# A Review of Pool-Boiling Processes Based on Bubble-Dynamics Parameters
## 3 Mechanical Analysis of the Bubbles
\begin{table}
\begin{tabular}{c c c c} \hline  & **[PERSON] [46] Model** & **[PERSON] [52] Model** & **[PERSON] [55] Model** \\ \hline \(F_{s}\) & \(F_{s}=-2\pi R_{b}\sigma\text{sin}^{2}\varphi\) & \(F_{s}=-2\pi R_{b}\sigma\text{sin}^{2}\varphi\) & \(F_{s}=-2\pi R_{b}\sigma\text{sin}^{2}\varphi\) \\ \(F_{b}\) & \(F_{b}=\frac{4}{3}\pi R_{b}^{3}\mathcal{G}(\rho_{l}-\rho_{v})\) & \(F_{b}=\frac{4}{3}\pi R_{b}^{3}\mathcal{G}(\rho_{l}-\rho_{v})\) & \(F_{b}=\frac{1}{3}\rho_{l}\pi R_{b}^{3}(2+3\cos\varphi-\cos^{3}\varphi)\) \\ \(F_{cp}\) & \(F_{cp}=\frac{2\rho_{v}}{R_{cur}}\pi R_{b}^{2}\sin^{2}\varphi\) & \(F_{p}=(\frac{2\rho_{v}}{R_{cur}}+\Delta p_{v})\pi R_{b}^{2}\) & \(F_{cp}=\frac{2\rho_{v}}{R_{cur}}\pi R_{b}^{2}\sin^{2}\varphi\) \\ \(F_{d}\) & \(F_{d}=-\frac{2}{2}\rho_{l}\pi R_{b}^{2}\Big{(}\frac{dR_{b}}{dt}\Big{)}^{2}\) & \(F_{d}=-\frac{C_{d}}{2}\rho_{l}\pi R_{b}^{2}\Big{(}\frac{dR_{b}}{dt}\Big{)}^{2}\) & \(F_{d}=-\frac{1}{4}\rho_{l}\pi R_{b}^{2}(1+6\sin^{2}\varphi)\Big{(}\frac{dR_{b}} {dt}\Big{)}^{2}\) \\ \(F_{i}\) & \(F_{i}=-\rho_{l}\pi R_{b}^{3}\frac{d^{2}R_{b}}{dt^{2}}\) & \(F_{i}=-\frac{4}{3}\rho_{l}\pi R_{b}^{3}\frac{d^{2}R_{b}}{dt^{2}}\) & \(F_{i}=-\rho_{l}\pi R_{b}^{3}\sin^{2}\varphi\frac{d^{2}R_{b}}{dt^{2}}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Bubble-force equations for three types of detachment. In all of the above models, the researchers used Archimedes' principle for the calculation of buoyancy, i.e., \(F_{b}=\rho_{l}gV\). It should be noted that Archimedes' law can only be applied to the bubbles after they have detached from the wall, and for the bubbles that have not yet been detached Archimedes' law does not apply because the part of the bubble that is in contact with the wall is not subjected to the pressure of the fluid [58]. [PERSON] et al. [59] calculated the pressure on the bubble surface by spherical coordinate integration, and considered the effect of the micro-liquid layer at the bottom of the bubble on the overall pressure. The receding contact angle \(\alpha\) was taken in their calculations, which was obtained by photographing the state of the bubble before it detached. The forces on the bubble in their mechanical model include the surface tension \(F_{S}\), the liquid pressure \(F_{P}\), the gravity \(F_{G}\), the momentum change force \(F_{e}\), the internal pressure of the bubble \(F_{m}\), the instantaneous expansion force acting on the boundary of the bubble \(F_{a}\), which is oriented perpendicular to the boundary of the bubble, and the combined force acting on the bubble, which was denoted as \(F_{Z}\). The comparison between [PERSON] et al. 's [52] proposed force model and that of [PERSON] et al. [59] is shown in Figure 9, and the computational equations of various forces are shown in Table 3, where \(H\) denotes the height from the wall to the liquid surface, \(\beta\) is the ratio of the area of the micro-fluid layer to the area of the bottom of the bubble, \(q_{l}^{\prime\prime}\) denotes the heat flux at the gas-liquid interface around the bubble, and \(S_{s}\) denotes the area of the gas-liquid interface, with the negative sign indicating that the direction is along the direction of gravity. In its assumptions, the radius of the micro-fluid layer is approximately twice the radius of the dry portion. Figure 8: Bubble growth images and mechanical calculations, \(T_{w}=113\ ^{\circ}\mathrm{C}\), \(p=1\) bar [57]. Figure 9: [PERSON] et al.’s [52] proposed force model (**a**) vs. [PERSON] et al.’s [59] model (**b**). It can be seen from the equation that the bubble force is also affected by the liquid level height \(H\) in [PERSON] et al. 's model, which provides a new insight for the design of the boiling system. [PERSON] et al. (2018) also pointed out that the boiling heat transfer coefficient increases significantly with the reduction in liquid height when the liquid height is reduced to a certain value. [PERSON] et al.