(52e) Predicting Particle Impact Velocities in Vibrationally Fluidized Granular Flows Using the Discrete Element Method

            Vibratory finishing is widely used to deburr, polish, burnish, harden and clean metal, ceramic and plastic parts. In a tub vibratory finisher, a container filled with granular media is oscillated by an eccentric rotating shaft so that it develops a vibrationally-fluidized circulatory bulk flow of the media that is largely two-dimensional.  The media have both a large-scale bulk flow velocity and a local impact velocity during each vibration cycle.  Therefore, work-pieces that are entrained in the flowing media are subjected to the repetitive, high-frequency impacts of the surrounding particles.

            Within a vibratory finisher, the erosive wear and plastic deformation of workpieces are largely affected by the velocity, frequency, and direction of the impacts with particles.  Discrete element modeling (DEM) is a numerical method that is often used to solve problems involving transient dynamics of systems comprising a large number of moving bodies that interact with each other.  Generally, DEM simulations give reasonable predictions of bulk flow velocity and volume fraction in both fluidized and non-fluidized flows. However, only a small number of studies have focused on the local behavior of the media in granular flows and some of these have investigated vibrationally-fluidized flows.  Most of these studies have been about modelling, and the predicted collision scale quantities, such as impact velocity, collision frequency or impact energy, were not validated experimentally. Therefore, it is not known if the commonly used approaches and parameters in DEM simulations yield correct predictions of the impact velocity.  This remains as a significant limitation since the impact forces that govern erosion, wear and fracture in granular flows are fundamentally linked to the particle impact velocity and kinetic energy.

            The main objective of this research was to compare previous experimental measurements of particle impact and bulk flow velocities with those values predicted by numerical simulations using the discrete element method (DEM).  The sensitivity of these predictions to the input values of the coefficients of restitution, friction and rolling resistance was also studied in a simplified model consisting of a single layer of particles. 

            The commercial software package, EDEM (DEM Solutions Inc. 2012), was used to perform the DEM simulations.  The Hertz-Mindlin contact model was selected to calculate contact forces between the particles.  In this model both the elastic and plastic deformation of particles were considered in the normal and tangential directions considering the coefficients of restitution, friction and rolling resistance. 

            It was of interest to study the sensitivity of the predicted bulk flow velocity and impact velocity to the contact coefficients used in the DEM.  To begin, three simple case studies were considered using an analytical model of two-body collision: 1- collision of a moving disk with a fixed disk; 2- collision of a moving disk with a moving wall; 3- collision of two moving disks.  In each case, the equations of the linear and angular rebound velocities of the disks were derived as functions of the contact parameters.  The sensitivity of the rebound velocities to the contact parameters was determined by differentiating the velocities with respect to the coefficients.    It was found that the coefficient of friction had the greatest effect on the tangential and angular velocities.  However, the normal velocity was only a function of the coefficient of restitution.

            The sensitivity analysis was then extended to a simplified DE model of the flow within the tub finisher consisting of a single-layer of particles moving between parallel sheets of glass.  Design of experiments (Taguchi method) was used to reduce the number of required runs of the model to 25 in order to investigate the relative sensitivity of the velocity predictions to the 6 contact parameters under consideration.  Analyzing the output of these 25 model runs with MINITAB^TMled to the conclusion that the coefficient of friction between the particles and the tub wall plays the most important role in determining the impact velocities.  A pin-on-disk tribometer was used to measure the coefficients of friction between two steel balls, two porcelain balls, and between these balls and the polyurethane tub wall., he corresponding coefficients of restitution were measured using a high-speed laser displacement sensor.

           The subsequent DE modelling sought to mimic the setup used in the earlier experimental study of bulk flow and impact velocities.  First, the mechanical properties including shear modulus, Poisson's ratio and density of the steel balls, polyurethane wall of the tub and the glass partitions were input to the software.  The glass partitions were assumed to be frictionless.  The tub geometry was imported to the EDEM interface and its motion was defined according to the experimental measurements  made using accelerometers.  The sinusoidal motions were introduced to simulate 19 s of real-time operation.  The simulation time step was set to be 25% of the Rayleigh critical time step which is defined as the time taken for a shear wave to propagate through a solid particle, and therefore was a function of the particle diameter, density and shear modulus.  The simulation time step for 6.3 mm diameter steel balls was calculated to be 2.17×10^-6s.  The data acquisition rate was 94 Hz which was twice the frequency of the tub vibration (47 Hz); i.e. the data was saved every 0.0105 s.  The instantaneous velocities of particles (impact velocities) were acquired at each time step in a 9×9×6.3 mm bin in two different locations.  The mean impact velocity was the average of the absolute values of the instantaneous velocities of the particles.  The data acquisition was started 9 s after the tub began to vibrate in order to reach a steady flow.   

            Different numbers of particle layers were modeled between the glass partitions  (1, 2, 4 and 8 layers).  were simulated to study the differences between the simulation results of the single-layered and the multi-layered granular flows. Finally, the DE model was used to simulate the actual experimental setup, including the laser displacement sensor probe that was used to measure the impact and bulk flow velocities.  In order to decrease the simulation time in this full model, only half of the geometry was modeled using a symmetry boundary made of the same material as the particles.  The symmetry boundary was assumed to be frictionless to satisfy the symmetry conditions.  The paper will present the comparisons of the measured impact and bulk flow velocities in a tub vibratory finisher with the measured values for two granular media – steel and porcelain balls having a diameter of 6.3 mm.