# A Review of Pool-Boiling Processes Based on Bubble-Dynamics Parameters
## 4 Bubble Growth and Detachment Model
### Bubble Growth in Non-Uniform Boiling
#### 4.2.1 Thermal Boundary Layer Bubble Growth Modeling  
Bubbles in non-uniform boiling are not homogeneous spheres, which are usually calculated as truncated spheres, and the heat in a thermal system usually comes from the superheated wall at the bottom of the bubble. In this regard, [PERSON] [75] modified the uniform-growth model proposed by [PERSON] and [PERSON] [70] to incorporate the effects of superheat. This model takes a uniform thickness \(\sqrt{\pi\alpha_{l}t}\) based on a \"thin boundary layer\" approximation of the interfacial heat balance. At uniform boiling, each heat-flux vector points to the internal bubble interface. On the contrary, in non-uniform boiling, partial heat is transferred to the bubble interface as a temperature potential (\(T_{max}-T_{stl}\)) and the other part of the heat is transferred to the liquid as a temperature difference (\(T_{max}-T_{LB}\)). In other words, there is a temperature peak \(T_{max}\) which exists in the boundary layer, which  
Figure 12: Different growth modes of heterogeneous nucleating bubbles [66]. (**a**,**c**) boundary layer evaporation, (**b**) microlayer evaporation, (**d**) combined microlayer and superheated liquid evaporation. decreases towards the bubble interface and the liquid. \(T_{max}\) is considered to be equal to the superheated surface temperature \(T_{w}\) and the heat flux \(q_{Lb}\) to the liquid body is assumed to be equal to the wall heat flux. The factor \(b\) is proposed to modify the interference of the interface curvature on the temperature gradient, usually between one and \(\sqrt{3}\), and the recommended value \(b=\pi/2\) is obtained by comparing with the experimental data. The energy equation in non-uniform boiling was corrected to Equation (22) to obtain the bubble radius in Equation (23). \[\rho_{v}h_{lv}\frac{dR}{dt}=b\bigg{[}k_{L}\frac{T_{w}-T_{sat}}{\sqrt{\pi a_{L} t}}-q_{Lb}\bigg{]} \tag{22}\]  
\[R=b\frac{2}{\sqrt{\pi}}Ia\sqrt{a_{L}t}\bigg{[}1-\frac{q_{w}\sqrt{\pi a_{L}t}}{ 2k_{L}(T_{w}-T_{\text{sat}})}\bigg{]} \tag{23}\]  
A relaxation phenomenon exists in all fields of physics, whereby when the system is out of equilibrium, it requires a delay time to return to its initial state, and this process shows exponential regression. [PERSON] (1976) launched a study on the growth model of bubbles with the help of this concept. As the bubble expands radially, superheated fluids accumulate around the bubble up to a certain height \(y\) measured from the heated surface. The boundary layer around the bubble up to this height \(y\) is assumed to have a uniform thickness and is referred to as the \"relaxation layer\". [PERSON] also assumed that the liquid superheat in the \"relaxation layer\" drops exponentially with time from the initial maximum (\(T_{w}\)), as in Equation (24). [PERSON] used the bubble detachment time \(t_{d}\) as the characteristic time in Equation (24) based on the assumption that bubble growth followed the relaxation law. This correlation performed an energy balance for partially heated bubbles and incorporated the time-dependent superheating term into the relationship. Its calculation result (Equation (25)) is similar to that of the [PERSON] (1979) model of uniform boiling, and his model was also multiplied by the factor \(b^{*}\) defined by Equation (26), which indicates the proportion of the bubble surface area covered by the superheated liquid layer. \[\Delta T=\Delta T_{w}\exp\left(-\sqrt{\frac{t}{t_{d}}}\right) \tag{24}\]  
\[R=b^{*}\sqrt{\frac{12}{\pi}}Ia\exp\left(-\sqrt{\frac{t}{t_{d}}}\right)\sqrt{a _{L}t} \tag{25}\]  
\[b^{*}=\frac{2.7183R_{d}\rho_{v}h_{lv}}{\sqrt{12/\pi}\sqrt{\rho_{1}c_{Pl}k_{L} \Delta T_{w}\sqrt{t_{d}}}} \tag{26}\]  
The \"relaxed boundary layer\" model proposed in Equation (25) predicted all the data well in all types of experiments, with deviation errors within 15% for a wide range of superheat and low- and medium-pressure conditions ([PERSON] and [PERSON] (1976).