Experimental study and numerical simulation of particles trajectories in a flexible-electrode-type electrostatic separator

The novel device used in this study is a free fall electrostatic separator equipped with flexible electrodes. This special feature enabled the experimentation of different electric field configurations, avoiding as much as possible the impacts of particles on electrode walls, which is the major drawback of standard free-fall electrostatic separators. The present work was focused on the numerical modelling and simulating the trajectories of charged insulating particles. The study was aimed at contributing to a more in-depth understanding of the various physical phenomena that occur during electrostatic separation. An accurate numerical model of particle movement was developed. The results of the numerical simulations were validated using the experimental data obtained with an appropriate image acquisition tool.


Introduction
The considerable increase of plastic waste, mainly non-biodegradable, has become a critical problem in the world, and for a better preservation of the environment, electrostatic separation is gaining more and more importance, especially for the simplicity and effectiveness this technology [1] - [4].
The free-fall electrostatic separator is one of the most efficient and simple device used to sort plastic waste mixtures.The separator used in this study is equipped with flexible electrodes, this modification of electrode form allows to avoid the impacts of the particles with the electrode wall, and also to intensify the electric field in the electrostatic separation zone, so that to improve the efficiency of the separator [5] - [7].
The objective of this study was to model and numerically simulate the behaviour of charged particles in this separator by varying several parameters such as the shape of the electrodes, and distance between them.The results obtained from the simulation were compared with experiments, carried out using a high-speed camera, which made possible the observation of the trajectories for different electrode configurations and operating conditions.This approach has provided a deep understanding of different phenomena that occur during the separation process, and has also enabled the validation of the numerical model developed by the authors.

Experimental set-up
The electrostatic separator used is equipped with two flexible copper plate electrodes (Figure 1), each of them fitted with 4 threaded rods which serve both to fix them in vertical position and to adjust the interelectrode distance.It is thus possible to obtain a multitude of electrode configurations.
Throughout the present study, the granular mixture obtained from shredding various plastic waste was introduced in the separator as a mono-layer curtain, at mid-distance between the upper edges of the electrodes, so that to limit the interactions between the particles and to ensure that their trajectories are determined only by the gravitational, electrical and air drag forces that act on each of them.Table 1.Different values of the adjustable inter-electrode distances D. The experiments were carried out for two electrode configurations (Table 1), using PP (polypropylene) particles of two different sizes (Figure 2): -Size 1: 1.4 mm < Ø < 2.8 mm -Size 2: 2.8 mm < Ø < 5 mm The high voltage applied to one electrode is 25 kV, while the other was connected to the ground.
Prior to being introduced in the separator, the particles were corona charged by using a grounded metal-belt-conveyor electrode and a wire-type electrode connected to the high-voltage supply (-15 kV).This solution (not illustrated here) ensured very good electric charge reproducibility, which was essential for the present study.The charge mass ratios for the two configurations were -10.3 nC/g and -10.6 nC/g, respectively.
The particle trajectories are recorded using a high-speed camera (Phantom VEO-E 310L).

Numerical modelling and simulation of insulating particle trajectories
The insulating particles, previously charged by the corona effect, fall freely under the effect of gravity to reach the effective separation zone of intense electric field.In this zone they are subjected to gravity, electric field and air drag forces [8] - [10].
The force of gravity   ⃗⃗⃗ acting on a particle is: where m is the mass of the particle and g = -9.81m/s 2 is the gravity constant.
The electric field force   ⃗⃗⃗ exerted on a charged insulating particle in the separation zone can be expressed as: where Q is the electrical charge of the particle and Ex (x, y); Ey (x, y) components of the electric field vector.
The formula of air drag force   ⃗⃗⃗ is: where vx and vy are the x and y components of the particle velocity  ; rp: radius of the particle; Cf ≈0.47: the drag coefficient; ρ=1.204 kg/m 3 : the mass density of the air.The calculation of the intensity of the electric field E as the gradient of the electric potential V is carried out by numerically solving Poisson's equation: Particle motion is governed by gravitational, electrical and air drag forces, According to Newton's law: where  is the acceleration of the particle.To calculate the velocity v and position p of the particle at each time step ∆, the differential equations of motion are numerically solved with Euler-Cromer method:

Results and discussion
For the first electrode configuration, the experimental and simulation data represented in figure 3 show that the trajectory of the small-radius particle is dominated by the electric field force.At first the particle is strongly accelerated towards the electrode of opposite polarity.Then, due to the shape given to the electrodes, the electric field intensity diminishes, and so does the electric force; thus, the particle is attracted to the electrode without colliding with it.
In the case of the larger particle, the movement is slightly dominated by the force of gravity.Nevertheless, the shape of the electrode facilitates the attraction of the particle to the electrode connected to the positive high voltage, so that the particle could be collected in the right compartment.
In the second configuration (Fig. 4), despite the fact that the distance D1 between the upper edges of the electrodes is the same, the effect of the electric field forces is weaker.The change of the electrode shape, by modifying the other characteristic distances D2, D3, and D4 (Table 1), produced a diminution of the electric field intensity in the active zone of the separator.The small particle pursues a quasi-optimal trajectory and eventually reaches a good location in the collector; while the movement of the large particle is too much affected by the force of gravity, and falls into the middling compartment of the collector.
The computed results are in good agreement with those obtained experimentally, which means that the program is sufficiently accurate to perform several numerical simulations, in order to optimize the shape of the electrodes and operational conditions of the separator and avoid the necessity of experimentally testing a multitude of configurations.

Conclusions
This work has contributed to the study of the behaviour of insulating particles in a newly designed free fall electrostatic separator with flexible electrodes.The comparative analysis of numerical simulation and experimental results has highlighted the following conclusions: 1) The high-speed camera allows in-depth study of the various phenomena that occur during the electrostatic process.The accuracy of the proposed numerical model, which considers the main electrical and mechanical forces governing particle motion, is confirmed by the good agreement with experimental observations.
2) The flexibility of the electrodes allows a clear improvement of the separation efficiency, in terms of both recovery and purity of the products.The electrode shape can be optimized based on the results of systematic numerical simulations of particle trajectories.

Figure 2 .
Figure 2. PP particles used in the experiments.

Figure 3 .
Figure 3. Experimental and simulation trajectories for 2 sizes of PP particles at 25 kV (1 st configuration)