## 3 Effective Thermal Conductivity Analysis
### Solid Conduction + Contact Area Conduction + Solid Conduction  
Contact area is the contact region between two adjacent spheres, usually caused by external pressures acting onto the spheres or their own weight in a packed bed. It is closely related to the elasticity of the sphere's material and usually increases in size with external load. It has been pointed out by many researchers that it is quite important to take the contact area thermal conduction into account at high solid-to-fluid thermal conductivity ratios (\(\lambda\geq 10^{3}\)) [11]. Hence, in this part, we consider the third type of effective thermal conductivity caused by heat transfer through a packed pebble bed with finite contact areas. The contact area between two adjacent spheres can be calculated by a model presented by [PERSON] using Hertzian deformation [17]. The third type of effective thermal conductivity caused by finite contact area conduction can be described by the following correlation:  
\[\frac{\lambda_{\varepsilon}^{c}}{\lambda_{s}}=\left[\frac{3(1-\mu_{p}^{2})}{4 E_{s}}fR\right]^{1/3}\cdot\frac{1}{0.5315}\left(\frac{N_{A}}{N_{L}}\right) \tag{3}\]  
where \(\lambda_{\varepsilon}^{c}\) is the third type effective thermal conductivity due to the solid conduction-contact area conduction-solid conduction heat transfer process; \(r_{c}=\left[\frac{3(1-\mu_{p}^{2})}{4E_{s}}fR\right]^{1/3}\) is the radius of the contact area between two spheres; \(\mu_{p}\) is the Poisson ratio; \(E_{s}\) is the Young modules; \(f\) is the collinear force acting on the spheres; \(R\) is the radius of the graphite sphere; \(S\) is the constant related to the structure of packed bed; \(N_{A}\) and \(N_{L}\) are the number of spheres per unit area and unit length in the packed bed, respectively. [PERSON] [17] studied three different close-packing ordered arrangements (simple cubic packing, face-centered cubic packing, and body centered cubic packing) and provided values of the structural parameters. Based on the study, a randomly packed bed can be considered depending on porosity. The third type effective thermal conductivity calculated by Equation (3) is shown in Figure 5, using parameters of the SANA experiment under helium atmosphere. As we can see, the type effective thermal conductivity shows a slight decline when temperature rises due to the graphite material characteristics. Figure 4: Effective thermal conductivity due to the solid conduction-gas conduction-solid conduction heat transfer process. Because there are three heat transfer processes that coexist in the packed bed, the total effective thermal conductivity of the pebble bed can be considered as the summation of the above three types of thermal conductivities:  
\[\lambda_{e}^{t}=\lambda_{e}^{r}+\lambda_{e}^{g}+\lambda_{e}^{c} \tag{4}\]  
the calculated total effective thermal conductivity and the measured values in SANA experiment under helium conditions are shown in Figure 6. As we can see, the calculated effective conductivity and the measured data fit well. Among these three heat transfer mechanisms, heat transfer by radiation plays a dominating role as temperature rises, while heat transfer through gas conduction and contact area conduction contributes to the major portion in the lower temperature zone. Figure 5: Effective thermal conductivity due to the solid conduction-contact area conduction-solid conduction heat transfer process. Figure 6: Total effective thermal conductivity and the experimental data of the pebble bed in SANA experiment under helium conditions. The calculated total effective thermal conductivity and the measured values in SANA experiment under nitrogen conditions are shown in Figure 7. As we can see, the total effective thermal conductivity under nitrogen conditions is a bit lower than that under helium conditions. However, the difference only comes from the second type of effective thermal conductivity due to different heat conduction properties of these two gases.