Commun. Comput. Phys., 10 (2011), pp. 823-843. |
Particle Collisions in a Lumped Particle Model Omar al-Khayat ^{1*}, Are Magnus Bruaset ^{1}, Hans Petter Langtangen ^{2} 1 Computational Geosciences, CBC, Simula Research Laboratory, P.O. Box 134, NO-1325 Lysaker, Norway; and Department of Informatics, University of Oslo, P.O. Box 1080, Blindern, NO-0316 Oslo, Norway.2 Department of Informatics, University of Oslo, P.O. Box 1080, Blindern, NO-0316 Oslo, Norway; and Center for Biomedical Computing, Simula Research Laboratory, P.O. Box 134, NO-1325 Lysaker, Norway. Received 29 January 2010; Accepted (in revised version) 26 November 2010 Available online 13 June 2011 doi:10.4208/cicp.290110.261110a Abstract 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 $C_D$ that influences each particle in the flow. This coefficient is given by $C_D=(1- \phi)^{-n}C_D^*$, where $\phi$ is the volume fraction of particles, $C_D^*$ is the drag coefficient for a single particle, and $n \simeq 4.67$ for creeping flow. We obtain an approximation for $C_D/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. AMS subject classifications: 76M28, 70F35Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Lumped particle model, particle collision, hindered settling. *Corresponding author. Email: omark@simula.no (O. al-Khayat), arem@simula.no (A. M. Bruaset), hpl@simula.no (H. P. Langtangen) |