# Fluid Dynamics in a Continuous Pump-Mixer
## 3 Results
### Modelling  
Parameter Analysis  
A standard approach according to Equation (1) by [PERSON] [25] for stirred batch vessels was utilized because Equation (4) by [PERSON] et al. [6] predicts increasing Sauter mean diameters at increased impeller speeds. This is due to relatively large flow rates where the energy dissipation at larger impeller speed decreased significantly due to the flow rate. The drawback of Equation (1) by [PERSON] [25] is that no flow rate is considered as it was developed for batch vessels and a modified approach led to Equations (9) and (10). The dependencies of the correlation constants on impeller speed are depicted in Figure 12 and corresponding values and confidence intervals are given in Table 3.  
Figure 11: (**a**) Sauter mean diameters at measurement position \(MP_{1}\) at a feed phase ratio of 30 vol.-%. (**b**) Measured DSD at different impeller speed at measurement position \(MP_{1}\), a feed phase ratio of \(\varphi_{F}\) = 30 vol.-%, and a total flow rate \(\dot{V}_{tot}\) = 1.33 L/min. The lines are a guide to the eye. All data of Exp. 1 for \(N=375\) and 425 rpm were used to fit Equation (9) for those impeller speeds. For \(N=400\) rpm, the Sauter mean diameters and process parameters were taken from Exp. 2, because the measured Sauter mean diameters had less fluctuations in the data. Only the set for \(\varphi_{F}=20\) and \(N=400\) rpm was excluded, since the Sauter mean diameters were found to be sensitive in this region and had larger deviations between Exp. 1 and Exp. 2. The ordinate, \(A_{0}\), and the slope, \(A_{1}\), at each impeller speed showed approximately similar values. A larger variation was found with \(B_{(\dot{V}_{tot})}\) where the slope \(B_{1}\) and the offset \(B_{0}\) varies at different impeller speeds and their values were the largest for \(N=425\) rpm. The slope of \(B_{(\dot{V}_{tot})}\) is questionable because \(B_{1}\) had an extremely large confidence interval (changing from positive to negative values). We can conclude that the coefficients for \(A_{(\dot{V}_{tot})}\) only have a small dependency on \(N\) and \(\dot{V}_{tot}\), strongly aligning with the Hinze-Kolmogorov theory [18] where \(A\) only depends on the impeller type. The coefficients \(B_{0}\) and \(B_{1}\) are more sensitive to the process parameters, especially at small flow rates (see Figure 12b). Nevertheless, when calculating the mean coefficients for the slope and the offset of \(A_{(\dot{V}_{tot})}\) and \(B_{(\dot{V}_{tot})}\) (see Table 3), one overall equation can sufficiently describe the data. \[ \begin{array}{c}\frac{d\alpha_{2}}{D}=A_{(\dot{V}_{tot})}\bigg{(}1+B_{(\dot {V}_{tot})}\varphi_{F}\bigg{)}We_{T}^{-0.6}=\\ \Big{(}1297\;\text{s}\cdot\text{m}^{-3}\cdot\dot{V}_{tot}+0.077)\cdot\Big{[}1+ \Big{(}-22,589\;\text{s}\cdot\text{m}^{-3}\cdot\dot{V}_{tot}+3\Big{)}\varphi_{ F}\Big{]}We_{T}^{-0.6}\end{array}  \tag{11}\]  
or as an apparatus independent form using the mean residence time \(\overline{\tau}\):  
\[ \begin{array}{c}\frac{d\alpha_{2}}{D}=\Big{(}A_{0}\frac{1}{\overline{\tau}} +A_{1}\Big{)}\cdot\Big{(}1+\Big{(}B_{0}\cdot\frac{1}{\overline{\tau}}+B_{1} \Big{)}\varphi_{F}\Big{)}We_{T}^{-0.6}=\\ \Big{(}33\;\text{s}\cdot\frac{1}{\overline{\tau}}+0.077)\cdot\Big{[}1+\Big{(}- 575\;\text{s}\cdot\frac{1}{\overline{\tau}}+3\Big{)}\varphi_{F}\Big{]}We_{T}^ {-0.6}\end{array}  \tag{12}\]  
A plot of Equation (11) and the parity plot are shown in Figure 13a,b. It shows that the new correlation is capable of describing all of the experimental data. Larger deviations were especially found at higher values of \(\varphi_{F}\) at an impeller speed of \(N=400\) rpm as discussed above (see Figure 11b). The good quality of the correlation can also be seen in the parity plot in Figure 13b where 152 of 155 experimentally measured Sauter diameters can be described within a 15% error range. The three values outside the 15% error range are caused by an accumulation of the error range of the shadowgraphic probe and the fitting.