Development of a granular normal contact force model based on a non-Newtonian liquid filled dashpot

Abstract

Normal contact force models often suffer from a weak prediction of collisions between particles. They regularly fail to predict an adequate energy restitution behavior with increasing normal impact velocity. In particular, most non-linear models predict a net attraction force between two impacting particles near the end of a collision, which is unrealistic according to reported results. Such limitations have provided the impetus for the development of a normal contact force model that better predicts the unfolding of a collision between two particles. This model comprises a Hertz elastic force and a dissipative force that is evaluated by the motion of a non-Newtonian liquid in a dashpot. The model parameters are set using experimental restitution data for particle/particle and particle/wall contacts. In the current work, the measurement of energy restitution for particle/wall collision was carried out using several materials over a wide range of impact velocities, whereas particle/particle collision data were obtained from the literature. Model predictions for microscopic (e.g. particle velocity) and macroscopic (e.g. collision time) quantities are presented and compared with those from other non-linear models and experimental data. The model is observed to adequately predict the coefficient of restitution and to decrease the attraction force at the end of a collision.

Introduction

Due to the importance of processes involving solids, a high percentage of recent research activities has focused on the flow of particles [1], which in many cases is governed by the collisions among them. A better understanding of particle impact in the granular bed facilitates the design of more efficient unit operations, thus improving the throughput and the quality of the final product [2]. A typical example of this can be seen in the pharmaceutical industry, with for instance mixing systems such as tumbling blenders, where improved knowledge of the particle behavior as these particles collide with their neighbors and the blender wall is known to help design systems that have greater throughput, consume less power and have smaller failure rate. Beside experimental efforts to investigate particulate material beds (e.g., [3], [4]), numerical investigations have progressed due to advances in high performance computing [5]. In these studies, both continuum and discrete models have been developed. The first is a Eulerian approach that considers powder as a fluid (e.g. [6], [7], [8]), and the second is a Lagrangian approach that treats particles as discrete entities. Among the discrete models, the discrete element method (DEM), initially introduced by Cundall and Strack [9], has been widely applied to investigate solids motion (e.g. [5], [10], [11], [12], [13]). It has been shown to provide valuable insight into phenomena occurring in the granular bed by modeling each particle and its interaction with the neighboring particles and the solid parts of the equipment.

In the DEM, the motion of each particle is subjected to Newton's second law of motion. It is a time-driven soft-particle method that allows any two colliding particles to interpenetrate so as to mimic their deformation upon impact. The total applied force may take into account gravity, drag, buoyancy, particle/particle and particle/wall contact, and cohesive terms such as the electrostatic, Van der Waals and capillary forces. Given the particle size in DEM simulations (generally in the order of one millimeter), the non-contact cohesive forces are often neglected so that only gravity and contact forces are considered. Particle contacts can be described via contact mechanics and modeled by the finite element method (FEM) [14], [15], [16]. Considering there can be millions and possibly billions of particles in a small rig, the FEM approach remains too computationally intensive. To overcome this problem, several simplified force models have been proposed, some of which will be described in Section 2.

Although these simplified force models decrease the simulation time, they are nonetheless subject to limitations. For instance, most of the current models require input parameters that can hardly be measured directly (e.g. spring and damping coefficients) [17]. In addition, some normal contact force models cannot yield accurate values of the normal coefficient of restitution (CoR) when the particle impact velocity changes. The CoR is defined as the ratio of the relative velocities after and before collision. The models that are able to predict the correct behavior of the CoR fails to approximate it accurately in comparison with experimental data. Furthermore, many of these models predict a net attraction force at the end of a collision, which is unrealistic. The details of such drawbacks will be described and further discussed in Section 2.

The aim of this work is to develop a normal contact force model that alleviates the stated shortcomings. The model parameters are adjusted to accurately predict particle/particle as well as particle/wall interactions. A wide range of experiments were carried out to measure the normal impact on a flat metal plate of spherical particles made from a wide variety of materials. These experiments enabled the estimation of the parameters inherent to the proposed model. Experimental data from the literature [17], [18], [19] were also used to assess the quality of the model in the case of two particles collisions.

The paper is organized as follows. In Section 2, the most common contact force models are reviewed and their limitations are discussed. In Section 3, the experimental procedure and materials used for the particle/wall contact experiments are described. In Section 4, the proposed collision model is introduced and model parameters are obtained for several materials in the case of particle/particle and particle/wall collisions. In Section 5, the model is assessed by means of experimental data and results obtained with other non-linear models. Finally, Section 6 provides concluding remarks.