An accurate and efficient algorithm to model the agglomeration of macroscopic particles

Highlights

• We present a novel numerical approach to compute particle interaction.

• Van der Waals, electrostatic, and collisional forces and plastic particle deformation is accounted for.

• This algorithm allows to predict accurately and efficiently particle agglomeration.

• Imposed constrain forces prevent spurious velocity fluctuations and potential particle overlapping.

Abstract

The agglomeration of particles during the handling of powders results in caking, lumping or the local accumulation of electrostatic energy which represents a serious hazard to the operational safety of industrial facilities. In the case of dry powders the attraction in-between particles can be mainly attributed to van der Waals and electrostatic forces. Nonetheless, due to the challenges related to the small size and distance of relevant particles and the optical density of powder flows the detailed physical mechanisms of their interaction are so far little investigated. In this paper we present a novel numerical approach which is based on an algorithm developed by Erleben [1] in the field of computer graphics. This algorithm is extended to compute binary and multiple particle interaction with each other and solid surfaces. Therein, besides van der Waals and electrostatic forces also collisional forces and plastic particle deformation is accounted for. The herein presented results demonstrate that this algorithm allows to predict accurately and efficiently whether particles agglomerate or separate depending on their kinetic parameters. In particular, the imposed constrain forces prevent spurious velocity fluctuations and potential particle overlapping in statically overdetermined systems. Simulated test cases reveal how electrostatic and van der Waals forces lead to the growth of structures in case the particle restitution coefficient is sufficiently low.

Introduction

The agglomeration of particles has a wide range of consequences to the handling of powders in an industrial context such as caking or lumping. Adhesive forces in-between particles and between particles and component surfaces can be attributed to various physical origins. In the case of dry powders van der Waals, electrostatic, aerodynamic or magnetic forces may be of importance whereas wet powders are additionally subjected to capillary forces. If electrostatic forces are dominant particle agglomeration also leads to the local accumulation of electrostatic energy which represents a serious hazard to the operational safety of industrial facilities.

A wide range of literature has been contributed to the settling of particles on surfaces. For example, Guha [2] reviews the experimental and numerical efforts regarding the physical mechanisms responsible for the deposition of particles. More specifically, correlations between on the one hand deposition rates and on the other hand the particle relaxation time, diffusion, turbopheresis, surface roughness, electrostatic charges, and thermal diffusion are established. However, this review focuses on airborne particles and does not tackle the actual deposition process, i.e. the forces acting during the contact of a particle with another object. In a similar fashion, Narayanan et al. [3] identified different deposition patterns on the surface of a channel depending on the particle relaxation time. In their theoretical model a particle was assumed to be absorbed by the wall once it was at a distance of less than one particle radius. Subsequently the particle was removed from the simulation. Liu et al. [4], even including magnetic forces in their numerical study of a square duct flow, also assumed a particle to be absorbed once it touched the wall. Also the authors of the present paper and co-workers conducted an amount of numerical investigations with focus on the behavior of charged particles [5], [6], [7], [8], [9], [10], [11]. In all of these studies the assumption of binary particle interaction was taken which prevents the establishment of a connected agglomeration of several particles.

The development of wall layers due to triboelectric charges in a fluidized bed was recently observed by Sippola et al. [12] through a combined experimental and numerical study. Their simulation included electrostatic and contact forces and enabled the prediction of the amount of stable layers. However, their model did not predict a completely static structure at the wall which resulted in a small residual velocity of particles belonging to an immobile deposit. As regards connected agglomerations, Xu et al. [13] set up a simulation by extending a commercial code. They computed collision probabilities and the outcome of collisions based on a Monte-Carlo simulation. If the statistical analysis predicted coalescence they replaced both particles with a new particle of a larger diameter. By doing so, and by neglecting detailed physical mechanisms, they were able to compute the growth of agglomeration in a flow of an industrial scale. Recently, Ray et al. [14] implemented a more detailed concept of wall layer formation in an Eulerian-Eulerian framework. They determined if a particle placed in a certain distance from the wall would have enough electrostatic force acting on it such that friction could resist dropping due to gravity. The distance beyond which this force is lower than a given threshold was assumed to be the wall layer thickness which satisfactorily correlated to their experimental observations.

However, if a deposit or an agglomeration forms the details of the underlying process become very complicated. One complication arises from the fact that particles in contact pose contact forces on each other. Thus, if a large number of particles is in contact with each other the equilibrium state is statically over-determined which makes it difficult so solve for. Additionally, not only forces but effects such as induced charges on each others surface or charge transfer between particles may play an important role. Nonetheless, due to the challenges related to the small size and distance of relevant particles and the optical density of powder flows the detailed physical mechanisms of their interaction are so far little investigated. Only recently it was feasible to experimentally visualize pairs of individual particles in proximity and to observe their orbiting motion [15]. Regarding the deposition process, Cooper et al. [16] developed and experimentally validated a model that predicts particle adhesion in aqueous and dry environments. Their simulation accounted for the in-detail particle and substrate surface morphology and mechanical properties but was limited to the adhesion of single particles.

A fundamental theoretical contribution to the numerical modeling of rigid spheres subject to a non-overlapping constraint was presented by Stewart [17], [18] and Maury [19], [20]. They demonstrated that their algorithms are capable to handle a large amount of particles being in contact. The focus of their work was to establish a stable and robust numerical approach. Thus, they did not require a detailed description of the contact forces which are responsible for deposit formation during industrial powder operations.

Song and Mehrani [21] proposed a phenomenological model of wall layer development on a gas-solid fluidization column due to electrostatic charges based on their experimental investigations. The model indicated that particle migration towards the wall was due to image and electrostatic forces. More specifically, they predict layers to develop in alternating polarity since e.g. a negatively charged wall layer would attract the positively charges particles from the bulk of the bed.

To sum up, to the best of our knowledge no comprehensive theoretical model of the particle deposition process was formulated yet. However, in the field of computer graphics an efficient algorithm appeared [1] which enables the efficient and accurate tracking of the dynamics of a large number of bodies. Therein, particular attention was paid to the accurate representation of contact forces and the avoidance of penetration of different bodies. In this paper we extend the algorithm of Erleben [1] to compute binary and multiple particle interaction accounting besides for van der Waals and electrostatic forces also for collisional forces and plastic particle deformation. The paper is organized as follows: the forces acting on a particle and the fundamentals of one or two particles in contact are summarized in section 2. Section 3 details the newly proposed numerical scheme to track multiple particles. In section 4 qualitative results are presented followed by the conclusions in section 5.