being the frequency and \(\sigma\) the electrical conductivity.  
For a harmonic field solution with a uniform magnetic field, the value of the integration constant \(C\) is given in SI-units as  
\[C=j\pi f\mu_{0}H_{o} \tag{11}\]  
with the magnetic field constant \(\mu_{0}=4\pi 10^{-7}\frac{N}{A^{2}}\).  
Since the distribution of the electric field in circumferential direction \(\mathbf{E}_{\varphi}\) is determined inside the sphere, one can calculate energy dissipation per unit volume by the Joule effect:  
\[p_{ind}=\frac{1}{2}\sigma\|\mathbf{E}_{\varphi}^{2}\| \tag{12}\]  
As a result, the total electric power generated inside \(n\) spheres is calculated by integrating over the sphere volume:  
\[P_{ind,tot}=\sum_{i=1}^{n}\int_{V}p_{ind}dV \tag{13}\]  
Equation (13) reflects the core assumption of this model, which was also employed by [PERSON] et al. [12]. It indicates that each sphere is electrically insulated from other spheres. In addition, the bulk of power induced inside a sphere comes only from the interaction with the external magnetic field and not from neighboring spheres. Therefore, each sphere is treated as electromagnetically independent.