# Induction Heating of a Fluidized Pebble Bed: Numerical and Experimental Analysis
## 4 Results and Discussion
### Experimental Results for PtH Efficiency and Heat Flux Density  
To verify the proposed PtH concept and to provide experimental data for the model validation, the experiments have been conducted with a constant input power of \(P_{td,PC}=5.3\,kW\)  
Figure 9: Comparison of simulation and experimental results of the total induced power \(P_{td,tot}\) and the HTF outlet temperature \(T_{f,out}\) with one cross-sectional temperature profile solution for d = 30 mm, \(w_{f}\) = 0.62 m/s.  
at the power converter and an air flow rate that is regulated from \(\dot{V}_{f}=25\;Nm^{3}h^{-1}\) to \(\dot{V}_{f}=150\;Nm^{3}h^{-1}\) (at reference conditions: \(T_{N}=273.15\;K\) and \(p_{N}=101,325\;Pa\)). A correction factor proposed by [PERSON] et al., of 0.85 is used for the flow rate to address the near-wall channeling effect, indicating that 15% of the flow rate does not thermally interact with the pebble bed [31]. In addition, Table 2 gives an overview of the geometry variation and the materials used for the test section.  
The experimental investigation and verification of the inductively heated pebble bed concept is conducted at steady state conditions. The concept evaluation focuses on the energetic efficiency \(\eta_{PHH}\) and heat flux density \(\dot{q}_{tot}\). The definition of the PtH efficiency is based on First Law considerations taking into account electrical and thermal power input and heat output introduced in Figure 1  
\[\eta_{PtH}=\frac{\dot{Q}_{th,out}}{P_{el,PC}\;\eta_{PC}}=\frac{\dot{Q}_{th,out} }{P_{el,in}}=\frac{\dot{m}_{f}\Big{(}h_{f,out}\Big{(}T_{f,out}\Big{)}-h_{f,in} \Big{(}T_{f,in}\Big{)}\Big{)}}{P_{el,coll}+P_{ind,tot}} \tag{15}\]  
where \(\dot{m}_{f}\) is the mass flow rate of the air flow and \(h_{f}\) the temperature dependent enthalpy at the inlet and the outlet of the heater. The electrical power \(P_{el,in}\) at the inductor terminals is measured to be approximately constant at 5 kW during all experiments. The heat flux density \(\dot{q}_{tot}\) is given in \(W/cm^{2}\)  
\[\dot{q}_{tot}=\frac{\dot{Q}_{th,out}}{A_{HTF}}=\frac{\dot{m}_{f}\Big{(}h_{f, out}\Big{(}T_{f,out}\Big{)}-h_{f,in}\Big{(}T_{f,in}\Big{)}\Big{)}}{n\;\pi\;d^{2}} \tag{16}\]  
which is the thermal output related to the heat transfer area \(A_{HTF}\) that is in direct contact to the total number of spheres \(n\).  
The Reynolds number \(R_{e}=d\;w_{f}v_{f}^{-1}\) containing the particle diameter \(d\), flow velocity \(w_{f}\) and kinematic viscosity \(v_{f}\) is an appropriate parameter since it contains information about the geometry and the flow conditions, which both influence the heater performance. The key performance indicators \(\dot{q}_{tot}\) and \(\eta_{PtH}\) presented in Figure 8 increase with growing Reynolds numbers and with larger sphere diameters. The main reason for this increase is the improved convectional heat transfer to the air due to the higher Reynolds and thus higher Nusselt numbers.  
The investigation of the particle diameter shows an improvement of heat flux density and PtH efficiency for larger particle diameters. Results for the PtH efficiency from [12] support this finding demonstrating a higher electrical resistance of the pebble bed and thus a higher efficiency for growing particle size (1.8 mm \(<d<\) 7.65 mm). Larger spheres have a higher electrical resistance to the induced eddy currents due to the larger circumference of the equator. According to [PERSON]'s law, a higher resistance results in a higher electrical power induced inside the sphere. The discussion from Section 4.4 further explains the relationship between particle size and efficiency, accounting for the effects of temperature dependence.  
Altogether, the achieved efficiency above 80% at a high heat flux density shows the potential of this PtH concept and proves the applicability for gas heating applications. In particular, high \(Re\) numbers above 8000 are favorable for the energy-efficient operation of this PtH concept at high power density.