In this work, Eulerian transport equations are derived for a polydispersed cloud of droplets in a turbulent carrier flow, which take into account the effects of polydispersion and coalescence. The approach is an extension of the Direct Quadrature Method Of Moments (DQMOM) formalism initially proposed by Marchisio and Fox (2005, J. Aerosol Sci., 36, pp. 43–95). In the initial DQMOM approach of Marchisio and Fox, the effects of polydispersion and coalescence can only be accounted for in the mass balance equations. By combining the DQMOM approach and the joint fluid-particle pdf approach of Simonin (1996, Von Karman Lecture Notes), Eulerian transport equations can be written in the frame of the DQMOM formalism for the velocity, agitation and fluid-particle covariance, which quantities are required to predict the behavior of a cloud of droplets in a turbulent flow. It is formally shown that the Eulerian transport equations in the DQMOM framework are the same equations as those in the multi-class Eulerian approach, except that now there is a collision term in the equations. The collision term due to coalescence can be easily expressed due to the assumptions made in the DQMOM framework, and it is shown that it couples the transport equations of the different classes and dispersed phase statistics, according to the change of number, mass and momentum during coalescence.
New developments of the Extended Quadrature Method of Moments to solve Population Balance Equations
