## 2 Investigations Focusing on Overall Accumulated-Bed Characteristics
### Investigations with Single-Sized and Single-Signed Solid Particles
#### 2.1.3 Modeling Studies  
For the purpose of modeling the important quantities characterizing DBF behavior, a predictive model focusing on \(H_{\text{b}}\), which is one key factor indicating bed shape characteristics, was empirically established by [PERSON] et al. [30, 39, 61, 73]. By using [PERSON]'s \(\pi\) theorem [75], the empirical model was established with the consideration of various parameters (see Equation (1)). \[\frac{H_{\text{b}}}{D_{\text{in}}}=k_{1}(\phi)^{a_{1}}\left(\frac{d_{\text{p}}} {D_{\text{in}}}\right)^{b_{1}}\left(\frac{d_{\text{n}}}{D_{\text{in}}}\right)^ {c_{1}}\left(\frac{H_{\text{n}}}{d_{\text{n}}}\right)^{d_{1}}\left(\frac{\Delta \rho}{\rho_{\text{p}}}\right)^{e_{1}}\left(\frac{\varepsilon V_{\text{p}}}{D_{ \text{in}}^{3}}\right)^{f_{1}}\left(\frac{\rho_{\text{p}}d_{\text{n}}V_{\text{ T}}}{\mu_{\text{f}}}\right)^{g_{1}}\left(\frac{\
u_{\text{ip}}}{V_{\text{T}}} \right)^{h_{1}} \tag{1}\]  
In Equation (1), the left side represents the similarity of geometry. For the right side, except the constant term \(k_{1}\), the first term uses the particle sphericity (\(\phi\)), which was determined by [PERSON]'s equation (see Equation (2)) with the particle's volume-equivalent diameter (\(d_{\text{ev}}\)) and measured pressure drop (\(\Delta P\)), to evaluate the effect of particle shape (esp. for non-spherical particles); the characteristic dimensional similarities are represented by the second, third, and fourth terms; buoyancy influence is considered in the fifth term; while the sixth term points out volumetric similarities; the seventh term signifies the particle-jet Reynolds number with particle terminal velocity \(V_{\text{T}}\)[76]; and the eighth term represents particle falling intensity. Furthermore, Table 2 lists the empirical indices in Equation (1) obtained by regression. \[\frac{\Delta P}{H_{\text{b}}}=150\frac{\left(1-\varepsilon\right)^{2}}{ \varepsilon^{3}}\frac{\mu_{\text{f}}}{\left(\phi d_{\text{ev}}\right)^{2}}U_{ \text{f}}+1.75\frac{\left(1-\varepsilon\right)}{\varepsilon^{3}}\frac{\rho_{ \text{f}}}{\phi d_{\text{ev}}}U_{\text{f}}^{2} \tag{2}\]  
Figure 5: Effect of particle shape on mound height (\(H_{\text{n}}\) = 720 mm, \(d_{\text{n}}\) = 30 mm). (a) SS particle, and (b) Al\({}_{2}\)O\({}_{3}\) particles [23, 29]. where \(U_{I}\), \(\rho_{I}\), and \(\mu_{I}\) are the fluid superficial velocity, density, and viscosity; \(\varepsilon\) stands for the bed voidage (=\(\frac{V_{\rm P}-m_{\rm P}/\rho_{\rm F}}{V_{\rm P}}\)). By employing this model, relatively good agreement between the experimental and predicted particle bed heights was found (see Figure 6a). However, according to the quantitative analyses of the parametric impacts based on this model, it was recognized that, in general, the experimental results could not be predicted well for non-spherical particles [29]. This may indicate the inadequate estimation of the fictional particle-particle drag effect for non-spherical particles. Further, to attain more valuable knowledge on DBF behavior in reactor accidents, as shown in Figure 7, extrapolation of the predictive results was performed for spherical particle cases. It was found that within the fuel density range, an initial decrease but a gradual upward trend for \(H_{\rm b}\) could be observed as the particle density increased. Possibly, this is because a larger frictional particle-particle drag resisted the lateral sedimentation of falling particles in cases of larger-density fuel particles [29].