(1995) discrete element simulation of granular flow in 2d and 3d

Pergamon Chemical Engineering Science, Vol. 50, No. 6, pp. 967 987, 1995 Copyright 1995 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0009 2509/95 $9.50 + 0.00

0009-2509(94)00467-6

DISCRETE ELEMENT SIMULATION OF GRANULAR FLOW IN 2D AND 3D HOPPERS: DEPENDENCE OF DISCHARGE RATE AND WALL STRESS ON PARTICLE INTERACTIONS

P. A. LANGSTON, U. TLrZON and D. M. HEYES t Department of Chemical and Process Engineering, University of Surrey, Guildford, GU2 5XH, U.K.

(Received 29 July 1994; accepted for publication 5 October 1994)

Abstract Discrete element Newtonian dynamics simulations have been carried out of filling and discharge under gravity of non-cohesive discs (in two dimensions) and spheres (in three dimensions) from model hoppers. The current model improves that developed previously by us (Langston et al., 1994) in several respects. We introduce a continuous and gradual hopper filling method, a more realistic normal-tangential interaction between the particles, particle size polydispersity, and the model is extended from two to three dimensions (3D). The hopper discharge rate has been computed as a function of material head height, outlet size and the hopper half-angle. The model results are, in general, in very good agreement with established literature empirical predictions. The hopper wall stresses have been compared in the static state after filling and in the dynamic state during discharge. Generally there is encouraging agreement with predictions from the continuum differential slice force balance method, with significant improvements over our previous work. We have also observed, for the first time in a discrete element simulation of hoppers, the appearance of rupture zones within the material and associated wall stress peaks where the rupture zones intersect with the hopper wall. We consider that the current model is more successful than the previous one because the particle interactions include a much greater level of frictional "engagement" at low loads, with less variation at high loads.

1. INTRODUCTION

Storage and flow of granular materials in hoppers is important in many process engineering applications in, for example, the coal, food, pharmaceutical, ce- ment and chemical industries. Problems encountered include erratic flow or blockage, dead zones and material ageing, segregation, attrition, wall adhesion, wear of the hopper wall, hopper collapse and dust explosions. Our current understanding of the different flow regimes in such systems is largely empirical as the granular state is rather difficult to characterise, it being neither liquid nor solid, but sharing some aspects of both. As a result the key determining processes in consolidated granular flow are still poorly understood. In the low solids fraction high shear rate and compact near-static high solids fraction extremes, kinetic theory and soil mechanics approaches, respectively, give a good description of the macroscopic behaviour of the granular state. However, the low shear rate or quasistatic high solids fraction granular state found in hoppers is still poorly understood from a fundamental (i.e., granule scale) level. A range of theoretical techniques has been employed in the analysis of hopper flow. There is the class of analytical treatments, for example, Beverloo's equation for pre- dicting mass discharge rate (Beverloo et al., 1961) and Janssen's differential slice force balance method for wall stress. The flow of cohesionless granular mater-

t Corresponding address: Department of Chemistry, Uni- versity of Surrey, Guildford, GU2 5XH, U.K.

ials in conical hoppers has been modelled by Nguyen et al. (1979) using a perfectly plastic continuum representation of the material satisfying the Mohr- Coulomb yield criterion. The finite element (FE) method has also been used to model wall stresses by, for example, Ragneau (1993). The problem with these "continuum" methods is that, in the main, they ignore the effects of microstructure of the bed and rely on an assumed (often over-simplistic) constitutive equation. Most of the bulk properties are assumed to be constant across the system and independent of particle properties such as shape, size and friction, with the velocity and stress distributions within the flowing bulk being assumed to follow a certain functional form. These treatments are prescriptive rather than predictive, as far as certain phenomena (for example, rupturing of the material) are concerned.

As a result, the so-called discrete element (DE) modelling methods have become popular in recent years because they start from the basis of the individual particles (and their physical characteristics) as separate entities in the model. This has the advantage that no global assumptions are required on the material such as steady-state behaviour or uniform consti- tuency. The Monte Carlo DE method is the most basic of these models and involves producing a series of random particle displacements which typically make little attempt to follow a particle collision se- quence and ensuing correlated motion from realistic equations of motion. The MC method has been applied to, for example, settling powders (Snyder and

967

968 P. A. LANGSTON et al.

Ball, 1994) and vibrated powders (Barker and Mehta, 1992). The granular dynamics (GD) method also incorporates the discreteness of the material by treating each particle explicitly and in addition attempts to generate a realistic description of the time evolution of the assembly. The GD method has attracted increasing attention since the pioneering work of Cundall and Strack (1979), who first applied this approach to powders. The technique is, in fact, essentially the same as the molecular dynamics (MD) method (out of which it grew) in which the particle trajectories are evolved in discrete time steps using a numerical inte- gration of Newton's equations of motion. The main difference between GD and MD is that particle interactions are not treated as being perfectly elastic but dissipative by virtue of particle friction and kinetic energy loss that takes place as a result of the relative motion of the particles at contact. This drives the assembly to a static state where the particles are essentially stationary and have zero "kinetic energy" (or zero "granular" temperature) unless agitated or continually subjected to some external force field (e.g., gravity). As with molecular dynamics, the GD method does have practical limitations, including a restriction on the shape of the particle to something idealised (e.g., a sphere) and finite sample sizes. Rarely, however, has this prevented the model from giving useful insights into the essential physical processes for a particular particle assembly.

The GD method has been applied to shear flow of non-cohesive assemblies by Thompson and Grest (1991), Campbell and Potapov (1993), Bashir and Goddard (1991) and Turner and Woodcock (1993), vibrated piles of powder (Hermann, 1992), flow down an inclined chute (P6schel, 1993) and hoppers by Thornton (1991), Ristow (1992), Yoshida (1993) and Tanaka and Tsuji (1993), who in the latter case com- bined the GD method with the FE method to include air flow around the particles. Hopper flow is a parti- cularly severe test of the GD method as the normal and frictional loads on the particles in the asymptotic stress region of the hopper can be quite extreme. In addition, the material during discharge at different times can undergo high load slow ("quasi-static") motion and highly "thermalised" high shear lean phase flow, so the algorithm must be able to accommodate stably the transition between these two extremes.

The choice of the particle-particle interaction is an important feature in a DE method. Most particles in a granular assembly will form contacts with several other particles.The contact interactions have elastic, kinetic energy damping and frictional components. For the elastic part, some groups (e.g., Thornton, 1991) have used near-exact contact mechanics with equations based on the Hertz force-displacement relation employing realistic values for the elastic moduli [see Johnson (1985)]. Using this approach, the dynamics of the system are followed at the microcontact level. This is in the spirit of the TRUBAL program developed by Cundall and co-workers, who considered

very stiff particle interactions. Such a model is slow to converge and reliant on the choice of ad hoc damping parameters to subdue spurious assembly vibrations. In any real granular materials, contact interactions between particles are quite complicated and rather poorly characterized due to the uncertainties, associated with, for example, particle shape, surface roughness, hardness. The conventional contact mechanical approach to assembly simulation relies on simplifying assumptions such as perfectly smooth surfaces, spherical particle shape and single point contacts which are, in fact, not appropriate for most of the real materials. Although individual contacts, where the particles are at specified relative orientations, are quite "hard" and incompressible on the scale of the mean particle diameter, the mean interaction as a function of particle centre-centre separation will in fact be rather soft on the particle distance scale. This is because the granules can reorientate and slide relative to one another to accommodate an applied load.

As an alternative to the contact mechanical approach, which would be impracticable to implement routinely with the current generation of computers, we explore a range of analytical forms for continuous normal force interactions between particles based on the assumption that the material behaviour is determined by "effective" contact interactions which act on the scale of the nominal particle size in an assembly rather than on the (much smaller) dimensions of a single asperity contact region. We make no attempt to model the assembly on the asperity distance and time scale, but on the scale of the assembly particles themselves. This results in much "softer" spherically symmetric effective interactions, which we consider could be an adequate description of the many body interactions in an assembly that arise predominantly from geometric effects. Consequently, the assembly friction is largely decoupled from individual particle microcontact friction, although we adopt similar functional relationships between the tangential and normal forces in the assembly to describe an "effective" frictional behaviour. We are not the first to adopt relatively soft "notional" interactions for granular assemblies, for example, Sakaguchi and Ozaki (1993), Thompson and Grest (1991), Campbell and Potapov (1993) and Walton and Braun (1986) modelled granular shear flow with arbitrary (relatively soft) Hooke's law spring constants. However, we believe that we are the first to compare the significance of the different analytic forms for the assembly and to make a system- atic study of the sensitivity of the computed material behaviour to the analytic form of the contact interactions.

We consider three analytical forms of the normal interaction laws between the particles. The first is a Hertzian interaction which is proportional to the elastic modulus, E*, of the two particles and R is the contact radius of curvature,

Fs . . . . . = ~E*xf lR(tr -- r) 3/2 (1)

where tr is the particle diameter and r is the separation

Discrete element simulation of granular flow

distance between the centres of the two particles. The Hertzian force varies as the 3/2 power of the overlap displacement. A continuous normal interaction is also considered which has a quite different dependence on the extent of particle overlap,

where n is an arbitrary index, e = amg/n and e set the and force and distance scales. The coefficients are chosen so that the force is equal to the gravitational force, rag, when the particle centre-to-centre separation equals a. We also consider a Hooke's law spring interaction with an effective value of the bulk stiffness, k,

FNs= k(a - r). (3) As seen in Fig. 1, these three different normal force interaction laws have a distinctly different dependence on the separation distance, r. (The forces are normalised so that unit forces equals mg, the product of the particle mass times gravitational acceleration.) Hence, adjustments of the arbitrary constants can achieve quantitative agreement over only a very limited range of r. In all three cases, the working values for the input parameters result in normal stiffnesses that are quite "soft" when compared with real elastic contacts between particles. For example, the values used of the "effective" bulk modulus E in Fig. 1 are in a range which is a factor of 104-105 times smaller than the Young's moduli of real particles. The

969

respective normal stiffnesses arc given by the deriva- tives ( - ~F/dr) which result in

3 FNa~t~ ks .... - 2 (a - r~ - 2E* x/g*(tr - r) 1/2 (4)

kc, (n+ 1) FNc'=~n(n+ l)(~)" = r r 2 (5)

FN s ks = = k . (6)

(o" - - r )

The advantage of the continuous interaction analytic form of eq. (2) is that it becomes appreciably stiffer with greater overlap, (a - r), much more so than those of eqs (1) and (3) which manifest comparatively modest increases in stiffness in the same separation region. As the particles impinge more for separations less than a the correcting force increases more dramatically, which reduces excessive particle overlap under high loads; a severe problem in the hopper configuration.

In our previous report (Langston et al., 1994) we described the results of gravitational discharge of discs from a 2D wedge-shaped hopper. We have improved the model in several ways. The method of filling the hopper is made more realistic. Rather than allowing a "block" of material to fall under gravity en masse until it collides with the hopper base, wc have developed a gradual filling method in which the

Continuous, Hertz and Spring Reactions

E

300

200

100

h%

0 .8 I . I

'\ % , '.Q E=15000 ' !

"-. K. ooo X ',. I i

\,, ~ : n=144

X X : n=3e. . , ",\, /X !i

. . . . . ~ ' ~ " ~ ,

.9 1 .0

r I cr

Fig. 1. A comparison between the elastic component of the normal force, FN(r), for the (a) continuous interaction, eq. (1), (b) Hertz interaction, eq. (2) and (c) Hooke's law spring, eq. (3), for parameter values (in

reduced units) given on the figure.


model granules are introduced into the top of the hopper a few particles at a time and allowed to fall under gravity. This is closer to the manner in which granular material is delivered into real hoppers, for example, by pneumatic conveying or conveyor belt. The growing bed in the model therefore has a longer time to relax and become better consolidated. We show that this filling procedure leads to more fully developed internal and wall stresses.

It is well known that the packing of hard disks (or spheres in 3D) in a column can manifest long-range structural correlations down the column as a result of crystal-like packing arrangements of the particles (Meakin and Jullien, 1991). Two 2D discs have a tendency to form crystal-like structures [seen in Langston et al. (1994)], which for example could lead to spurious stress distributions near the walls. Campbell and Potapov (1993) also showed that uniform discs can give rise to packing arrangements that build up or- dered stress patterns. In order to minimise this effect we have made our sample slightly polydisperse. In addition the model has been extended to consider discharge from a 3D conical hopper. We also make a more detailed investigation of the effect of the form of the particle contact interactions on the computed properties. These simulations with thousands of particles are rather "small scale" when compared with the millions or billions of particles found in full-scale silos. However, we nevertheless consider that the simulations can give useful insights into the underlying phys- ics of the processes involved and which eventually could lead to improvements in the design of hoppers.

2. SIMULATION DETAILS

A description of the technical aspects of the simulation method was given in Langston et al. (1994). In that study N two-dimensional mono-sized discs were allowed to fall under gravity into a wedge-shaped hopper, allowed to settle to a near-static state, and then discharged under gravity when the hopper base was opened. The particles started in "mid-air" from a lattice configuration each with a random velocity scaled to an input granular "temperature". The particles were allowed to fall en masse into the closed hopper. Translational and rotational motion of the particles were incremented using a numerical integra- tion procedure in which assembly configurations were generated at time intervals of At. The contact forces consisted of normal and tangential components, each of which had a "viscous" dissipation term proportional to the component of the velocity in that direction. The elastic part of the normal interaction was of the continuous interaction form [eq. (2) above with n = 36] and there was a tangential frictional interaction of the Mindlin form (although it operated on a much larger distance scale up to the point of gross sliding than originally intended for microcontacts). The same interaction forms were deemed to apply between the particle and wall, though with independently chosen parameters.

The equations of motion for the translational and rotational motion we used were

= F /m + g (7)

~b = T / I (8)

where v is the particle translational velocity vector, m is the particle mass, F is the sum of the contact forces on the particle, g is the acceleration due to gravity, 09 is the angular velocity vector, T is the torque and I is the moment of inertia. The contact force consists of a normal interaction, Fn,

F, = FN -- CnVn, (9)

where FN is the elastic part of the normal interaction, given by eqs (1)-(3), cn is a normal interaction damping coefficient and if n is the unit vector in the direction between the centres of the two particles, then vn = (v. n)n. The second term on the right-hand side of eq. (9) represents the contact damping.

There is also a tangential interaction, Ft, the frictional component of which is given by

Fr =/~FN(1 -- (1 -- IcStl/C~max) a/2) (10)

where fit is the total tangential displacement between the two surfaces from the point where they initially came into contact. If Ic5~1 > 6ma, then gross sliding is deemed to have started and the frictional force as- sumes a constant value given by Amonton's law, Fr = I~FN, where/~ is the coefficient of friction. The analytic form of eq. (I0) was first proposed by Mindlin and Deresiewicz (1953) for micromechanical contacts, although our choice of 6max is typically several orders of magnitude larger than that for a typical micromechanical contact. The total tangential interaction force at any time is

Ft = FT -- c,v~. (11)

The second term on the right-hand side of eq. (11) is the contact damping force proportional to the tangential component of the relative velocity, vt = v n. The damping coefficients cn = 0.5 and ct =0.5 were chosen empirically on the basis of simulations of small numbers of particles. The overlap of the particles along the lines of their centres is equal to cSn = ac - r, where ac is a nominal particle diameter. In the case of the continuous interaction, the particles have no well-defined "diameter" as their distance of approach will depend quite significantly on the normal load (especially for low loads < m#). It is reasonable to assume that frictional engagement between the particles (which can have a significant influence on the behaviour of the assembly) will start to take place at separations greater than a in the interaction curve (i.e., for normal loads less than rag). There is a "tail" to the interaction that extends beyond r = a, so that the value of cr should be somewhat less than ac, the interaction cut-off and the commencement of tangential and normal contact. In contrast, for the Hertzian and Hooke's law interactions we have only one possible choice for ao which is crc = or, because for separations r in excess of a their two functional forms


demand there is no interaction between the particles. The ability to have tr~ > o is a point in favour of the continuous interaction analytic form.

Turning now to the tangential displacement between two particles in contact, it is important to specify the relationship between the normal and tangential compliances. As we argued earlier, the frictional engagement between the particles and associated tangential displacements are key factors in determining the static state and the kinematics of granular materials in hoppers. The magnitude and functional form of the coupling between the normal, ~,, and tangential, fit, displacements is therefore a matter of some importance. Mindlin and Dere- siewicz (1953) formulated this relationship using Hertzian contact mechanics. They derived the analytic form of an elastic-plastic tangential displacement curve up to the point of "slip" or gross sliding. The stiffness of the "Mindlin" curve is determined by the tangential displacement at the onset of gross sliding, 3~.~, because this scales the entire Mindlin curve. The duration of frictional engagement is determined by the magnitude of 6m,~" In our earlier model, the average tangential compliance, CF = 6=,J#FN, was chosen so that Ce was a constant during the simulation and independent of FN. Therefore, we had 6=,~ oc FN, i.e.,

3max = CvlaFN (12)

which because FN is non-linear in 6, meant that 6m,~ was non-linear in 6,. However, in the contact mechanics equations of Hertz, and Mindlin and Deresiewicz, Cv is dependent on the normal load. Using these equations it is possible to show that at the limit of gross sliding for elastic spherical particles, the ratio of tangential displacement to normal displacement is a constant, i.e.,

~m~ = fiR3. (13)

971

where fir is a constant. This result, which we believe is new, is derived in the Appendix. The analysis gives

(2 - v) 3R =/~2(1 -- v) (14)

where v is Poisson's ratio. For example, for an elastic material with/~ = 0.6 and v = 0.3 we have fir = 0.73. Therefore, the model based on contact mechanics reveals a quite different normal load dependence for 3m,x than given by eq. (12). The two prescriptions for 3m,x [i.e., eqs (12) and (13)] are compared in Fig. 2, adopting trc = 1.2tr. The figure shows that in the case of eq. (12) used in our previous publication, the level of frictional engagement (measured approximately by the magnitude of 6m.~) is quite small compared to the diameter of the particle, and therefore likely to lead to relatively shortlived contacts for normal loads less than ~ 10mg. For normal loads in excess of ca. 40 rag, 3mx increases dramatically. This will lead to a large mean tangential compliance and again to weakly developed frictional engagement. In contrast, the scheme embodied in eqs (13) and (14) exhibits a 6m,~ with a much smaller normal load variation,

0.15o for n = 36 and 144, even at low loads of rag. This latter variation we consider is more realis-

tic as geometric/frictional coupling between neigh- bouring particles is likely to occur even under conditions of minimal normal load. Therefore in this study, we use the prescription 6m,~ = 3R6,, where 6R is a constant for each simulation.

The GD algorithm is, without any special program- ing, ~ N 2 in computer time. As in Langston et al. (1994), we use neighbourhood lists to pre-eliminate distant interactions and therefore reduce the amount of computer time used. The program periodically es- tablishes a list of near neighbours which contains all those pairs that could, but do not necessarily, interact (i.e., r ~< ao, where we have chosen a o = 1.55a~). When

b

E ~o

0.6

0.5

0.4

0.3

0.2

0.1

0

. . . . . . . . i . . . . . . . .

/

constant compliance - - / -

. . . . . . . i . . . . . . .

10 100 FN / mg

Fig. 2. The dependence of the maximum tangential displacement, tS=.x, as a function of the elastic component of the normal load, FN for the situation of a constant tangential compliance, eq. (12) and for

a continuous interaction with eq. (13) and tr c = 1.2o.

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the neighbour lists needs to be recompiled, it would still be inefficient to test all combinations of particles in the system. To avoid this here, the hopper was divided into square zones in 2D and cubic zones in 3D. Each particle i was considered in turn, and all other particles j in the same zone and immediately surrounding zones were considered as potential interaction partners. If the distance from i to j was less than the outer cut-off, ao, then j was added to the list of possible interacting neighbours for particle i. (The zone lengths were made equal to 1.55trc, the radius of the outer cut-off in the neighbour list scheme.) For each contact the tangential displacement 6z (the accu- mulation of incremental displacements over past time steps) was recorded. When recompiling the particle neighbour lists it was necessary to determine whether particles i and j were already in contact (i.e., had already built up a tangential displacement). If so, the value for fit was transferred to that of the contact in the new neighbour list.

It is important to choose the magnitude of the time step in the simulation carefully. It needs to be as large as possible in order to cover a fixed period of time in the least number of time steps. However, if the time step is too large the algorithm becomes inaccurate and can even become unstable. Ultimately, the only sure test is to discover empirically if the algorithm is stable and leads to consistent results with small vari- ations near to a particular value of the time step. There are several possible criteria in the literature that enable a trial time step to be established. It is possible to base the time step on the natural period of oscilla- tion of the equivalent spring with the same mean stiffness, k, as the interaction being used. The time step using this criterion (Thompson and Grest 1991; Zhang and Campbell, 1992) is

Ate ~ O. ln /m/k . (15)

For 0.1-1cm diameter particles and a realistic G ~ 109 N m -2 the Hertz interaction gives At ~< 5x 10 -5 reduced time units. Using a more typical value of G ~ 105 N m -2 for these simulations then At increases to ~< 5 10 -3 reduced time units, which is comparable to the value that we have used here for the CI (n = 36 - 144) and the Hooke's law (k = 500 reduced units). Another criterion, used in classical engineering modelling, is to base the time step on the Rayleigh wave speed through the solid spherical particles. This criterion would be appropriate if the dynamics on the asperity level were being followed. As we make no attempt to follow the behaviour of the particles on this time scale, we consider that the Rayleigh wave speed is not pertinent to our model. However it is interesting to compare the value if the critical time step obtained by this method with that derived from the eq. (15). The critical time step is given for Hertz interaction law,

Ate = (nR/~t ) (p /G) 1/2 (16)

where 0.90 < ct < 0.95 (Johnson, 1985). The ratio,

P. A. LANGSTON et al.

E /p , is relatively material independent (e.g., 2.6 _ 0.1 x l0 T m 2 s -2 for iron and glass). For 1 cm diameter particles, p~10akgm -3 and G~109Nm -2 we At ~< 5 x 10 -4 reduced units which is significantly smaller than the value we are using (0.001).

The normal elastic interactions, FN, were taken from all three of the analytic forms of eqs (1)-(3). The exponent n in Langston et al. (1994) was fixed at a value of 36 which was somewhat arbitrary at that stage. Here we explore a much wider range of n values to determine the effect they have on the behaviour of the model hopper. Particle size polydispersity has been introduced in the model. The particle size distribution is uniformly distributed in the range (1 -6 )a ~ a, where 6 ~ 0.1 and a is the maximum particle diameter. In the computation of the interaction for two specific particles with independently assigned diameters, the reference particle diameter, tr, used in eqs (1)-(3) was set to the arithmetic mean of those of the two particles. Figure 3 compares particle

(a) monosized

(b)

polydisperse

Yl ) x Fig. 3. Comparison between a 2D static assembly in the vicinity of the base of a flat-bottomed hopper for the cases of (a) monosized disks and (b)a polydisperse mixture with

10% polydispersity as discussed in the text.

Discrete clement simulation of granular flow

packing in the lower part of a 2D hopper for t5 = 0 (mono-sized particles) and for 6 = 0.1. Clearly, polydispersity reduces the tendency to form "microcrystal- lites" and gives rise to a more amorphous structure which is closer to that found in real granular systems. In Fig. 4 wc compare the en masse filling method with that of the new gradual filling scheme (for mono-sized particles) which wc use exclusively in this work.

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The 3D model is very similar to the 2D version. It simulates uniform non-cohesive spheres filling a closed cone-cylinder hopper by the "gradual" filling method, allowing a settling of the assembly to a near- static state and then the material is allowed to discharge under gravity through an opened orifice at the bottom of the hopper. The main difference is in the treatment of the tangential contact interactions, as the

(a) en Masse

(b) Gradua l

t=O t=10 t=40 t=60

start fill -static discharge

O00000000( oooooooooo ~oo o~oo oo c

%

~ 0

o

NNN

t=O t=SO t=ISO t=160

start fill -static discharge

Fig. 4. Snapshots during two different hopper filling methods (a)en masse as in our previous work (Langston et al., 1994) and (b) using the gradual filling mechanism used throughout this report.


possibility of spin and roll is open to a 3D contact but is geometrically excluded for 2D disks. Our model for the contact friction extends the previous 2D Mind- lin-like case. The contact tangential displacement fit is now a 3D vector. The algorithm goes through the following steps. The contact coordinates rc and velocity Vc were calculated using

vc = v + to x (re -- r) (17)

where v is the velocity of the centre of mass. The additional contact tangential displacement in the current time interval, 6', was computed from the relative velocities of the two particles at contact. This was added to the displacement vector from previous time steps,

6,(t + At) = 6,(0 + ~'. (18)

Then 6, was projected onto the plane perpendicular to the line of the particle centres, but maintaining the same magnitude. [If the magnitude of 6, exceeded 6max (i.e., gross sliding occurs) then the tangential displacement was set to this value]. Therefore, the new value of 6, was assigned to this "projected" quantity. Then the magnitude of the friction force was calculated from the magnitude of ~t using the Mindlin-like tangential force-displacement relationship of eq. (10), with the direction of the friction force vector being along the displacement vector. Unlike the original Mindlin treatment we made no allowance for contact hyster- esis effects along loading and unloading paths. The above algorithm neglects the effects of particle spin (rotation about the line of the particle centres). There are two effects here. Torsional friction will set up a moment about this axis, but as the area of contact for spheres is likely to be small, the moment should also be relatively small. Spin could also affect the direction and magnitude of the tangential displacement vector, fir. It was considered that these are likely to be second-order effects for spherical particles and

at present poorly quantified for arbitrary granular materials. Consequently at this stage of the model's development the effects of contact spin have not been included. The contact forces were summed vectorially for each particle and linear acceleration calculated at each time step. The moment, G, acting on a particle from each tangential force F was calculated from the vector product of the contact vector relative to the centre of the particle, R, and force G = R. Ft. These were vectorially summed for all contacts on the particle and the angular accelerations then calculated. The Verlet algorithm was used to calculate the particle motion as described in our previous report.

All quantities are quoted in particle orientated units, e.g., a for length and m for mass. The energy scale for the continuous interaction (e) of eq. (2) set so that ~, = amo/n and unit force is equivalent to mo. Time is in units of (a/ff) ~/2.

3. RESULTS AND DISCUSSION

3.1. Two-dimensional hopper The results of a series of simulations using the 2D

hopper program, incorporating the model refine- ments described above, are reported is this section. They were based on similar hopper geometries to those used in Langston et al. (1994). The principal parameters are listed in Table 1. As in the previous study, the quantities investigated primarily were discharge rate, static wall stress and dynamic wall stress. The graphical output shows the wall stresses, the fraction discharged and discharge rate as a function of time. The normal and tangential wall stresses are represented by scaled lines in the direction the forces act on the wall. (The tangential wall stress lines have been reorientated by 10 for clarity.) The wall stresses are shown at a near-static situation prior to hopper opening and at a moment during discharge soon after opening the orifice. The continuum differential slice force balance predictions, given for comparison, are

Table 1. Details of the hopper simulations. To convert to real time units, the reduced times should be multiplied by (d/a) 1/2. For example, for 1 cm discs falling vertically, 100 reduced

time units correspond to 3.2 s.

Quantity Symbol 2D 3D

Hopper diameter D/a 17 Orifice diameter B/o 10 Half-angle (deg) ~ 90, 19 Number of particles N 2000 Diameter dispersion 6 0.1 Internal friction tt 0.6 Wall friction /~ w 0.3 Particle-particle displacement ratio fir 0.73, 0.073 Particle-wall displacement ratio 6R 0.37, 0.037 Normal damping ratio c, 0.5 Tangential damping ratio cf 0.5 Normal interaction index n 36, 72, 144 Interaction truncation acla 1.2 Time step At 0.001 Time orifice opened topn 150.0 Time simulation stopped qtop 200.0

14 9

90, 19 8000

0.1 0.6 0.3

0.73, 0.073 0.37, 0.037

0.5 0.5

36, 72, 144 1.2 0.00l

50.0 75.0


represented by the letters, A, for the active state and, P, for the passive state in the figures. (These coincide when the material coefficient of friction equals the wall coefficient of friction). The symbol H in the figures represents the hydrostatic equivalent stress. We also go beyond our previous work, in analysing the magnitude and distribution of particle contact forces within the bed, as well as making an assessment of the true level of compaction of the material.

As the particles become harder, they overlap less. This is evident in Fig. 5, which shows the structure of mono-sized continuous interaction particles at the bottom of a fiat-bottomed hopper. Three cases are shown for which n = 36, 72 and 144. The circles shown are the nominal particle size based on a diameter of a. Clearly, in the case of n = 36 there is significant overlap of the particles, whereas for n = 144 no perceptible overlap can be distinguished. This leads to a difference in the height of the bed in the hopper when all other factors (number of particles,

(a) n=36

(b) n=72

(C) n=144

Fig. 5. The static assembly near the base of a flat bottomed hopper for three values of the continuous potential index

(a) n = 36, (b) n = 72 and (c) n = 144.

975

hopper dimensions) are the same. The packing height is greater for larger n, because the lower particle overlap results in a lower average bulk number density (Ntr2/Ar).

Figure 6 shows the process of filling and discharging a fiat-bottomed 2D hopper with the system parameters listed in Table 1. We used a continuous interaction force [eq. (2)] as in the previous publication, with n = 36. The maximum tangential displacement ratio, 6s, was given a value of 0.73 for the particle-particle interaction and a value of 0.37 for the particle-wall interaction. The internal and wall coefficients of friction had the values p = 0.6 and Pw = 0.3, respectively (we used /~ = 0.3 and Pw = 0.3 in our previous work). It is more generally found in real hoppers that the internal friction is greater than the wall friction, which ensures that passive wall stress exceeds the active stress. The magnitude of the internal coefficient of friction partially incorporates particle shape effects so we consider that it was not unreasonable to choose a value p = 0.6. We also chose trc = 1.2a for computing tr,. In this simulation which includes internal (particle-particle) and wall (particle-wall) friction it can be seen that the discharge flow rate is reasonably constant as the hopper empties with a magnitude quite close to the Beverloo prediction. (The flowing density, p f, required in the Beverloo equation was taken to be the nominal value Na2/A, multiplied by an overlap correction estimated from the force interaction curve using the mean load on the particle in the asymptotic stress region. This gives values of the solids fraction 0.83 + 0.01 for n = 36 ~ 144.)

In Fig. 6 for funnel flow, the static wall stresses are in between the active and passive continuum theory predictions, which is a considerable improvement on the results of our previous study [see Langston et al. (1994)] where the static stress was approximately equal to the hydrostatic stress. It should also be noted that wall friction is acting downwards on the wall over the entire length which is in accord with continuum theoretical assumptions, unlike the earlier simulations. The dynamic wall stresses are larger than the static stresses and close to the passive predictions which is also in agreement with continuum theory.

In Fig. 7 we show the results of simulations carried out with a mass flow hopper. The discharge rate and wall stress are, in this case, also in good agreement with theoretical predictions. The discharge rate has increased in accord with the Rose and Tanaka correlation for hopper half-angle (Rose and Tanaka, 1959). The hopper wall stresses in the vertical section follow similar trends to the FF hopper above. In the wedge and cone sections in the static state the stresses are in-between the active and passive states which is reasonable. In the discharge state the wall stress at the top of the wedge/cone is very large, which is a mani- festation of the so-called "switch stress" [see Nedder- man (1992)]. This part of the hopper is most likely to experience a high switch stress soon after discharge has commenced.

976

(a)

H H

H H

P. A. LANGSTON etal.

H H

H H

H

(b)

H H

H H

H H

H H

H

q-I

1

0

0

0

0

0 re. 0 ,0 ' " 120.

(c)

160. 200.

50.

40. W

e 30.

20.

10.

0 .0 O. 0 ~l '120.

t t

(d)

160. 2OO.

C. I. n=3 6

Fig. 6. A 2D granular discharge event from a flat-bottomed hopper using the continuous interaction eq. (2) with n = 36 showing the wall stresses in (a) near static and (b) discharge states, (c) the discharge fraction, W s and (d) the discharge rate, I4". The parameters used in the simulation were D = 17, B = 10, = 90 , /~ = 0.6, #~ = 0.3, ~, = 0.5, 6R = 0.73. The normal wall stresses are compared with Janssen's differential slice method (Nedderman, 1992) which leads to active, A, passive, P and hydrostatic H states which are also shown on (a) and (b). The theoretical discharge rate, We based on the Beverloo equation is shown on (d) as

a dotted line. For (a) FNM = 123, FrM = 14.4, (b) FNM = 213, FT~ = 50.0 and for (d) We = 28.

The wall stresses indicate that the prescription of eq. (12) which written alternatively is fro,, ~ FN (~rc _ ~.)-, 1) used in our previous publication gives rise to states with unrealistically low tangential frictional forces at the contacts. As the maximum loads in these model hoppers are in excess of lOOmg, the 6m,~ values will be greater than 0.5or leading to too large a tangential compliance. The frictional stresses within the material and at the walls would have been poorly developed. This is because for a given tangential displacement the frictional stress decreases as 6m,, increases. The new prescription, 6,.,. ~ 6, leads to a more uniformly distributed range of &m,. within a fairly narrow band for all normal loads, and there-

fore a more fully developed tangential friction force for a given 6t. The wall stresses and discharge rates turned out to be relatively insensitive to the value of 6R which was reduced from 0.73 to 0.073. As illus- trated in Fig. 8, a reduction in the value of ~, reduces the value of ~ma, = 6,&R for.a given normal load. This has the effect of increasing the initial tangential stiffness and reducing the maximum value of the tangential displacement before gross sliding occurs.

We have also explored the effect of modifying the value of the continuous interaction index n, which determines the effective particle normal stiffness. Simulations were undertaken in 2D for granular discharge from the FF hopper for n = 36, 72 and

Discrete element simulation of granular flow 977

~Hp P (a) (b)

H H

\ \

~ P P P \ ~

1.0

O.

O.

O.

0.2

O. Ol ~i 0.0 120.

(c)

160. 200. t

50 (d)

40 . . . . . . . W

e 30

20

10

0.0 0.0 120. 160, 200.

t

C. I. n=36

Fig. 7. As for Fig. 6, except the mass flow case was considered with = 19 . (a) FN~ = 124, Ft , = 17.6, (b) FNu = 242, Fr , = 63.1 and for (d) We = 40.

144. Generally, there was no significant difference in the discharge rate or hopper wall stresses between these cases. For the discharge rate, this indicates that the true level of consolidation in the outlet zone (the most important material parameter influencing discharge rate) is relatively insensitive to the value of n, as we confirmed above. We do, however, observe some increase in static normal wall stress at the base of the hopper for n = 144 as shown in Fig. 9 indicating that the material tends towards the hydrostatic limit in this region.

In Fig. 10 we compare the 2D hopper behaviour for three forms of the normal interaction. We show the tangential displacement vectors (normalised to the maximum displacement prior to gross sliding) and wall stress profiles in a tall 2D bin with a flat bottom and a central slot orifice. Figure 10(a) and (b) are for a continuous interaction force curve with a value of n = 36 and ac = 1.2a. In Fig. 10(a), the tangential displacement vectors are shown for 2500 contacts soon after the start of discharge from the bin. Figure 10(b)

shows the particle packing arrangement during this flow and the corresponding profiles of the normal and shear stresses at the silo wall. The figure reveals evidence of "dislocations" in the velocity flow within the material which are observed experimentally in consolidated granular materials and commonly referred to as "rupture" zones. In comparing Fig. 10(a) and (b), there can be seen a coincidence of the position of the wall normal stress peaks and where the rupture zones within the flow field intersect with the hopper walls. Rupture zones can be produced by continuum analyses in the past based on the theories of soil plasticity (Drescher, 1991) where the reconciliation of the infinite shear strain rate resulting during rigid-plastic failure with energy dissipation equations (Drucker, 1959) necessitates the formulation of plastic zones of finite size within the flowing material. Rup- ture zones have also been observed experimentally during the flow of sand in silos (Bransby and Blair- Fish, 1973). In this model for the particle interactions using ac = 1.2a, the effective range of the frictional

CES 50:6-F


....... indicates gross sliding

r..

A B C _ -

axA ~maxB 6maxC

h ~ f

6 t

A & B : 6max ~ $n (6RA < 6RB ~ 6maxA < 6maxB )

C : 6ma x ~ F N leads to underdeve loped s t ress

Fig. 8. The effect of varying tangential gross sliding limit, fro,,, on the tangential force at a given value of the tangential displacement, fit. Curves A and B correspond to 6,,a, oc 6, with 6~ being greater in case B. In

curve c we have 6max oc F. which leads to an underdeveloped stress.

P

P _A_ HP

PH

__A_ P H H

1.0

0.8

0.6

0.4

0.2

0.0 ~(, 0,0 120.

(a)

FI H

H H

H H

H H

H H

H H

H H

H H

(c)

160, 200.

50.

(b)

AHP .AHP

A HP

H H

P

__A P

(d)

H H

H H

H

40.

30.

20.

10.

0 .0 ,,. o.d'~2o.

C. I. n=144

W e

160. 200.

t

Fig. 9. As for Fig. 6, except that n = 144.

Discrete element simulation of granular flow 979

6TR vectors

(a)

T :'~.5:~":'~9'!:~ :~!::

- .,. ,tn

2D

CI n=36

G =1.2 C

(b)

wal l stress

H H H

H H H

(c)

CI n=36

=1.064 C

(d)

H H

H H

H H I M - - H

H - - H

H

(e)

r -7 ~,~:~" .~

;:.~%~. ' ...5 +~

LLa i , "~" t ~ Z

Hertz

E=50000

(f)

Fig. 10. Internal tangential displacements and wall stresses for 2D fiat-bottomed hoppers with the following particle interactions: (a), (b) continuous interaction with n = 36 and ac = 1.2c;; (c), (d) continuous interaction with n = 36 and a cut-off, ;c = 1.0646, which corresponds to a force cut-off criterion of 0.1 toO;

(e), (f) the Hertzian interaction with E = 50,000 and ac = a.

engagement is some 10-20% of ~ at all normal loads. As a result, the continuous normal interaction law gives rise to an extended range for the bulk frictional engagement within the material which goes beyond the notional particle size. The rather large normal and tangential contact deformations result in a greater degree of connectivity within the bulk which we sug- gest is responsible for the rupture zones observed during flow. It is believed that the shape of the wall stress profile in this case reflects the overall rotation of

the flow direction in the vessel as particles move from the near vertical to the converging flow regimes.

In Fig. 10(c) and (d) the physical conditions of (a) and (b) are repeated, but this time with a continuous interaction cut-off based on a force criterion rather than a pre-specified separation. The maximum force is O.lm 9, which for n = 36 corresponds to a separation, r equal to 1.064~. This severely limits the range of possible 6, as the contact is deemed not to have formed until the particles overlap more. Therefore


. . . . the maximum duration of frictional engagement on relative displacement, using eq. (13) covers a much smaller range. In contrast to Fig. 10(a) and (b), the flow field is seen to be more homogeneous and the matching wall stress profile consistent with almost independently flowing particles experiencing vari- ations in stress on a particle distance scale rather than on that of the hopper. No evidence of rupture zones was found either for n = 144 and r equal to 1.016~r which again corresponds to a truncation force of O.lmo.

We have also examined the effects of changing the analytic form of the norma~ interaction. The Hertz interaction force of eq.(1) with an effective bulk modulus of elasticity E = 50,000, and ~r c = ~r, has the same slope as the n = 36 continuous interaction at r = 0.88a. The normal loads at r = 0.88cr are typical of those found in the asymptotic stress region of the

hopper. (The mean normal load per particle reaches an asymptotic value at approximately the same depth as that of the wall stress, as seen in Fig. 11). The continuous interaction has quite a different shape to the Hertz interaction for larger separations, however. The CI rises more steeply at close separations and therefore has more of a "tail" than the Hertzian interaction. Figure 10(e) and (f) show the simulation results obtained using the Hertz interaction model. The flow field is even more homogeneous than in the case of Fig. 10(c) and (d). It would appear therefore that the smaller the available range of 6, and hence 6max then the less well developed is the bulk friction within the material.

3.2. Three-dimensional hopper We now consider the behaviour of model 3D hop-

pers. A cross-section down the middle of the hopper is

2D C I n=36

(a) Stat i c

Max (M) = 123 .

Average

Contact

Force

(b) D ischarge

Max (M) = 213 .

Fig. 11. Internal normal contact forces for a continuous interaction assembly with n = 36 and ~ = 1.2 in a 2D flat-bottomed hopper. The horizontal fines are proportional to the average normal elastic load in each "slice", with the maximum normal elastic load in the hopper denoted by M: (a) static, maximum FN = 123

and (b) discharge, maximum FN = 213.


shown. Discs with a radius appropriate to the sliced sphere are shown. In Fig. 12 the 3D model static and discharge wall stresses are shown for a simulation in which the internal and wall friction coefficients were set to zero (i.e.,/~ = ttw = 0). It was expected that the system would behave like a fluid and give hydrostatic stresses and discharge rates. The agreement with con- t inuum theory is excellent in the static case, e.g., the hydrostatic stress is very close to poh, apart from very near to the base of the hopper where it is slightly smaller than the continuum prediction. In this limit- ing case the wall stress at the bottom of the hopper during discharge was somewhat lower than the con- t inuum prediction. We attribute these departures from the continuum predictions to the finite size of the

981

particles, when compared with the dimensions of the hopper, which leads to, for example, a dilation of the material assembly in the region of the orifice in flow and consequently a falling off of the pressure there. The discharge rate was found to be lower than the hydrostatic prediction as a result. In 2D the calculated and continuum theory initial discharge rates were 60 and 100, respectively. In 3D these quantities were 210 and 400, respectively.

Figure 13 shows the wall stresses and discharge rates from a 3D flat-bottomed hopper simulation with particle and wall friction. The discharge rate of ~ 90 is reasonably constant for the period simulated, and which is statistically indistinguishable from the prediction of the 3D Beverloo equation, 94, based on

3D CI n=36 zero-fr ict ion

-" eL - {~ -, 0 ) 0-) . (D

I!

H

H

H

H

H

(a) Static

H H

H

H

={_ o

i __H

_ _ H

H

H

H

(b) Discharge

H H

H

Fig. 12. Wall stress for zero friction 3D funnel flow hopper and 6 = 0.1 polydispersity; (a) near-static fill and (b) discharging state. Cross-sectional views through hopper centre are shown. H is the equivalent

hydrostatic wall stress D = 14, B = 9, ~ = 90 , # = #w = 0 and y = 0.5.

982

N


(a)

p H / A P H

p / A P

(b)

i ' P S

A P H A P H A P A P

. A P

I .0,

0.8

0.6

0.4

0.2

0.0 ,,. 0.0" '45.

(c)

60. 75.

(d)

100

8 0 . - - w- 60 ,,. e

40

20.

0.0 0.0 ' '45. 150

t t

3D C. I. n=36

Fig. 13. Granular discharge from a 3D FF hopper: (a) wall stresses in near static and (b) discharge states; (c) discharge fraction and (d) discharge rate. Key: as for Fig. 12 except # = 0.6 and #w = 0.3, We = 94.

a true (i.e., overlap corrected) bulk solids fraction of 0.53 in the vicinity of the base of the hopper which was computed from the simulation data. The wall stresses broadly follow the 2D simulation behaviour, except that in general they vary more smoothly with height, probably because the wall element areas are larger in the 3D case. The hopper wall stresses of the equivalent mass flow hopper are shown in Fig. 14. In the vertical section they are similar to those of the FF hopper. In the cone section in the static state the stresses are, as to be expected, in between the active and passive states. As for the 2D hopper the discharge wall stress at the top of the cone section is very large in 3D. The results for discharge rate and wall stress are also in good agreement with their corresponding continuum theories. The discharge rate has increased in accord with the Rose and Tanaka correlation for hopper half-angle.

The above simulations were carried out for relatively large orifice/hopper diameter ratios (10/17 in 2D

and 9/14 in 3D). We also carried out simulations using lower ratios (e.g., B = 5) for both the funnel and mass flow hoppers in 2D and 3D which show very similar behaviour to that reported above. In fact, the model behaviour is largely independent of outlet size ratio. Empirical evidence (Nedderman, 1992) shows that B > 6 is necessary to obtain stable flow for real granular systems, however, because the particles are idealised here as discs and spheres, the Beverloo equation is applicable to much lower values of B. For example, in the previous study (Langston et al., 1994) regular flow continued down to B = 3.5 and only stopped for B = 2 with mono-sized disk particles.

In Fig. 15 we consider a 3D cylindrical vessel with particles interacting via the continuous interaction law, n = 36 and trc = 1.2a. In contrast to the corresponding 2D system, the flow field at the mid-plane of the cylinder does not provide any visual evidence of velocity discontinuities or rupture zones. Although it should be noted that visualisation of this effect is


(a)

~3 _&_H PS

H S A P H S A P H

H H

H

(b)

s / ^ PH S l ^ P H S

H H

983

1.0

0 .8

O.

O.

O.

o.o;,j

(c) (d)

2OO

60. 75.

a60

120

80.

40.

O. 0 .0

W e

i - 45. 60. 75.

t t

3D C. I. n=36

Fig. 14. Granular discharge from a 3D MF hopper: (a) wall stresses in near static and (b) discharge states; (c) discharge fraction and (d) discharge rate. Key: as for Fig. 13 except ct = 19 . We = 137.

inherently more difficult in 3D. Histograms of the normalised values of the contact tangential displacements (where the value of unity indicates gross sliding) taken from an annular cross-section marked on Fig. 15(a) are shown in Fig. 15(b). In comparing the tangential displacement histograms corresponding to the static [Fig. 15(b)] and flowing states [Fig. 15(c)], it is clear that only a proportion of the contacts exceed the gross-sliding limit during flow while a substantial number retain the near-static tangential displacements values. The corresponding Hertz interaction hopper is given in Fig. 15(d)-(f). In this case the number of contacts with a given 6,/6max is lower for the Hertz interaction hopper, although it shows a similar trend during discharge, of a bimodal distribution of contacts centred around di,/6,,ax ~ 0.2 and 6,/&m,, ~ 1.0. This suggests that there could be re- gions of small and large deformation in the material under flow, although there is no visual evidence for rupture zones in the 3D model using the present method of representation.

A series of simulations was carried out, computing the dependence of the discharge rate on the orifice diameter in 2D and 3D for FF and MF hoppers. As can be seen on Fig. 16, the simulation and continuum theory predictions are in very good agreement, reflect- ing the non-linear dependence of the flow rate on B. The data shown are for the continuous interaction with n = 36. The discharge rate, was statistically indistinguishable for n = 144, which is consistent with the usual assumption that the flow rate is dominated by the (dilated) state of the material in the vicinity of the orifice. In this region of the hopper the material is experiencing quite low normal loads so that details of the particles interactions (e.g., magnitude of maximum displacements) should be relatively insensitive to the value of n.

4. CONCLUSIONS We have developed further a discrete element

model of the filling and discharging of granular material from hoppers. Realistic values for discharge rate,

984

Ca)


CI n=36 Hertz E=50000

. (d) ~ AHPS ~AH%ps I~ '~o~ i A HPS

~ a ~ A I~ S t A ~ S ~ I A PH S- / A PH S ~-~] /, A P H S I A P H Z A P H

~I~, A P H ! ^ P H /, A P ~ I / i A P H /, A P J ?~~ A P

A P A P ~ A P ~ I A P

o ~

(b) N C 400 C 160

2001

40. 100 l 0 " 0 [ . . . . . 0 . 0

0 , 0 0 , 2 0 I ~ 0 .6 0, 8 1 ,0 . . . . . 0.0 0.2 0.,4 0.6 0.8 1.0

at la max 6 t / 6ma x (c) N C 500 iI 200I 4o~ ( f ) NC 160

3G ~ D i s c h a r g e 12el ~ 2OO 80 "I 1001 -

40. 0. OL 0 1 0 b i ~ ; . ~ b " ~ b . 8 I~ . 0 0 . 0

00 b2 b . b6 b8 ' t0

max 6 t / ~max

Fig. 15. 3D granular filled and discharging states of continuous interaction, n = 36, and Hertzian, E* = 50,000 systems. The number of contacts Nc in the boxed cylindrical volume with a specified relative tangential displacement are shown in (b) and (c) for static and discharging of the CI material; (e) and (f) are

the corresponding figures for the Hertzian normal contact interaction.

static average bulk density, static hopper wall stress and discharge wall stress in two and three dimensions have been obtained using an improved "effective" inter-granule force law. In particular, a new gradual filling method leads to a much improved wall stress profile of the static bed.

We have explored different analytic forms for the granule-granule interaction, and conclude that even at low normal loads it is important that the particles can develop a significant tangential displacement (some 10-20% of the particle's diameter) before start- ing to slide. Such an interaction takes into account, in some mean field sense, the faceted shape and roughness of most granular materials. As a result, this model has produced, for the first time in any DE

simulation of slow shear granular flows, discontinu- ous velocity slip planes within the bed or "rupture zones", which are features found in real granular materials under discharge.

The discharge rates are rather insensitive to the stiffness of the interaction, indicating that the material is under negligible load at the orifice. We compared different forms for the interaction laws, which showed that if allowance is made for a continuously varying stiffness with normal load, then extended frictional engagement at low normal loads gives rise to a higher degree of connectivity within the bulk which results in collective motion, rupture zones and associated stress spikes on the hopper wall. In contrast, the Hertz interaction, which shows little variation of stiffness

60

40

20


Discharge rate v orifice size 2D hopper

|

MF simulation , / / / t / v FF simulation

MF empirical / / FF empirical /

/ . / e/.'/~ 4 8 12

B

(a)

985

160

120

~: 80

Discharge rate v orifice size 3D hopper

40

0 ~- ' 0 12

. , . . , . /

/ " MF simulation / / v FF simulation / / -' MF empirical / / - FF empirical /,//~

4 8

(b)

B

Fig. 16. Comparison of simulated discharge rates with empirical predictions of Beverloo and Rose and Tanaka. The simulation results are using the CI force interaction choosing n = 36. The results for, --- 144

are statistically indistinguishable.

over a large load range, results in largely independent particle motion and the absence of rigid-plastic failure.

Acknowledgement--The authors would like to thank the Engineering and Physical Sciences Research Council of Great Britain for the funding provided under the Specially Promoted Programme in Particle Technology.

a

A, B Cr

NOTATION

contact radius area

hopper orifice size average tangential compliance at onset of sliding

CFO

Cn

Ct

CN CNo

D E E*

F.

initial value of tangential compliance at zero load damping ratio or fraction of critical damping for normal overlap damping ratio or fraction of critical damping for tangential displacement average normal compliance initial value of normal compliance at zero load hopper width in 2D, diameter in 3D Young's modulus of elasticity reduced modulus of elasticity (defined in text) total normal force from particle interaction curve

986

Fs

FNm F, Fr Fr~ 9 G G* m

n

N Nc r R R* t topen

tstop V W We w~


elastic normal force from particle interac- MD tion curve MF maximum normal contact force P total tangential force Mindlin-like friction force maximum tangential contact force gravitational acceleration moment vector of a particle reduced shear modulus (defined in text) particle mass index in normal continuous interaction curve number of particles number of contacts separation of particle centres particle radius ( = a/2) or radial vector reduced particle radius (defined in text) time time hopper orifice opened measured from the start of the filling process time simulation stops translational velocity of a particle mass discharge rate empirical prediction of W mass fraction discharged by the hopper

Greek letters hopper half-angle

6' tangential displacement at contact in a time step

6, accumulated tangential displacement at contact

6m~ maximum tangential displacement at contact before sliding

6. normal displacement at contact 6a the ratio 6max/~. Aa range of particle diameters At time step e energy constant in normal continuous inter-

action equation p particle-particle coefficient of friction Pw particle-wall coefficient of friction v Poisson's ratio PB static bulk density pf flowing bulk density a nominal particle diameter ac cut-off separation for normal interaction

curve ao outer separation for neighbour lists ~o angular velocity vector of particle

Abbreviations 2D 3D A CI DE FF GD H MC

2 dimensions 3 dimensions active state stress continuous interaction discrete element funnel flow granular dynamics simulation hydrostatic equivalent stress Monte Carlo simulation

molecular dynamics simulation mass flow passive state stress

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APPENDIX: COUPLING MINDLIN FRICTION EQUATION TO CONTINUOUS NORMAL INTERACTION

The contact equations of Hertz and Mindlin [see Johnson (1985)] for interactions between noncohesive elastic spherical particles can be used to reveal that the force--displacement relationships in the normal and tangential direction are related. The Hertz equation for normal displacement is as follows:

where

3FN x~ 2/3 6,, = \ 4(E, R,,12) ) (A1)

3FNR*y/3 a = \~- / (A2)

I/R* = 1/R1 + 1/R2. (A3)

The Mindlin equation gives the following expression for the tangential displacement for the onset of gross sliding

6,,,~x = 3#F~/(16G*a). (A4)

987

Combining these three equations gives

3# F 2/3 [4E*'~ 1/3 3=.x = 1" -~ ~3--~) (A5)

6m.x/ 5. = #E* /4G* . (A6)

For two particles of the same material,

E E* (A7)

2(1 - v 2)

G G* = - - (A8)

2(1 - v)

E G = 2(1 + v--~" (A9)

Hence, combining eqs (A6)--(A9) gives eq. (14) in the main text

6m,x (2 - v) fir = ~ = #2(1 -- v)" (A10)

This equation can be used to calculate the ratio of tangential displacement to normal displacement at the onset of gross sliding. This ratio (dependent on material properties only) is an input parameter in our model and scales the tangential displacement curve onto the normal displacement. Mindlin derived his expression for tangential displacement assuming a ratio for initial tangential compliance to normal compliance. The above equations show that the ratio of total compliance, from initial load to gross sliding, is a constant. Continuing the above analysis, if at the onset of gross sliding,

Cv = 3=,x/Fv = 3raax/I-tFF (A 11)

CN = 6n/FN (A12)

then therefore,

(2 - v) CR = Cr/CN = ~R/# (A13)

2(1 - v).

For a typical value of v = 0.3, then CR = 1.214 and 6R = 1.214#. The initial tangential compliance, Cro, can be calculated by differentiation of the Mindlin equation. We have

Fr = #Fs(1 - (1 - [~r]/~m,~) 3/2) (AI4)

dFr 3#FN,. 2--~maxl I --I~Fl/l~max) 1/2 (A15)

d6r

/{dFF'~ =

(AI6)

(1995) discrete element simulation of granular flow in 2d and 3d

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