# A Review of Pool-Boiling Processes Based on Bubble-Dynamics Parameters
## 3 Mechanical Analysis of the Bubbles
[50] proposed that the static force wall-contact pressure \(F_{cp}\) as well as the first term of the unsteady resistance \(F_{du}\) should be neglected. Therefore, the second coefficient of \(F_{du}\) was changed from 1.5 to 2, and the wall-contact pressure \(F_{cp}\) was modified to obtain the mechanical equilibrium on the bubble detachment in the pool-boiling, as shown in Equation (15):  
\[\frac{4}{3}(\rho_{l}-\rho_{v})\pi R_{b}^{3}g-2\pi\rho_{l}R_{b}^{2}(\frac{d^{2 }R_{b}}{dt^{2}})^{2}-2\pi R_{b}\sigma\sin^{2}\varphi=0 \tag{15}\]  
[PERSON] et al. [51] modified [PERSON] and [PERSON]'s model by proposing a different growth-force equation and considering the drag force caused by the wake of the bubble after it leaves. They proposed that as the bubble grows, the vapor in the bubble and the liquid expelled around the bubble will exert a growth or inertial force on the bubble. They modeled this inertial force on the growth rate of the bubbles and developed a dynamic-boiling model that takes into account the effects of evaporation action at the contact line and heat transfer from the thermal boundary layer, and which also takes into account the change in the contact angle during the growth of the bubbles. [PERSON] et al. [52] made some modifications to the coefficients and considered the Marangoni stress acting on the bubbles, which is a stress gradient caused by the temperature difference, with the greater the liquid subcooling, the greater the effect of Marangoni force. [PERSON] et al. [53] and [PERSON] et al. [54] on the other hand, proposed that for bubbles in boiling liquid the effect of contact pressure can be neglected. [PERSON] [55] performed further calculations on the volume of the truncated sphere and modified the drag coefficient. Among these models, [PERSON], [PERSON] and [PERSON]'s models are most widely used for reference. The specific relational equations they proposed are listed in the Table 2, where \(R_{cur}=5\), \(R_{b}\) is the radius of curvature at the bottom of the bubble, and \(C_{d}\) is the drag coefficient, which is obtained by the rate of growth of the radius of the bubble [56], whose expression is \(C_{d}=5360\mu_{l}^{-0.79}/\big{[}\rho_{l}R(dR/dt)\big{]}^{0.79}\). [PERSON] et al. [57] took high-speed video and infrared thermometry to modify these formulae by quantifying the forces. They conducted a series of experimental studies on the growth and detachment of bubbles under boiling conditions on the horizontal pool. By contouring the liquid-gas interface, they calculated the external forces acting on the bubbles. These forces versus time during bubble growth and detachment are shown in Figure 8. [PERSON] et al. argue that the force-balance approach relies on predicting, for example, the rate of change of bubble momentum, which is very small, to give a basis for the viscous drag \(F_{d}\) and inertial force \(F_{i}\) of the bubble, leading to experiments with very high demands on the precision of the measurements. This relationship makes it difficult to obtain the desired results for bubble-detachment-radius prediction by force-balance methods.