Article pubs.acs.org/IECR
Modeling of Particle Flow and Sieving Behavior on a Vibrating Screen: From Discrete Particle Simulation to Process Performance Prediction K. J. Dong,† B. Wang,†,‡ and A. B. Yu*,† †
Laboratory for Simulation and Modelling of Particulate Systems, School of Materials Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia ABSTRACT: This paper presents a numerical study of particle flow and sieving behavior on a vibrating screen. The particle flow is modeled by means of discrete element method (DEM) at a particle scale. The DEM model is first verified by the good agreement between the numerical and experimental results in terms of the distribution of percentage passing of different sized particles along the screen deck. The effects of variables such as incline angle, vibrating frequency, and amplitude are then studied with special reference to the overall screen efficiency and distribution of the passing rates for different sized particles. The performance of such a screening process is shown to be related to particle flow on the screen such as the structure of particle bed, particle velocities, and particle− particle and particle−deck interactions. In particular, it is demonstrated that the sieving performance at the process equipment scale can be linked with the particle−deck collisions obtained from the DEM simulations, facilitated by the well-established probability theory. Finally, based on the simulation data, a mathematical model is proposed to link the particle−deck collisions with the three variables considered. The mathematical model can satisfactorily estimate the process performance in terms of weight percentage passings, particularly when the effect of particle−particle interactions is not significant.
1. INTRODUCTION Screening or sieving is an important operation for the separation of particles according to sizes. It is widely used in industries ranging from the traditional mineral and metallurgical sectors to the contemporary fast-growing food and pharmaceutical engineering sectors.1−4 Hundreds of millions of tonnes of particulate materials are subjected to industrial screening each year,5 and a better understanding of the screening behavior can lead to better process design and control under different conditions, which is of great economic significance. The studies on screening date back to the middle of the last century or even earlier.6,7 The early experimental studies revealed that the particle flow in a screening process is complicated, and the effects of operational conditions such as the vibration amplitude and frequency on the sieving performance are complex and condition-dependent.6 Attempts have been made to predict the sieving performance based on the phenomenological observations that follow the established chemical reaction kinetics, such as the “first-order kinetics”.7 For instance, such a treatment has been discussed for batch5 and continuous8,9 sieving, with the sieving rate modeled for different particle sizes, and the effects of certain operational variables have also been investigated. Due to its phenomenological background, such a mathematical model may encounter difficulties for general use. For example, it was found that oversized particles can benefit the sieving of near-mesh sized particles, which would be problematic to be represented in the traditional theoretical formulation.5,8 More fundamental models to describe the passing of particles through screen meshes have been developed based on the probability theory.10−12 Such models have also been extended to take into account the shapes of particles and meshes.13 Although widely accepted and applied, these models use the particle-scale information such as the attempts of a particle to pass © 2013 American Chemical Society
meshes, which are not easy to obtain experimentally. In addition, they do not consider the effect of particle−particle interactions which, sometimes referred to as the percolation or stratification of small particles in a particle bed, will affect the passing of particles through a particle bed, before interacting with a screen deck.14 Soldinger15,16 proposed an integrated model for both the particle stratification and passing processes. Both the sieving rate and stratification rate were modeled together. Stratification rate was found to diminish with a larger bed thickness and the high proportion of fine particles plugging the void space between large particles. The passage of a particle through the screen was found to be more probable for larger beds, due to the reduced relative movement between the bottom layer of particles and screen. Such a model is still based on certain assumptions of particle motion that need further examination. In principle, the bulk behavior of particles in a system depends on the collected outcomes of the interactions between individual particles and between particles and boundary walls. Therefore, it is helpful to investigate the screening phenomenon on a particle scale. However, such an investigation is difficult to be carried out with the conventional research techniques. In recent years, granular dynamic simulations based on the so-called discrete element method (DEM) have been widely used to overcome this difficulty. This method has been applied in the study of particle flows in various industrial Special Issue: Multiscale Structures and Systems in Process Engineering Received: Revised: Accepted: Published: 11333
December 15, 2012 February 26, 2013 February 28, 2013 February 28, 2013 dx.doi.org/10.1021/ie3034637 | Ind. Eng. Chem. Res. 2013, 52, 11333−11343
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Figure 1. Screen process in the present simulation: (a) schematic of the setup; (b) typical snapshot showing the motion of particles in a screening process, with particles colored by their sizes and vectors showing their velocities.
processes and has been proven to be effective as reviewed by Zhu et al.17,18 However, its application to the screening operations is few and preliminary. For example, in the early work of Li et al.,14,19 DEM was used to simulate the particle motion on static meshes or vibrating sieves, but the simulations were done under simple conditions, e.g., two-dimensional DEM was used. More recently, Dong et al.20,21 studied the effects of vibration conditions on banana screens and combinations of a sieve bend and low-head screen, which is relevant to coal preparation. On the other hand, Cleary et al.22,23 simulated a large-scale double layer banana screening process related to ore preparation. These studies demonstrate the capability of DEM to deal with complicated geometries and vibrational conditions. However, more fundamental, systematic studies are necessary in order to generate information that can be generally used to improve the design and control of sieving processes under different conditions. This paper presents an effort in this direction, with special reference to vibrating inclined screens. The DEM approach used is first verified by comparing the simulated results with the experimental ones under similar conditions. The effects of some key variables on the screening performance in terms of the partition curve and distribution of percentage passing along the
Table 1. Parameters Used in the Present Simulation (Base Value and Varying Range for Parametric Study in Brackets) feed particle size feed rate (particles/s) incline angle θ (deg) vibration frequency f (Hz) vibration amplitude A (mm) vibration direction
3.7 3.15 2.50 100 200 400 11 [5, 17] 20 [10, 30] 2.5 [1.0, 3.0]
2.00 1000
1.68 1400
1.40 2400
1.20 4000
sinusoidal, along the direction 45° to the horizontal line.
screen deck are then studied by a series of controlled numerical experiments. The particle flow is also analyzed aiming to establish some useful relationships between microscopic, particle-scale behavior, and the macroscopic, process performance.
2. SIMULATION METHOD 2.1. Discrete Element Method. In DEM, an explicit numerical scheme is used to trace the motion of individual particles in a considered system according to their mutual interaction.17 For particle i, its translational motion and rotational motion are respectively determined by 11334
dx.doi.org/10.1021/ie3034637 | Ind. Eng. Chem. Res. 2013, 52, 11333−11343
Industrial & Engineering Chemistry Research mi
dvi = dt
Article
∑ (Fijn + Fijs) + mi g j
Ii (1)
dωi = dt
∑ (R ij × Fijs − μr R i|Fijn|ω̂i ) j
(2)
where vi, ωi, mi, and Ii are the translational and angular velocities, mass, and moment of inertia of particle i, respectively; g is the gravitational acceleration; Rij is the vector pointing from the center of particle i to its contact point with particle j; and Fijn and Fijs are the normal and tangential contact forces respectively, which can be given as17,24,25
and
⎤ ⎡2 Y Y R̅ ξn3/2 − γn R̅ ξn (vij·n̂ ij)n̂ ij⎥ Fijn = ⎢ 2 ⎦ ⎣ 3 1 − σ̃2 1 − σ̃ (3)
Fijs = −μs Fijn [1 − (1 − min(ξs , ξs ,max )/ξs ,max )3/2 ]ξs̑
(4)
where Y is Young’s modulus; σ̃ is the Poisson ratio; γn is the normal damping coefficient; μs is the sliding friction coefficient; R̅ = RiRj/(Ri + Rj), and Ri and Rj are the radii of particles i and j,
Figure 2. Weight percentage passing as a function of deck position for different sized particles: solid points and lines, numerical results; and hollow points, experimental results.8
Figure 4. Averaged velocity parallel to the screen deck for near-mesh sized (□, d = 3.15 mm) particles and undersized particles (▲, d = 1.20 mm) on the first quarter of the deck as a function of (a) vibration frequency; (b) vibration amplitude; and (c) incline angle.
Figure 3. Particle bed volume along the deck as a function of the deck length for different operational conditions: (a) vibration frequency; (b) vibration amplitude; and (c) incline angle. 11335
dx.doi.org/10.1021/ie3034637 | Ind. Eng. Chem. Res. 2013, 52, 11333−11343
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respectively; ξs,max = μs[(2 − σ̃)/2(1 − σ̃)]ξn; n̂ij = (Ri − Rj)/ |Ri − Rj|; and ξs is the total tangential displacement during a contact, with ξŝ = ξs/|ξs|. The second term of the torque results from the rolling resistance between two contacting particles due to elastic hysteretic losses or viscous dissipation, where μr is the rolling friction coefficient and ω̑ i = ωi/|ωi|.17,26 The DEM model as described above has been used in our previous studies (see refs 20 and 21 for example), and the algorithm for computation is also documented in the literature.27,28 Note the tangential damping force is omitted here to reduce the computational cost. This term only works when ξs is below the Coulomb frictional limit (ξs,max) prior to macroscopic sliding,17 which is usually useful in the simulation of static or quasi-static systems, but not dynamic systems with intensive particle−wall and particle−particle collisions. Moreover, for the present system, it was found that the rolling friction plays a more significant role in dissipating the rotational energy.
As is well-established, particle−wall interactions can be calculated according to the same equations, with the radius of a wall is assumed to be infinitely large. In our DEM code, a complicated boundary is considered to be composed of smooth planes and edges and vertexes shared by two or several planes, with a rigorous protocol to judge particle−vertex, particle−edge, and particle−plane collisions. In addition, each of these walls can move, including rotating and vibrating. These treatments are implemented by use of the object-oriented programming (OOP) in the code to facilitate the simulation of complex screen decks vibrated under different conditions. They have been used in our previous studies of different processes in mineral processing.20,21,29,30 2.2. Simulation Conditions. The screen deck used in the simulation is schematically shown in Figure 1a. The deck consists of a flat surface uniformly perforated with square apertures of size 3.5 mm × 3.5 mm spaced 3.0 mm apart. The left side of the deck is the feed end, while the right side is the discharge end. The length, width (in the x direction), and thickness of the deck are 600, 26, and 2 mm, respectively. Periodic boundary conditions are applied in the x direction (front and rear) to reduce the computational effort. Table 1 lists the operational conditions used in the present work, which are based on the experimental work of Standish and Meta.8 However, to be computationally efficient, particles of smaller undersize (
