Solving GPBE
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Recently, I am trying to understand the state of the art for solving GPBE. However, I have no experience in this continum approach. So, I wrote a mini-paragraph and posted it here asking for OpenFOAMers to help discuss remaining issues and miss understandings of the literature. Thanks in advance.
- The generalized population balance equation (GPBE) (Marchisio and Fox, 2013) describes the evolvement of particle number density function in dimensions of time, space, velocity and particle properties, such as size. However, it is infeasible to directly solve this integral-differential equation due to its high dimensionality and the unclosed term such as particle collisions.
- To reduce the transport equation into only time and space dimensions, both the velocities and particle properties should be integrated out. This results in at least four dimensions of moments. High order multivariable moment-inversion algorithm used to find the weights and abscissas is very challenging (Fox, 2008). One approach to solve this problem is assuming that the velocity depends on particle size (Buffo et al., 2013). Then, for a specific particle size, the velocity distribution is integrated out and result in a macro scale averaged particle velocity conditioned on particle size. This macro-scale velocity is then solved using hydrodynamic models such as two-fluid model or multi-fluid model (Fan et al., 2004; Buffo and Marchisio Daniele, 2014). Note that, the momentum equation of the hydrodynamic model used in the two-fluid model or multi-fluid model is derived from Boltzmann equation which doesn’t include the effect of particle size evolution. Solving GPBE without these assumptions is rarely seen due to its high dimensionality.
