## 3 Materials and Methods
### Governing Equations  
A population balance model can be described by a transport equation (describing the particles entering/leaving a prescribed control volume), which is derived based on the Boltzmann statistical transport equation. In particular, the bubble number density transport equation is often referred to as the population balance equation [7]:  
\[\frac{\partial}{\partial t}n(\overset{\rightarrow}{x},V_{b},t)+\frac{\partial }{\partial z}\left[n(\overset{\rightarrow}{x},V_{b},t)u_{b}(\overset{ \rightarrow}{x},V_{b})\right]+\frac{\partial}{\partial V_{b}}\left[n( \overset{\rightarrow}{x},V_{b},t)\frac{\partial}{\partial t}V_{b}(\overset{ \rightarrow}{x},V_{b})\right]=S(\overset{\rightarrow}{x},V_{b},t) \tag{1}\]  
In Equation (1), \(n(\overset{\rightarrow}{x},V_{b},t)\) is the bubble number density distribution function, which is assumed to be continuous and specifies the probable number density of bubbles at a given time \(t\), in the spatial range \(d(\overset{\rightarrow}{x})\) about a position \(\overset{\rightarrow}{x}\), with bubble volume between \(V_{b}\) and \(V_{b}+d(V_{b})\); \(u_{b}\) is the local velocity of bubble volumes between \(V_{b}\) and \(V_{b}+d(V_{b})\) at time \(t\); \(S(\overset{\rightarrow}{x},V_{b},t)\) is a source term, which reads as follows:  
\[S(\overset{\rightarrow}{x},V_{b},t)=S_{c}+S_{b}+S_{ph}+S_{p}+S_{m}+S_{r} \tag{2}\]  
In Equation (2), \(S_{c}\), \(S_{b}\), \(S_{ph}\), \(S_{p}\), \(S_{m}\), and \(S_{r}\) are the bubble source/sink terms due to coalescence, breakup, phase change, pressure change, mass transfer, and reaction, respectively. In particular, the source term related to coalescence, \(S_{c}\), and the source term related to breakup, \(S_{b}\), read as follows:  
\[S_{c}=\frac{1}{2}\int_{0}^{\infty}\Gamma(V_{b}-{V_{b}}^{\prime},V_{b}^{\prime} )f(\overset{\rightarrow}{x},V_{b}-{V_{b}}^{\prime},t)f(\overset{\rightarrow }{x},{V_{b}}^{\prime},t)d{V_{b}}^{\prime}-f(\overset{\rightarrow}{x},V_{b},t )\int_{0}^{\infty}\Gamma(V_{b},{V_{b}}^{\prime})f(\overset{\rightarrow}{x},{ V_{b}}^{\prime},t)d{V_{b}} \tag{3}\]  
\[S_{b}=\int_{\mathbb{P}}^{\infty}m({V_{b}}^{\prime})\Omega({V_{b}}^{\prime})P( V_{b},{V_{b}}^{\prime})f(\overset{\rightarrow}{x},{V_{b}}^{\prime},t)d{V_{b}}^{ \prime}-\Omega(V_{b})f(\overset{\rightarrow}{x},V_{b},t) \tag{4}\]  
In Equation (3), the first term is the birthrate of bubbles of volume \(V_{b}\) due to coalescence of bubbles of volume \(V_{b}-{V_{b}}^{\prime}\) and \(V_{b}^{\prime}\); the second term is the death rate of bubbles of volume \(V_{b}\) due to coalescence with other bubbles; \(\Gamma(V_{b},{V_{b}}^{\prime})\) is the coalescence rate between bubbles of volume \(V_{b}\) and \({V_{b}}^{\prime}\). Conversely, in Equation (4), the first term is the birthrate of bubbles of volume \(V_{b}\) due to the breakup of bubbles having a volumn larger than \(V_{b}\); the second term is the death rate of bubbles of volume \(V_{b}\) due to breakup; \(\Omega(V_{b})\) is the breakup rate of bubbles of volume \(V_{b}\), \(m({V_{b}}^{\prime})\) is the mean number of daughter bubbles produced by the breakup of a parent bubble of volume \(V_{b}\)', and \(P(V_{b}\), \(V_{b}\)') is the probability density function of daughter bubbles produced upon the breakup of a parent bubble with volume \(V_{b}\)'. Neglecting the interfacial mass transfer, the reaction contribution, the phase variations, and the changes due to the pressure gradients, the coalescence and breakup rates are the only phenomena occurring. In addition, under the assumption of a steady-state, from Equation (4) follows Equation (5):  
\[u_{b,i}\frac{\partial n_{i}}{\partial z}=S_{i}(\overset{\rightarrow}{x},V_{b} )=S_{c}+S_{b} \tag{5}\]  
Equation (5) can be written in terms of the bubble diameter:  
\[u_{b,i}\frac{\partial n_{i}}{\partial z}=S_{i}(\overset{\rightarrow}{x},d_{ eq})=S_{c}+S_{b} \tag{6}\]  
\(S_{c}\) and \(S_{b}\) are computed by modeling the bubble breakup and coalescence rates.