Two-equation population balance model for turbulence-induced coalescence and breakage physics in bubbly flows

The current article introduces a generic mathematical framework for developing a two-equation population balance model (PBM) for multiphase flows. The benefit of such a model is obvious: traditional PBMs routinely employ population bin equations easily numbering north of fifteen or twenty; the current approach drastically reduces the number to just two. The main premise of the PBM is the assumption of a lognormal distribution for the particle size. The first and second moments of the number density provide the required input to completely specify the particle size distribution when the volume fraction of the dispersed phase is known. While there is an instance of a similar approach existing in the literature, the current investigation differs significantly on a couple of important points. First, a new, mathematically rigorous strategy has formulated a considerably distinct source term that accounts for the all important coalescence physics. The second important highlight of this study is a new Gaussian quadrature procedure that has been custom developed for the improper integrals that arise naturally in the coalescence and breakage kernel terms in the integrodifferential equations describing the PBM. This integration procedure is critical in significantly enhancing the computational efficiency of the model. A detailed computational investigation further reveals that a normal numerical integration procedure can fritter away all the practical advantages of the model, thereby emphasizing the significance of the custom-developed integration technique. The paper has employed four different coalescence kernels within the new two-equation PBM framework. In addition to applying a well-known breakage kernel, the study has also developed a new, simplified, turbulence-induced, empiricism-based breakage kernel for utilization with the new PBM. Comparative studies of the PBM with the different kernels produced reasonably good agreement with existing experimental data in the literature. The paper concludes by enumerating guidelines for further research in the field.