# Induction Heating of a Fluidized Pebble Bed: Numerical and Experimental Analysis
## 4 Results and Discussion
### Idealized Parameter Study for \(\mu(T_{S})\)  
With the validated model, the design space for the inductively heated pebble bed concept can be explored using only the electromagnetic model deduced in Section 2.1 by neglecting thermal losses by convection and radiation (\(\eta_{th}=1\)). The objective is to identify energy-efficient designs with high volumetric power density for the inductively heated pebble bed concept. In contrast to Section 4.1, this study is idealized, which means that it discusses the maximum achievable heat flux density with the associated PtH efficiency. The maximum PtH efficiency is obtained when thermal losses to the ambient are neglected and consequently the entire induced heat inside the PB is transferred to the air flow (\(\dot{Q}_{th,out}=P_{ind,tot}\)):  
\[\eta_{PtH,max}=\eta_{ind}\eta_{th,max}=\eta_{ind}=\frac{P_{ind,tot}}{P_{el,oil} +P_{ind,tot}} \tag{17}\]  
The formulation for \(P_{el,oil}\) is obtained from [PERSON] et al. [21]. Under the assumption \(\eta_{th,max}=1\), Equation (16) becomes  
\[\dot{q}_{tot,max}=\frac{P_{ind,tot}}{A_{HTF}} \tag{18}\]  
Instead of using quantities given in Equation (16), a simple procedure to implement the formulation for \(P_{ind,tot}\) has been found without loss of accuracy. This formulation applies scalar quantities for the power density in the bulk \(\dot{q}_{ind,PB}\) together with the volume of the pebble bed \(V_{PB}\) and is expressed as:  
\[P_{ind,tot}=\dot{q}_{ind,PB}V_{PB} \tag{19}\]  
The volumetric power density is given using Equations (3) and (6) as:  
\[\dot{q}_{ind,PB}=0.75H_{o}^{2}\omega\mu_{0}F \tag{20}\]  
where \(\omega\) is the angular frequency given by \(\omega=2\pi f\) and \(F\) is the power transmission factor characterizing the effectiveness of energy dissipation in the spheres by using the imaginary part of the \(\psi\) function from Equation (6):  
\[F=-\text{Im}\{\psi\} \tag{21}\]  
Finally, the volume of the pebble bed is expressed as:  
\[V_{PB}=\frac{\pi}{4}D_{PB}^{2}H_{PB}(1-\varepsilon) \tag{22}\]  
using the void fraction \(\varepsilon\), the outer diameter \(D_{PB}\) and the height of the pebble bed \(H_{PB}\), which are both specified in Table 2. For the void fraction the common value of \(\varepsilon=0.40\) is applied for further calculations. To investigate the performance of the PtH concept in relation to changing geometry and material properties of the used ferromagnetic steel, the heat transfer area \(A_{HTF}\) and the solid temperature \(T_{Solid}\) have been chosen as variables for the first parameter study. The heat transfer area \(A_{HTF}=\eta_{max}\ \pi\ d^{2}\) includes both, the number of spheres and their diameter \(d\) as geometric parameters, whereby the maximum number of spheres \(n_{max}\) is predetermined to fill the entire bed height \(H_{PB}\). The solid temperature is a suitable variable parameter for the material since both, electrical material properties \(\sigma\) and \(\mu\) are functions of temperature which have been already shown in Figure 10.  
Figure 11 shows results for the PtH efficiency \(\eta_{PtH,max}\) and for the power transmission factor \(F\) from Equation (21) as functions of bed temperature and HTF area. The temperature dependence of the electrical material properties during heating has a significant influence on the factor \(F\), which expresses the power density given by Equation (20). The power density \(\dot{q}_{ind}\), in turn, dictates the PtH efficiency according to Equation (17). This is the reason for the strong relationship between \(F\) and \(\eta_{PtH,max}\) from Figure 11 for the influencing variables \(T_{Solid}\) and \(d\). The influence of the temperature reveals maxima at 580 \({}^{\circ}\)C for diameters larger than 10 mm. These maxima are typical for ferromagnetic steel due to opposing temperature curves for \(\sigma\) and \(\mu\) from Figure 10 where the abrupt drop in magnetic permeability prevails near the Curie temperature.