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Particle Segregation of Inclined Plane Flows Brian Lawney Clarkson University, Potsdam, NY 13699-6261 Advisor: Dr. Hayley H. Shen Department of Civil and Environmental Engineering, Clarkson University, Potsdam, NY Abstract Particle segregation is an important feature of non-uniform sheared granular flows, in which large particles rise to the top of the flow regime. By changing the flow’s spatial composition, segregation has significant consequences on the behavior of mixed particle systems. In an attempt to better understand the effects of this mechanism of particle transport, experimental and computational studies have been developed to model the dynamics of various geometrical arrangements. One particular problem of interest is the inclined plane flow. Thus far, research on this topic has linked segregation rate to such factors as relative particle size ratio, angle of incline, and shear rate. Other works on mixed flow have cited the importance of boundary conditions on the rate of segregation, but none have studied this in particular. By utilizing an existing numerical model modified for this specific incline geometry, this problem can be examined in detail. Thus, future work will focus on studying the relationship between boundary roughness and segregation rate. If time permits, additional studies will be performed to determine if there exists a relationship between relative large particle concentration and segregation rate. I. Introduction: The study of segregation phenomena is of particular importance in applications where the features of non-uniform granular flow are of interest. Many processes involving granular material are conducted with particles of varying sizes, and an accurate model of the mechanics of the system may be advantageous. The change in the even dispersion of different size particles may have some important consequences. In certain applications, the result of this flow property may be undesirable, as it affects the homogeneity of the mixture. In yet other processes, the separation of different sized particles could be an integral part of system operations. It is for these reasons, among others encountered in many disciplines, that the mechanics of non-uniform granular flows must be studied. Particle segregation on an incline is characteristic of a non-uniform, gravity driven granular flow. In systems with varying particle sizes, it has been shown that larger particles will generally segregate to the top of the flow regime, while smaller particles will remain at the bottom. This topic has been studied in various ways, typically with the system being excited with periodic vibration [1, 2], sheared in a cylindrical geometry [3], or gravity driven on an inclined slope [4]. Both experimental [2, 3] and numerical computational methods [1, 4] have been developed to model the behavior of the particular systems. In the study of segregation, two possible explanations for the process have been proposed. The first theory, supported by computer simulations performed by Jullien and Meakin, [1] contends that during flow movement, conical shaped voids develop underneath large particles as they are displaced vertically. It was found that grains of smaller size would translate or “avalanche” into the space, and subsequently build up underneath the larger particles. Thus, in each successive excitation, larger particles would displace upwards, against gravity. In their simulations, vertical periodic shaking provided the mechanism for particle transport. The second theory, studied in a report by Knight et al. [2], proposes that the mechanism of segregation is a convection process, highly dependent upon the boundary conditions of the flow. In their experiments, large particles translated to the top of the flow in a fashion similar to a convection cycle. Depending upon the relative size of the grains, it was found that the large particles would eventually return downward (with gravity) a short distance before being forced to the top again. When convective processes were controlled in experimentation, no segregation occurred, suggesting the proposed mechanism. The authors of the report also assert that friction and/or inelastic collision of the flow particles with the boundaries is essential to observing convective segregation behavior. They cite the periodic boundary conditions of Jullien and Meakin’s [1] model as a possible reason for the simulation’s inability to observe convection in the flow. In the experiments of Khosropour and Zirinsky [3], glass beads of various sizes were sheared between two glass cylinders to study the convective theory of segregation. It was found that the larger beads did indeed rise to the top of the flow regime, and the smaller, more uniform assortment of particles at the bottom displayed typical convection cycles of rising and falling. Their findings supported the assertion that boundary conditions were critical in the development of segregation, but it was concluded that the role of frictional boundaries warranted further consideration. In perhaps a more relevant (in terms of geometry and method) study, Hirshfeld and Rapaport [4] performed a two-dimensional numerical simulation of non-uniform inclined plane flow. In their report, the relationship between slope angle and segregation rate, and the temporal and spatial dependence on segregation was investigated. It was reported that larger differences in relative particle sizes and higher shear rates contributed to faster segregation. For the proposed study, a numerical simulation code developed by Babić [5] will be utilized. In the time since its initial formulation, the code has been modified to investigate specific behavior of shear granular systems, including the case of inclined plane flows, which will be the focus of future work. Currently, the model has been altered to study non-uniform flows in other geometric configurations. The specifics of this program and the physical model will be discussed in the methodology section. A common theme expressed in much of the literature was the need for further study into the role of boundary conditions on segregation. Thus, it may be of interest to study the segregation rate under different roughness conditions imposed by the incline, and possibly side walls. By varying the size of the static particles composing the incline, and including additional frictional losses due to “walls”, it will be possible to investigate this. Additionally, both [4] and [1] determined a general relationship between relative particle size and segregation rate. Both reports found that a critical size ratio of greater than 2 is required to observe appreciable segregation. It may also be interest to determine a relationship between segregation rate and the concentration of large particles. By controlling the amount of large particles (in relation to smaller ones), it may be possible to change the segregation effects of the flow. If a relationship exists, it could have significant implications. II. Methodology The numerical code formulated by Babić [5] was developed as a calculation tool for simulating the behavior of various two-dimensional discrete particle systems. By the inclusion of additional code through further development, the program has become quite modular, and has been used to study several different granular flow cases. Thus far, it has been used in the analysis of couette, channel, and chute (inclined) flows [6]. This numerical simulation will be used in the segregation study proposed in this report. The numerical code consists of three main parts. A simple flow chart, appearing in Babić’s original report [6] is given in figure 1. From figure 1, it can be seen that the program framework is adaptive and can be suited to particular needs of study. To describe the program in more detail, discussion of specific algorithms is necessary. Model Initialization and preparation: In the Input/Initialization unit, the model is prepared. Material properties, physical constants, initial conditions, and boundary conditions are read from an input file, and a computational grid is defined. Walls and other features of the model are implemented. The model space is then subdivided into small cells into which only one particle can fit, generally determined by the size of the particles. This is important in defining where each particle is, and where its closest neighbors are. Improper sizing of the cells will cause problems within the searching subroutine, and particles will fail to be detected, resulting in severe overlaps. Thus, for non-uniform flow, the cell dimensions must be defined by the smallest particles present. By some simple trigonometry, it can be shown that the length and width of the cell (2W), is 2 Rs , where Rs is the radius of the smallest particle. Recent work by Bastien and Ji [7] suggests that problems may be encountered with overlapping in the initial packing arrangement, so some changes may need to be implemented to avoid this behavior. Model calculation cycle: Start Input and Initialization Unit -input physical properties -input or define problem specification -input of prepare initial conditions -input or prepare boundary conditions Main Unit Repeat: {-detect all current contacts; -calculate all current forces; -sum all forces and moments acting on each particle; -integrate equations of motion for all particles to obtain new positions and velocities; -update wall positions; -update statistical summations} Until: {Specified end condition is met} The main unit of the code controls the particle neighbor searches, force calculations, and the subsequent particle movement. In the search process, the program defines a neighborhood of cells around each particle (based upon the grid that was created), and constructs a list of possible particle and/or boundary contacts. With this list in mind, the distances between neighboring particles are found, and force calculations are made if the particles are in contact. With the known forces, appropriate equations of motion are applied to find the movement of each grain. This cycle of searching, calculating forces, and moving the particles repeats for each time step until the program is complete. The specific details of the kinematic equations are provided in the model development section. Model Development (formulated by Babić [6]): Output and Statistics Unit -print full printout at specified times -perform time-averaging of interesting variables and print results Stop Figure 1. Flow Chart-from reference [6] In the analysis of the system, a soft-particle approach is used to model the collisions that occur. Unlike the hard-particle model, which assumes instantaneous collisions, the soft particle technique features collisions that have finite durations. Because of this relation, appropriate time steps need to be chosen so that short contacts are not missed. This is addressed in the input and initialization of the program. Collisions are modeled as viscoelastic, with elastic and damping forces in both the normal and tangential directions. The method of analysis is the same for contact between moving particles, and contact with walls. A schematic is given in figure 3. Rs W W Figure 2. Cell size determination Figure 3. Binary contact model [6] Because of the nature of frictional forces acting as moments, the particles in the model can rotate as well as translate. It should be noted that when particles are sliding past each other, the tangential damping contribution (Cs) is ignored. Each particle shown in the simple binary collision model has its own spring constant which contributes to the overall constants shown. To find this overall stiffness, we recall that elastic elements in series add as follows: Figure 4. Collision model [6,7] Position vectors, defining the centers of particles A and B, are defined in the given coordinate system. rA= (x A, yA) rB= (xB, yB) By inspection, the unit vector k lies along a line connecting the centers of the two particles, and is given as, K eq 1 1 k1 k 2 1 (1) It is in this manner that the normal and tangential (shear) spring coefficients are calculated. The expression for the damping terms can obtained in a similar fashion. k rB rA (cos , sin ) rB rA (3) The unit vector t, perpendicular to k as shown, is defined as C eq 1 1 C1 C 2 1 (2) t ( sin , cos ) (4) To derive an appropriate equation of motion to describe the system, a simple binary collision is given in figure 4. Locations PA and PB are points of intersection between the line in the direction of vector k and the circumference of the respective disks. The relative velocity of PA with respect to PB can be written as, VAB (rA rB ) ( R A A RB B )t (5) By integrating this result, the relative positions of the two particles can be obtained. For the purposes of calculating the spring and damping forces in the normal and tangential directions, VAB can be broken into components. To do so, a dot product is taken between VAB and k to find the normal relative velocity, and VAB and t for the tangential velocity. Total force: The total force exerted on each particle is the sum of the elastic and viscous force contributions. n V AB k ( x A x B ) cos ( y A y B ) sin (6) q V AB t ( x A x B ) sin ( y A y B ) cos ( R A A R B B ) (7 ) Elastic (spring) force calculation: For each time step Δt, the normal and tangential spring forces are, S n Fn Dn S s Fs Ds (15) (16) Resolving these forces into x and y components, Fx A S n cos S s sin M A S s R A (19) (20) (21) (22) F y A Sn sin S s cos (17) (18) Fn K n nt Fs K s qt (8) (9) Fx B Fx A F y B Fy A The total spring force is the sum of these incremental spring forces over the time of the entire contact. In equation form, M B S s RB It should be noted that the signs are opposite due to the force convention of Newton’s third law. Equations of motion: By applying Newton’s second law for linear and rotational systems, the equations of motion can be found, and subsequently the dynamics of each particle. For translation, ( Fn ) T ( Fn ) T 1 Fn ( Fs ) T ( Fs ) T 1 Fs Viscous damping calculation: (10) (11) The damping force associated with particle contact is proportional to the relative velocities of particles A and B. Therefore, Dn Cn n Ds C s q (12) (13) mr F mg (23) Frictional effects: As noted earlier, the tangential damping is removed when the particles are sliding. To account for this, a maximum tangential spring force is defined as: where mg is the weight force of the particle. Naturally, for an accurate model, this weight force should be specific for the size of particle in question. For rotation, I M (24) ( Fs ) m ax Fn (14) where μ is the friction coefficient of the particles. During a contact, if Fn develops a magnitude greater than (Fs)max, Fn is equated to (Fs)max and the tangential damping is ignored. The forces and moments acting on the particles are assumed to be acting for a time step of Δt from t NFor constant linear and angular 1/2 to tN+1/2 [8]. acceleration over this time period, equations (23) and (24) can be integrated to obtain, F r N 1 2 r N 1 2 N N 1 2 N 1 2 M I m g t t (25) April-May 2004: Alter existing code to model non-uniform flows and output desired information June-July 2004: Using modified simulation, run trials relating to the study of segregation rate vs. boundary roughness. If time permits, run trials for secondary study. Collect and organize data. August-September: Analyze and report data. Begin initial draft of thesis October-November: final thesis Complete rough draft and N (26) Performing a second integration, equations for particle position are given to be, r N 1 r N r N 1 2 t N 1 N N 1 2 t III. Future Work (27) (28) V. References [1] R. Jullien, P. Meakin, and A. Pavlovitch (1992). “Three Dimensional Model for ParticleSize Segregation by Shaking”, Physical Review Letters. Vol. 69, 640-643. [2] Knight, H.M. Jaeger, and S.R. Nagel (1993). “Vibration Induced Size Separation in Granular Media: The Convection Connection”, Physical Review Letters. Vol. 70, 3728-3731. [3] R. Khosropour, J. Zirinsky, H.K. Pak, and R.P. Behringer (1997). “Convection and Size Segregation in a Couetter Flow of Granular Material”, Physical Review E. Vol. 56, 44674473. The primary focus of study will be the examination of the relationship between boundary roughness and segregation rate. To accomplish this, the current computer model will be updated to include new code which will introduce non-uniform particles to the system. Currently, this task has been done for other flow cases at Clarkson University, but not for the incline geometry. Study of current research on this task will aid in efficient and accurate implementation of changes. To study the problem, trials will be performed with varying wall roughness, and the output data will be collected. Thus, it may become necessary to develop additional code which will yield desired results. The author has already added code which will aid in the output of velocity profile information. With the proper data, it may be possible to develop a quantitative relation between boundary roughness and segregation rate. Additionally, frictional losses associated with side walls will be added, to determine if it has any bearing on the two dimensional simulation. A secondary problem of interest, if time permits, would concern the relationship of large particle concentration on segregation rate. Based on the amount of large particles contained in a mixed flow, it may be possible to optimize or diminish the rate of segregation. By utilizing the same computational model, the relative concentration of large particles will be varied to investigate its effects. IV. Schedule For the proposed study, a tentative timeline: [4] D. Hirshfeld, D.C. Rapaport (1997). “Molecular Dynamics studies of Grain Segregation in Sheared Flow”, Physical Review E. Vol. 56, 2012-2018. [5] Babić, M., H.H. Shen, H.T. Shen (1990) “The Stress Tensor in Granular Shear Flows of Uniform, Deformable Disks at High Solids Concentrations”, Journal of Fluid Mechanics. Vol. 219, 81-118. [6] Babić, M. (1988) “Discrete Particle Numerical Simulation of Granular Material Behavior”, Report No. 88-11. Department of Civil and Environmental Engineering, Clarkson University, Potsdam, NY. [7] Bastien, C (2003). “Simulating the Behavior of Granular Materials in 2-D Shear Flow”.

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