## 4 Development of a New Bubble Size Model  
For the development of a bubble size correlation reflecting the bubble breakup and coalescence mechanisms, the major governing parameters for stable bubble size in the flowing channel are obtained from the one-dimensional bubble number density equation below. \[\frac{\partial n}{\partial t}+\frac{d}{dz}(nv_{p})=\phi_{B}-\phi_{C}+\phi_{PN}- \phi_{PC} \tag{27}\]  
where \(n\), \(t\), \(v_{p}\), and \(z\) are the bubble number density, time, average bubble velocity weighted by the bubble number, and axial position along the flow direction, respectively. \(\phi_{B}\) and \(\phi_{PN}\) are the increased rates of bubble number density by bubble breakup and bubble nucleation, respectively. \(\phi_{C}\) and \(\phi_{PC}\) are the decreased rates of bubble number density due to bubble coalescence and bubble condensation, respectively. The steady-state fully developed flow condition is defined as a constant bubble number density along the flow direction. Because the bubble number density in the flow direction remains constant, the second term on the left-hand side of Equation (27) is zero. Furthermore, the bubble nucleation (\(\phi_{PN}\)) and condensation (\(\phi_{PC}\)) terms are zero because of the adiabatic flow. Here, one of the mechanisms for increasing the bubble number density is through the turbulent impact between bubbles and turbulent eddies (\(\phi_{\rm TI}\)), causing the bubbles to break. Conversely, the mechanism for reducing bubble number density is through collapsing of the bubbles, owing to the random collision between the bubbles (\(\phi_{RC}\)). Another bubble coalescence mechanism is wake entrainment (\(\phi_{\rm WE}\)), which dominates in small-diameter tubes. When large and small bubbles coexist in the flow channel, the small bubbles are accelerated and merge into large bubbles, owing to the reduced drag in the wake region behind the large bubbles. Therefore, assuming the steady-state fully developed condition of an adiabatic flow, the bubble number density transport equation is simplified as follows:  
\[\phi_{TI}-\phi_{RC}-\phi_{WE}=0 \tag{28}\]  
The modeling for each term in Equation (28) is summarized in Table 2. Finally, the bubble size of a bubbly flow under steady-state fully developed conditions can be obtained from the modeling of bubble breakup and coalescence as:  
\[ \begin{split} D_{b}&=\left\{K_{B}\Big{(}1-\frac{K_ {C}}{K_{B}}D_{b}^{2}\Big{(}\frac{\alpha}{\rho_{c}}\Big{)}^{-1.5}\varepsilon \Big{)}\right\}^{0.6}\\ &\times\left\{\ln\frac{\gamma_{B}(1-\alpha)\varepsilon^{\frac{1}{3 }}}{\alpha(\gamma_{\rm{KC}}\varepsilon^{\frac{1}{3}}+\gamma_{\rm{WBE}}\varepsilon ^{\frac{1}{3}}(1-\alpha)^{\frac{1}{3}}(\frac{\varepsilon-\alpha}{\rho_{c}} \varepsilon^{\frac{1}{3}}))\upsilon_{J}^{\frac{2}{3}}(\alpha_{\rm{max}}-\alpha )}\right\}^{-0.6}\end{split}\begin{split}\Big{(}\frac{ \alpha}{\rho_{c}}\Big{)}^{0.6}\varepsilon^{0.4}\end{split}  \tag{29}\]  
where \(\alpha_{max}\) is the maximum void fraction. The remaining variables (\(K_{B}\), \(K_{C}\), \(\gamma_{B}\), \(\gamma_{RC}\), \(\gamma_{WE}\)) are the constants of bubble breakup and coalescence terms, which are adjustable parameters to be determined experimentally. However, implementation of the above equation directly into the CFD code is still complicated. Considering the dependence of variables appearing in Equation (29), the equation can be simplified as: