## 3 Dynamic Heterogeneity Investigation into Debris Bed Safety Analysis
### Results and Discussion
#### 3.3.3 Cooling Characteristics Analysis  
The purpose of this section is to investigate the effects of the alternative heterogeneous configurations' transformations (divided into PC and BC effects in this research) on the debris bed's cooling safety assessment, including the heat removal efficiency of the debris bed (taking the average \(\mathrm{Nu}_{p}\) of the debris bed as the criteria in this research) and the safety margin of stable particle cooling (measured by the occurrence of particle re-melting accidents). Since the debris bed's cooling was still under development after 10 s, the following transient behaviors of the bed's decay heat removal are shown in Figures 11 and 12. Figure 11 demonstrates the flow pattern of the liquid sodium from 10 s to 100 s in three debris bed models. The decay heat generated from the debris particles induced the buoyancy-driven convective flow upwards to remove the debris bed's heat. In the meantime, the liquid sodium adjacent to the debris bed was drawn into the porous area due to the pressure drop from the left convective flow, facing the flow resistance offered by the drag force as the obstacle to the flow circulation inside the debris bed. Subsequently, the upward convective flow brought the decay heat to the outlet of the calculating domain  
Figure 11: Liquid sodium’s flow pattern for 10–100 s (vertical cross-section of the calculation domain): (**a**) Homogeneous model, (**b**) Mixed model, and (**c**) Stratified model. Figure 12: Liquid sodium’s temperature distribution for 10–100 s (vertical cross-section of the calculation domain): (**a**) Homogeneous model, (**b**) Mixed model, and (**c**) Stratified model. (center of the top surface), and it sank to the lower part afterward. Ultimately, the natural circulation in the clockwise flow pattern was achieved at 100 s for all debris bed models. Figure 12 presents the temperature distribution of the liquid sodium from 10 s to 100 s among three debris bed models. Corresponding to the flow pattern, the liquid sodium's temperature gradient was gradually developed from the debris bed to the outlet, and, in the end, the maximum temperatures of the liquid sodium (\(T_{f,Max}\) [K]) in Figure 13 also approached stability without liquid sodium boiling in all the three models (Sodium's boiling temperature is 1153 K at 1 atm [39]). Therefore, the subsequent investigation into the PC and BC effects on the cooling assessment was based on the information collected at 100 s.  
As for the PC effect, firstly, it could affect the flow around the heat-generating particles (Fuel particles in the Mixed model; Homogeneous particles in the Homogeneous model), and the resulting flow velocity distributions collected from where the heat-generating particles were located are shown in Figure 14, including the radial direction and the vertical direction of the debris bed. \(\left|\overset{\rightarrow}{V}_{f(p)}\right|\) [m/s] stands for the flow velocity where the heat-generating particles' centroids are located, and \(R_{p}\) [m] and \(H_{p}\) [m] mean the heat-generating particles' radial and vertical locations, defined as below:  
\[R_{p}=\sqrt{x^{2}+y^{2}}, \tag{33}\]  
\[H_{p}=z, \tag{34}\]  
where \(x\) [m], \(y\) [m], and \(z\) [m] stand for the particle's coordinates in the calculating domain. Figure 14a shows that \(\left|\overset{\rightarrow}{V}_{f(p)}\right|\) presented the declined trend of the inward flow velocity in both Mixed and Homogeneous models, since \(\left|\overset{\rightarrow}{V}_{f(p)}\right|\) along the radial direction was suppressed by the flow resistance (drag force) in the porous area. The average \(\left|\overset{\rightarrow}{V}_{f(p)}\right|\) (\(\left|\overset{\rightarrow}{V}_{f(p)}\right|_{Atg. }\) [m/s]) in the radial direction also showed similar values between the two models (listed in Table 9).