## 2 Model Description and Methods
### Bubble Population Balance Model (PBM)  
The size of the bubbles distributed in the reactor varies depending on the location. This inhomogeneity is the key reason why the population balance model should be applied to describe the system.  
Let \(n(V,t)\) represent the number density of bubbles of volume \(V\) at time \(t\). The population balance equation is:  
\[\frac{\partial}{\partial t}[n(V,t)]+\
abla\cdot\left[\overset{\rightarrow}{u }_{g}n(V,t)\right]=B_{ag}-D_{ag}+B_{br}-D_{br} \tag{10}\]  
where \(D_{ag}\), \(B_{ag}\) and \(D_{br}\), \(B_{br}\) are the death, birth rate of bubbles by reason of aggregation and death, birth rate of bubbles by reason of breakage, respectively. These rates can be expressed as:  
\[B_{ag}=\frac{1}{2}\int_{0}^{V_{p}}Q(V_{p}-V_{q},V_{q})n(V_{p}-V_{q},t)n(V_{q}, t)dV_{q} \tag{11}\]  
\[D_{ag}=\int_{0}^{\infty}Q(V_{p},V_{q})n(V_{p},t)n(V_{q},t)dV_{q} \tag{12}\]\[B_{br}=\int_{V_{p}}^{\infty}g\big{(}V_{p},V_{q}\big{)}n\big{(}V_{q},t\big{)}dV_{q} \tag{13}\]  
\[D_{br}=n\big{(}V_{p},t\big{)}\frac{1}{2}\int_{0}^{V_{p}}g\big{(}V_{p},V_{q}\big{)} dV_{q} \tag{14}\]  
where \(Q\big{(}V_{p},V_{q}\big{)}\) is the specific aggregation rate and \(g\big{(}V_{p},V_{q}\big{)}\) is the specific breakage rate. In this study, the [PERSON] and Svendsen model [26] was adopted to express the specific breakage rate as:  
\[g\big{(}V_{p},V_{q}\big{)}=0.9238\big{(}1-\alpha_{g}\big{)}\Bigg{(}\frac{ \epsilon}{\bar{q}_{p}^{5}}\Bigg{)}^{\frac{1}{5}}\int_{\xi_{min}}^{1}\frac{(1+ \xi)^{2}}{\xi^{11/3}}\exp\Bigg{(}-\frac{12\Big{(}f^{\frac{3}{3}}+(1-f)^{\frac{ 3}{3}}-1\Big{)}\sigma}{\beta\rho_{l}\epsilon^{\frac{2}{3}}d_{p}\,^{\frac{5}{3} }\xi^{\frac{11}{3}}}\Bigg{)}d\xi \tag{15}\]  
This model considers the integral over the size of eddies \(\lambda\) hitting the bubble with a diameter of \(d\). \(\xi\) is the dimensionless size of eddies \(\xi=\lambda/d_{p}\). \(f=V_{q}/V_{p}\), \(\alpha_{g}\), \(\sigma\), and \(\beta=2.047\) are the size fractions of the bubbles, gas volume fraction, interfacial tension, and constant, respectively.  
The general aggregation kernel, using the Luo aggregation model [26], is defined as the formation rate of a particle in consequence of the collision of two particles having a volume of \(V_{p}\) and \(V_{q}\) each:  
\[Q\big{(}V_{p},V_{q}\big{)}=\omega\big{(}V_{p},V_{q}\big{)}P\big{(}V_{p},V_{q} \big{)} \tag{16}\]  
\[\omega\big{(}V_{p},V_{q}\big{)}=\frac{\pi}{4}\big{(}d_{p}+d_{q}\big{)}^{2}n_{p }n_{q}\overline{u}_{pq} \tag{17}\]  
\[P\big{(}V_{p},V_{q}\big{)}=\exp\Bigg{(}-c_{1}\frac{\Big{[}0.75\big{(}1+x_{pq} ^{2}\big{)}\Big{(}1+x_{pq}^{3}\big{)}\Big{]}^{\frac{1}{2}}}{\Big{(}\frac{\rho _{l}}{\rho_{g}}+0.5\Big{)}^{\frac{1}{2}}\big{(}1+x_{pq}\big{)}^{3}}We_{pq}^{0 5}\Bigg{)} \tag{18}\]  
where \(\omega\big{(}V_{p},V_{q}\big{)}\) is the frequency of collision and \(P\big{(}V_{p},V_{q}\big{)}\) is the probability that the collision results in coalescence. \(\overline{u}_{pq}\) is the characteristic velocity of collision of two bubbles with diameters \(d_{p}\) and \(d_{q}\) and number density \(n_{p}\) and \(n_{q}\), respectively.  
\[\overline{u}_{pq}=\Big{(}\overline{u}_{p}^{2}+\overline{u}_{q}^{2}\Big{)}^{0.5} \tag{19}\]  
\[\overline{u}_{p}=1.43(\epsilon d_{p})^{1/3} \tag{20}\]  
\(c_{1}\) is a constant of order unity, \(x_{pq}=d_{p}/d_{q}\), and the Weber number is defined as follows:  
\[We_{pq}=\frac{\rho_{l}d_{p}\big{(}\overline{u}_{pq}\big{)}^{2}}{\sigma} \tag{21}\]