### Theory and Modelling  
Based on the model of isotropic turbulence developed by [PERSON] [45] the following expression (Equation (1)) was derived for the maximum stable droplet diameter in a fully turbulent flow at high Reynolds numbers:  
\[\overline{u}^{2}=C_{1}(\overline{\epsilon}\cdot d_{\text{max}})^{2/3} \tag{1}\]  
The mean relative velocity \(\overline{u}\), caused by turbulent pressure fluctuations in the direction of the flow, is proportional to the mean energy dissipation \(\overline{\epsilon}\) and the maximum stable droplet diameter \(d_{\text{max}}\) and a correlation coefficient \(C_{1}\). Based on this theory, [PERSON] [46] derived a Weber number \(We_{(d)}\) for the maximum stable droplet diameter:  
\[We_{(d)}=\frac{E_{kin}}{E_{\sigma}}=\frac{\rho_{c}\cdot\overline{u}^{2}\cdot d }{\sigma_{d,c}} \tag{2}\]  
Herein \(E_{kin}\) is the turbulent kinetic energy and \(E_{\sigma}\) is the surface energy, with the density of the continuous phase \(\rho_{d}\), the mean relative velocity \(\overline{u}\), the drop diameter \(d\) and the interfacial tension \(\sigma_{d,c}\). If the turbulent kinetic energy and the surface energy are equal, the droplet is stable. According to [PERSON] a droplet remains stable at a Weber number  
Figure 2: Endoscopic probe at laboratory scale. The mounting flange is adaptable to DN40 or larger. \(We_{(d)}\leq 1\), otherwise it breaks up into one or more smaller droplets [46]. Substituting Equation (1) in Equation (2) gives:  
\[We_{(\mathrm{dmax,crit})}=\frac{\rho_{c}C_{2}\bar{\epsilon}^{\,2/3}d_{\mathrm{max }}^{5/3}}{\sigma_{d,c}} \tag{3}\]  
and rearranged for \(d_{\mathrm{max}}\) is:  
\[d_{\mathrm{max}}=C_{2}\cdot We_{(\mathrm{dmax,crit})}^{-0.6}\left(\frac{\sigma _{d,c}}{\rho_{c}}\right)^{0.6}\bar{\epsilon}^{-0.4}=C_{2}\left(\frac{\sigma_{ d,c}}{\rho_{c}}\right)^{0.6}\bar{\epsilon}^{-0.4} \tag{4}\]  
with the correlation coefficient \(C_{2}\). In stirred tanks, \(\bar{\epsilon}\) is independent of the liquid properties according to the theory of isotropic turbulence:  
\[\bar{\epsilon}=KN^{3}D^{2} \tag{5}\]  
Herein \(N\) is the stirrer speed, \(D\) is the impeller diameter and \(K\) is a correlation coefficient that depends on the impeller design. With Equations (4) and (5) a basic model for droplet breakup in stirred vessels by [PERSON] and [PERSON] [47] is as follows:  
\[\frac{d_{\mathrm{max}}}{D}=C_{3}\left(\frac{\rho_{c}N^{2}D^{3}}{\sigma_{d,c}} \right)^{-0.6}=C_{3}We_{T}^{-0.6} \tag{6}\]  
This results in a proportionality of \(d_{\mathrm{max}}\propto We_{T}^{-0.6}\), whereby \(We_{T}\) is the Weber number of the tank and isotropic turbulence is assumed. Nowadays, stirred tank drop diameters are usually described by the Sauter mean diameter (see Section 2.4). In the work of [PERSON] [48], the mean diameters, e.g., \(d_{50}\) or the Sauter mean diameter \(d_{32}\) are linearly related to the maximum stable diameter via a correlation coefficient \(C_{4}\)[49]:  
\[d_{32}=C_{4}d_{\mathrm{max}} \tag{7}\]  
This approach is much more practical for describing the droplet size, since the Sauter mean diameter is proportional to the specific surface area being essential for describing mass transfer processes. The equations above only describe droplet breakage and are only valid at low holdup where coalescence is negligible. As technically relevant phase fractions are significantly higher, drop-drop interactions will occur, resulting in coalescence. As to that, Equation (6) is extended by a linear function considering the influence of the phase fraction [50]:  
\[\frac{d_{32}}{D}=f_{(\varphi)}C_{4}We_{T}^{-0.6}=C_{4}(1+C_{5}\varphi)We_{T}^ {-0.6}=C_{4}(1+C_{5}\varphi)We_{T}^{C_{5}} \tag{8}\]  
In this commonly used approach, the correlation coefficient \(C_{4}\) reflects the impeller design, while \(C_{5}\) is for coalescence. The exponent for the Weber number in Equation (6) is valid in systems without coalescence, and some authors found \(C_{9}\) as a function of the phase fraction \(\varphi\)[14; 17]. [PERSON] et al.