# A Review of Pool-Boiling Processes Based on Bubble-Dynamics Parameters
## 4 Bubble Growth and Detachment Model
### Bubble Growth in Uniform Boiling  
The second-order non-linear type of ordinary differential equation, as shown in Equation (17), for the bubble when uniformly heated can be obtained from the mass conservation equation and momentum conservation equation of the bubble. These equations are grounded in the following assumptions. 1. [label=()]
2. Bubbles grow in an infinite medium and that their growth has spherical symmetry;
3. The surrounding fluid is Newtonian fluid;
4. The bubbles are assumed to be unaffected by external forces;
5. The temperature and pressure inside the bubble are uniform;
6. Due to mass transfer at the interface, the liquid around the bubble is assumed to flow at a velocity of \(u_{l}=(1-\rho_{v}/\rho_{l})\frac{dR}{dt}\);
7. There is no translational or rotational motion in bubble growth. \[\frac{p_{v}-p_{los}}{(1-\rho_{v}/\rho_{l})\rho_{l}}=R\frac{d^{2}R}{dt^{2}}+ \frac{3}{2}\bigg{(}\frac{dR}{dt}\bigg{)}^{2}+\frac{2\sigma}{(1-\rho_{v}/\rho_{ l})\rho_{l}R}+\frac{4\mu_{l}R}{(1-\rho_{v}/\rho_{l})\rho_{l}}\frac{dR}{dt} \tag{17}\]  
The homogeneous nucleation of bubbles was categorized into three distinct phases by [PERSON] [66]: surface tension growth, inertial growth, and thermal diffusion growth, as shown in Figure 11.  
Figure 10: One complete bubble cycle during boiling at \(p=1\) bar, \(\Delta T=9.4\) K [65]. In the surface tension growth phase, the initial equilibrium radius of the bubble is \(R_{0}\), which is obtained from the pressure difference between inside and outside the bubble, and the surface tension equilibrium is \(p_{v}-p_{l\infty}=2\sigma/R\), i.e., the Young-Laplace equation. The temperature of the bubble is equal to the temperature of the surrounding liquid, i.e., \(T_{v}=T_{L\infty}\). At this point, the velocity and acceleration are so tiny that its inertial terms are negligible, and the dynamic growth of the bubble is driven by the pressure difference. In the inertial growth phase of the bubbles (\(R\approx 10\,R_{0}\)), the bubble radius is significantly larger than the initial radius \(R_{0}\). Therefore, the third term on the right-hand side of surface tension Equation (17) becomes negligible and it is the inertia of the liquid that dominates the growth of the bubbles. It needs to be noted that the time scales of the first and second phases are in the order of microseconds. [PERSON] et al. [67] and [PERSON] et al. [68] confirmed the weakening of the dynamic effect after \(50\)\(\upmu\)s (from experimental measurements) and \(100\)\(\upmu\)s (from numerical analysis), respectively. During the thermal-diffusion-growth phase, phase change appears on the bubble interface and its thermal boundary layer, called the \"cooling effect\" [69]. As a result, the vapor temperature reduces from the initial superheat to saturation temperature, so that the vapor pressure \(p_{v}\) equals the system pressure \(p_{l\infty}\). This makes the left term of Equation (17) zero, the dynamic effect no longer drives bubble growth, and it is the temperature difference \(T_{l\infty}-T_{v}\) that drives bubble growth. Although the above differential equation does not have an analytical solution, each of the individual growth stages in it has an approximate solution or a numerical solution. [PERSON] [66] neglected the surface tension phase as well as the vapor-liquid density ratio (\(\rho_{v}\)/\(\rho_{l}\)), and assumes that the bubble grows isothermally to obtain a Rayleigh solution (Equation (18)) to inertia-controlled growth, which shows that radius is proportional to time. \[R=\sqrt{\frac{2}{3}\frac{p_{v}-p_{l}}{\rho_{l}}}t \tag{18}\]  
The energy equation needs be coupled with the kinetic Equation (17) in order to combine the heat transfer with the bubble growth problem. The association between Equations (17) and (18) is realized using the pressure difference (\(p_{v}-p_{l\infty}\)) in Equation (17), e.g., the well-known Clausius-Clapeyron equation (Equation (19)), which allows us to relate the pressure difference term to the liquid superheat term (\(T_{l\infty}-T_{v}\)).