## 3 Investigations Focusing on Flow Regime Characteristics
### Investigations with Mixed-Sized Spherical Particles
#### 3.3.3 Modeling Studies  
In order to predict the flow regimes for DBF behavior with mixed-sized spherical particles, there were two modeling approaches attempted by [PERSON] et al. [69]. The first one attempted to estimate the general hydraulic performance of mixed-sized particle mixtures by directly employing Equation (5) (i.e., the base model) with certain equivalent particle diameters, including four mean diameters (the mean number diameter, mean length diameter, area mean diameter (\(d_{\text{a}}\), and volume mean diameter (\(d_{\text{v}}\))),  
\[\overline{d}_{\text{p}}=\frac{\sum d_{\text{p},\,j}^{+1}h_{\text{p},\,j}}{ \sum d_{\text{p},\,j}^{+}h_{\text{p},\,j}} \tag{9}\]  
as well as a volume-equivalent diameter (\(d_{\text{evp}}\)) [81]:  
\[d_{\text{evp}}=\left[\frac{\sum\bigl{[}V_{\text{b},\,j}\bigl{(}1-\varepsilon_{ j}\bigr{)}\bigr{]}}{\sum\bigl{[}V_{\text{b},\,j}\bigl{(}1-\varepsilon_{j} \bigr{)}/d_{\text{p},\,j}^{3}\bigr{]}}\right]^{1/3} \tag{10}\]  
In Equation (9), \(\overline{d}_{\text{p}}\) stands for the mean diameter of particles, \(r\) is the distribution type (\(r\) = 0, 1, 2, and 3 represent, respectively, number, length, area, and mean volume); \(n_{\text{p},\,j}\) and \(d_{\text{p},\,j}\) are, respectively, the number and diameter of the \(j\)-th sized particles for the particle mixtures. While in Equation (9), \(\varepsilon_{j}\) and \(V_{\text{b},\,j}\) represent the porosity and volume of particle beds constituted by the \(j\)-th sized particles, respectively.  
In light of detailed analysis, as shown in Figure 15c, it was validated that through utilizing the base model with \(d_{\text{a}}\), overall, results can be better predicted and attained in contrast to other mean diameters.  
On the other hand, the second approach referred to the aforementioned extended model for non-spherical particles, attempting to develop an extension correlation, \(K_{\text{MS}}\), with consideration of the flow regime characteristics in cases of mixed-sized particles. Therefore, the extended model was developed as [69]:  
\[I=K_{\text{MS}}\left[K_{\text{NS}}K_{\text{B}}\times\frac{\left(\frac{v_{\text{ PV}}}{v_{\text{c}}}\right)^{a}\left(\frac{H_{\text{W}}^{2}}{DW}\right)^{b}}{ \left(\frac{\rho_{\text{p}}-\rho_{\text{t}}}{\rho_{\text{t}}}\right)^{c}\left( \frac{\mu_{\text{t}}V_{\text{T}}}{\rho_{\text{t}}}\right)^{d}}\right] \tag{11}\]  
where  
\[K_{\text{MS}}=1.0+p_{1}\frac{(1-\Omega)^{p_{2}}}{(Re)^{p_{3}}} \tag{12}\]  
In Equation (12), \(p_{1}\), \(p_{2}\), and \(p_{3}\) are empirical constants (see Table 3); \(\Omega\) is the extent of convergence of particle size distribution determined by using [PERSON]'s equation with \(d_{\text{v}}\) to characterize the effective sphericity of the particle mixture. Considering the rationale of the establishment of \(K_{\text{MS}}\), it can be easily deduced that in single-sized spherical cases, \(\Omega\) equals 1.0, so \(K_{\text{MS}}\) becomes 1.0; while for mixed-size spherical particles, \(\Omega\) is smaller than 1.0, so \(K_{\text{MS}}\) becomes larger than 1.0, namely diminishing the effective impact of particle inertia, which corresponds well with the experimental observations.  
With the rational and appropriate \(K_{\text{MS}}\), an extended regime map can be attained (see Figure 15d). In addition, to further validate the rationale of these two modeling approaches, the prediction results were also analyzed in accordance with the fundamental parametric effects (such as the effects of particle density and size and the effect of volumetric mixed ratio) on the flow regime transitions [69] observed in the experiments. Further, the extension potential of the model for application under more realistic accidental situations was again further confirmed.