## 3 Materials and Methods
### Coalescence Rate Modeling
#### 3.2.1 Collision Frequency  
We suppose that the collision frequency \(\vartheta_{ij}\) is constituted by three physical phenomena:  
1. turbulent fluctuations, \(\vartheta_{ij}\)T (turbulent fluctuations collision frequency);
2. buoyancy, \(\vartheta_{ij}\)B (buoyancy collision frequency);
3. laminar shear, \(\vartheta_{ij}\)L (laminar shear collision frequency).  
We assume that the contributions of the three mechanisms are cumulative; therefore, the overall \"collision frequency\" reads as:  
\[\vartheta_{ij}=\vartheta_{ij}^{T}+\vartheta_{ij}^{B}+\vartheta_{ij}^{L} \tag{8}\]  
A. Turbulent Fluctuations  
The turbulent fluctuations collision frequency is written as a function of the bubble density number, velocity, and size [10]:  
\[\vartheta_{ij}^{T}=n_{i}n_{j}A_{ij}(u_{b,T,i}+u_{b,T,j})^{1/2} \tag{9}\]  
where \(A_{ij}\) is the collision cross-sectional area of the bubbles, defined as follows [9]:  
\[A_{jj}=\frac{\pi}{4}(r_{b,i}+r_{b,j})^{2} \tag{10}\]where \(r_{b,i}\) and \(r_{b,j}\) are the radii of the bubbles. \(n_{i}\) and \(n_{j}\) are the bubble density numbers:  
\[n_{i}=\frac{\varepsilon_{C}r_{c}^{2}H_{c}}{\frac{1}{6}d_{eq,i}^{3}}x_{i} \tag{11}\]  
where \(H_{c}\) and \(r_{c}\) are the column height and radius, respectively; \(x_{i}\) is the bubble density number; \(\varepsilon_{G}\) is the gas holdup. Then, \(\varepsilon_{G}\) can be computed, for example, by the correlation of Hughmark [11]:  
\[\varepsilon_{G}=\frac{1}{2+\left(\frac{0.35}{U_{G}}\right)\left(\frac{\mu_{c }r}{72}\right)^{1/3}} \tag{12}\]  
In Equation (9), the bubble velocities (\(u_{b,T,i}\) and \(u_{b,T,j}\)) are computed as a function of the energy dissipation rate, \(\varepsilon\), and the bubble equivalent diameter [12]:  
\[u_{t,i}=1.4\varepsilon^{1/3}d_{eq,i}^{1/3} \tag{13}\]  
The energy dissipation rate, \(\varepsilon\), is computed as follows [13]:  
\[\varepsilon=U_{G}g \tag{14}\]  
where \(g\) is the gravitational acceleration.  
B. Buoyancy  
Bubbles with different sizes have different rise velocities and, thus, collisions may occur. The buoyancy collision frequency is given by [14]:  
\[\theta_{ij}^{B}=n_{i}n_{j}A_{jj}(u_{b,B,i}+u_{r,B,j}) \tag{15}\]  
The bubble rise velocities (\(u_{b,B,i}\) and \(u_{b,B,j}\)) are evaluated as a function of the bubble sizes and the physical properties of the system [15]:  
\[u_{b,B,i}=\left[2.14\frac{\sigma}{\rho_{L}d_{eq,i}}+0.505 gd_{eq,i}\right]^{1/2} \tag{16}\]  
C. Laminar Shear  
At low \(U_{G}\), the bubbles can be considered uniformly distributed in the cross-sectional area of the column. When \(U_{G}\) increases, they tend to move toward the center of the column causing an annular region of downward liquid flow. Collisions between bubbles located in regions characterized by different liquid velocities may occur. The laminar shear collision frequency is given by [14]:  
\[\theta_{ij}^{S}=n_{i}n_{j}\frac{4}{3}\left(\frac{d_{eq,i}}{2}+\frac{d_{eq,j}} {2}\right)\left(\frac{dU_{b,S}}{dr}\right) \tag{17}\]  
The last term indicates the bubble shear stress rate [8] and reads as follows:  
\[\left(\frac{dU_{b,S}}{dr}\right)\approx\frac{0.787(gd_{c}U_{G})^{1/3}}{\frac{ d_{c}}{2}} \tag{18}\]  
where \(d_{c}\) is the diameter of the bubble column.