(untitled) (Communications in Computational Physics, 2011)
This paper presents an extension of the lumped particle model in [1] to include the effects of particle collisions. The lumped particle model is a flexible framework for the modeling of particle laden flows, that takes into account fundamental features, including advection, diffusion and dispersion of the particles. In this paper, we transform a binary collision model and concepts from kinetic theory into a collision procedure for the lumped particle framework. We apply this new collision procedure to investigate numerically the role of particle collisions in the hindered settling effect. The hindered settling effect is characterized by an increase in the effective drag coefficient CD that influences each particle in the flow. This coefficient is given by CD =(1−φ)−nC∗D, where φ is the volume fraction of particles, C∗D is the drag coefficient for a single particle, and n ≃ 4.67 for creeping flow. We obtain an approximation for CD/C∗D by calculating the effective work done by collisions, and comparing that to the work done by the drag force. In our numerical experiments, we observe a minimal value of n = 3.0. Moreover, by allowing high energy dissipation, an approximation for the classical value for creeping flow, n = 4.7, is reproduced. We also obtain high values for n, up to n=6.5, which is consistent with recent physical experiments on the sedimentation of sand grains.