On the use of the population balance equation for the turbulent transport of polydispersed inertial particles

This paper presents a probability density function (PDF) form of the population balance equation (PBE) for polydispersed particles in turbulent flows and, for the first time, it includes explicit consideration of inertial effects in the PDF-PBE formulation. This model has wide application but has particular potential for understanding and improving liquid fuel spray in diesel engines. In this first step towards modelling inertial particles, the number density is taken as a function of particle size (volume) and velocity as well as space and time. Inertial effects are quantified through the Stokes number, leading to accurate modelling of the different trajectories that are followed by particles with different sizes. To treat these effects, a new affordable numerical approach is proposed which we call the binned Stokes method. In this approach, particles accelerate due to fluid dynamics forces associated with an averaged numerical Stokes number in each bin and hence the dimensionality of the model is determined by the number of such bins. A test case is constructed involving the turbulent jet dispersion of particles ranging in size from 1 μm to 100 μm. The results confirm the ability of the approach to accurately model the particle dispersion using as few as six Stokes bins thus offering the potential to greatly reduce the computational cost of the PDF-PBE equations for inertial particles.