(444j) A Quadrature-Based Model for Polydisperse Laminar Bubbly Flows
In the proposed approach, the disperse phase is assumed to be mono-kinetic, due to the moderate Stokes number of bubbles. Because of this assumption, only the first-order velocity moments are considered, while p moments in size are transported. The major novelty of this contribution consists in the solution algorithm, in which the moment method is formulated as a function of the mean gas-phase velocity, and of the deviation of the bubble size velocity from the mean. This allows a scheme that ensures the boundedness of the numerical solution, and, in particular, of the volume fraction, to be formulated. Key element of the algorithm is the treatment of the moment advection term, which is split into two stages. First moments of the disperse phase are advected with respect to the deviation velocity with respect to the mean using kinetic fluxes. Then, a continuity and momentum equation for the disperse phase, similar to those used in traditional two-fluid solvers are solved and coupled to the equations for the liquid phase. These equations are solved with an iterative procedure (Spalding, 1980), supplemented by a bounded scheme based on the Multi-dimensional Universal Limiter for Explicit Solution (MULES) (Weller, 2006) for the volume fraction. Notable difference, with respect to the traditional two-fluid model is the definition of the momentum exchange term, which is function of the bubble size distribution in the proposed approach, while it relies on a mean bubble size diameter in the traditional two-fluid model. Once the updated mean volume fraction and mean velocity are found, size and joint size-velocity moments are updated as a consequence of the mean transport. The effect of the force term on velocity moments is included by directly integrating the ordinary differential equation for the velocity abscissae, as discussed in Fox (2008).
The proposed numerical approach was tested considering two bubble columns studied in the literature. A mono-disperse case (Pfleger et al., 1999) was used to verify the approach in the limit case of monodisperse bubbles, to ensure the solution reproduces the results of the equivalent two-fluid model. Results are in agreement with the predictions of the two-fluid model implementation available in OpenFOAM 4 (OpenFOAM, 2016). A polydisperse case (Díaz et al., 2008) was also considered, obtaining results in agreement with their experimental data for what concerns the gas mean axial velocity and hold-up.
The research discussed in this work was supported by the National Science Foundation of the United States, under the SI2 â SSE award NSF â ACI 1440443.
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