(161h) A Eulerian-Lagrangian Hybrid Model for the Simulation of the Droplet Size Distribution of Liquid-Liquid Emulsions in Stirred Tank Reactors

The mixture of two or more immiscible liquids is referred to as emulsion, which is commonly used in many industries such as food, pharmaceutical, cosmetic, chemical and petroleum. Predicting the droplet size distribution (DSD) is crucial in particulate flows as it controls the mass transfer and the heat transfer. Commonly a population balance equation (PBE) is employed to describe the breakup and coalescence of individual droplets; however, considering the different drift velocities for the droplets with different sizes, would require unacceptably high computational resources. Therefore, we employ a Lagrangian-Eulerian hybrid model, which enables the efficient evaluation of the poly-disperse liquid-liquid drag force form the local distribution of the different droplet diameters (Schneiderbauer et al., 2016). The latter can be obtained by tracking statistically representative droplet trajectories (parcels) for each droplet diameter class following the dispersed phase velocities stemming from a two-fluid model simulation. Furthermore, a semi-empirical correlation is used to compute the local equilibrium droplet size distribution (log-normal distribution. This correlation is developed based on the Kolmogorov turbulence theory and the Prandtl mixing length (Farzad et al., 2017). If a specific droplet is much larger than the mean droplet size given from the DSD it might be prone to breakup. In this work, we employ the D₉₀ for this threshold, which can be computed in each computational cell from the corresponding local DSD. If a droplet is larger than D₉₀, each droplet represented by the parcel will break into two daughter droplets, where the diameter of the first daughter droplet is given by a random number following the log-normal distribution. Modelling coalescence is developed based on the work of Coulaloglou and Tavlarides (1977). Since collisions between the Lagrangian parcels are not resolved the following strategy is proposed to compute effect of coalescense. First, we introduce a specific number of diameter classes. For each of this diameter classes, we are able to compute the corresponding volume fraction from mapping the data coming from the Lagrangian parcels to the Eulerian grid used for the TFM solution. Second, based on these “imaginary coalescence partners” given from this binning, we are able to compute the individual rates of coalescence. Then the volume of created droplet is stored into a new bin is corresponding to its new diameter class. Finally, we employed bus-stop model to unload the stored volume of the created droplets into an appropriate parcel available in the computational cell (Schneiderbauer et al., 2016).

 We compare the predictions of the proposed hybrid approach with the experimental data of Boxall et al. (2010) and Wang and Calabrese (1986). Simulation results show good agreement with the experimental results for the droplet size distribution.

REFERENCES

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