# A Review of Pool-Boiling Processes Based on Bubble-Dynamics Parameters
## 2 Bubble Nucleation
### Heterogeneous Nucleation  
Rough surfaces with pits, crevices, and cracks are more likely to retain gas and induce bubble nucleation. The heating effect is also stronger than for flat regions, as shown in Figure 3[23]. The superheating around the bubbles is not uniform because of the thermal boundary layer and micro-liquid layer, making the growth of bubbles in non-homogeneous boiling more complex than in homogeneous boiling. It is widely recognized that bubble formation requires cavities capable of trapping gas. [PERSON] [24] simplified the cavity to a conical cavity, where the condition for trapping the gas requires the contact angle to be greater than the cone angle of the cavity, i.e., \(\theta>\pi-2\alpha\), as shown in Figure 4. [PERSON] [25] synthesized the influence of microstructure in cavities and extended the bubble-trapping criterion to various shapes, proposing that the gas-trapping surface should satisfy \(\theta>\pi-\alpha-\cos^{-1}(1/S)\), where \(S\) is the ratio of the wetted area to the projected area of the cavity surface, i.e., \(S=S_{\text{actual}}/S_{\text{projected}}\). The [PERSON] criterion is in agreement with the [PERSON] criterion when the cavity is a tapered cavity. Both equations indicate that hydrophobic surfaces (large contact angles) are more likely to trap gas and form nucleation sites than hydrophilic surfaces (small contact angles). The formation of nucleation sites requires not only pre-existing gas, but also a certain degree of superheat to induce nucleation growth of the bubbles. A combination of force equilibrium and thermodynamic equilibrium is currently commonly used to calculate the required superheat of the bubble embryo, i.e., it is predicted by Equation (3) obtained by combining the Young-Laplace equation and the Clausius-Clapeyron equation, as shown in Equations (1) and (2), where \(\Delta v=\frac{1}{\rho_{\text{v}}}-\frac{1}{\rho_{\text{l}}}\approx\frac{1}{ \rho_{\text{v}}}\). \[R(p_{\text{v}}-p_{l})=2\sigma \tag{1}\]  
Figure 4: Schematic of the gas entrapment model proposed by [PERSON] [24]. Figure 3: Pits and crevices are more likely to entrap gas [23]. \[\frac{dp}{dT}=\frac{\Delta s}{\Delta v}=\frac{h_{lb}}{T\Delta v} \tag{2}\]  
\[T_{l}-T_{v}=\frac{2\sigma T_{l}}{h_{lb}\rho_{v}R} \tag{3}\]  
[PERSON] [26] argued that the size of the cavity that satisfies the nucleation condition should be an interval rather than the specific size predicted by Equation (1). Therefore, a criterion related to the liquid properties and to the wall temperature was proposed. As shown in Figure 5, the bubble embryo will nucleate successfully when the ambient temperature reaches the bubble temperature, i.e., \(T_{l}\geq T_{v}\) when the relative height to the wall surface \(h=h_{b}\). The liquid temperature Equation (4) was obtained by solving the transient one-dimensional heat-transfer equation, and the bubble temperature was calculated by Equation (3). Thus, [PERSON] arrived at a quadratic equation whose solution bounds the size interval of the nucleation cavity, as shown in Equation (5). \[T_{l}(y)=(T_{w}-T_{\infty})\bigg{(}1-\frac{y}{\delta_{l}}\bigg{)}+T_{\infty} \tag{4}\]  
\[\{r_{c,min},r_{c,max}\}=\frac{\delta_{l}\sin\varphi}{2(1+\cos\varphi)}\frac{ \Delta T_{w}}{(\Delta T_{w}+\Delta T_{sub})}\times\left[1\mp\sqrt{1-\frac{8 \sigma T_{S}(1+\cos\varphi)(\Delta T_{w}+\Delta T_{sub})}{h_{lb}\rho_{v} \delta_{l}\Delta T_{w}^{2}}}\right] \tag{5}\]  
In Equation (5), the angle \(\varphi\) is the shape angle formed by the embryonic bubble and the surface measured from the liquid side when the bubble temperature is equal to the liquid temperature, as shown in Figure 5. The shape angle is related to surface tension and hydrophilicity, and the more hydrophilic the smaller the angle \(\varphi\). It is important to note that during boiling, cavities may still settle after activation to produce bubbles, resulting in a rapid rise in local surface temperature.