## 5 Modeling the Specific Critical Flow of Heat and Replenishment
### Dryout Model
#### 5.1.2 End of the Process  
The end of the dryout process is reached when the time variation of the volume of liquid displaced from the porous medium to the tray is zero; that is \(dV_{d}/dt=0\). This is equivalent to the limit of the thermal power required for the formation of a vapor bubble from the smallest pores; that is \(\lim W_{tf,r_{p}-0}\). These conditions also imply:  
\[\Delta T_{ef}=\lim_{r_{p}\to 0}\frac{1}{r_{p}}\frac{2\sigma T_{pl}}{\rho_{v} \Delta\Pi_{lv}} \tag{25}\]  
Once \(r_{p}\) and \(T_{pl}\) are known, \(\Delta T_{ef}\) is calculated; subsequently:  
\[\dot{W}_{dr,f}=C_{1}\varepsilon A\kappa\Delta T_{ef} \tag{26}\]  
where \(C_{1}\) is the fraction of the minimum diameter pore surface area and \(\kappa\) is the heat transfer coefficient at the plate-porous medium interface.  
The experimental results indicate that the porous media based on bronze powder reach dryout when the volume of liquid accumulated in the tray is at maximum; however, the temperature of the plate continues to increase monotonically, which is explained by the evolution \(r_{p}\to 0\) and \(\Delta T_{ef}\rightarrow\infty\).  
Actually, \(r_{p,min}>0\) and the state of the system, when condition (25) is reached, is defined by the following values of the significant magnitudes of the problem:  
\(t=7800\) s, \(T_{pl}=256\)\({}^{\circ}\)C, \(T_{mp}=101\)\({}^{\circ}\)C, \(\Delta T_{ef}=155\)\({}^{\circ}\)C, \(T_{cb}=57\)\({}^{\circ}\)C,  
\(V_{dt}=609\times 10^{-6}\) m\({}^{3}\), \(e_{cs}=5.4\times 10^{-2}\) m, \(S=36\%\)\(\sigma=33.2\times 10^{-3}\) Nm\({}^{-1}\), \(\rho_{v}=23.25\) kgm\({}^{-3}\), \(\Delta h_{v}=1676.25\times 10^{3}\) Jkg\({}^{-1}\), \(\dot{W}_{el}=350\) W, \(\dot{W}_{c,pl}=35\) W, \(\dot{W}_{tf}=316\) W  
By substituting these values in Equation (21), the results are:  
\(r_{p,min}\geq 0.4\times 10^{-6}\) m \(\geq 0.0004\) mm  
This value of the minimum pore radius is lower than the value determined by the Faghri relation, which explains why the final state of dryout in the B+DS porous medium corresponds to a high saturation value. Now, it is necessary to calculate the fraction of the active pore area when the final state of dryout has been reached.  
The use of Equation (26) with a value of \(\alpha\) greater than 2000 Wm\({}^{-2}\)K\({}^{-1}\) (lower limit of nucleated boiling) gives \(\dot{W}_{dr,f}\geq 1978C_{1}\) W. The value 314 W is obtained by using the thermal power balance approach with a corresponding power per unit of heating surface area of 17,793 Wm\({}^{-2}\) and \(\alpha\cong 2500\) Wm\({}^{-2}\)K\({}^{-1}\). According to this new value, a power of \(\dot{W}_{dr,f}\geq 2472C_{1}\) W is obtained which corresponds to the balance value of \(C_{1}=0.127\); in other words, the fraction of the pore area that corresponds to a state of total dryout is 12.7%.  
From the state of the system and the calculated pore radius value, it can be concluded that the plate-porous medium interface is a superheated steam layer from which heat is transferred to the porous medium practically only by thermal radiation. The range of values of thermal power applied to the porous media corresponding to the beginning of the dryout (i) and the total dryout state (f), calculated by the thermal power balance approach, are shown in Table 8.