Modelling of particle stratification in jigs by the discrete element method

Pergamon 0892-6875(98)00033-8

Minerals Engineering, Vol. 11, No. 6, pp. 511-522, 1998 © 1998 Elsevier Science Ltd

All rights reserved. Primed in Great Britain 0892--6875/98 $19.00-tO.00

MODELLING OF PARTICLE STRATIFICATION IN JIGS BY THE DISCRETE ELEMENT METHOD

B.K. MISHRA and S.P. MEHROTRA

Department of Materials and Metallurgical Engineering Indian Institute of Technology, Kanpur, India. E-mail: [email protected]

(Received 7 August 1997; accepted 27 January 1998)

ABSTRACT

This paper attempts to review the theories and mathematical models which have been advanced to explain the jigging process. The existing literature on mathematical modelling and quantitative analysis of jigging reveal that very little is known about the stratification of particles in a jig bed. It is realized that a microscopic model is the one that is most suited to study stratification in jigs. A new modelling approach based on Newtonian mechanics is used to study the stratification behavior of particles in a batch jig. In this approach, the motion of solid particles is treated using the discrete element method (DEM). Since in DEM, the motion of the particles is computed from the forces acting on them, the effect of the fluid is easily incorporated through the buoyancy and drag forces. For illustration purposes, a jig bed consisting of 400 particles of two different densities is simulated. Preliminary results show that the model predicts the stratification of particles reasonably well. © 1998 Elsevier Science Ltd. All rights reserved

Keywords Mineral processing; gravity concentration; modelling; simulation

~TRODUCTION

In jigging the pulsation of the particle bed by a current of water results in stratification of mineral particles of different specific gravity. A jig operates in a cyclic manner where one cycle consists of four stages, namely, inlet, expansion, exhaust, and compression. In the inlet stage the bed lifts up en-masse. Near the end of the lift stroke the particles at the bottom of the bed start falling resulting in loosening of the bed which, in turn, causes its expansion or dilation. During the third and fourth stages of the jig cycle, the particles resettle through the fluid, and the bed collapses back to its original volume. The pulsation and suction is repeated to bring about stratification with respect to specific gravity across the bed height.

Many factors affect the stratification in a jig, as shown in Figure 1. Ideally one may want to formulate a simple and practically applicable mathematical model of the jigging process correlating all the variables. However, in spite of several efforts by a large number of researchers such a model has remained elusive. This is due to the fact that although jigging is one of the oldest mineral processes, the basic principles involved are not yet well understood. With the revival of interest in gravity concentration techniques in general, and jigging in particular, once again there is a concerted effort to improve our understanding of the jigging process. The present paper is an attempt in that direction.

511

512 B.K. Mishra and S. P. Mehrotra

This paper consists of two sections. In the first section, a review of literature on mathematical modelling and quantitative analysis of physical phenomena leading to stratification is presented. The second section deals with a numerical technique known as the discrete element method (DEM) which is used to simulate the stratification of particles in a jig bed. Preliminary results of simulations are presented and discussed.

Ope ra t i ng var iables: • Amplitude and

frequency of pulsation • Bed thiokncss • Hutch water • Feed quality

_.L_~'r

I1' I

P e r f o r m a n c e Variables: • Concentration profile • Partition index • Separation time • Energy input

Fig. 1 Representation of a jig showing various factors that influence the process.

J IGGING THEORIES

Commonly, the jigging process is described by theories as simple as the single particle theory to those as complex as stochastic theory. The classical theory considers the principles of equal settling of particles, interstitial currents, acceleration, and suction of the fluid and the corresponding response of the bed. There are three major effects that contribute to the stratification in jigs: (i) hindered settling, (ii) differential acceleration, and (iii) consolidated trickling [1-3], It is generally assumed that the solid-fluid mixture in jigging is so thick that it is essentially a loosely packed bed of solids with interstitial fluid, instead of a fluid carrying a large number of suspended solid particles. In such thick slurries, the settling ratio of heavy to light minerals is greater than that for free settling conditions. Hindered settling has a marked effect on separation of coarse minerals.

The initial acceleration during settling of particles in a suspension can be derived based on a simple force balance as

(.~t.t) =(1 - P__f)g (1) Ps

where v is instantaneous velocity of a settling particle, pf and Ps are the densities of fluid and solid respectively, t is time, and g is acceleration due to gravity. It is thus evident that the initial acceleration depends only on the ratio of solid and liquid densities and is independent of particle size. Hindered settling and differential acceleration combined together result in a bed in which small particles are on top of large particles. The large particles in some regions rest on each other in such a way that passage ways exist between them large enough to allow the trickling of small particles through them. This aspect is difficult to quantify by simple settling theory.

In the potential energy theory proposed by Mayer [4], the cause of stratification in a jig bed is related to the difference between potential energies of the system before and after stratification. It postulates that an unstratified bed of particles of different densities and sizes is an unstable system under gravity potential which strives to move towards more stable (stratified) configuration by redistributing the particles within the bed in a manner such that there is a reduction in the overall potential energy of the bed. The change

Particle stratification in jigs 513

in potential energy, which is represented by the changing centre of gravity of the bed, corresponds to the minimum potential energy only when complete stratification is achieved. Mayer did not believe that the jig stroke supplies the flow energy which causes stratification. According to him this supplied energy only releases the potential energy stored in the granular mixture and, hence, is not directly responsible for stratification. The potential energy theory gives no fundamental description of the dynamics involved in the approach of the system to stable configuration and therefore it cannot completely describe the rate of separation.

The potential energy theory has been extended by several investigators to incorporate the dispersive effects arising due to particle-particle collision. Vetter et al. [5] have argued against the postulate that claims that the material in the jig bed is stratified once the potential energy is minimized, simply because it does not describe the remixing of the material due to its natural dispersion within the bed. Drawing an analogy with diffusion mixing within homogenized fluidized beds, a mathematical formulation has been proposed to describe dispersion of particulate materials with different densities but uniform size and shape. The particle motion in the jig bed is modelled in a probabilistic manner by interpreting the conventional hydrodynamic force balance on a particle as a stochastic differential equation with particle-particle interactions included as a noise term. The final dispersion equation, in which concentration C i is represented as conditional probability, is formulated as

acj _ + < 2 )

at ay ay 2

where k~ is a drag coefficient and k D is a coefficient which is a measure of dispersive mixing of particles in the jig bed. One of the major limitations of this formulation is that it is valid only for particles of uniform size and shape. For a real jigging process the formulation fails because of this assumption.

In one of the major contributions to the quantitative analysis of stratification phenomena in jigging, Tavares and King [6] recognized that the ideal stratification, as visualized by Mayer in his potential theory, would never be achieved because of ever present dispersive forces which disturb the already stratified bed by random motion. The stratification achieved in actual practice represents a balance between the stratification forces and dispersive forces. Extending the potential theory, the separation performance of a jig has been determined by calculating the density profiles of the stratified bed, which results from the jigging action, using the actual composition of the bed. Considering a dynamic state of equilibrium of the bed, where the stratification flux for each particle type i is balanced by the corresponding diffusive flux, it is shown that

dCi(h)- aC,(h)[p,--p(h)] i :1 ,2 .... n (3) dh

where h is a normalized height, and a is the stratification coefficient defined as

. _ ug v , nb

D (4)

In the above equation Vp is the volume of the particle, u is the specific mobility due to particle-fluid interactions, H b is the bed height, D is a diffusion coefficient and g is the acceleration due to gravity. Solution of n coupled differential equations, represented by eq.3, gives the equilibrium density distribution profile, which may be used as a measure of the overall performance of the jigging process. One of the advantages of this procedure, as compared to the traditional procedure using partition curves for performance evaluation and simulation of the jigging process, is that only a single parameter, or, needs to be estimated from experimental data. However, one of the main limitations is that it can be applied only to mono-size feed.


In all the jigging theories discussed above, the role of water motion on stratification has essentially been ignored. Assuming that the separation and stratification of particles in a jig are directly related to bed expansion which, in turn, is governed by water pulsation and physical properties of particles, Jinnouchi et

al. [7,8] have modelled the air pulsated jigging process, viewing it as a simple mechanical system. In this system, the air chamber in the jig is an energy storage device (represented by a spring), the water mass that oscillates is an inertial element, the bed plate is a frictional element, and the particulate bed is another inertial element. Variations in pressure in the air chamber and motion of water in the jig are modelled using an equation of state for the air, the unsteady state Bernoulli's equation, and unsteady state material balances for air and water. The bed is visualized as an unsteady state fluidized bed. The set of differential equations describing this mechanical system is solved numerically to determine the air chamber pressure, fluid velocity, and bed porosity. One of the attractive features of this model is that it is able to establish that the physical dimensions of the jig have a large bearing on how water motion will relate to the air valve settings. Based on this analysis, a method to control the wave pattern of water pulsation has been developed. The limitation of this model, however, is that there is no attempt to describe the stratification of particles in the jig.

A major contribution to the understanding of air pulsated jigs has come from a series of publications by Lyman and co-workers [9-14] based on well designed experimental studies and elaborate quantitative analysis of experimental data. The primary effort has been towards identifying a single parameter that has a key influence on bed stratification which, in turn, depends on jig operating parameters, feed characteristic and the jig size. This parameter is found to be the total energy dissipated in the jig bed within a cycle. It is claimed that any setting of jigging parameters that produces the same energy dissipation in the bed achieves similar stratification results. The parameters which appear to influence the energy dissipation most strongly are the frequency of pulsation and operating air pressure. The main feature of this analysis is that the bed stratification mechanism takes into consideration the role of operating parameters, and the air water behavior in the jigging process.

Owing to the complexity and the large number of operating variables involved in the jigging process, a few investigators have attempted to develop empirical models to describe the process kinetics. Assuming that the bed stratification is most sensitive to the jigging time, these models attempt to express stratification as a function of jigging time using empirical equations. Karantzavelos and Frangiscos [15] have proposed a first order model to represent the relationship between the jigging time and several parameters of the jigging process. Recognizing that the jig bed stratification is significantly affected by a number of parameters affecting the behavior of water in the jigging chamber, Rong and Lyman [10] have proposed a power function equation to represent the relationship between the bed stratification indices and the jigging time.

A NEW APPROACH TO JIG MODELLING

It is clear from the above discussion that much of mathematical modelling work has been done to describe the jigging process in its entirety. These models range from the simplest based on Stoke's settling laws, to those that attempt to predict pressure and velocities in the jig chamber utilizing the unsteady state Bournouli's equation. Yet, as with many other mineral processing operations, there has always been a search for the best model, that will become the most effective tool for process analysis and control. In the jigging context, a model that describes the flow of water, as well as the solid particles, would be more revealing. In the recent past similar models have been developed to describe the axial motion of the slurry in tumbling mills [16].

In the proposed model, the motion of solid materials is treated by a numerical tool known as the discrete element method and the corresponding effect of the fluid is considered through buoyancy and drag forces. Although the model is numerically intensive, it makes no assumption with regard to the manner in which the particles in the jig bed stratifies. This approach is also particularly justified at a time when the power of modern day computers is growing at an exponential rate.

Keeping the above modelling philosophy in mind, an attempt is made here to simulate the stratification of


particles by applying the discrete element method. At the very outset, the behavior of the fluid is represented by a drag coefficient that is computed from the properties and the physical state of the jig bed. The motion of the particles is calculated by performing a force balance on each individial particle. The model equation in its most simple form may be written as

ni'y=~_, F-Fb-F a (5)

where j$ is the acceleration in the y-direction, F b and F d are the buoyancy and the drag force respectively,

EF is all other applied forces, such as the contact and other body forces. The position of the particle is determined from its acceleration and when this treatment performed on all the particles, en masse behavior of the bed results.

Discrete Element Method (DEM)

Models based on the discrete element method are primarily devised to simulate any problem dealing with discontinuum behavior of particle systems. Mineral processing systems invariably deal with particulate material systems of varying nature. Quantitative models characterizing these processing units often treat the particulate system as a continuum for mathematical simplicity. DEM allows modelling of hundreds of thousands of particles at the cost of significant computational work to maintain independence of each particle; it is this computational requirement that limits its acceptance. Nevertheless, DEM results are more accurate and give better insight to the micro mechanical processes occurring at the particulate level.

In the DEM, the position of each particle is tracked incrementally by applying Newton's second law of motion and a force displacement law. Time steps are chosen which are so small that no new contacts are generated in the course of the motion of the particle in that time step. Hence, the resultant force acting on any particle over one time step is made up of forces arising exclusively from the contacts that the particle shared in that time step. The success of the programs depends on an efficient scheme of identifying contacts and keeping track of them as time progresses. In order to achieve this, the whole experimental space is divided into a uniform cubical grid. The identity of each particle is mapped into these grids. In the case where particles are considered as spherical elements, it can at most have entries in eight cubes, if the cube dimension is kept larger than that of the sphere. Thus, by checking each cube where the element has entry, all potential and existing contacts can be identified. In calculating the contact forces, the elements are allowed to overlap. These contacts are represented by a pair of springs and dashpots in both normal and shear directions. This numerical scheme is described in detail by Cundail and Strack [17]

The main calculation is carried out under a time-increment loop. In the i ~ time step, the contact velocity v', between particles A and B, is the relative velocity,

v =v a - v n (6)

where 9~ and ~ are the velocities of the contact point on A and B respectively. Ifi ffi s the contact

unit normal then

(7)

are the contact normal and shear velocities respectively. The incremental forces in the spring and dashpot are computed from these velocities and added to their previous values as

516 B .K . Mishra and S. P. Mehrotra

Sni _ 9 ~ S n i- 1 ( k~) t )~ ln i

i ~ S i- 1 ~)s i S . = ,, - ( k i l t ) .

(8)

where, S~ and S~ are the normal and shear forces due to the spring, ~ is a rotation tensor that rotates

the unit vector ~ , i - I to /~i , k is the stiffness of the spring, and fit is the time step. Similarly, the

dashpot forces in the normal and shear direction are given by

L~ = -[~,,7~ (9) Ds = - D,v,

Here, 13 n, and 13 s, are the normal and shear damping coefficients determined from the coefficient of restitution, as described in Ting and Corkum [18]. Thus the total normal and shear forces acting on element A due to its contact with element B at time step i are

F i i i - ' -S D,, n n ~ F s i i i

"- Ss ÷ D s

(lo)

respectively.

Mineral particles differ in size and specific gravity. In jigging the particles are separated based on their density. DEM allows particle density and specific gravity as inputs and based on these properties the mass and moment of inertia of the particles are computed, which are used in the calculation cycle. Collision between particles is controlled by the coefficient of restitution. Similarly, the coefficient of friction is used to allow for particles to slide past one another under limiting conditions. Thus, the model is able to predict the actual behavior of particles as it only considers physical and material properties. This method has been successfully used for complete analysis of ball mills [19].

The effect of fluid on the motion of particles is computed in a simplified manner. When a particle enters the fluid its Reynolds number R e is computed. Then the drag coefficient is determined using the Abraham equation

Ca=0.28( 1 +9.06)2 (11)

where C d is the drag coefficient. The above equation is generally valid for R e < 105. For higher Reynolds number a constant value of 0.44 as drag coefficient is chosen. The force on a particle due to viscous drag is given by

Fa=O.5×CaxpFv2xa (12)

where F d is the force due to drag, v is the velocity of the particle, pf is the fluid density, and A is its area of cross section. Similarly, the buoyancy force is given by


Fb=mxgxPi (13) Ps

where F b is the buoyancy force acting on a particle of mass m and density Ps. These forces are added to the right hand side of equation 5 and the positions of the particles are found by numerical integration as discussed earlier. This model also allows computation of the energy expended during jigging.

NUMERICAL RESULTS

A two dimensional computer program based on the DEM algorithm was developed to study the stratification of particles in a jig bed. A total of 435 disks of different specific gravity and size were used for the simulation. The width of the bed was kept at 2-m. The simulation was done with an amplitude and frequency of 0.14-m and 14 cycles per second respectively. The result of the simulation is shown in Figure 2 by means of snapshots taken at the beginning and end of the simulation. Particles are perfectly separated in about 10 seconds of jigging time. In the past, DEM has been used by Mishra and Chakraborty [20], and Beck and Holtham [21] to study jigging problems. However, this approach has some drawbacks concerning the dynamic state of the bed. Simulation of a two dimensional system means that sorting mechanisms such as consolidation trickling cannot be modelled as there is no connection between inter-particle pore spaces in the two dimensional compacted bed through which small particles could trickle. Moreover, the idealized water model is used. It is now well established that the water motion in a jig is considerably more complex [9]. Thus, there arises a need for a three dimensional model.

Particle types 3 Density(Kg/m 3) 3500

2200 1500

Size (m) 0.031 0.0325 0.051

Amplitude (rn) 0.14 Frequency (Hz) 2.23

Fig.2 Snapshots showing initial and final stages of jigging.

A three dimensional computer program based on the DEM algorithm discussed earlier was developed. A separate auxiliary program was written to randomly generate particles within a specified volume. The main program takes the operating parameters such as amplitude and frequency of the pulsating fluid as input. The program also allows input of other pertinent data concerning the particle properties and the geometry of the jig bed. The program runs for a specified number of cycles and at the end of each run the particles are allowed to settle. Once the particles are settled, then the entire bed is scanned to determine the variation in concentration along the bed height.


In a simpler case, the 3d-model was tested to study the stratification in a bed consisting of 400 mono-size particles. Two types of particles, of density 1.5 g/cm 3 and of 2.5 g/cm 3, were used in equal proportion. The particles were pulsated in a 16-cm diameter jig column where the particles occupied a bed height of approximately 16 cm. Pulsation of the jig bed was carried out by applying a sinusoidal jig cycle where the amplitude and frequency were fixed at 18 cm and 2.5 Hz respectively. A graphic interface was provided for visualization of numeric data. Two snapshots of the graphical display are shown in Figure 3. These snapshots correspond to the beginning and end of the cycle. It is observed that in about 10 cycles complete separation was achieved. The graphical interface helps in identifying unusual motion of the bed due to system resonance. This is a problem inherent to the DEM model, where the bed resonates when the frequency of oscillation is comparable to the system frequency. The analysis of this system to predict resonance for a given set of operating conditions is quite unwieldy particularly when the number of particles is large. However, this feature can be observed through continuous data visualization.

I Density(kg/m3) ' 1500, 2500 [ Radius (m) : 0.0095 No. of Particles: 400

Start End

p,O,O)

Fig.3 Result of 3-d simulation: snapshots showing initial and final stages of jigging.

There are various ways in which the degree of stratification can be quantified. One way is to determine the difference in the centre of gravity between the assembly of particles of different densities. In the case of only two types of particles, the difference in the centre of gravity is half the total bed height when the particles are assumed to be perfectly separated. On the other hand, at the beginning of the stratification, when the particles are mixed, the difference in the centre of gravity is close to zero. As jigging progresses the difference in the centre of gravity tends to increase from zero. Figure 4 shows the variation in the difference in centre of gravity (CG-difference) as jigging progresses for two typical situations. In one situation jigging was performed continuously until the end, while in the other, the process was stopped a cycle earlier allowing the particles to settle. In either situation, the centre of gravity of the assembly of heavier and lighter particles was measured. The difference was calculated to determine the CG-difference of the system.

Several observations can be made from Figure 4 with regard to the dynamics of the process. First, the centre of gravity of the lighter fraction fluctuates more than that of the heavier fraction. In other words, the peak- to-peak distance is higher for the lighter fraction in comparison to that of the heavier fraction. Second, during continuous jigging, the CG-difference is always overestimated. At the time of measurement, if the bed is in a dilated state then the CG-difference is in fact higher than the true one that corresponds to a compacted state. For this reason, in order to determine the CG-difference attained by the system, jigging was stopped a cycle earlier. As can be seen in the right-side plot in Figure 4, after about 15 seconds the


bed is settled. The CG-difference measured is close to that of the theoretical value. Finally, in either case the CG-difference tends to increase, showing the progress of stratification.

Fig.4

Y

i

I C

i ~bc

0 5 10 15 20 0 S 10 15 20 25

Variation in the difference of center of gravity of two types of particles during jigging with and without settling.

The dynamics of the stratification process are quite complex. At the initial stage, the en-masse motion of the bed is small but there is mutual adjustment of particles for stratification. This leads to increase in the difference in the center of mass. Towards the middle of the process lighter particles loosen from the surrounding heavier ones and this causes the bed to be disturbed. When this happens, the difference in the centre of mass is maximum. Towards the end of the stratification, the heavier portion of the bed moves very little, therefore, the difference in the centre of mass tends to remain constant. At this stage, pulsation is stopped and the bed is brought to a static state.

The motion of the lighter fraction in a jig bed is quite crucial, particularly in the context of continuous operation. The location of the splitter depends on the extent to which these particles are distributed across the bed. Figure 5 shows one such distribution during jigging. As can be seen from the Figure, the light fraction is well distributed across the bed at the initial stage (t=0.256 seconds). If one considers the concentration of the lighter particles at a fixed bed height (for example, three quarter distance above the bed height), this concentration must continuously increase during jigging. At the end (t=10.24 seconds), as shown in the Figure, the lighter particles are concentrated above a bed height of 11 cm showing a good degree of separation.

0.16

0 .14

~ 0.12

0.1

--~mO.O8 i .~ 0.06

m O.04

0.02

i

- .u - t . lo .24 |

I I I I

0.2. 0.4 O.B O.B 1

Concentration

Fig.5 Variation in the concentration of light particles through the jig bed.


At the end of the simulation the particle bed is scanned to determine the concentration of particle types at different bed heights. This result is shown in Figure 6. It is seen here that the separation curve is quite similar to those found in actual practice. It is observed that there is some amount of misplaced material in the heavy fraction. In other words, the lighter particles are still trapped in the heavier fraction. Between a bed height of 0.06-m and 0.1-m the particles are mixed. It is felt that by varying the amplitude and the frequency of pulsation better separation may be achieved.

i 0= =

1

0.9

0.8

0.7

0.6 0.5

0.4

0.3

0.2

0.1

0

- . -= - - light

*1- | I !

0 0.05 0.1 0.15 0.2

Bed height [m]

Fig.6 Variation in the concentration of particles through the jig bed.

DISCUSSION

In this paper several mathematical models aimed at describing the jigging process are reviewed. Modelling of the stratification process is not a formidable task as long as a good knowledge of the physics of the stratification process is available. In particular, an understanding of the fluid flow pattern through the porous structure of the jig bed is central to the manner in which the bed stratifies. Fluid flow of this nature draws analogies to many of the main classes of fluid flow that have been extensively studied in other disciplines of engineering. In fact, it can be considered as flow through a fixed bed because the fluid has to find its way through the porosity of the bed, and it is this porosity that keeps changing with time that eventually leads to a stratified bed. At the same time, it does share some properties with a fluidized bed where the velocity of the fluid has a significant effect on the motion of the particles comprising the bed. The fact that the fluid has to flow through a bed whose properties change across the length of the bed makes the problem very complex. The solution to this problem rests on a massive numerical effort, coupling the discrete element method that models the motion and interaction of particles, with a model that can describe the motion of fluid through the particle bed. While the discrete element formulation (DEM) to model the particle interaction problem is completed, the model for the analysis of the fluid is under progress.

The major drawback of the 3d-DEM approach is its enormous computational requirement. In this regard, the time step of integration should be as high as possible while maintaining numerical stability and accuracy. The parameters that influence the time step are the contact stiffness and the mass of the particle as given below

8t=0.2 m ~ (14)

This can be derived considering a single particle connected to the ground with a normal spring. It is


recommended that one-tenth of this critical time step be used for integration. Several simulations were carried out where the contact stiffness was varied between 400000 to 200000 N/m. It was observed that in each case the CG-difference and the concentration profile of the bed were quite comparable. Thus the lowest feasible value for the stiffness was chosen. In addition instead of one-tenth, one-fourth of the critical time step was used without losing any accuracy.

Although the fluid model is simplistic in nature, the result of the simulation is consistent with experimental observation in a model jig. This model is being modified to incorporate a more rigorous treatment for the motion of fluid by using MAC technique. This technique has been used in conjunction with DEM to predict the motion of fluid in ball mills [16]. Also the model can only be accurate if a good knowledge of the governing principles of the process is available. This can only be achieved by careful experimentation. To this end, a jig consisting of a 30-cm column for the particle bed is being constructed. The jig is being instrumented to have provision for measuring the density profile and pressure drop across the bed, air pressure, and hutch water flow. It is hoped that these detailed measurements will help in validating the DEM model that can be eventually used for simulation and control purposes.

ACKNOWLEDGEMENT

The authors gratefully acknowledge the financial support provided under SERC scheme of the department of science and technology (DST) of the Government of India.

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Modelling of particle stratification in jigs by the discrete element method