Particle collision modeling – A review
- a Centre for Research in Computational and Applied Mechanics, University of Cape Town, Rondebosch, South Africa
- b Centre for Minerals Research, University of Cape Town, Rondebosch, South Africa
- Received 26 October 2010
- Accepted 17 March 2011
- Available online 14 April 2011
Abstract
Over the past 100 years particle collision models for a range of particle inertias and carrier fluid flow conditions have been developed. Models for perikinetic and orthokinetic collisions for simple, laminar shear flows as well as collisions associated with differential sedimentation are well documented. Collision models developed for turbulent flow conditions are demarcated on the one side with the model of Saffman and Turner (1956) associated with particles exhibiting zero inertia and on the other side with the model of Abrahamson (1975) for particle velocities that are completely decorrelated from the carrier fluid velocities. Various attempts have been made to develop universal collision models that span the entire range of inertias in a turbulent flow field. It is a well-accepted fact that models based on a cylindrical as opposed to a spherical formulation are erroneous. Furthermore, the collision frequency of particles exhibiting identical inertias are not negligible. Particles exhibiting relaxation times close to the Kolmogorov time scale of the turbulent flow are subject to preferential concentration that could increase the collision frequency by up to two orders of magnitude. In recent years the direct numerical simulation (DNS) of colliding particles in a turbulent flow field have been preferred as a means to secure the collision data on which the collision models are based. The primary advantage of the numerical treatment is better control over flow and particle variables as well as more accurate collision statistics. However, a numerical treatment places a severe restriction on the magnitude of the turbulent flow Reynolds number. The future development of more comprehensive and accurate collision models will most likely keep pace with the growth in computational resources.
Highlights
► Particle collision is an important sub-process in a range of natural and industrial processes. ► This paper reviews particle collision models developed over the past 100 years. ► Models range from simple laminar shear flows to ‘universal’ collision models for turbulent flow.
Keywords
- Collision;
- Modeling;
- Turbulence
1. Introduction
Particle collision constitutes an important sub-process in a wide range of natural occurring as well as industrial processes where the agglomeration and/or breakup of particles is of importance. These processes involve a continuous phase (liquid or gas) and one or more dispersed phases (solid and/or liquid and/or gas). Where the continuous phase is a liquid, as is common in mineral processing, the dispersed phase/s may be a solid (particle) and/or a liquid (droplet) and/or a gas (bubble).
Natural processes characterized by particle collision range from planetary formation from protoplanetary nebula Champney et al. (1995) to the formation of rain drops in clouds Pinsky et al. (2000).
Particle collision is relevant to many industrial processes. Examples include, amongst others, the aggregation of solid particles in flocculation/sedimentation Balthasar et al. (2002), the rate of coalescence of droplets/bubbles in liquid and gas dispersions Kamp et al., 2001 and Narsimhan, 2004, the interaction between particles and bubbles in froth flotation Schubert, 1999, Bloom and Heindel, 2002 and Deglon, 2005, the secondary nucleation of crystals in crystallization ten Cate et al. (2001) and soot formation in furnaces Balthasar et al. (2002).
Particle collision is particularly relevant to mineral processing as turbulent multiphase systems are common and many sub-processes are controlled/influenced by turbulent collision. Most of the examples quoted previously are relevant to common unit operations in mineral processing (e.g. thickening, solvent extraction, froth flotation, crystallization).
This paper is an effort to present the various approaches used to model dispersed phase collision as it pertains to industrial processes. The discussion commences with the presentation of the general collision modeling approach with special attention being given to the definition and interpretation of concepts central to the modeling approach. This is followed by a listing and discussion of appropriate collision models with emphasis being placed on formulations for collision frequency. The expressions for collision frequency are augmented, in the case of turbulent flow, with formulations of velocity fluctuations.
2. Collision modeling
Smoluchowski (1917), using a population balance approach, formulated the following expression to quantify the agglomeration of particles due to fluid agitation
Analytical solutions to Eq. (1) exist where the collision kernel takes on a simple form, such as a constant value for instance. However, in most practical applications the kernel takes on a significantly more complex form that depends, to a large extent, on the flow and particle kinematics.
The primary collision mechanisms are listed in Table 1 where the Stokes number, St, contrasts the particle relaxation time, τi, with that of the smallest scales of fluid motion, τη. In the case of fully developed turbulent flow the latter will constitute the Kolmogorov micro time scale, where ν is the fluid kinematic viscosity and ϵ the dissipation rate of turbulent kinetic energy per unit mass.