# A Review of Pool-Boiling Processes Based on Bubble-Dynamics Parameters
## 4 Bubble Growth and Detachment Model
### Bubble Growth in Uniform Boiling
The liquid temperature at the bubble interface is equal to the vapor temperature in thermodynamic equilibrium, which is provided by the transient energy equation without an internal heat source (Equation (20)), and the boundary conditions at the bubble interface take the second type of boundary conditions (heat flux is known). Thus, the solutions of  
Figure 11: Different growth stages of homogeneous nucleation [66]. Equations (17) and (19), with the proper initial assumptions, are able to model the growth of bubbles, and these equations can be solved by the numerical calculations  
\[\Delta P=P_{v}-P_{l\infty}=\frac{\rho_{v}h_{lv}}{T_{sat}}(T_{l\infty}-T_{v}) \tag{19}\]  
\[\frac{\partial T}{\partial t}+\frac{(1-\rho_{v}/\rho_{l})R^{2}}{t^{2}}\frac{dt }{dt}\frac{\partial T}{\partial r}=\frac{1}{r^{2}a_{l}}\frac{\partial}{ \partial r}\bigg{(}r^{2}\frac{\partial T}{\partial r}\bigg{)} \tag{20}\]  
\[k_{l}\bigg{(}\frac{\partial T}{\partial r}\bigg{)}_{r=R}=\rho_{v}h_{lv}\frac{ dR}{dt} \tag{21}\]  
Since the numerical solution does not provide an explicit expression for the bubble radius and the kinetic effect of the bubble is only important for a tiny time interval, the complicated inception phase (i.e., surface tension growth and inertial growth) has been ignored and an approximate analytical solution is provided for the asymptotic phase in which the radius of the bubble is proportional to the square root of time (i.e., \(R\propto t^{0.5}\)). Moreover, the asymptotic solution could be calculated based on Equation (21) if the temperature gradient (\(\partial T/\partial r\))\({}_{r=R}\) is known. A comparison of correlations for uniform boiling [70; 71; 72; 73; 74; 75] was conducted and shown in Table 4. In the following correlations, \(Ja\) is the Jacob number and the expression is \(Ja=\rho_{l}c_{pl}\Delta T_{v}/\rho_{v}h_{lv}\), \(\alpha_{l}\) is the thermal diffusivity and the expression is \(\alpha_{l}=\lambda/\rho c_{pl}\). These correlations were compared with pool-boiling experiments by [PERSON] et al. [65]. It was found that for bubble growth at atmospheric pressure, the prediction curve of the formula proposed by [PERSON] et al. [70] is most consistent with the experimental curves, with relative errors between 18.5% and 31.6%; for the conditions at 0.5 bar and 0.15 bar, these homogeneous models are difficult to coincide with the experimental curves. The growth rate of bubbles is mainly determined by the \(Ja\) number and the thermal diffusivity of the liquid in uniform growth models from Table 4. Early studies considered the bubble growth rate to grow linearly with the square root of time, while later studies mostly took the form of dimensionless numbers to describe the bubble growth rate in uniform growth models. \begin{table}
\begin{tabular}{c c} \hline
**Author** & **Mathematical Model** \\ \hline [PERSON] and [PERSON] [70] & \(R=\sqrt{4/\pi}l\sqrt{\alpha_{l}t}\) \\ [PERSON] and [PERSON] [69] & \(R=\sqrt{12/\pi}l\sqrt{\alpha_{l}t}\) \\ [PERSON] and [PERSON] [68] & \(R=\sqrt{\pi}l\sqrt{\alpha_{l}t}\) \\ [PERSON] [71] & \(R=\sqrt{12/\pi}l\sqrt{\alpha_{l}t}\{h_{lv}/\left[h_{lv}-(c_{pl}-c_{p})\Delta T \right]\}\) \\ \hline [PERSON] and [PERSON] [72] & \(r_{1}=\frac{t}{3}\left[(1+\sqrt{\pi})^{3/2}-3(1+\sqrt{\pi})^{\frac{1}{2}}+2 \right]\), \(\psi=\left[1+\sqrt{\pi}(\frac{1}{\sqrt{1-N}}-1)\right]\), \(N=\frac{c_{pl}\Delta T}{h_{lv}}\) \\ \hline [PERSON] [73] & \(r^{+}=\frac{A^{+}}{(B^{+})^{2}}\), \(R^{+}=\left(\frac{12}{\pi}\alpha_{l}\right)^{1/2}\), \(A^{+}=\left(\frac{2}{3}\frac{h_{lv}\rho_{v}(T_{v}-T_{tot})}{\rho_{l}T_{sat}} \right)^{1/2}\),\(t^{+}=\left(\frac{A^{+}}{B^{+}}\right)^{2}t\) \\ \hline \end{tabular}
\end{table} 
Table 4: Growth curves of bubbles in uniform boiling.