## 3 Investigations Focusing on Flow Regime Characteristics
### Investigations on the Effect of Coolant Boiling Caused by Falling Debris
#### 3.5.3 Modeling Studies  
Considering the boiling resulting from the falling particles, an effective power, \(P_{\text{eff}}\), was taken into account to heat the liquid and generate the boiling [72]:  
\[P_{\text{eff}}=P_{\text{g}}-P_{\text{w}} \tag{13}\]  
where \(P_{\text{g}}\) is the effective power to generate bubbles (or gas) by heated particles from the overheated water, and \(P_{\text{w}}\) stands for the effective power needed to heat the water to the boiling point from a subcooled status during particle discharging:  
\[P_{\text{g}}=3\pi\lambda\frac{(T_{\text{P}}-T_{\text{B}})(1-\varepsilon)(d_{ \text{n}})^{2}H_{\text{w}}}{(d_{\text{p}})^{2}} \tag{14}\]  
\[P_{\text{w}}=C_{\text{w}}P_{\text{w}}V_{\text{w}}\frac{(T_{\text{B}}-T_{\text{ w}})}{t} \tag{15}\]  
where \(T_{\text{B}}\) represents the liquid boiling point, \(T_{\text{w}}\) is the initial water temperature, and \(C_{\text{w}}\) is the water's specific heat. Then, the effective bubble generation rate, \(Q_{\text{ge,eff}}\), can be derived:  
\[Q_{\text{g, eff}}=\left\{ \begin{array}{ll}\frac{P_{\text{eff}}}{\rho_{\text{ w}}l_{\text{g}}},&\text{if }P_{\text{eff}}>0\\ 0,&\text{if }P_{\text{eff}}\leq 0\end{array} \right. \tag{16}\]  
Figure 20: DBF flow-regime characteristics with boiling liquid surrounding the falling particles (Al\({}_{2}\)O\({}_{3}\) spheres, \(d_{\text{p}}\) = 0.5 mm, \(d_{\text{n}}\) = 30 mm, and \(T_{\text{w}}\) = 368 K) (Arrows representing the visually-observed flow directions). (**a**) Regime II (\(T_{\text{p}}\) = 473 K); (**b**) Regime III (\(T_{\text{p}}\) = 523 K); and (**c**) Regime IV (\(T_{\text{p}}\) = 673 K) [72]. where \(h_{\text{lg}}\) signifies the latent heat for boiling water. To characterize the boiling effect on restraining the particle-induced pool convection, a dimensionless quantity, \(\psi\), was defined [72]:  
\[\psi=\frac{Q_{\text{g,eff}}}{v_{\text{pr}}A_{\text{n}}}=\frac{Q_{\text{g,eff}}}{v _{\text{pr}}\pi(d_{\text{n}}/2)^{2}} \tag{17}\]  
Similar to the above model extension, an extension scheme, \(K_{\text{HP}}\), accounting for the boiling effect caused by the falling debris, was proposed for incorporation into the previous model:  
\[I=K_{\text{HP}}\left[K_{\text{MS}}K_{\text{NS}}K_{\text{B}}\times\frac{\left( \frac{v_{\text{pr}}}{v_{\text{c}}}\right)^{a}\left(\frac{H_{\text{BW}}^{2}}{DW} \right)^{b}}{\left(\frac{\rho_{\text{p}}-\rho_{\text{p}}}{\rho_{\text{I}}} \right)^{c}\left(\frac{h_{\text{p}}V_{\text{r}}}{\alpha_{\text{I}}}\right)^{d}}\right] \tag{18}\]  
where  
\[K_{\text{HP}}=\frac{1}{1+r_{1}\psi^{r_{2}}{Re_{\text{g}}}^{r_{3}}} \tag{19}\]  
where \(Re_{\text{g}}\) is the gas Reynolds number accounting for the boiling effect on centralizing the particle accumulations, and \(r_{1}\), \(r_{2}\), and \(r_{3}\) are the empirical constants given in Table 3. When \(\psi\) or \(Re_{\text{g}}\) trends to zero (such as \(Q_{\text{g,eff}}=0\) or \(v_{\text{pr}}\) trends to \(\infty\)), indicating a negligible bubbling effect (e.g., minimal or non-intensive bubble generation or dominant particle-flow-induced convection), the value of \(K_{\text{HP}}\) approaches 1, which restores the model to the base model that is suitable for non-bubbling cases. On the other hand, for cases where there is significant boiling during heated particle discharging, \(K_{\text{HP}}\) is a positive value lower than 1 to consider the effects of boiling surrounding the falling particles on a regime transition to larger-numbered ones. Therefore, the proposed \(K_{\text{HP}}\) was qualitatively reasonable. Figure 15e shows the regime map with the consideration of \(K_{\text{HP}}\). It is seen that the model prediction and experimental observation show good agreement within the parameter ranges considered.