## 5 Modeling the Specific Critical Flow of Heat and Replenishment
### Dryout Model
#### 5.1.1 Beginning of the Process  
Once the previous thermal state has been reached, the formation of the first vapor bubbles will occur in those parts of the heated plate-saturated porous medium interface where the larger diameter capillaries are located. The bubble radius is calculated as a function of the superheating \(\Delta T\) to be approximately:  
\[r\cong\frac{2\sigma T_{pc}}{\rho_{v}\Delta t_{p}\Delta T} \tag{21}\]  
Bubbles of radius \(r^{*}>r\) are in liquid that is superheated, and they can continue to grow. The thermal power at onset of dryout is calculated as:  
\[\dot{W}_{dr,i}=C\epsilon Av_{Dy}\rho_{v}\Delta t_{lv}\big{(}T_{pc}\big{)} \tag{22}\]  
where \(C\) is the fraction of the pore surface area of greatest diameter through which the flow of liquid displaced to the tray occurs by capillary action, and \(v_{Dy}\) is the Darcian velocity of the vapor through the two-phase zone, defined as:  
\[v_{Dy}=\frac{k_{rv}\cdot k^{\prime}}{\mu_{v}}\frac{dp_{c}}{dz} \tag{23}\]  
Figure 20: Volume of liquid accumulated in the tray and pressure in the S+DS porous medium: \(z=3.364\) cm to the heating plate, piezometer 4.  
where \(p_{c}=2\sigma cos\theta/r_{p}\) and \(r_{p}>r^{*}>r=\frac{2\sigma T_{s}}{\rho_{c}\Delta\Pi_{tr}\Delta T_{sf}}\), which is the Thomson equation for nucleated boiling bubble formation.  
For particles with a diameter greater than 1 mm, the equation proposed by [PERSON] [40] is used:  
\[r_{p}=\frac{1}{2}exp\big{(}0.01\sum N_{i}Ind_{p,i}\big{)} \tag{24}\]  
where \(N_{i}\) is the percent of the total mass of particles with a diameter equal to less than \(d_{p,i}\). Additionally, \(dp_{c}/dz=p_{c}/e_{cs}\) where \(e_{cs}\) is the thickness of the saturated layer of the porous medium.  
As an example, the conditions of the process when the dryout begins for the B+DS porous medium are as follows:  
\(t_{i,dr}=2400\) s, \(T_{pl}=102\)\({}^{\circ}\)C, \(T_{mp}=100\)\({}^{\circ}\)C, \(T_{cb}=19\)\({}^{\circ}\)C, \(V_{d}=1.4\times 10^{-4}\) m\({}^{3}\),  
\(e_{cs}=0.12\) m, \(S=85\%\), \(\dot{W}_{eq}\cong 0\), \(\dot{W}_{cpl}=43\) W, \(\dot{W}_{tf}\cong 232\) W, \(\dot{W}_{d}\cong 275\) W,  
\(\sigma=46.415-0.13124\)\(T(^{\circ}\)C) mNm\({}^{-1}\),  
\(k^{\prime}=3.92\times 10^{-11}\) m\({}^{2}\), \(k_{r0}=0.62\), \(\mu_{v}=12.3\times 10^{-6}\) m\({}^{2}\)s\({}^{-1}\)  
By substituting these values, the results are:  
\(\sigma=33.3\times 10^{-3}\) Nm\({}^{-1}\), \(\Delta T_{ef}=2\)\({}^{\circ}\)C, \(r^{*}=r_{p}\gg 3\times 10^{-6}\) m, \(\frac{d}{dz}p_{c}=0.573\frac{1}{r_{p}}\)  
Now, the calculation of \(\dot{W}_{dr,i}\) requires adopting values of \(C\) and \(r_{p}\), taking into account the conditions imposed for \(N_{i}\) and \(r_{p}\). It is known that the smallest significant diameter of a particle is \(161.9\times 10^{-6}\) m and the minimum porous radius is 0.034 mm (evaluated by using \(r_{p,min}\cong 0.21d_{p}\), Faghri [41]) greater than 0.003 mm. By introducing these values to Equation (23), \(v_{Dv}=0.0336\) ms\({}^{-1}\) is obtained. Afterwards, these values are substituted in Equation (22) and \(\dot{W}_{dr,i}=288\)C W. The thermal power transferred by unit area of the heating surface is \(16,650\)C Wm\({}^{-2}\); this value is in the region of nucleate boiling of water at 100 \({}^{\circ}\)C from a horizontal heated surface, according to [PERSON] et al. [42].  
Taking \(C\approx 1\), \(\dot{W}_{dr,i}=288\) W is obtained. This power must be the sum of the required power to increase the temperature of the porous medium, \(\dot{W}_{mp}=232\) W, and bubble generation, of which the value should be 56 W. The theoretical calculation gives 57 W.