# Fluid Dynamics in a Continuous Pump-Mixer
## 1 Introduction
Modeling a DSD in general is usually performed via semi-empirical approaches based on the theory of isotropic turbulence of [PERSON] [17] and [PERSON] [18]. For liquid-liquid systems, the most of it is related to stirred-batch vessels [6; 19; 20; 21; 22] and in rare cases with pump-mixers, as by [PERSON] et al. [6]. Most of the literature refers to highly diluted dispersions and only a few studies relate to higher phase fractions [22]. The most common approach [2; 23; 24] initially developed by [PERSON] [25] for high-phase fractions up to \(\varphi\leq 30\) vol.-% is given as:  
\[\frac{d_{32}}{D}=f_{(\varphi)}C_{4}We_{T}^{-0.6}=C_{4}(1+C_{5}\varphi)We_{T}^ {-0.6}=C_{4}(1+C_{5}\varphi)We_{T}^{C_{\varphi}} \tag{1}\]  
Equation (1) relates the Sauter mean diameter, \(d_{32}\), for a given impeller diameter, \(D\), and the Weber number, \(We_{T}\), of the tank. It is modified by an exponent, \(C_{9}\), which can be theoretically derived as \(C_{9}=-0.6\)[22] in a droplet breakage domain. At higher phase fractions, \(\varphi\), increased coalescence leads to other values [26]. It should be noted here that \(\varphi\) describes the volume-based fraction in a batch tank as:  
\[\varphi=\frac{V_{d}}{V_{d}+V_{c}} \tag{2}\]  
where \(V_{d}\) is the total volume of the dispersed, and \(V_{c}\) is the volume of the continuous phase. For ideally mixed batch vessels, the integral holdup \(\overline{\varphi}\) equals the phase fraction \(\varphi\). The parameters \(C_{4}\) and \(C_{5}\) depend on the geometry of the vessel, the coalescence tendency of the chemical system, and especially on the impeller type and were derived in our previous publication [22]. The definition of the Weber number is as:  
\[We_{T}=\left(\frac{\rho_{c}N^{2}D^{3}}{\sigma_{d,c}}\right) \tag{3}\]It depends on the impeller speed, \(N\), the impeller diameter, \(D\), the continuous phase density, \(\rho_{c}\), and the interfacial surface tension between the dispersed and continuous phase, \(\sigma_{d,c}\). Equation (1) is valid for stirred batch vessels but can also be adapted for pump-mixers due to the similar apparatus design. [PERSON] et al. [6] used the approach in Equation (1) with a quadratic function for the phase fraction and introduced the volume flow rate via the mean residence time \(\overline{\tau}\) as an exponent in an exponential function shown in Equation (4). \[\frac{d_{32}}{D}=C_{4}\Big{(}1+C_{5}\varphi_{F}+C_{6}\varphi_{F}^{2}\Big{)}We_ {T}^{C_{9}}\cdot e^{C\gamma\overline{\tau}}\text{ with }\overline{\tau}=\frac{V_{T}}{\dot{V}_{c}+\dot{V}_{d}} \tag{4}\]  
\(V_{T}\) is the tank volume. \(C_{6}\) is introduced to represent the quadratic trend of the droplet diameters over the phase fraction and \(C_{7}\) to model the volume flow as mean residence time \(\overline{\tau}\). Since this is a continuous apparatus, there is often a difference between the feed phase fraction \(\varphi_{F}\) and the actual integral holdup \(\overline{\Phi}\). Therefore, it should be noted that in the analysis the authors refer to the feed phase ratio \(\varphi_{F}\), which can be significantly different from the holdup. The feed phase ratio is defined as:  
\[\varphi_{F}=\frac{\dot{V}_{d}}{\dot{V}_{d}+\dot{V}_{c}}=\frac{\dot{V}_{d}}{ \dot{V}_{tot}} \tag{5}\]  
The mean residence time \(\overline{\tau}\) is calculated here using the tank volume \(V_{T}\) and the volume flows \(\dot{V}_{c}\). and \(\dot{V}_{d}\) in steady state. The authors do not use the theoretical value for the Weber exponent, instead, they correlated \(C_{9}\) to be \(-0.85125\). Available literature provides a certain insight into the behavior of pump-mixers. However, predictive simulations of fluid dynamic (flow rates, flow patterns, local energy dissipation) and hydrodynamic behavior (local droplet size distributions, local and integral holdup) are not yet possible. Another problem is that some publications use pumps to operate multiphase pump-mixers [14; 16], so there is a change in residence time, dispersing properties, and the resulting DSD.