(444j) A Quadrature-Based Model for Polydisperse Laminar Bubbly Flows | AIChE

# (444j) A Quadrature-Based Model for Polydisperse Laminar Bubbly Flows

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## Authors

Iowa State University
Iowa State University
Iowa State University
A quadrature-based moment method for the simulation of laminar polydisperse bubbly flow is developed and implemented into OpenFOAM®as part of the OpenQBMM project (OpenQBMM, 2017; Passalacqua et al., 2017). The proposed approach aims at addressing one key limitation of previously adopted quadrature-based methods coupled to gas-liquid systems (Sanyal et al., 2005, 1999): the use of the local mean velocity of the gas phase to advect bubbles of different sizes. The proposed approach represents an evolution of the mass-velocity moment method proposed in Yuan et al., (2014), which presented numerical difficulties when the continuous phase disappeared and the disperse phase became continuous.

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.

Acknowledgments

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.

References

DÃ­az, M.E., Montes, F.J., GalÃ¡n, M.A., 2008. Experimental study of the transition between unsteady flow regimes in a partially aerated two-dimensional bubble column. Chemical Engineering and Processing: Process Intensification 47, 1867â€“1876. doi:10.1016/j.cep.2007.10.012

Fox, R.O., 2008. A quadrature-based third-order moment method for dilute gas-particle flows. Journal of Computational Physics 227, 6313â€“6350. doi:10.1016/j.jcp.2008.03.014

OpenFOAM, 2016. The OpenFOAM Foundation [WWW Document]. OpenFOAM. URL http://openfoam.org/

OpenQBMM, 2017. An open-source implementation of Quadrature-Based Moment Methods [WWW Document]. URL https://www.openqbmm.org/

Passalacqua, A., Laurent, F., Madadi-Kandjani, E., Heylmun, J.C., Fox, R.O., 2017. An open-source quadrature-based population balance solver for OpenFOAM.

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Sanyal, J., VÃ¡squez, S., Roy, S., Dudukovic, M.P., 1999. Numerical simulation of gasâ€“liquid dynamics in cylindrical bubble column reactors. Chemical Engineering Science 54, 5071â€“5083. doi:10.1016/S0009-2509(99)00235-3

Spalding, D.B., 1980. Numerical Computation of Multi-phase fluid flow and heat transfer, in: Taylor, C. (Ed.), Recent Advances in Numerical Methods in Fluids. Pineridge Press.

Weller, H.G., 2006. Bounded Explicit and Implicit Second-Order Schemes for Scalar Transport.

Yuan, C., Kong, B., Passalacqua, A., Fox, R.O., 2014. An extended quadrature-based mass-velocity moment model for polydisperse bubbly flows. The Canadian Journal of Chemical Engineering 92, 2053â€“2066. doi:10.1002/cjce.22006