This paper presents a probability density function (PDF) form of the population balance equation (PBE) for polysized and polyshaped droplets and solid particles in turbulent flows. A key contribution of this paper lies in the inclusion of an explicit consideration of the inertial effects and the shape of particles in the PDF-PBE formulation. The number density is taken as a function of droplet or particle size (volume) and shape as well as space and time. Potentially, other particle properties could also be included in the formulation. Inertial effects are quantified through the Stokes number, leading to accurate modelling of the different trajectories that are followed by droplets and/or particles with different sizes and shapes. To treat these effects, a new affordable approach is proposed and referred to as the method of Stokes binning. Here, the inertial dispersed elements are accelerated due to fluid dynamic forces associated with an averaged Stokes number in each bin. The model is validated against two data sets. The first data set includes a series of numerical test cases involving the injection of polyshaped droplets ranging in size from 1 to 50 into a turbulent jet resulting in inlet Stokes numbers ranging from 0.03 to 75.2. The second data set consists of an experimental case focusing on the dispersion of 60 and 90 spherical droplets in a turbulent round jet, resulting in inlet Stokes numbers of 53 and 122, respectively. The results confirm the ability of the approach to accurately model the polysized and polyshaped droplet dispersion using as few as eight Stokes bins. This approach has the potential to greatly reduce the computational cost of modelling the evolution of inertial droplets and particles in turbulent flows.
- multiphase and particle-laden flows